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− | < | + | Published in ''Comput. Methods Appl. Mech. Engrg.'' Vol. 195, pp. 6750-6777, 2006<br /> |
− | + | doi: 10.1016/j.cma.2004.10.018 | |
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==Abstract== | ==Abstract== | ||
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The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk forming processes. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal forming problems and mixing processes are shown. | The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk forming processes. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal forming problems and mixing processes are shown. | ||
− | ''' | + | '''Keywords''': Bulk forming processes, stabilized finite element method, particle method, particle finite element method, mixing processes. |
==1 INTRODUCTION== | ==1 INTRODUCTION== | ||
− | The development of efficient and robust numerical methods for analysis of bulk forming problems has been a subject of intensive research in recent years <span id= | + | The development of efficient and robust numerical methods for analysis of bulk forming problems has been a subject of intensive research in recent years <span id=‘citeF-1’></span>[[#cite-1|[1-7]]]. Many of these problems require the solution of incompressible fluid flow situations (such as in mould filling problems) whereas in other cases (such as forging, rolling, extrusion, etc.) the numerical method must be able to account for the quasi/fully incompressible behaviour induced by the large plastic deformation. The solution of these problems has motivated the development of the so called ''stabilized numerical methods'' overcoming the two main sources of instability in the analysis of incompressible continua, namely those originated by the high values of the convective terms in fluid flow situtions and those induced by the difficulty in satisfying the incompressibility condition. |
− | Different approaches to solve both type of problems in the context of the finite element method (FEM) have been recently developed <span id= | + | Different approaches to solve both type of problems in the context of the finite element method (FEM) have been recently developed <span id=‘citeF-8’></span>[[#cite-8|[8]]]. Traditionally, the underdiffusive character of the Galerkin FEM for high convection flows has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations <span id=‘citeF-8’></span>[[#cite-8|[8,9]]]. |
− | A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum and using implicit and explicit algorithms ''ad hoc'' such as artificial compressibility schemes <span id= | + | A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum and using implicit and explicit algorithms ''ad hoc'' such as artificial compressibility schemes <span id=‘citeF-10’></span>[[#cite-10|[10]]] and preconditioning techniques <span id=‘citeF-11’></span>[[#cite-11|[11]]]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms in fluid flows are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods <span id=‘citeF-8’></span>[[#cite-8|[8,9,12]]]. A general class of Galerkin FEM has been developed where the standard Galerkin variational form is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme. Among the many FEM of this kind we can name the Streamline Upwind Petrov Galerkin (SUPG) method <span id=‘citeF-8’></span>[[#cite-8|[8,13-16]]], the Galerkin Least Square (GLS) method <span id=‘citeF-8’></span>[[#cite-8|[8,17]]], the Taylor-Galerkin method <span id='citeF-18'></span>[[#cite-18|[18]]], the Characteristic Galerkin method <span id='citeF-19'></span>[[#cite-19|[19]]] and its variant the Characteristic Based Split (CBS) method <span id='citeF-20'></span>[[#cite-20|[20,21]]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|[22]]] and the Subgrid Scale (SS) method <span id='citeF-23'></span>[[#cite-23|[23]]]. A good review of these methods can be found in <span id='citeF-24'></span>[[#cite-24|[24]]]. Extensions of the CBS and SS methods to treat incompressible problems in solid mechanics are reported in <span id='citeF-25'></span>[[#cite-25|[25,26,58]]] and <span id='citeF-32'></span>[[#cite-32|[27-29]]], respectively. |
− | In this paper a different class of stabilized FEM for quasi and fully incompressible fluid and solid materials applicable to a wide range of bulk forming problems is presented. The starting point is the modified governing differential equations of the continuous problem formulated via a finite calculus (FIC) approach <span id='citeF-35'></span>[[#cite-35| | + | In this paper a different class of stabilized FEM for quasi and fully incompressible fluid and solid materials applicable to a wide range of bulk forming problems is presented. The starting point is the modified governing differential equations of the continuous problem formulated via a finite calculus (FIC) approach <span id='citeF-35'></span>[[#cite-35|[30,31]]]. The FIC method is based in invoking the classical balance (or equilibrium) laws in a domain of ''finite size''. This introduces naturally additional terms in the differential equations of infinitesimal continuum mechanics which are a function of the balance domain dimensions. The new terms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via the standard Galerkin FEM. One of the main advantages of the FIC formulation versus other alternative approaches (such as mixed FEM, etc.) is that it allows to solve incompressible fluid problems using low order finite elements (such as linear triangles and tetrahedra) with equal order approximations for the velocity and pressure variables <span id='citeF-32'></span>[[#cite-32|[32-35]]]. The FIC formulation has been successfully used for analysis of fully or quasi incompressible solids <span id='citeF-36'></span>[[#cite-36|[36,37]]]. |
The layout of the paper is the following. In the next section the basic FIC equations for incompressible flow problems formulated in an Eulerian frame are presented. The finite element discretization is introduced and the resulting discretized equations are detailed. A fractional step scheme for the transient solution is presented. | The layout of the paper is the following. In the next section the basic FIC equations for incompressible flow problems formulated in an Eulerian frame are presented. The finite element discretization is introduced and the resulting discretized equations are detailed. A fractional step scheme for the transient solution is presented. | ||
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The following sections outline the FIC formulation for analysis of quasi/fully incompressible solids using a Lagrangian description and linear triangles and tetrahedra. An explicit algorithm for integrating in time the equations of motion of elasto-plastic solids in large-strain problems involving frictional contact is described. Examples of application to a casting problem and some bulk forming processes are presented. | The following sections outline the FIC formulation for analysis of quasi/fully incompressible solids using a Lagrangian description and linear triangles and tetrahedra. An explicit algorithm for integrating in time the equations of motion of elasto-plastic solids in large-strain problems involving frictional contact is described. Examples of application to a casting problem and some bulk forming processes are presented. | ||
− | In the last part of the paper a Lagrangian formulation for fluid flow analysis is presented as a straightforward extension of the formulation for solid mechanics. The procedure, called the ''Particle Finite Element Method'' (PFEM) <span id='citeF- | + | In the last part of the paper a Lagrangian formulation for fluid flow analysis is presented as a straightforward extension of the formulation for solid mechanics. The procedure, called the ''Particle Finite Element Method'' (PFEM) <span id='citeF-38'></span>[[#cite-38|[38-40]]], treats the mesh nodes in the fluid and solid domains as dimensionless particles which can freely move, an even separate from the main fluid domain, representing, for instance, the effect of liquid drops. A finite element mesh connects the nodes defining the discretized domain where the governing equations are solved in the standard FEM fashion. The main advantage of the Lagrangian flow formulation is that the convective terms do not enter in the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes. The final examples show that the PFEM is a promising method to solve mould filling and casting problems, material mixing processes and many other bulk metal forming problems involving the interaction between solids and fluids which can be treated with the same formulation. |
==2 FIC EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLOWS== | ==2 FIC EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLOWS== | ||
− | The FIC governing equations for a viscous incompressible fluid can be written in an Eulerian frame of reference as <span id='citeF- | + | The FIC governing equations for a viscous incompressible fluid can be written in an Eulerian frame of reference as <span id='citeF-30'></span>[[#cite-30|[30-34]]] |
− | + | '''Momentum''' | |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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|} | |} | ||
− | + | '''Mass balance''' | |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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In Eqs.(7) and (8) <math display="inline">t_i^p</math> and <math display="inline">v_j^p</math> are surface tractions and prescribed velocities on the boundaries <math display="inline">\Gamma _t</math> and <math display="inline">\Gamma _u</math>, respectively, <math display="inline">n_j</math> are the components of the unit normal vector to the boundary and <math display="inline">\sigma _{ij}</math> are the total stresses given by <math display="inline">\sigma _{ij}=s_{ij}-\delta _{ij}p</math>. | In Eqs.(7) and (8) <math display="inline">t_i^p</math> and <math display="inline">v_j^p</math> are surface tractions and prescribed velocities on the boundaries <math display="inline">\Gamma _t</math> and <math display="inline">\Gamma _u</math>, respectively, <math display="inline">n_j</math> are the components of the unit normal vector to the boundary and <math display="inline">\sigma _{ij}</math> are the total stresses given by <math display="inline">\sigma _{ij}=s_{ij}-\delta _{ij}p</math>. | ||
− | Eqs. (1) and (2) are obtained by invoking the classical balance equations in fluid mechanics in a domain of finite size and retaining higher order terms <span id='citeF- | + | Eqs. (1) and (2) are obtained by invoking the classical balance equations in fluid mechanics in a domain of finite size and retaining higher order terms <span id='citeF-30'></span>[[#cite-30|[30]]]. The <math display="inline">h_i's</math> in above equations are characteristic lengths of the domain where the balance of momentum and mass is enforced. In Eq.(7) these lengths define the domain where equilibrium of boundary tractions is established. In the discretized problem the <math display="inline">h_{i}</math> coincide with a typical element dimension, as described in Section 5. Note that by making <math display="inline">h_i =0</math> in these equations the standard infinitesimal form of the fluid mechanics equations is recovered <span id='citeF-8'></span>[[#cite-8|[8,9,24]]]. |
− | Eqs.(1)–(8) are the starting point for deriving stabilized FEM for solving the incompressible Navier-Stokes equations using equal order interpolation for the velocity and pressure variables <span id='citeF- | + | Eqs.(1)–(8) are the starting point for deriving stabilized FEM for solving the incompressible Navier-Stokes equations using equal order interpolation for the velocity and pressure variables <span id='citeF-32'></span>[[#cite-32|[32-35]]]. Application of the FIC formulation to meshless analysis of fluid flow problems using the finite point method can be found in <span id='citeF-42'></span>[[#cite-42|[42]]]. |
− | + | '''Remark'''. In most metal forming processes the viscosity <math display="inline">\mu </math> is a non linear function of the strain rate and the yield stress of the material (which also depends on the temperature) <span id='citeF-8'></span>[[#cite-8|[8,43,44]]]. This dependence adds another non linearity to the problem. | |
− | + | ||
− | In most metal forming processes the viscosity <math display="inline">\mu </math> is a non linear function of the strain rate and the yield stress of the material (which also depends on the temperature) <span id='citeF-8'></span>[[#cite-8|8]]. This dependence adds another non linearity to the problem. | + | |
− | + | ||
− | + | ||
− | In Eqs.(1)–(7) <math display="inline">\Omega </math> and <math display="inline">\Gamma _t</math> respectively denote the actual volume and the boundary of the domain where the governing equations are solved at each instant of the forming processes. This domain is considered here to be fixed in space (Eulerian approach). Moving free surfaces, such as in the case of mold filling processes, can be modelled by using standard level set and volume of fluid (VOF) techniques <span id='citeF- | + | '''Remark'''. In Eqs.(1)–(7) <math display="inline">\Omega </math> and <math display="inline">\Gamma _t</math> respectively denote the actual volume and the boundary of the domain where the governing equations are solved at each instant of the forming processes. This domain is considered here to be fixed in space (Eulerian approach). Moving free surfaces, such as in the case of mold filling processes, can be modelled by using standard level set and volume of fluid (VOF) techniques <span id='citeF-47'></span>[[#cite-47|[47,48]]]. An alternative is to use an arbitrary Lagragian-Eulerian (ALE) description or even a fully Lagrangian description to follow the motion of the particles during the forming process. The Lagrangian description is typical in the case of solid mechanics problems (Section 5). A particular Lagrangian formulation for fluid flow problems involving large motion of the free surface, named the Particle Finite Element Method, is presented in Section 8. |
===2.1 Stabilized integral forms=== | ===2.1 Stabilized integral forms=== | ||
− | From the momentum equations it can be obtained <span id='citeF- | + | From the momentum equations it can be obtained <span id='citeF-32'></span>[[#cite-32|[32-34]]] |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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with <math display="inline">i,j,k=1,n_d</math>. In Eqs.(21) and (22) <math display="inline">\delta c_i</math> and <math display="inline">\delta \pi _i</math> are appropriate weighting functions and the <math display="inline">\rho </math> and <math display="inline">\tau _i</math> weights are introduced for convenience. | with <math display="inline">i,j,k=1,n_d</math>. In Eqs.(21) and (22) <math display="inline">\delta c_i</math> and <math display="inline">\delta \pi _i</math> are appropriate weighting functions and the <math display="inline">\rho </math> and <math display="inline">\tau _i</math> weights are introduced for convenience. | ||
− | We note that accounting for the convective and pressure gradient projections enforces the consistency of the formulation as it ensures that the stabilization terms in Eqs.(19–22) have a residual form which vanishes for the “exact” solution. Neglecting these terms lowers the accuracy of the numerical solution and it makes the formulation more sensitive to the value of the stabilization parameters as shown in <span id='citeF- | + | We note that accounting for the convective and pressure gradient projections enforces the consistency of the formulation as it ensures that the stabilization terms in Eqs.(19–22) have a residual form which vanishes for the “exact” solution. Neglecting these terms lowers the accuracy of the numerical solution and it makes the formulation more sensitive to the value of the stabilization parameters as shown in <span id='citeF-34'></span>[[#cite-34|[34,36,37]]]. |
==3 FINITE ELEMENT DISCRETIZATION== | ==3 FINITE ELEMENT DISCRETIZATION== | ||
− | We choose <math display="inline">C^\circ </math> continuous linear interpolations of the velocities, the pressure, the convection projections <math display="inline">c_i</math> and the pressure gradient projections <math display="inline">\pi _i</math> over three node triangles (2D) and four node tetrahedra (3D) <span id='citeF-8'></span>[[#cite-8|8]]. The linear interpolations are written as | + | We choose <math display="inline">C^\circ </math> continuous linear interpolations of the velocities, the pressure, the convection projections <math display="inline">c_i</math> and the pressure gradient projections <math display="inline">\pi _i</math> over three node triangles (2D) and four node tetrahedra (3D) <span id='citeF-8'></span>[[#cite-8|[8]]]. The linear interpolations are written as |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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|} | |} | ||
− | where the sum goes over the number of nodes of each element <math display="inline">n</math> (<math display="inline">n=3/4</math> for triangles/tetrahedra), <math display="inline">\bar {(\cdot )}^k</math> denotes nodal variables and <math display="inline">N^k</math> are the linear shape functions <span id='citeF-8'></span>[[#cite-8|8]]. | + | where the sum goes over the number of nodes of each element <math display="inline">n</math> (<math display="inline">n=3/4</math> for triangles/tetrahedra), <math display="inline">\bar {(\cdot )}^k</math> denotes nodal variables and <math display="inline">N^k</math> are the linear shape functions <span id='citeF-8'></span>[[#cite-8|[8]]]. |
Substituting the approximations (23) into Eqs.(19–22) and choosing the Galerking form with <math display="inline">\delta v_i =q=\delta c_i=\delta \pi _i =N^i</math> leads to following system of discretized equations | Substituting the approximations (23) into Eqs.(19–22) and choosing the Galerking form with <math display="inline">\delta v_i =q=\delta c_i=\delta \pi _i =N^i</math> leads to following system of discretized equations | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\displaystyle {M}\dot{\bar {v}} + {H} \bar {v} - {G}\bar {p}+{C}\bar {c}={f}</math> | + | | style="text-align: center;" | <math>\displaystyle {\bf M}\dot{\bar {\bf v}} + {\bf H} \bar {\bf v} - {\bf G}\bar {\bf p}+{\bf C}\bar {\bf c}={\bf f}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (24a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (24a) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\displaystyle {G}^T \bar {v} + \hat{L}\bar {p}+{Q}\bar {\boldsymbol \pi }={0}</math> | + | | style="text-align: center;" | <math>\displaystyle {\bf G}^T \bar {\bf v} + \hat{\bf L}\bar {\bf p}+{\bf Q}\bar {\boldsymbol \pi}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (24b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (24b) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\displaystyle \hat {C}\bar{v}+ {M}\bar {c}={0}</math> | + | | style="text-align: center;" | <math>\displaystyle \hat {\bf C}\bar{\bf v}+ {\bf M}\bar {\bf c}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (24c) | | style="width: 5px;text-align: right;white-space: nowrap;" | (24c) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\displaystyle {Q}^T \bar {p} + \hat {M}\bar {\boldsymbol \pi }={0}</math> | + | | style="text-align: center;" | <math>\displaystyle {\bf Q}^T \bar {\bf p} + \hat {\bf M}\bar {\boldsymbol \pi}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (24d) | | style="width: 5px;text-align: right;white-space: nowrap;" | (24d) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{M} \displaystyle{1\over \Delta t} (\bar{v}^{n+1}-\bar{v}^{n}) + {H}^{n+\theta }\bar{v} ^{n+\theta } - {G}\bar{p}^{n+\theta }+ {C}^{n+\theta }\bar {c}^{n+\theta }={f}^{n+\theta }</math> | + | | style="text-align: center;" | <math>{\bf M} \displaystyle{1\over \Delta t} (\bar{\bf v}^{n+1}-\bar{\bf v}^{n}) + {\bf H}^{n+\theta}\bar{\bf v} ^{n+\theta} - {\bf G}\bar{\bf p}^{n+\theta}+ {\bf C}^{n+\theta}\bar {\bf c}^{n+\theta}={\bf f}^{n+\theta}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (25a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (25a) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{G}^T \bar {v}^{n+\theta }+\hat {L}^{n+\theta } \bar {p}^{n+\theta }+{Q}\bar {\boldsymbol \pi }^{n+\theta }={0}</math> | + | | style="text-align: center;" | <math>{\bf G}^T \bar {\bf v}^{n+\theta}+\hat {\bf L}^{n+\theta} \bar {\bf p}^{n+\theta}+{\bf Q}\bar {\boldsymbol \pi}^{n+\theta}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (25b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (25b) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\hat {C}^{n+\theta } \bar {v}^{n+\theta }+ {M} \bar {c}^{n+\theta }={0}</math> | + | | style="text-align: center;" | <math>\hat {\bf C}^{n+\theta} \bar {\bf v}^{n+\theta}+ {\bf M} \bar {\bf c}^{n+\theta}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (25c) | | style="width: 5px;text-align: right;white-space: nowrap;" | (25c) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{G}^T \bar {p}^{n+\theta }+\hat {M}^{n+\theta }\bar {\boldsymbol \pi }^{n+\theta }={0}</math> | + | | style="text-align: center;" | <math>{\bf G}^T \bar {\bf p}^{n+\theta}+\hat {\bf M}^{n+\theta}\bar {\boldsymbol \pi}^{n+\theta}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (25d) | | style="width: 5px;text-align: right;white-space: nowrap;" | (25d) | ||
|} | |} | ||
− | where <math display="inline">{H}^{n+\theta }={H} (\bar{v}^{n+\theta })</math>, <math display="inline">\bar{v}^{n+\theta }</math> are the velocities evaluated at time <math display="inline">n+\theta </math> and the parameter <math display="inline">\theta \in [0,1]</math>. The direct monolitic solution of Eqs.(25) is possible using an adequate iterative scheme. However, we have found more convenient to use a fractional step method as described in the next section. | + | where <math display="inline">{\bf H}^{n+\theta}={\bf H} (\bar{\bf v}^{n+\theta})</math>, <math display="inline">\bar{\bf v}^{n+\theta}</math> are the velocities evaluated at time <math display="inline">n+\theta </math> and the parameter <math display="inline">\theta \in [0,1]</math>. The direct monolitic solution of Eqs.(25) is possible using an adequate iterative scheme. However, we have found more convenient to use a fractional step method as described in the next section. |
===3.1 Fractional step method=== | ===3.1 Fractional step method=== | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{M} \displaystyle{1\over \Delta t} (\tilde{v}^{n+1}-\bar{v}^{n}) + {H}^{n+\theta }\bar{v} ^{n+\theta } - \alpha {G}\bar{p}^{n} + {C}^{n+\theta }\bar {c}^{n+\theta }={f}^{n+\theta }</math> | + | | style="text-align: center;" | <math>{\bf M} \displaystyle{1\over \Delta t} (\tilde{\bf v}^{n+1}-\bar{\bf v}^{n}) + {\bf H}^{n+\theta}\bar{\bf v} ^{n+\theta} - \alpha {\bf G}\bar{\bf p}^{n} + {\bf C}^{n+\theta}\bar {\bf c}^{n+\theta}={\bf f}^{n+\theta}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (26) | | style="width: 5px;text-align: right;white-space: nowrap;" | (26) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{M} \displaystyle{1\over \Delta t} (\bar{v}^{n+1}-\tilde{v}^{n+1})- {G}(\bar{p}^{n+1}-\alpha \bar {p}^{n})={0}</math> | + | | style="text-align: center;" | <math>{\bf M} \displaystyle{1\over \Delta t} (\bar{\bf v}^{n+1}-\tilde{\bf v}^{n+1})- {\bf G}(\bar{\bf p}^{n+1}-\alpha \bar {\bf p}^{n})={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (27) | | style="width: 5px;text-align: right;white-space: nowrap;" | (27) | ||
|} | |} | ||
− | In above equation <math display="inline">\tilde{v}^{n+1}</math> is a predicted value of the velocity at time <math display="inline">n+1</math> and <math display="inline">\alpha </math> is a variable whose values of interest are zero and one. For <math display="inline">\alpha =0</math> (first order scheme) the splitting error is of order <math display="inline"> 0 (\Delta t)</math>, whereas for <math display="inline">\alpha =1</math> (second order scheme) the error is of order <math display="inline">0 (\Delta t^2)</math> <span id='citeF- | + | In above equation <math display="inline">\tilde{v}^{n+1}</math> is a predicted value of the velocity at time <math display="inline">n+1</math> and <math display="inline">\alpha </math> is a variable whose values of interest are zero and one. For <math display="inline">\alpha =0</math> (first order scheme) the splitting error is of order <math display="inline"> 0 (\Delta t)</math>, whereas for <math display="inline">\alpha =1</math> (second order scheme) the error is of order <math display="inline">0 (\Delta t^2)</math> <span id='citeF-45'></span>[[#cite-45|[45]]]. |
Eqs.(26) and (27) are completed with the following three equations emanating from Eqs.(25b-d) | Eqs.(26) and (27) are completed with the following three equations emanating from Eqs.(25b-d) | ||
Line 480: | Line 461: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{G}^T\bar{v}^{n+1}+\hat {L}^n \bar{p}^{n+1}+{Q}\bar {\boldsymbol \pi }^{n}= {0}</math> | + | | style="text-align: center;" | <math>{\bf G}^T\bar{\bf v}^{n+1}+\hat {\bf L}^n \bar{\bf p}^{n+1}+{\bf Q}\bar {\boldsymbol \pi}^{n}= {\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (28a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (28a) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\hat {C}^{n+1} \bar{v}^{n+1} + {M} \bar {c}^{n+1}= {0}</math> | + | | style="text-align: center;" | <math>\hat {\bf C}^{n+1} \bar{\bf v}^{n+1} + {\bf M} \bar {\bf c}^{n+1}= {\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (28b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (28b) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{Q}^T \bar{p}^{n+1}+ \hat {M}^{n+1} \bar {\boldsymbol \pi }^{n+1}= {0}</math> | + | | style="text-align: center;" | <math>{\bf Q}^T \bar{\bf p}^{n+1}+ \hat {\bf M}^{n+1} \bar {\boldsymbol \pi}^{n+1}= {\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (28c) | | style="width: 5px;text-align: right;white-space: nowrap;" | (28c) | ||
|} | |} | ||
− | The value of <math display="inline">\bar{v}^{n+1}</math> obtained from Eq.(27) is substituted into Eq.(28a) to give | + | The value of <math display="inline">\bar{\bf v}^{n+1}</math> obtained from Eq.(27) is substituted into Eq.(28a) to give |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 512: | Line 493: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{G}^T\tilde{v}^{n+1}+ \Delta t {G}^T {M}^{-1} {G} (\bar{p}^{n+1} - \alpha \bar{p}^n)+ \hat {L}^n {p}^{n+1}+{Q}\bar {\boldsymbol \pi }^n={0} </math> | + | | style="text-align: center;" | <math>{\bf G}^T\tilde{\bf v}^{n+1}+ \Delta t {\bf G}^T {\bf M}^{-1} {\bf G} (\bar{\bf p}^{n+1} - \alpha \bar{\bf p}^n)+ \hat {\bf L}^n {\bf p}^{n+1}+{\bf Q}\bar {\boldsymbol \pi}^n={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (29) | | style="width: 5px;text-align: right;white-space: nowrap;" | (29) | ||
|} | |} | ||
− | The product <math display="inline">{G}^T {M}^{-1} {G}</math> can be approximated by a laplacian matrix, i.e. | + | The product <math display="inline">{\bf G}^T {\bf M}^{-1} {\bf G}</math> can be approximated by a laplacian matrix, i.e. |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{G}^T {M}^{-1} {G}={1\over \rho } {L}\quad \hbox{with }L^{ab}=\int _{ \Omega ^e} {\boldsymbol \nabla }^T N^a {\boldsymbol \nabla } N^b d\Omega </math> | + | | style="text-align: center;" | <math>{\bf G}^T {\bf M}^{-1} {\bf G}={1\over \rho } {L}\quad \hbox{with }L^{ab}=\int _{ \Omega ^e} {\boldsymbol \nabla }^T N^a {\boldsymbol \nabla } N^b d\Omega </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (30) | | style="width: 5px;text-align: right;white-space: nowrap;" | (30) | ||
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The steps of the fractional step scheme chosen here are: | The steps of the fractional step scheme chosen here are: | ||
− | + | '''Step 1''' | |
The fractional nodal velocities <math display="inline">\tilde {v}^{n+1}</math> can be explicitely computed from Eq.(26) by | The fractional nodal velocities <math display="inline">\tilde {v}^{n+1}</math> can be explicitely computed from Eq.(26) by | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\tilde{v}^{n+1} = \bar{v}^{n} - \Delta t {M}^{-1}_d [\tilde{H}^{n}\bar{v}^{n} - \alpha {G} \bar{p}^n + {C}^{n}\bar{c}^{n} - {f}^{n}]</math> | + | | style="text-align: center;" | <math>\tilde{\bf v}^{n+1} = \bar{\bf v}^{n} - \Delta t {\bf M}^{-1}_d [\tilde{\bf H}^{n}\bar{\bf v}^{n} - \alpha {\bf G} \bar{\bf p}^n + {\bf C}^{n}\bar{\bf c}^{n} - {\bf f}^{n}]</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (31) | | style="width: 5px;text-align: right;white-space: nowrap;" | (31) | ||
|} | |} | ||
− | where <math display="inline">{M}_d</math> is the diagonal form of | + | where <math display="inline">{\bf M}_d</math> is the diagonal form of <math display="inline">{\bf M}</math> obtaining by lumping the row terms into the corresponding diagonal terms. |
<br/> | <br/> | ||
− | '''Step 2''' Compute <math display="inline">\bar{p}^{n+1}</math> from Eq.(29) as | + | '''Step 2''' Compute <math display="inline">\bar{\bf p}^{n+1}</math> from Eq.(29) as |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 558: | Line 539: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar{p}^{n+1}= -[\hat{L}^n + {\Delta t \over \rho } {L}]^{-1} [{G}^T\tilde{v}^{n+1} - \alpha {\Delta t \over \rho }{L}\bar{p}^n +{Q} \bar{\boldsymbol \pi }^{n}] </math> | + | | style="text-align: center;" | <math>\bar{\bf p}^{n+1}= -[\hat{\bf L}^n + {\Delta t \over \rho} {\bf L}]^{-1} [{\bf G}^T\tilde{\bf v}^{n+1} - \alpha {\Delta t \over \rho}{\bf L}\bar{\bf p}^n +{\bf Q} \bar{\boldsymbol \pi }^{n}] </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (32) | | style="width: 5px;text-align: right;white-space: nowrap;" | (32) | ||
|} | |} | ||
− | '''Step 3''' Compute <math display="inline"> \bar{v}^{n+1}</math> explicitely from Eq.(28a) as | + | '''Step 3''' Compute <math display="inline"> \bar{\bf v}^{n+1}</math> explicitely from Eq.(28a) as |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 570: | Line 551: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar {v}^{n+1}=\tilde{v}^{n+1}+ \Delta t {M}_d^{-1} {G} (\bar {p}^{n+1}- \alpha \bar {p}^n) </math> | + | | style="text-align: center;" | <math>\bar {\bf v}^{n+1}=\tilde{\bf v}^{n+1}+ \Delta t {\bf M}_d^{-1} {\bf G} (\bar {\bf p}^{n+1}- \alpha \bar {\bf p}^n) </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (33) | | style="width: 5px;text-align: right;white-space: nowrap;" | (33) | ||
Line 577: | Line 558: | ||
<br/> | <br/> | ||
− | '''Step 4''' Compute <math display="inline"> \bar{c}^{n+1}</math> explicitely from Eq.(28b) as | + | '''Step 4''' Compute <math display="inline"> \bar{\bf c}^{n+1}</math> explicitely from Eq.(28b) as |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 584: | Line 565: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar{c}^{n+1}=- {M}_d^{-1}\hat {C}^{n+1}\bar{v}^{n+1} </math> | + | | style="text-align: center;" | <math>\bar{\bf c}^{n+1}=- {\bf M}_d^{-1}\hat {\bf C}^{n+1}\bar{\bf v}^{n+1} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (34) | | style="width: 5px;text-align: right;white-space: nowrap;" | (34) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar{\boldsymbol \pi }^{n+1}=- \hat {M}_d^{-1} {Q}^T \bar {p}^{n+1} </math> | + | | style="text-align: center;" | <math>\bar{\boldsymbol \pi }^{n+1}=- \hat {\bf M}_d^{-1} {\bf Q}^T \bar {\bf p}^{n+1} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (35) | | style="width: 5px;text-align: right;white-space: nowrap;" | (35) | ||
|} | |} | ||
− | where <math display="inline">\hat {M}_d</math> is the lumped form of <math display="inline">\hat {M}</math>. A standard diagonal lumping procedure based in summing up the terms of each row has been used. | + | where <math display="inline">\hat {\bf M}_d</math> is the lumped form of <math display="inline">\hat {\bf M}</math>. A standard diagonal lumping procedure based in summing up the terms of each row has been used. |
Note that all steps can be solved explicitely except for the computation of the pressure in Eq.(32) which requires to invert the sum of two laplacian matrices. This can be effectively performed using an iterative solution scheme such as the conjugate gradient method. | Note that all steps can be solved explicitely except for the computation of the pressure in Eq.(32) which requires to invert the sum of two laplacian matrices. This can be effectively performed using an iterative solution scheme such as the conjugate gradient method. | ||
− | Above algorithm has improved stabilization properties versus the standard segregation methods due to the introduction of the laplacian matrix <math display="inline">\hat {L}</math> in Eq.(32). This matrix emanates from the FIC formulation. | + | Above algorithm has improved stabilization properties versus the standard segregation methods due to the introduction of the laplacian matrix <math display="inline">\hat {\bf L}</math> in Eq.(32). This matrix emanates from the FIC formulation. |
− | The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities <math display="inline">\tilde{v}^{n+1}</math> in Eq.(31). The prescribed velocities at the boundary are applied when solving for <math display="inline">\bar{v}^{n+1}</math> in step 3. The prescribed pressures at the boundary are imposed by making <math display="inline">\bar{p}^n</math> equal to the prescribed pressure values when solving Eq.(32). | + | The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities <math display="inline">\tilde{\bf v}^{n+1}</math> in Eq.(31). The prescribed velocities at the boundary are applied when solving for <math display="inline">\bar{\bf v}^{n+1}</math> in step 3. The prescribed pressures at the boundary are imposed by making <math display="inline">\bar{\bf p}^n</math> equal to the prescribed pressure values when solving Eq.(32). |
===3.2 Stokes flow=== | ===3.2 Stokes flow=== | ||
− | Many metal forming processes can be simulated with the assumption that the convective terms are negligible (Stokes flow) <span id='citeF-8'></span>[[#cite-8|8]]. The Stokes FIC formulation can be readily obtained simply by neglecting the convective terms in the Navier-Stokes formulation presented earlier. This also implies neglecting the convective stabilization terms in the momentum equations and, consequently, the convective projection variables are not larger necessary. Also the intrinsic time parameters <math display="inline">\tau _i</math> take now the simpler form (see Eq.(12)): | + | Many metal forming processes can be simulated with the assumption that the convective terms are negligible (Stokes flow) <span id='citeF-8'></span>[[#cite-8|[8,43,44]]]. The Stokes FIC formulation can be readily obtained simply by neglecting the convective terms in the Navier-Stokes formulation presented earlier. This also implies neglecting the convective stabilization terms in the momentum equations and, consequently, the convective projection variables are not larger necessary. Also the intrinsic time parameters <math display="inline">\tau _i</math> take now the simpler form (see Eq.(12)): |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 632: | Line 613: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\begin{array}{l}\displaystyle {M}\dot{\bar {v}} + {K}\bar{v} - {G}\bar {p}={f}\\ \displaystyle {G}^T \bar {v} + \hat{L}\bar {p}+{Q}\bar {\boldsymbol \pi }={0}\\ \displaystyle {Q}^T \bar {p} + \hat {M}\bar {\boldsymbol \pi }={0} \end{array} </math> | + | | style="text-align: center;" | <math>\begin{array}{l}\displaystyle {\bf M}\dot{\bar {\bf v}} + {\bf K}\bar{\bf v} - {\bf G}\bar {\bf p}={\bf f}\\ \displaystyle {\bf G}^T \bar {\bf v} + \hat{\bf L}\bar {\bf p}+{\bf Q}\bar {\boldsymbol \pi }={\bf 0}\\ \displaystyle {\bf Q}^T \bar {\bf p} + \hat {\bf M}\bar {\boldsymbol \pi }={\bf 0} \end{array} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (37) | | style="width: 5px;text-align: right;white-space: nowrap;" | (37) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\left[\begin{matrix}{K}&-{G}&{0}\\ -{G}^T & - \hat{L} &-{Q}\\ {0}& -{Q}^T&-\hat {M}\\\end{matrix}\right]\left\{\begin{matrix}\bar {v}\\ \bar {p}\\ \bar {\boldsymbol \pi }\\\end{matrix}\right\}= \left\{\begin{matrix}{f}\\ {0}\\{0}\\ \end{matrix}\right\} </math> | + | | style="text-align: center;" | <math>\left[\begin{matrix}{\bf K}&-{\bf G}&{\bf 0}\\ -{\bf G}^T & - \hat{\bf L} &-{\bf Q}\\ {\bf 0}& -{\bf Q}^T&-\hat {\bf M}\\\end{matrix}\right]\left\{\begin{matrix}\bar {\bf v}\\ \bar {\bf p}\\ \bar {\boldsymbol \pi }\\\end{matrix}\right\}= \left\{\begin{matrix}{\bf f}\\ {\bf 0}\\{\bf 0}\\ \end{matrix}\right\} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (38) | | style="width: 5px;text-align: right;white-space: nowrap;" | (38) | ||
|} | |} | ||
− | The system is symmetric and always positive definite and therefore leads to a non singular solution. This property holds for ''any interpolation function'' chosen for <math display="inline">\bar {v},\bar {p}</math> and <math display="inline">\bar {\boldsymbol \pi }</math>, therefore overcoming the Babuŝka-Brezzi (BB) restrictions <span id='citeF-8'></span>[[#cite-8|8]]. | + | The system is symmetric and always positive definite and therefore leads to a non singular solution. This property holds for ''any interpolation function'' chosen for <math display="inline">\bar {\bf v},\bar {\bf p}</math> and <math display="inline">\bar {\boldsymbol \pi }</math>, therefore overcoming the Babuŝka-Brezzi (BB) restrictions <span id='citeF-8'></span>[[#cite-8|[8]]]. |
− | A reduced velocity-pressure formulation can be obtained by elliminating the <math display="inline">\bar \pi </math> variables from the last row of Eq.(38) <span id='citeF- | + | A reduced velocity-pressure formulation can be obtained by elliminating the <math display="inline">\bar \pi </math> variables from the last row of Eq.(38) <span id='citeF-37'></span>[[#cite-37|[37]]]. |
The FIC formulation for Stokes flows is applicable for analysis of quasi/fully incompressible solids. An analogous formulation based on solid mechanics concepts is derived in Section 6. | The FIC formulation for Stokes flows is applicable for analysis of quasi/fully incompressible solids. An analogous formulation based on solid mechanics concepts is derived in Section 6. | ||
Line 663: | Line 644: | ||
The effect of temperature can be easily introduced by solving the equation for heat transport coupled to the fluid flow equations. | The effect of temperature can be easily introduced by solving the equation for heat transport coupled to the fluid flow equations. | ||
− | The equation of balance of heat is written in the FIC formulation as <span id='citeF- | + | The equation of balance of heat is written in the FIC formulation as <span id='citeF-30'></span>[[#cite-30|[30]]] |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 687: | Line 668: | ||
|} | |} | ||
− | In above <math display="inline">\phi </math> is the temperature, <math display="inline">c</math> and <math display="inline">k</math> are the specific heat and the thermal conductivity of the material, respectively, <math display="inline">Q</math> is the heat source and <math display="inline">h_j</math> are the characteristic length distances which are typical of the FIC formulation <span id='citeF- | + | In above <math display="inline">\phi </math> is the temperature, <math display="inline">c</math> and <math display="inline">k</math> are the specific heat and the thermal conductivity of the material, respectively, <math display="inline">Q</math> is the heat source and <math display="inline">h_j</math> are the characteristic length distances which are typical of the FIC formulation <span id='citeF-30'></span>[[#cite-30|[30]]]. |
− | Eq.(39a) is completed with the Dirichlet and Neuman boundary conditions for the heat problem. For details see <span id='citeF-35'></span>[[#cite-35|35]]. | + | Eq.(39a) is completed with the Dirichlet and Neuman boundary conditions for the heat problem. For details see <span id='citeF-35'></span>[[#cite-35|[35]]]. |
− | The convective velocities <math display="inline">v_i</math> in Eq.(39b) are provided by the solution of the fluid flow problem. As usual in metal forming processes, the heat source <math display="inline">Q</math> is a function of the mechanical work generated in the flow of the material during the forming process. The temperature field affects in turn the flow viscosity via its dependence with the yield stress which is very sensitive to the temperature changes. The solution of the heat transfer equation is therefore fully coupled with that of the fluid flow problem <span id='citeF- | + | The convective velocities <math display="inline">v_i</math> in Eq.(39b) are provided by the solution of the fluid flow problem. As usual in metal forming processes, the heat source <math display="inline">Q</math> is a function of the mechanical work generated in the flow of the material during the forming process. The temperature field affects in turn the flow viscosity via its dependence with the yield stress which is very sensitive to the temperature changes. The solution of the heat transfer equation is therefore fully coupled with that of the fluid flow problem <span id='citeF-30'></span>[[#cite-30|[30,46]]]. |
===4.2 Treatment of the free boundary motion=== | ===4.2 Treatment of the free boundary motion=== | ||
− | For the treatment of the free boundary we use a standard volume of fluid (VOF) technique, also known as pseudo-concentration method or level set technique <span id='citeF-22'></span>[[#cite-22|22]]. In the VOF method the motion of the free boudary is followed by solving the following transport equation (written using the FIC formulation) | + | For the treatment of the free boundary we use a standard volume of fluid (VOF) technique, also known as pseudo-concentration method or level set technique <span id='citeF-22'></span>[[#cite-22|[22,47,48]]]. In the VOF method the motion of the free boudary is followed by solving the following transport equation (written using the FIC formulation) |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 721: | Line 702: | ||
In above <math display="inline">\beta </math> is an auxiliary variable which takes a value equal to one on the free surface and zero elsewhere. The solution of Eq.(40a) in time in the discretized fluid-air domain allows to track the free surface which is characterized by the contours of <math display="inline">\beta </math> which have a unit value. | In above <math display="inline">\beta </math> is an auxiliary variable which takes a value equal to one on the free surface and zero elsewhere. The solution of Eq.(40a) in time in the discretized fluid-air domain allows to track the free surface which is characterized by the contours of <math display="inline">\beta </math> which have a unit value. | ||
− | The underlined term in Eq.(40a) emerges from the FIC formulation <span id='citeF- | + | The underlined term in Eq.(40a) emerges from the FIC formulation <span id='citeF-33'></span>[[#cite-33|[33]]]. Again, the <math display="inline">\bar h_j</math>'s in Eq.(40a) are characteristic length distances of the order of the element size. This term introduces the necessary stabilization in the transient solution of Eq.(40a). |
===4.3 Fully coupled algorithm for mould filling problems=== | ===4.3 Fully coupled algorithm for mould filling problems=== | ||
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* Compute the new position of the free surface <math display="inline">(\beta ^{n+1})</math> solving Eq.(40a). | * Compute the new position of the free surface <math display="inline">(\beta ^{n+1})</math> solving Eq.(40a). | ||
− | A number of implicit versions of the algorithm are possible and they all involve an iteration loop within each time step until convergence for the flow variables, the temperature and the free surface position is found. For details see <span id='citeF- | + | A number of implicit versions of the algorithm are possible and they all involve an iteration loop within each time step until convergence for the flow variables, the temperature and the free surface position is found. For details see <span id='citeF-47'></span>[[#cite-47|[47,48]]]. |
==5 COMPUTATION OF THE CHARACTERISTIC LENGTHS== | ==5 COMPUTATION OF THE CHARACTERISTIC LENGTHS== | ||
Line 745: | Line 726: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{h}=h_s {{v}\over {v}}+h_{c} {{\boldsymbol \nabla } v\over \vert{\boldsymbol \nabla }v\vert }</math> | + | | style="text-align: center;" | <math>{\bf h}=h_s {{\bf v}\over {v}}+h_{c} {{\boldsymbol \nabla } v\over \vert{\boldsymbol \nabla }v\vert }</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (41a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (41a) | ||
|} | |} | ||
− | where <math display="inline">v=\vert {v}\vert </math> and <math display="inline">h_s</math> and <math display="inline">h_{c}</math> are the “streamline” and “cross wind” contributions given by | + | where <math display="inline">v=\vert {\bf v}\vert </math> and <math display="inline">h_s</math> and <math display="inline">h_{c}</math> are the “streamline” and “cross wind” contributions given by |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 757: | Line 738: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>h_s=\max ({l}^T_j {v})/{v} </math> | + | | style="text-align: center;" | <math>h_s=\max ({l}^T_j {\bf v})/{v} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (41b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (41b) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>h_{c}=\max ({l}^T_j {\boldsymbol \nabla }v)/ \vert {\boldsymbol \nabla }v\vert \quad , \quad j=1,n_s</math> | + | | style="text-align: center;" | <math>h_{c}=\max ({\bf l}^T_j {\boldsymbol \nabla }v)/ \vert {\boldsymbol \nabla }v\vert \quad , \quad j=1,n_s</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (41c) | | style="width: 5px;text-align: right;white-space: nowrap;" | (41c) | ||
|} | |} | ||
− | where <math display="inline">{l}_j</math> are the vectors defining the element sides (<math display="inline">n_s=6</math> for tetrahedra). | + | where <math display="inline">{\bf l}_j</math> are the vectors defining the element sides (<math display="inline">n_s=6</math> for tetrahedra). |
− | As for the free surface equation the following value of the characteristic length vector <math display="inline">\bar{h}</math> in Eq.(40a) has been taken | + | As for the free surface equation the following value of the characteristic length vector <math display="inline">\bar{\bf h}</math> in Eq.(40a) has been taken |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 781: | Line 762: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar{h} =\bar h_s {{v}\over {v}}+\bar h_c {{\boldsymbol \nabla }\beta \over \vert {\boldsymbol \nabla }\beta \vert } </math> | + | | style="text-align: center;" | <math>\bar{\bf h} =\bar h_s {{\bf v}\over {v}}+\bar h_c {{\boldsymbol \nabla }\beta \over \vert {\boldsymbol \nabla }\beta \vert } </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (42a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (42a) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar h_c = \max [{l}_j^T {\boldsymbol \nabla }\beta ] {1\over \vert {\boldsymbol \nabla }\beta \vert } \quad ,\quad j=1,2,3</math> | + | | style="text-align: center;" | <math>\bar h_c = \max [{\bf l}_j^T {\boldsymbol \nabla }\beta ] {1\over \vert {\boldsymbol \nabla }\beta \vert } \quad ,\quad j=1,2,3</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (42b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (42b) | ||
|} | |} | ||
− | The cross-wind terms in eqs.(41a) and (42a) account for the effect of the gradient of the solution in the stabilization parameters. This is a standard assumption in most “shock-capturing” stabilization procedures <span id='citeF-8'></span>[[#cite-8|8]]. | + | The cross-wind terms in eqs.(41a) and (42a) account for the effect of the gradient of the solution in the stabilization parameters. This is a standard assumption in most “shock-capturing” stabilization procedures <span id='citeF-8'></span>[[#cite-8|[8,24]]]. |
==6 FIC FORMULATION FOR QUASI/FULLY INCOMPRESSIBLE SOLIDS== | ==6 FIC FORMULATION FOR QUASI/FULLY INCOMPRESSIBLE SOLIDS== | ||
Line 804: | Line 785: | ||
===6.1 Equilibrium equations=== | ===6.1 Equilibrium equations=== | ||
− | As usual the governing equations of solid mechanics are written in a Lagrangian reference frame. Following the arguments of Section 2 the equilibrium equations for a solid are written using the FIC technique as <span id='citeF- | + | As usual the governing equations of solid mechanics are written in a Lagrangian reference frame. Following the arguments of Section 2 the equilibrium equations for a solid are written using the FIC technique as <span id='citeF-37'></span>[[#cite-37|[37]]] |
<span id="eq-43"></span> | <span id="eq-43"></span> | ||
Line 919: | Line 900: | ||
where <math display="inline">K</math> is the bulk modulus of the material and <math display="inline">p^{av}</math> is the average value of the pressure over domain <math display="inline">V</math>. | where <math display="inline">K</math> is the bulk modulus of the material and <math display="inline">p^{av}</math> is the average value of the pressure over domain <math display="inline">V</math>. | ||
− | The value of <math display="inline">p^{av}</math> can be approximated as <span id='citeF- | + | The value of <math display="inline">p^{av}</math> can be approximated as <span id='citeF-37'></span>[[#cite-37|[37]]] |
<span id="eq-51"></span> | <span id="eq-51"></span> | ||
Line 974: | Line 955: | ||
|} | |} | ||
− | Eq.(54) expresses the limit incompressible behaviour of the solid. This equation is identical to Eq.(2) for incompressible flow problems and there arises from the mass continuity condition <span id='citeF- | + | Eq.(54) expresses the limit incompressible behaviour of the solid. This equation is identical to Eq.(2) for incompressible flow problems and there arises from the mass continuity condition <span id='citeF-30'></span>[[#cite-30|[30,32]]]. |
By combining Eqs. ([[#eq-43|43]]), ([[#eq-44|44]]), ([[#eq-47|47]]), ([[#eq-48|48]]) and ([[#eq-53|53]]) a mixed displacement–pressure formulation can be written as | By combining Eqs. ([[#eq-43|43]]), ([[#eq-44|44]]), ([[#eq-47|47]]), ([[#eq-48|48]]) and ([[#eq-53|53]]) a mixed displacement–pressure formulation can be written as | ||
Line 1,001: | Line 982: | ||
|} | |} | ||
− | From the observation of Eq.([[#eq-55|55]]) we can obtain after some algebra <span id='citeF- | + | From the observation of Eq.([[#eq-55|55]]) we can obtain after some algebra <span id='citeF-37'></span>[[#cite-37|[37]]] |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 1,074: | Line 1,055: | ||
* The stabilization terms have been neglected in Eq.(60). These terms were useful to derive the stabilized form of Eq.(61). However they are not longer necessary as the convective terms do not appear in the equilibrium equations. | * The stabilization terms have been neglected in Eq.(60). These terms were useful to derive the stabilized form of Eq.(61). However they are not longer necessary as the convective terms do not appear in the equilibrium equations. | ||
− | * The constitutive equation for the pressure (Eq.(61)) has been written in an incremental form. This is more convenient for non linear material behaviour typical of metal forming situations. Here, appropriate elasto-plastic or elasto-viscoplastic constitutive laws relating stresses and strains in incremental or rate forms must be used <span id='citeF-1'></span>[[#cite-1|1]]. | + | * The constitutive equation for the pressure (Eq.(61)) has been written in an incremental form. This is more convenient for non linear material behaviour typical of metal forming situations. Here, appropriate elasto-plastic or elasto-viscoplastic constitutive laws relating stresses and strains in incremental or rate forms must be used <span id='citeF-1'></span>[[#cite-1|[1-5,43,44]]]. |
===6.3 Finite element equations=== | ===6.3 Finite element equations=== | ||
Line 1,119: | Line 1,100: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{M} \ddot{\bar {u}} + {g} - {f}={0}</math> | + | | style="text-align: center;" | <math>{\bf M} \ddot{\bar {\bf u}} + {\bf g} - {\bf f}={\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (64a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (64a) | ||
Line 1,129: | Line 1,110: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{G}^T \Delta \bar {u} +{C} \Delta \bar {p}+ {L} \bar {p} + {Q}\bar {\boldsymbol \pi }= {0}</math> | + | | style="text-align: center;" | <math>{\bf G}^T \Delta \bar {\bf u} +{\bf C} \Delta \bar {\bf p}+ {\bf L} \bar {\bf p} + {\bf Q}\bar {\boldsymbol \pi }= {\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (64b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (64b) | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{Q}^T \bar {p}+ \bar {C} \bar {\boldsymbol \pi }= {0}</math> | + | | style="text-align: center;" | <math>{\bf Q}^T \bar {\bf p}+ \bar {\bf C} \bar {\boldsymbol \pi }= {\bf 0}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (64c) | | style="width: 5px;text-align: right;white-space: nowrap;" | (64c) | ||
Line 1,163: | Line 1,144: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{g}= \int _\Omega {B}^T {\boldsymbol \sigma } d\Omega </math> | + | | style="text-align: center;" | <math>{\textbf g}= \int _\Omega {B}^T {\boldsymbol \sigma } d\Omega </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (66) | | style="width: 5px;text-align: right;white-space: nowrap;" | (66) | ||
Line 1,170: | Line 1,151: | ||
is the internal nodal force vector and the rest of matrices and vectors are defined in the Appendix. Note that the expression of <math display="inline">{g}</math> of eq.(66) is adequate for non linear analysis. | is the internal nodal force vector and the rest of matrices and vectors are defined in the Appendix. Note that the expression of <math display="inline">{g}</math> of eq.(66) is adequate for non linear analysis. | ||
− | A four steps semi-explicit time integration algorithm can be derived as follows | + | A four steps semi-explicit time integration algorithm can be derived as follows: |
− | |||
− | '''Step | + | '''Step 1.''' Compute the nodal velocities <math display="inline">\dot{\bar {\textbf u}}^{n+1/2}</math> from Eq.(64a) |
− | + | ||
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 1,182: | Line 1,161: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math> \dot{\bar {u}}^{n+1/2} = \dot{\bar {u}}^{n-1/2}+\Delta t {M}^{-1}_d ({f}^n - {g}^n)</math> | + | | style="text-align: center;" | <math> \dot{\bar {\textbf u}}^{n+1/2} = \dot{\bar {\textbf u}}^{n-1/2}+\Delta t {\textbf M}^{-1}_d ({\textbf f}^n - {\textbf g}^n)</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (67a) | | style="width: 5px;text-align: right;white-space: nowrap;" | (67a) | ||
|} | |} | ||
− | + | '''Step 2'''. Compute the nodal displacements <math display="inline">\bar {\textbf u}^{n+1}</math> | |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 1,194: | Line 1,173: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar {u}^{n+1} = \bar{u}^n + \Delta t \dot{\bar {u}}^{n+1/2} </math> | + | | style="text-align: center;" | <math>\bar {\textbf u}^{n+1} = \bar{\textbf u}^n + \Delta t \dot{\bar {\textbf u}}^{n+1/2} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (67b) | | style="width: 5px;text-align: right;white-space: nowrap;" | (67b) | ||
|} | |} | ||
− | + | '''Step 3'''. Compute the nodal pressures <math display="inline">\bar {\textbf p}^{n+1}</math> from Eq.(64b) | |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 1,206: | Line 1,185: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar {p}^{n+1}=-[ {C} + {L}]^{-1} [\Delta t {G}^T \dot{\bar {u}}^{n+1/2}- {C} \bar {p}^n + {Q} \bar {\boldsymbol \pi }^n]</math> | + | | style="text-align: center;" | <math>\bar {\textbf p}^{n+1}=-[ {\textbf C} + {\textbf L}]^{-1} [\Delta t {\textbf G}^T \dot{\bar {\textbf u}}^{n+1/2}- {\textbf C} \bar {\textbf p}^n + {\textbf Q} \bar {\boldsymbol \pi }^n]</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (67c) | | style="width: 5px;text-align: right;white-space: nowrap;" | (67c) | ||
|} | |} | ||
− | + | '''Step 4'''. Compute the nodal projected pressure gradients <math display="inline">\bar {\boldsymbol \pi }^{n+1}</math> from Eq.(64c) | |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 1,218: | Line 1,197: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar {\boldsymbol \pi }^{n+1}=- \bar {C}_d^{-1} {Q}^T \bar {p}^{n+1}</math> | + | | style="text-align: center;" | <math>\bar {\boldsymbol \pi }^{n+1}=- \bar {\textbf C}_d^{-1} {\textbf Q}^T \bar {\textbf p}^{n+1}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (67d) | | style="width: 5px;text-align: right;white-space: nowrap;" | (67d) | ||
Line 1,230: | Line 1,209: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{g}^n =\int _{\Omega ^e} [{B}^T {\boldsymbol \sigma }]^n\,d\Omega </math> | + | | style="text-align: center;" | <math>{\textbf g}^n =\int _{\Omega ^e} [{B}^T {\boldsymbol \sigma }]^n\,d\Omega </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (68) | | style="width: 5px;text-align: right;white-space: nowrap;" | (68) | ||
Line 1,237: | Line 1,216: | ||
where the stresses <math display="inline">{\boldsymbol \sigma }^n</math> are obtained by consistent integration of the adequate (non linear) constitutive law. | where the stresses <math display="inline">{\boldsymbol \sigma }^n</math> are obtained by consistent integration of the adequate (non linear) constitutive law. | ||
− | Note that steps 1, 2 and 4 are ''fully explicit'' as a diagonal form of matrices <math display="inline">{M}</math> and <math display="inline">\bar{C}</math> has been chosen. The solution of step 3 requires invariably the inverse of a Laplacian matrix. This can be an inexpensive process using an iterative equation solution method (e.g. a preconditioned conjugate gradient method). | + | Note that steps 1, 2 and 4 are ''fully explicit'' as a diagonal form of matrices <math display="inline">{\textbf M}</math> and <math display="inline">\bar{\textbf C}</math> has been chosen. The solution of step 3 requires invariably the inverse of a Laplacian matrix. This can be an inexpensive process using an iterative equation solution method (e.g. a preconditioned conjugate gradient method). |
− | For the full incompressible case <math display="inline">K=\infty </math> and <math display="inline">{C}=0 </math> in all above equations. | + | For the full incompressible case <math display="inline">K=\infty </math> and <math display="inline">{\textbf C}=0 </math> in all above equations. |
− | The critical time step <math display="inline">\Delta t</math> is taken as that of the standard explicit dynamic scheme (see Section 6.5 and <span id='citeF-9'></span>[[#cite-9|9]]). | + | The critical time step <math display="inline">\Delta t</math> is taken as that of the standard explicit dynamic scheme (see Section 6.5 and <span id='citeF-9'></span>[[#cite-9|[9,37]]]). |
− | + | '''''Fully explicit algorithm''''' | |
A fully explicit four steps algorithm can be obtained by computing <math display="inline">\bar {p}^{n+1}</math> from step 3 in eq.(67c) as follows | A fully explicit four steps algorithm can be obtained by computing <math display="inline">\bar {p}^{n+1}</math> from step 3 in eq.(67c) as follows | ||
Line 1,252: | Line 1,231: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\bar{p}^{n+1}= - {C}_d^{-1}[\Delta t {G}^T \dot{\bar {u}}^{n+1/2} - ({C}_d - {L}) \bar{p}^n + {Q} \bar{\boldsymbol \pi }^n] </math> | + | | style="text-align: center;" | <math>\bar{\textbf p}^{n+1}= - {\textbf C}_d^{-1}[\Delta t {\textbf G}^T \dot{\bar {\textbf u}}^{n+1/2} - ({\textbf C}_d - {\textbf L}) \bar{\textbf p}^n + {\textbf Q} \bar{\boldsymbol \pi }^n] </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (69) | | style="width: 5px;text-align: right;white-space: nowrap;" | (69) | ||
|} | |} | ||
− | Note that the explicit algorithm is not applicable in the full incompressible limit as the solution of Eq.(69) breaks down for <math display="inline">K=\infty </math> and <math display="inline">{C}=0</math>. The explicit form can however be used with success in problems where quasi-incompressible regions exist adjacent to standard “compressible” zones. Examples of this kind are shown in the next section. In both cases the semi-explicit and fully explicit schemes gave identical results with important savings in both computer time and memory storage requirements obtained when using the explicit form. | + | Note that the explicit algorithm is not applicable in the full incompressible limit as the solution of Eq.(69) breaks down for <math display="inline">K=\infty </math> and <math display="inline">{\textbf C}=0</math>. The explicit form can however be used with success in problems where quasi-incompressible regions exist adjacent to standard “compressible” zones. Examples of this kind are shown in the next section. In both cases the semi-explicit and fully explicit schemes gave identical results with important savings in both computer time and memory storage requirements obtained when using the explicit form. |
===6.4 Thermal coupled effects=== | ===6.4 Thermal coupled effects=== | ||
Line 1,277: | Line 1,256: | ||
Note that the convective terms do not enter into Eq.(70) This also eliminates the need to stabilize the numerical solution. | Note that the convective terms do not enter into Eq.(70) This also eliminates the need to stabilize the numerical solution. | ||
− | The coupled thermal-mechanical problem requires the computation of the temperature <math display="inline">\phi </math> at each time step using a transient solution scheme <span id='citeF-49'></span>[[#cite-49| | + | The coupled thermal-mechanical problem requires the computation of the temperature <math display="inline">\phi </math> at each time step using a transient solution scheme <span id='citeF-49'></span>[[#cite-49|[47,48,50]]]. |
− | + | '''Remark'''. Above formulation is similar to that developed by Chiumenti ''et al.'' <span id='citeF-27'></span>[[#cite-27|[27,29]]] and Cervera ''et al.'' <span id='citeF-28'></span>[[#cite-28|[28]]] for analysis of incompressible problems in solid mechanics using a sub-grid scale approach. | |
− | + | ||
− | Above formulation is similar to that developed by Chiumenti ''et al.'' <span id='citeF- | + | |
===6.5 Computation of the intrinsic time parameter for quasi/fully incompressible solids=== | ===6.5 Computation of the intrinsic time parameter for quasi/fully incompressible solids=== | ||
Line 1,299: | Line 1,276: | ||
where <math display="inline">V^{(e)}</math> is the element volume (or the element area for 2D problems). This expression for <math display="inline">\tau ^{(e)}</math> does not take into account the element distorsions along a particular direction during the deformation process. | where <math display="inline">V^{(e)}</math> is the element volume (or the element area for 2D problems). This expression for <math display="inline">\tau ^{(e)}</math> does not take into account the element distorsions along a particular direction during the deformation process. | ||
− | The correct value of the shear modulus in the expression of <math display="inline">\tau ^{(e)}</math> is another sensitive issue as, obviously, for non linear problems the value of <math display="inline">G</math> will differ from the elastic modulus. This fact has been identified by Cervera ''et al.'' <span id='citeF- | + | The correct value of the shear modulus in the expression of <math display="inline">\tau ^{(e)}</math> is another sensitive issue as, obviously, for non linear problems the value of <math display="inline">G</math> will differ from the elastic modulus. This fact has been identified by Cervera ''et al.'' <span id='citeF-28'></span>[[#cite-28|[28]]] for non linear analysis of incompressible problems using linear triangles. |
A useful alternative to compute <math display="inline">\tau ^{(e)}</math> for explicit non linear transient situations is to make use of the value of the speed of sound in an elastic solid, defined by | A useful alternative to compute <math display="inline">\tau ^{(e)}</math> for explicit non linear transient situations is to make use of the value of the speed of sound in an elastic solid, defined by | ||
Line 1,313: | Line 1,290: | ||
|} | |} | ||
− | where <math display="inline">E</math> is the Young modulus. The stability condition for explicit dynamic computations is given by the Courant condition defined as <span id='citeF-8'></span>[[#cite-8|8]] | + | where <math display="inline">E</math> is the Young modulus. The stability condition for explicit dynamic computations is given by the Courant condition defined as <span id='citeF-8'></span>[[#cite-8|[8]]] |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
Line 1,345: | Line 1,322: | ||
===7.1 Aluminium casting simulation=== | ===7.1 Aluminium casting simulation=== | ||
− | A numerical simulation of an aluminium casting process is presented as a demonstration of the accuracy of the stabilized formulation. The computations are performed with the finite element code VULCAN where the stabilized FEM presented has been implemented [56]. | + | A numerical simulation of an aluminium casting process is presented as a demonstration of the accuracy of the stabilized formulation. The computations are performed with the finite element code VULCAN where the stabilized FEM presented has been implemented <span id='citeF-56'></span>[[#cite-56|[56]]]. |
− | The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Material behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mould has been modeled by a simpler thermo-elastic model <span id='citeF- | + | The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Material behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mould has been modeled by a simpler thermo-elastic model <span id='citeF-51'></span>[[#cite-51|[51]]]. Geometrical and material data were provided by the foundry RUFFINI. Figure 1 shows the finite element mesh used for the part and the cooling system. The full mesh, including the mould has 380.000 four node tetrahedra. The pouring temperature is 650<math display="inline">^\circ </math>C. Initial temperature for the mould is obtained through a thermal die-cycling simulation. Figure 2 shows the evolution of the mould temperature after 6 cycles. The cooling system has been kept at 20<math display="inline">^\circ </math>C. Filling evolution has been simulated as in a pressure die-casting process using the stabilized VOF technique described in <span id='citeF-49'></span>[[#cite-49|[47,48]]]. Figure 3 shows different time steps of the simulation. |
<div id='img-1'></div> | <div id='img-1'></div> | ||
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[ | + | |[[File:Draft_Samper_805805629_6310_Fig1cast.png|351px|Finite element discretization of the aluminium casting.]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 1:''' Finite element discretization of the aluminium casting. | | colspan="1" | '''Figure 1:''' Finite element discretization of the aluminium casting. | ||
Line 1,360: | Line 1,337: | ||
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[ | + | |[[File:Draft_Samper_805805629_9923_Fig2-cast.png|351px|Temperature die-cycling.]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 2:''' Temperature die-cycling. | | colspan="1" | '''Figure 2:''' Temperature die-cycling. | ||
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The final temperature field obtained after the filling simulation is taken as the initial condition for the solidification and cooling analysis. Temperature and liquid-fraction distributions during solidification are shown in Figures 4 and 5, respectively. The heat transfer coefficient takes into account the air-gap resistance due to the casting shrinkage during the solidification process. Figure 6 shows volumetric and von Mises deviatoric stress distributions in a x-y section. The figures also show the air-gap between the part and the mould, responsible of a non-uniform heat flux at the contact interface. | The final temperature field obtained after the filling simulation is taken as the initial condition for the solidification and cooling analysis. Temperature and liquid-fraction distributions during solidification are shown in Figures 4 and 5, respectively. The heat transfer coefficient takes into account the air-gap resistance due to the casting shrinkage during the solidification process. Figure 6 shows volumetric and von Mises deviatoric stress distributions in a x-y section. The figures also show the air-gap between the part and the mould, responsible of a non-uniform heat flux at the contact interface. | ||
− | Other examples of application of the stabilized formulation to casting problems can be found in <span id='citeF- | + | Other examples of application of the stabilized formulation to casting problems can be found in <span id='citeF-47'></span>[[#cite-47|[47-51]]]. |
<div id='img-3'></div> | <div id='img-3'></div> | ||
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{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[ | + | |[[File:Draft_Samper_805805629_9790_Fig6.png|351px|Aluminium casting. Stress-trace and von Mises deviatoric stress indicator during phase-change (plane xy).]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 6:''' Aluminium casting. Stress-trace and von Mises deviatoric stress indicator during phase-change (plane xy). | | colspan="1" | '''Figure 6:''' Aluminium casting. Stress-trace and von Mises deviatoric stress indicator during phase-change (plane xy). | ||
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|- | |- | ||
|[[Image:Draft_Samper_805805629-cylt-linia.png|300px|]] | |[[Image:Draft_Samper_805805629-cylt-linia.png|300px|]] | ||
− | |[[ | + | |[[File:Draft_Samper_805805629_8020_Fig12b.png|350px]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="2" | '''Figure 12:''' a) Definition of the line for comparison of pressure distribution b) Pressure distribution along the line <math>ABCDEA</math> | | colspan="2" | '''Figure 12:''' a) Definition of the line for comparison of pressure distribution b) Pressure distribution along the line <math>ABCDEA</math> | ||
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|[[Image:Draft_Samper_805805629-bext-geo.png|330px|]] | |[[Image:Draft_Samper_805805629-bext-geo.png|330px|]] | ||
|[[Image:Draft_Samper_805805629-bext4a-quad-nurbs-sprn9-eps.png|266px|]] | |[[Image:Draft_Samper_805805629-bext4a-quad-nurbs-sprn9-eps.png|266px|]] | ||
+ | |[[Image:Draft_Samper_805805629-bext4a_036-eps.png|279px|]] | ||
|- | |- | ||
− | | colspan="2"| | + | |(a) |
+ | |colspan="2" |(b) | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan=" | + | | colspan="3" | '''Figure 13:''' Backward extrusion a) geometry definition. Final deformed shape with effective plastic strain distribution; b) solution with quadrilaterals and mixed formulation; c) solution with triangles and the FIC algorithm |
|} | |} | ||
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|[[Image:Draft_Samper_805805629-bext4a_002-eps.png|330px|]] | |[[Image:Draft_Samper_805805629-bext4a_002-eps.png|330px|]] | ||
|[[Image:Draft_Samper_805805629-bext4a_014-eps.png|330px|]] | |[[Image:Draft_Samper_805805629-bext4a_014-eps.png|330px|]] | ||
− | + | |[[Image:Draft_Samper_805805629-bext4a_022-eps.png|330px|]] | |
− | + | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan=" | + | | colspan="3" | '''Figure 14:''' Backward extrusion – deformed shapes with effective plastic strain distribution at different stages of forming. Solution with triangles and the FIC algorithm |
|} | |} | ||
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{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[Image:Draft_Samper_805805629-hose-clip.png| | + | |[[Image:Draft_Samper_805805629-hose-clip.png|530px|A hose clamp ]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 15:''' A hose clamp | | colspan="1" | '''Figure 15:''' A hose clamp | ||
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{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[Image:Draft_Samper_805805629-p1222e_ini.png| | + | |[[Image:Draft_Samper_805805629-p1222e_ini.png|410px|A hose clamp – initial set-up ]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 16:''' A hose clamp – initial set-up | | colspan="1" | '''Figure 16:''' A hose clamp – initial set-up | ||
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{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[Image:Draft_Samper_805805629-p1222d_12-eps.png| | + | |[[Image:Draft_Samper_805805629-p1222d_12-eps.png|470px|]] |
− | |[[Image:Draft_Samper_805805629-p1222e_29-eps.png| | + | |- |
+ | |[[Image:Draft_Samper_805805629-p1222e_29-eps.png|440px|A hose clamp – deformed shapes at different stages of forming with distribution of effective plastic strain ]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | + | | '''Figure 17:''' A hose clamp – deformed shapes at different stages of forming with distribution of effective plastic strain | |
|} | |} | ||
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{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[Image:Draft_Samper_805805629-clamp-detail.png| | + | |[[Image:Draft_Samper_805805629-clamp-detail.png|460px|Detail of the deformed shape with finite element discretization and distribution of effective plastic strain ]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 18:''' Detail of the deformed shape with finite element discretization and distribution of effective plastic strain | | colspan="1" | '''Figure 18:''' Detail of the deformed shape with finite element discretization and distribution of effective plastic strain | ||
|} | |} | ||
− | + | '''Remark'''. The examples presented show that the FIC/FEM formulation is an effective procedure for solving bulk metal forming problems involving full or quasi-incompressible situations. The key advantage of the FIC approach versus more standard mixed FEM formulations is that it provides a natural theoretical framework for equal order finite element interpolations for the velocity and pressure variables, both in the context of implicit and explicit solution schemes. We note the simplicity and effectiveness of the full explicit algorithm as demonstrated in the examples presented. The FIC formulation reproduces also the best feature of the so called stabilized FEM method for incompressible problems, such as the CBS scheme <span id='citeF-20'></span>[[#cite-20|[20,21,25,26,58]]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|[22]]] and the subgrid scale method <span id='citeF-23'></span>[[#cite-23|[23]]] among others. | |
− | + | ||
− | The examples presented show that the FIC/FEM formulation is an effective procedure for solving bulk metal forming problems involving full or quasi-incompressible situations. The key advantage of the FIC approach versus more standard mixed FEM formulations is that it provides a natural theoretical framework for equal order finite element interpolations for the velocity and pressure variables, both in the context of implicit and explicit solution schemes. We note the simplicity and effectiveness of the full explicit algorithm as demonstrated in the examples presented. The FIC formulation reproduces also the best feature of the so called stabilized FEM method for incompressible problems, such as the CBS scheme <span id='citeF-20'></span>[[#cite-20|20]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|22]] and the subgrid scale method <span id='citeF-23'></span>[[#cite-23|23]] among others. | + | |
==8 LAGRANGIAN FLOWS. THE PARTICLE FINITE ELEMENT METHOD== | ==8 LAGRANGIAN FLOWS. THE PARTICLE FINITE ELEMENT METHOD== | ||
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The Lagrangian fluid flow equations can be simply obtained by noting that the velocity of the mesh nodes and that of the fluid particles are the same. Hence the the convective terms vanish in the momentum equations, while the rest of the fluid flow equations remain unchanged. The resulting governing equations have an identical form as those of the Stokes flow problem, with the motion of the flow particles being referred now to a Lagrangian coordinate frame. | The Lagrangian fluid flow equations can be simply obtained by noting that the velocity of the mesh nodes and that of the fluid particles are the same. Hence the the convective terms vanish in the momentum equations, while the rest of the fluid flow equations remain unchanged. The resulting governing equations have an identical form as those of the Stokes flow problem, with the motion of the flow particles being referred now to a Lagrangian coordinate frame. | ||
− | The FEM algorithms for solving the Lagrangian flow equations are very similar to those presented for incompressible solids in a previous section. Here we present a particular class of Lagrangian formulation called the ''particle finite element method'' (PFEM) <span id='citeF- | + | The FEM algorithms for solving the Lagrangian flow equations are very similar to those presented for incompressible solids in a previous section. Here we present a particular class of Lagrangian formulation called the ''particle finite element method'' (PFEM) <span id='citeF-38'></span>[[#cite-38|[38-41]]]. The PFEM treats the mesh nodes in the fluid and solid domains as dimensionless particles which can freely move an even separate from the main fluid domain representing, for instance, the effect of fluid drops. A finite element mesh connects the nodes defining the discretized domain where the governing equations are solved in the standard FEM fashion. |
The quality of the numerical solution depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur. | The quality of the numerical solution depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur. | ||
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|- | |- | ||
| style="text-align: center;" | <math> | | style="text-align: center;" | <math> | ||
− | + | \tilde{\bf v}^{n+1,i+1} = \bar{\bf v}^n - \Delta t {\bf M}^{-1}_d [{\bf K} \bar{\bf v}^{n} - {\bf G}\bar {\bf p}^{n} -{\bf f}^{n+1}] </math> | |
− | \tilde{v}^{n+1,i+1} = \bar{v}^n - \Delta t {M}^{-1}_d [{K} \bar{v}^{n} - {G}\bar {p}^{n} -{f}^{n+1}] </math> | + | |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (75) | | style="width: 5px;text-align: right;white-space: nowrap;" | (75) | ||
|} | |} | ||
− | ''Step 3.2.'' Compute <math display="inline">\bar {p}^{n+1,i+1}</math> from Eq.(32) for <math display="inline">\alpha =1</math>. | + | ''Step 3.2.'' Compute <math display="inline">\bar {\textbf p}^{n+1,i+1}</math> from Eq.(32) for <math display="inline">\alpha =1</math>. |
− | ''Step 3.3.'' Compute explicitely <math display="inline"> \bar{v}^{n+1,i+1}</math> from Eq.(33) with <math display="inline">\alpha =1</math>. | + | ''Step 3.3.'' Compute explicitely <math display="inline"> \bar{\textbf v}^{n+1,i+1}</math> from Eq.(33) with <math display="inline">\alpha =1</math>. |
''Step 3.4.'' Compute <math display="inline"> \bar{\boldsymbol \pi }^{n+1,i+1}</math> explicitely from Eq.(35). | ''Step 3.4.'' Compute <math display="inline"> \bar{\boldsymbol \pi }^{n+1,i+1}</math> explicitely from Eq.(35). | ||
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|- | |- | ||
| style="text-align: center;" | <math> | | style="text-align: center;" | <math> | ||
− | + | {\bf x}_j^{n+1,i+1} = {\bf x}_j^{n}+\bar {\bf u}_j^{n+1,i+1} \Delta t</math> | |
− | {x}_j^{n+1,i+1} = {x}_j^{n}+\bar {u}_j^{n+1,i+1} \Delta t </math> | + | |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (76) | | style="width: 5px;text-align: right;white-space: nowrap;" | (76) | ||
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</ol> | </ol> | ||
− | Above algorithm can be found to be analogous to the standard ''updated lagrangian'' scheme typically used in non linear solid mechanics problems <span id='citeF-8'></span>[[#cite-8|8]]. | + | Above algorithm can be found to be analogous to the standard ''updated lagrangian'' scheme typically used in non linear solid mechanics problems <span id='citeF-8'></span>[[#cite-8|[8,57]]]. |
Despite the motion of the nodes within the iterative process, in general there is no need to regenerate the mesh at each iteration. ''In the examples presented in the paper the mesh in the fluid domain has been regenerated at each time step''. A cheaper alternative is to generate a new mesh only after a prescribed number of converged time steps, or when the nodal displacements induce significant geometrical distorsions in some elements. | Despite the motion of the nodes within the iterative process, in general there is no need to regenerate the mesh at each iteration. ''In the examples presented in the paper the mesh in the fluid domain has been regenerated at each time step''. A cheaper alternative is to generate a new mesh only after a prescribed number of converged time steps, or when the nodal displacements induce significant geometrical distorsions in some elements. | ||
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{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\Delta t =\min (\Delta t_i ) \quad \hbox{with}\quad \Delta t_i ={\vert {v}\vert \over h_i^{\min }} </math> | + | | style="text-align: center;" | <math>\Delta t =\min (\Delta t_i ) \quad \hbox{with}\quad \Delta t_i ={\vert {\bf v}\vert \over h_i^{\min }} </math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (77) | | style="width: 5px;text-align: right;white-space: nowrap;" | (77) | ||
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===8.2 Treatment of contact between fluid and solid interfaces=== | ===8.2 Treatment of contact between fluid and solid interfaces=== | ||
− | The condition of prescribed velocities or pressures at the solid boundaries in the PFEM are applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. In some problems it is useful to define a layer of nodes adjacent to the external boundary in the fluid where the condition of prescribed velocity is imposed. These nodes typically remain fixed during the solution process. Contact between liquid particles and the solid boundaries is accounted for by the incompressibility condition which ''naturally prevents the liquid nodes to penetrate into the solid boundaries'' <span id='citeF- | + | The condition of prescribed velocities or pressures at the solid boundaries in the PFEM are applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. In some problems it is useful to define a layer of nodes adjacent to the external boundary in the fluid where the condition of prescribed velocity is imposed. These nodes typically remain fixed during the solution process. Contact between liquid particles and the solid boundaries is accounted for by the incompressibility condition which ''naturally prevents the liquid nodes to penetrate into the solid boundaries'' <span id='citeF-38'></span>[[#cite-38|[38-41]]]. |
===8.3 Generation of a new mesh=== | ===8.3 Generation of a new mesh=== | ||
− | One of the key points for the success of the PFEM is the fast regeneration of a finite element mesh at every time step on the basis of the position of the nodes in the space domain. In our work the mesh is typically generated using the so called extended Delaunay tesselation (EDT) <span id='citeF- | + | One of the key points for the success of the PFEM is the fast regeneration of a finite element mesh at every time step on the basis of the position of the nodes in the space domain. In our work the mesh is typically generated using the so called extended Delaunay tesselation (EDT) <span id='citeF-38'></span>[[#cite-38|[38,52]]]. The EDT allows one to generate non standard meshes combining elements of arbitrary polyhedrical shapes (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrary polyhedra in 3D) in a computing time of order <math display="inline">n</math>, where <math display="inline">n</math> is the total number of nodes in the mesh. The <math display="inline">C^\circ </math> continuous shape functions of the elements are obtained using the so called meshless finite element interpolation (MFEM) <span id='citeF-53'></span>[[#cite-53|[53]]]. |
===8.4 Identification of boundary surfaces=== | ===8.4 Identification of boundary surfaces=== | ||
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The PFEM requires the correct definition of the boundary domain. Sometimes, boundary nodes are clearly distinguished from internal nodes. In other cases, the total set of nodes is the only information available and the algorithm must recognize the boundary nodes. | The PFEM requires the correct definition of the boundary domain. Sometimes, boundary nodes are clearly distinguished from internal nodes. In other cases, the total set of nodes is the only information available and the algorithm must recognize the boundary nodes. | ||
− | Considering that the nodes follow a variable <math display="inline">h(x)</math> distribution, where <math display="inline">h(x)</math> is the minimum distance between two nodes, the following criterion has been used. For each two nodes (three nodes in 3D) a circle (a sphere in 3D) of radius equal to <math display="inline">\alpha h</math> containing the nodes is plotted. ''All nodes laying on a circle (sphere) with a radius greater than <math>\alpha h</math>, not containing any other node in the interior are considered as boundary nodes''. In practice, <math display="inline">\alpha </math> is a parameter close to, but greater than one. This criterion coincides with the Alpha Shape concept <span id='citeF- | + | Considering that the nodes follow a variable <math display="inline">h(x)</math> distribution, where <math display="inline">h(x)</math> is the minimum distance between two nodes, the following criterion has been used. For each two nodes (three nodes in 3D) a circle (a sphere in 3D) of radius equal to <math display="inline">\alpha h</math> containing the nodes is plotted. ''All nodes laying on a circle (sphere) with a radius greater than <math>\alpha h</math>, not containing any other node in the interior are considered as boundary nodes''. In practice, <math display="inline">\alpha </math> is a parameter close to, but greater than one. This criterion coincides with the Alpha Shape concept <span id='citeF-39'></span>[[#cite-39|[39,55]]]. |
The method also allows to identify isolated fluid particles (nodes) outside the main fluid domain. These particles are treated as part of the external boundary where the pressure is fixed to the atmospheric value. We note that the “flying particles” are in fact “points” ''with no mass'' which motion is followed by integrating the dynamic equations of continuum mechanics for known values of the mass, the initial velocity and acceleration, gravity body forces and a prescribed zero (atmospheric) pressure. | The method also allows to identify isolated fluid particles (nodes) outside the main fluid domain. These particles are treated as part of the external boundary where the pressure is fixed to the atmospheric value. We note that the “flying particles” are in fact “points” ''with no mass'' which motion is followed by integrating the dynamic equations of continuum mechanics for known values of the mass, the initial velocity and acceleration, gravity body forces and a prescribed zero (atmospheric) pressure. | ||
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Thermal-mechanical problems can be effectively treated with the PFEM. This requires solving for the temperature field at each time step, accounting for the couplings induced by the mechanical equations. The effect of temperature in the mechanical problem is introduced via the constitutive equation in the usual manner. | Thermal-mechanical problems can be effectively treated with the PFEM. This requires solving for the temperature field at each time step, accounting for the couplings induced by the mechanical equations. The effect of temperature in the mechanical problem is introduced via the constitutive equation in the usual manner. | ||
− | The form of the heat transfer equation is identical to that of Eq.(70). Recall that the Lagrangian formulation eliminates the convective term in the heat transfer equation and hence the FIC stabilization is here not needed <span id='citeF- | + | The form of the heat transfer equation is identical to that of Eq.(70). Recall that the Lagrangian formulation eliminates the convective term in the heat transfer equation and hence the FIC stabilization is here not needed <span id='citeF-41'></span>[[#cite-41|[41]]]. |
===Remark=== | ===Remark=== | ||
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===9.1 Sloshing problem=== | ===9.1 Sloshing problem=== | ||
− | The sloshing example shown in Figure 19 ilustrates the ability of the PFEM to simulate the flow of liquids within closed cavities with large motions of the free surface. This feature of the PFEM is essential to model the filling of moulds as shown in the next examples. More applications of the PFEM to sloshing problemas are reported in <span id='citeF- | + | The sloshing example shown in Figure 19 ilustrates the ability of the PFEM to simulate the flow of liquids within closed cavities with large motions of the free surface. This feature of the PFEM is essential to model the filling of moulds as shown in the next examples. More applications of the PFEM to sloshing problemas are reported in <span id='citeF-39'></span>[[#cite-39|[39]]]. |
<div id='img-19'></div> | <div id='img-19'></div> | ||
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;" | ||
|- | |- | ||
− | |[[ | + | |[[File:Draft_Samper_805805629_3102_Fig19.png|501px|PFEM results for a large amplitude sloshing problem]] |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
| colspan="1" | '''Figure 19:''' PFEM results for a large amplitude sloshing problem | | colspan="1" | '''Figure 19:''' PFEM results for a large amplitude sloshing problem | ||
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===9.2 Filling of moulds=== | ===9.2 Filling of moulds=== | ||
− | Figure 20 shows the results of the filling of a 2D mould cavity with a powder material using the PFEM. Note that the essential feature of the filling process are well reproduced. The finite element mesh used for the computation in a certain instant is shown in Figure 20b. This ilustrates the fact that the PFEM is, in fact, a blending of particle and finite element procedure. Other applications of the PFEM to powder compaction problems are reported in [57]. | + | Figure 20 shows the results of the filling of a 2D mould cavity with a powder material using the PFEM. Note that the essential feature of the filling process are well reproduced. The finite element mesh used for the computation in a certain instant is shown in Figure 20b. This ilustrates the fact that the PFEM is, in fact, a blending of particle and finite element procedure. Other applications of the PFEM to powder compaction problems are reported in <span id='citeF-57'></span>[[#cite-57|[57]]]. |
Figure 21 shows the different stages of the 3D simulation of the filling of a mould with a casted metal using the PFEM. The pictures show the evolution of the casting during the filling process. The metal flows through the vertical mould first and then fills the lower and upper lateral ducts, as expected. The gray colour intensities denote the different temperatures in the melt with a whiler colour indicating the cooler region. Note that the metal cools down as it progresses within the mould. | Figure 21 shows the different stages of the 3D simulation of the filling of a mould with a casted metal using the PFEM. The pictures show the evolution of the casting during the filling process. The metal flows through the vertical mould first and then fills the lower and upper lateral ducts, as expected. The gray colour intensities denote the different temperatures in the melt with a whiler colour indicating the cooler region. Note that the metal cools down as it progresses within the mould. | ||
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|- | |- | ||
|[[Image:Draft_Samper_805805629-Figure19_5.png|390px|]] | |[[Image:Draft_Samper_805805629-Figure19_5.png|390px|]] | ||
− | |[[Image:Draft_Samper_805805629-Figure19_6.png|390px| | + | |[[Image:Draft_Samper_805805629-Figure19_6.png|390px|]] |
− | |- | + | |- |
− | | colspan="2" | | + | |colspan="2" | (a) |
− | | | + | |- |
− | + | |colspan="2" | [[Image:Draft_Samper_805805629-cuadro25mesh.png|470px|]] | |
− | + | ||
− | + | ||
|- | |- | ||
− | | | + | |colspan="2" | (b) |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan=" | + | | colspan="2" | '''Figure 20:''' (a) Filling of a 2D mould with a powder material using the PFEM. (b) Finite element mesh used for the computations in a certain instant. The circles indicate the nodes at the external and internal free surfaces |
|} | |} | ||
+ | |||
<div id='img-22'></div> | <div id='img-22'></div> | ||
Line 1,716: | Line 1,690: | ||
|[[Image:Draft_Samper_805805629-fig4.png|300px|Filling of a 3D mould with a casted metal using the PFEM. Colours indicate the temperature values]] | |[[Image:Draft_Samper_805805629-fig4.png|300px|Filling of a 3D mould with a casted metal using the PFEM. Colours indicate the temperature values]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan="2" | '''Figure | + | | colspan="2" | '''Figure 21:''' Filling of a 3D mould with a casted metal using the PFEM. Colours indicate the temperature values |
|} | |} | ||
Line 1,724: | Line 1,698: | ||
|[[Image:Draft_Samper_805805629-picture1.png|240px|]] | |[[Image:Draft_Samper_805805629-picture1.png|240px|]] | ||
|[[Image:Draft_Samper_805805629-picture50.png|240px|]] | |[[Image:Draft_Samper_805805629-picture50.png|240px|]] | ||
+ | |[[Image:Draft_Samper_805805629-picture150.png|240px|]] | ||
|- | |- | ||
− | |||
|[[Image:Draft_Samper_805805629-picture300.png|240px|]] | |[[Image:Draft_Samper_805805629-picture300.png|240px|]] | ||
− | |||
|[[Image:Draft_Samper_805805629-picture600.png|240px|]] | |[[Image:Draft_Samper_805805629-picture600.png|240px|]] | ||
|[[Image:Draft_Samper_805805629-picture801.png|240px|Simulation of a casting process using the PFEM]] | |[[Image:Draft_Samper_805805629-picture801.png|240px|Simulation of a casting process using the PFEM]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan=" | + | | colspan="3" | '''Figure 22:''' Simulation of a casting process using the PFEM |
|} | |} | ||
Line 1,739: | Line 1,712: | ||
|[[Image:Draft_Samper_805805629-medit.500.png|351px|(cont.) Cut through the mesh]] | |[[Image:Draft_Samper_805805629-medit.500.png|351px|(cont.) Cut through the mesh]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan="1" | '''Figure | + | | colspan="1" | '''Figure 22:''' (cont.) Cut through the mesh |
|} | |} | ||
===9.3 Mixing problems=== | ===9.3 Mixing problems=== | ||
− | The PFEM is particularly suited for analyzing problems involving mixing of two fluids or that of a collection of particles in a fluid. This is very useful to model a wide class of material forming processes <span id='citeF-5'></span>[[#cite-5|5]]. | + | The PFEM is particularly suited for analyzing problems involving mixing of two fluids or that of a collection of particles in a fluid. This is very useful to model a wide class of material forming processes <span id='citeF-5'></span>[[#cite-5|[5,7]]]. |
A first example of this kind is shown in Figure 23. The problem represents the mixing of two fluids of different density. The PFEM allows to track the position of the fluid particles of lower density which mix within the second fluid. | A first example of this kind is shown in Figure 23. The problem represents the mixing of two fluids of different density. The PFEM allows to track the position of the fluid particles of lower density which mix within the second fluid. | ||
Line 1,752: | Line 1,725: | ||
|- | |- | ||
|[[Image:Draft_Samper_805805629-mixing1.png|600px|]] | |[[Image:Draft_Samper_805805629-mixing1.png|600px|]] | ||
+ | |- | ||
|[[Image:Draft_Samper_805805629-mixing2.png|600px|]] | |[[Image:Draft_Samper_805805629-mixing2.png|600px|]] | ||
|- | |- | ||
− | + | | [[Image:Draft_Samper_805805629-mixing3.png|350px|Analysis of the mixing of two fluids of different density with the PFEM]] | |
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | + | | '''Figure 23:''' Analysis of the mixing of two fluids of different density with the PFEM | |
|} | |} | ||
Figure 24 shows a last example of application of the PFEM to the mixing of a collection of particles within a container filled with a fluid of a higher density. Initially the particles are thrown into the container and mix within the fluid as shown. As time evolves the particles move up naturally towards the surface of the fluid due to their lower density. | Figure 24 shows a last example of application of the PFEM to the mixing of a collection of particles within a container filled with a fluid of a higher density. Initially the particles are thrown into the container and mix within the fluid as shown. As time evolves the particles move up naturally towards the surface of the fluid due to their lower density. | ||
− | The last two examples clearly show the possibilities of the PFEM for analysis of material mixing situations. Extensions of the approach accounting for thermal-coupled effects and chemical reactive situations are straightforward <span id='citeF- | + | The last two examples clearly show the possibilities of the PFEM for analysis of material mixing situations. Extensions of the approach accounting for thermal-coupled effects and chemical reactive situations are straightforward <span id='citeF-41'></span>[[#cite-41|[41]]]. |
<div id='img-26'></div> | <div id='img-26'></div> | ||
Line 1,768: | Line 1,742: | ||
|[[Image:Draft_Samper_805805629-Fig_mixing.png|578px|Mixing of particles in a fluid. Evolution of the particles during the mixing process]] | |[[Image:Draft_Samper_805805629-Fig_mixing.png|578px|Mixing of particles in a fluid. Evolution of the particles during the mixing process]] | ||
|- style="text-align: center; font-size: 75%;" | |- style="text-align: center; font-size: 75%;" | ||
− | | colspan="1" | '''Figure | + | | colspan="1" | '''Figure 24:''' Mixing of particles in a fluid. Evolution of the particles during the mixing process |
|} | |} | ||
Line 1,779: | Line 1,753: | ||
The authors thank the company QUANTECH ATZ, SA (www.quantech.es) for providing the code VULCAN for solution of the casting problem presented. Thanks are also given to MIKALOR SA for providing data for the hose clamp example. | The authors thank the company QUANTECH ATZ, SA (www.quantech.es) for providing the code VULCAN for solution of the casting problem presented. Thanks are also given to MIKALOR SA for providing data for the hose clamp example. | ||
− | + | ==BIBLIOGRAPHY== | |
<div id="cite-1"></div> | <div id="cite-1"></div> | ||
Line 1,785: | Line 1,759: | ||
<div id="cite-2"></div> | <div id="cite-2"></div> | ||
− | '''[2]''' J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (Eds), '' Numerical Analysis of Forming Processes'', Wiley, Chichester, 1984. | + | '''[[#citeF-2|[2]]]''' J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (Eds), '' Numerical Analysis of Forming Processes'', Wiley, Chichester, 1984. |
<div id="cite-3"></div> | <div id="cite-3"></div> | ||
− | '''[3]''' J. Huetnik and F.P.T. Baaijens (Eds), ''Simulation of Materials Processing: Theory, Methods and Applications, NUMIFORM 98'', Enschede, Países Bajos, A.A. Balkema, Rotterdam, 22–5 Jun. 1998. | + | '''[[#citeF-3|[3]]]''' J. Huetnik and F.P.T. Baaijens (Eds), ''Simulation of Materials Processing: Theory, Methods and Applications, NUMIFORM 98'', Enschede, Países Bajos, A.A. Balkema, Rotterdam, 22–5 Jun. 1998. |
<div id="cite-4"></div> | <div id="cite-4"></div> | ||
− | '''[4]''' M. Pietrzyk, J. Kusiak, J. Majta, P. Hartley and I. Pillinger, ''Metal Forming 2000'', A.A. Balkema, Rotterdam, 2000. | + | '''[[#citeF-4|[4]]]''' M. Pietrzyk, J. Kusiak, J. Majta, P. Hartley and I. Pillinger, ''Metal Forming 2000'', A.A. Balkema, Rotterdam, 2000. |
<div id="cite-5"></div> | <div id="cite-5"></div> | ||
Line 1,797: | Line 1,771: | ||
<div id="cite-6"></div> | <div id="cite-6"></div> | ||
− | '''[6]''' D.R.J. Owen, E. Oñate and B. Suarez (Eds.), ''Computational Plasticity VII. Fundamentals and Applications'', CIMNE, Barcelona 2003. | + | '''[[#citeF-6|[6]]]''' D.R.J. Owen, E. Oñate and B. Suarez (Eds.), ''Computational Plasticity VII. Fundamentals and Applications'', CIMNE, Barcelona 2003. |
<div id="cite-7"></div> | <div id="cite-7"></div> | ||
− | '''[7]''' S. Ghosh, J.C. Castro and J.K. Lee, ``Material Processing and Design: Simulation and ApplicatOñate,ions, NUMIFORM 2004, American Institute of Physics, 2004. | + | '''[[#citeF-7|[7]]]''' S. Ghosh, J.C. Castro and J.K. Lee, ``Material Processing and Design: Simulation and ApplicatOñate,ions, NUMIFORM 2004, American Institute of Physics, 2004. |
<div id="cite-8"></div> | <div id="cite-8"></div> | ||
Line 1,815: | Line 1,789: | ||
<div id="cite-12"></div> | <div id="cite-12"></div> | ||
− | '''[12]''' J.C. Heinrich, P.S. Hayakorn and O.C. Zienkiewicz, “An upwind finite element scheme for two dimensional convective transport equations”, ''Int. J. Num. Meth. Engng.'', '''11''', 131–143, 1977. | + | '''[[#citeF-12|[12]]]''' J.C. Heinrich, P.S. Hayakorn and O.C. Zienkiewicz, “An upwind finite element scheme for two dimensional convective transport equations”, ''Int. J. Num. Meth. Engng.'', '''11''', 131–143, 1977. |
<div id="cite-13"></div> | <div id="cite-13"></div> | ||
− | '''[13]''' A. Brooks and T.J.R. Hughes, “Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”, ''Comput. Methods Appl. Mech. Engrg'', '''32''', 199–259, 1982. | + | '''[[#citeF-13|[13]]]''' A. Brooks and T.J.R. Hughes, “Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”, ''Comput. Methods Appl. Mech. Engrg'', '''32''', 199–259, 1982. |
<div id="cite-14"></div> | <div id="cite-14"></div> | ||
− | '''[14]''' T.J.R. Hughes and M. Mallet, “A new finite element formulations for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems”, ''Comput Methods Appl. Mech. Engrg.'', '''58''', pp. 305–328, 1986. | + | '''[[#citeF-14|[14]]]''' T.J.R. Hughes and M. Mallet, “A new finite element formulations for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems”, ''Comput Methods Appl. Mech. Engrg.'', '''58''', pp. 305–328, 1986. |
<div id="cite-15"></div> | <div id="cite-15"></div> | ||
− | '''[15]''' M.A. Cruchaga and E. Oñate, “A finite element formulation for incompressible flow problems using a generalized streamline operator”, ''Comput. Methods in Appl. Mech. Engrg.'', '''143''', 49–67, 1997. | + | '''[[#citeF-15|[15]]]''' M.A. Cruchaga and E. Oñate, “A finite element formulation for incompressible flow problems using a generalized streamline operator”, ''Comput. Methods in Appl. Mech. Engrg.'', '''143''', 49–67, 1997. |
<div id="cite-16"></div> | <div id="cite-16"></div> | ||
− | '''[16]''' M.A. Cruchaga and E. Oñate, “A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces”, ''Comput. Methods in Appl. Mech. Engrg.'', '''173''', 241–255, 1999. | + | '''[[#citeF-16|[16]]]''' M.A. Cruchaga and E. Oñate, “A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces”, ''Comput. Methods in Appl. Mech. Engrg.'', '''173''', 241–255, 1999. |
<div id="cite-17"></div> | <div id="cite-17"></div> | ||
− | '''[17]''' T.J.R. Hughes, L.P. Franca and G.M. Hulbert, “A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations”, '' Comput. Methods Appl. Mech. Engrg.'', '''73''', pp. 173–189, 1989. | + | '''[[#citeF-17|[17]]]''' T.J.R. Hughes, L.P. Franca and G.M. Hulbert, “A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations”, '' Comput. Methods Appl. Mech. Engrg.'', '''73''', pp. 173–189, 1989. |
<div id="cite-18"></div> | <div id="cite-18"></div> | ||
Line 1,842: | Line 1,816: | ||
<div id="cite-21"></div> | <div id="cite-21"></div> | ||
− | '''[21]''' R. Codina and O.C. Zienkiewicz, “CBS versus GLS stabilization of the incompressible Navier-Stokes equations and the role of the time step as stabilization parameter”, ''Communications in Numerical Methods in Engineering'', '''18 (2)''', 99–112, 2002. | + | '''[[#citeF-21|[21]]]''' R. Codina and O.C. Zienkiewicz, “CBS versus GLS stabilization of the incompressible Navier-Stokes equations and the role of the time step as stabilization parameter”, ''Communications in Numerical Methods in Engineering'', '''18 (2)''', 99–112, 2002. |
<div id="cite-22"></div> | <div id="cite-22"></div> | ||
Line 1,851: | Line 1,825: | ||
<div id="cite-24"></div> | <div id="cite-24"></div> | ||
− | '''[[#citeF-24|[24]]]''' J. Donea and A. Huerta, | + | '''[[#citeF-24|[24]]]''' J. Donea and A. Huerta, ''"Finite element method for flow problems"'', J. Wiley, 2003. |
<div id="cite-25"></div> | <div id="cite-25"></div> | ||
Line 1,857: | Line 1,831: | ||
<div id="cite-26"></div> | <div id="cite-26"></div> | ||
− | '''[26]''' J. Rojek, O.C. Zienkiewicz, E. Oñate and E. Postek, “Advances in FE explicit formulations for simulation of metal forming processes”, ''J. of Material Processing Technol.'', '''119''', 41–47, 2001. | + | '''[[#citeF-26|[26]]]''' J. Rojek, O.C. Zienkiewicz, E. Oñate and E. Postek, “Advances in FE explicit formulations for simulation of metal forming processes”, ''J. of Material Processing Technol.'', '''119''', 41–47, 2001. |
<div id="cite-27"></div> | <div id="cite-27"></div> | ||
− | '''[27]''' M. Chiumenti, Q. Valverde, C. Agelet de Saracibar and M. Cervera, “A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations”, ''Computer Methods in Applied Mechanics and Engineering'', '''191''', ('''46'''), 5253–5264, (2002). | + | '''[[#citeF-27|[27]]]''' M. Chiumenti, Q. Valverde, C. Agelet de Saracibar and M. Cervera, “A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations”, ''Computer Methods in Applied Mechanics and Engineering'', '''191''', ('''46'''), 5253–5264, (2002). |
<div id="cite-28"></div> | <div id="cite-28"></div> | ||
Line 1,869: | Line 1,843: | ||
<div id="cite-30"></div> | <div id="cite-30"></div> | ||
− | '''[30]''' E. Oñate, Derivation of stabilized equations for advective-diffusive transport and fluid flow problems, ''Comput. Meth. Appl. Mech. Engng.'', Vol. '''151''', pp. 233–267, (1998). | + | '''[[#citeF-30|[30]]]''' E. Oñate, Derivation of stabilized equations for advective-diffusive transport and fluid flow problems, ''Comput. Meth. Appl. Mech. Engng.'', Vol. '''151''', pp. 233–267, (1998). |
<div id="cite-31"></div> | <div id="cite-31"></div> | ||
− | '''[31]''' E. Oñate, “Possibilities of finite calculus in computational mechanics”, ''Int. J. Num. Meth. Engng.'', '''60'''(1), 255–281, 2004. | + | '''[[#citeF-31|[31]]]''' E. Oñate, “Possibilities of finite calculus in computational mechanics”, ''Int. J. Num. Meth. Engng.'', '''60'''(1), 255–281, 2004. |
<div id="cite-32"></div> | <div id="cite-32"></div> | ||
Line 1,881: | Line 1,855: | ||
<div id="cite-34"></div> | <div id="cite-34"></div> | ||
− | '''[34]''' | + | '''[[#citeF-34|[34]]]''' E. Oñate, J. García, S.R. Idelsohn and F. Del Pin. FIC formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian approaches, Submitted to ''Comput. Methods Appl. Mech. Engng.'', July 2004. |
<div id="cite-35"></div> | <div id="cite-35"></div> | ||
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<div id="cite-36"></div> | <div id="cite-36"></div> | ||
− | '''[36]''' | + | '''[[#citeF-36|[36]]]''' E. Oñate, R.L. Taylor, O.C. Zienkiewicz and J. Rojek, “A residual correction method based on finite calculus”, ''Engineering Computations'', '''20''', (5/6), 629–658, 2003. |
<div id="cite-37"></div> | <div id="cite-37"></div> | ||
Line 1,899: | Line 1,873: | ||
<div id="cite-40"></div> | <div id="cite-40"></div> | ||
− | '''[40]''' S.R. Idelsohn, E. Oñate and F. Del Pin, “The Particle Finite Element Method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves”, ''Int. J. Num. Meth. Engng.'', (Submitted), 2004. | + | '''[[#citeF-40|[40]]]''' S.R. Idelsohn, E. Oñate and F. Del Pin, “The Particle Finite Element Method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves”, ''Int. J. Num. Meth. Engng.'', (Submitted), 2004. |
<div id="cite-41"></div> | <div id="cite-41"></div> | ||
Line 1,914: | Line 1,888: | ||
<div id="cite-45"></div> | <div id="cite-45"></div> | ||
− | '''[45]''' R. Codina, “Pressure stability in fractional step finite element method for incompressible flows”, ''J. Comput. Physics'', '''170''', 112–140, 2001. | + | '''[[#citeF-45|[45]]]''' R. Codina, “Pressure stability in fractional step finite element method for incompressible flows”, ''J. Comput. Physics'', '''170''', 112–140, 2001. |
<div id="cite-46"></div> | <div id="cite-46"></div> | ||
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<div id="cite-48"></div> | <div id="cite-48"></div> | ||
− | '''[48]''' | + | '''[[#citeF-48|[48]]]''' Codina, R., and Soto, O., “Finite element simulation of the filling of thin moulds”, ''Int. J. Num. Meth. Engng.'', '''46''', 1559–1573, (1999). |
<div id="cite-49"></div> | <div id="cite-49"></div> | ||
Line 1,929: | Line 1,903: | ||
<div id="cite-50"></div> | <div id="cite-50"></div> | ||
− | '''[50]''' M. Cervera, C. Agelet de Saracibar and M. Chiumenti, “ Thermo-mechanical analysis of industrial solidification processes”, ''Int. Journal for Num. Methods in Engineering'', Vol. '''46''', 1575-1591, (1999). | + | '''[[#citeF-50|[50]]]''' M. Cervera, C. Agelet de Saracibar and M. Chiumenti, “ Thermo-mechanical analysis of industrial solidification processes”, ''Int. Journal for Num. Methods in Engineering'', Vol. '''46''', 1575-1591, (1999). |
<div id="cite-51"></div> | <div id="cite-51"></div> | ||
− | '''[51]''' C. Agelet de Saracibar, M. Cervera and M. Chiumenti, “On the constitutive modeling of coupled thermo-mechanical phase-change problems”, ''International Journal of Plasticity'', Vol. '''17''', 1565-1622, (2001). | + | '''[[#citeF-51|[51]]]''' C. Agelet de Saracibar, M. Cervera and M. Chiumenti, “On the constitutive modeling of coupled thermo-mechanical phase-change problems”, ''International Journal of Plasticity'', Vol. '''17''', 1565-1622, (2001). |
<div id="cite-52"></div> | <div id="cite-52"></div> | ||
− | '''[52]''' N. Calvo, S.R. Idelsohn and E. Oñate, “The extended Delaunay tesselation”, ''Engineering Computations'', '''20''' (5/6), 583–600, 2003. | + | '''[[#citeF-52|[52]]]''' N. Calvo, S.R. Idelsohn and E. Oñate, “The extended Delaunay tesselation”, ''Engineering Computations'', '''20''' (5/6), 583–600, 2003. |
<div id="cite-53"></div> | <div id="cite-53"></div> | ||
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<div id="cite-56"></div> | <div id="cite-56"></div> | ||
− | '''[56]''' VULCAN. Finite element system for casting simulation, Quantech ATZ, Barcelona, Spain, (2003), (www.quantech.es). | + | '''[[#citeF-56|[56]]]''' VULCAN. Finite element system for casting simulation, Quantech ATZ, Barcelona, Spain, (2003), (www.quantech.es). |
<div id="cite-57"></div> | <div id="cite-57"></div> | ||
− | '''[57]''' J.C. Cante, J. Oliver and C. González, “Powder forming simulation with the particle finite element method”, ''Computational Mechanics'', Z.H. Yao, M.W. Yuan and W.X. Zhong (Eds.), CD-Rom Proceedings of the WCCM VI, Beijing, September 5–10, 2004. | + | '''[[#citeF-57|[57]]]''' J.C. Cante, J. Oliver and C. González, “Powder forming simulation with the particle finite element method”, ''Computational Mechanics'', Z.H. Yao, M.W. Yuan and W.X. Zhong (Eds.), CD-Rom Proceedings of the WCCM VI, Beijing, September 5–10, 2004. |
<div id="cite-58"></div> | <div id="cite-58"></div> | ||
− | '''[58]''' J. Rojek, E. Oñate and R.L. Taylor, “CBS based stabilization in explicit solid mechanics”. ''Int. J. Num. Mech. Engng.'' | + | '''[[#citeF-58|[58]]]''' J. Rojek, E. Oñate and R.L. Taylor, “CBS based stabilization in explicit solid mechanics”. ''Int. J. Num. Mech. Engng.'', '''66'''(10), 1547-1568, 2006. |
==APPENDIX== | ==APPENDIX== | ||
Line 1,959: | Line 1,933: | ||
In Eqs.(24) | In Eqs.(24) | ||
− | <span id="eq- | + | <span id="eq-A.1"></span> |
{| class="formulaSCP" style="width: 100%; text-align: left;" | {| class="formulaSCP" style="width: 100%; text-align: left;" | ||
|- | |- | ||
Line 1,965: | Line 1,939: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>{H}={A}+{K}+\hat {K} </math> | + | | style="text-align: center;" | <math>{\bf H}={\bf A}+{\bf K}+\hat {\bf K} </math> |
|} | |} | ||
− | | style="width: 5px;text-align: right;white-space: nowrap;" | ( | + | | style="width: 5px;text-align: right;white-space: nowrap;" | (A.1) |
|} | |} | ||
Line 1,977: | Line 1,951: | ||
{| style="text-align: left; margin:auto;width: 100%;" | {| style="text-align: left; margin:auto;width: 100%;" | ||
|- | |- | ||
− | | style="text-align: center;" | <math>\begin{array}{l} \displaystyle M_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a N^b d\Omega \right)\delta _{ij} \quad ,\quad A_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a ({u}^T {\nabla } N^b) d\Omega \right)\delta _{ij}\\ \\ \displaystyle {K}_{ij}^{ab} = \left(\int _{\Omega ^e} \mu {\boldsymbol \nabla }^T N^a{\boldsymbol \nabla }N^b d\Omega \right)\delta _{ij} \quad ,\quad {\boldsymbol \nabla } = \left[{\partial \over \partial x_1},{\partial \over \partial x_2},{\partial \over \partial x_3}\right]^T\\ \\ \displaystyle \hat{K}_{ij}^{ab} = \left({1\over 2} \int _{\Omega ^e} ({h}^T {\boldsymbol \nabla } N^a)(\rho {u}^T {\boldsymbol \nabla } N^b)d\Omega \right)\delta _{ij}\quad ,\quad {G}_{i}^{ab}=\int _{\Omega ^e} {\partial N^a \over \partial x_i}N^b d\Omega \\ \\ \displaystyle {C}= \left[\begin{matrix}{C}_1\\ {C}_2\\ {C}_3\\\end{matrix}\right]\quad ,\quad {C}_{ij}^{ab}=\left({1\over 2} \int _{\Omega ^e} [{h}^T {\boldsymbol \nabla } N^a]N^bd\Omega \right)\delta _{ij}\\ \\ \displaystyle \hat L^{ab}= \int _{\Omega ^e} ({\boldsymbol \nabla }^T N^a) [\tau ] {\boldsymbol \nabla } N^b d\Omega \quad ,\quad [\tau ]= \left[\begin{matrix}\tau _1 &0 &0 \\ 0 & \tau _2&0\\ 0&0& \tau _3\\\end{matrix}\right]\\ \\ \displaystyle {Q}= [{Q}_1,{Q}_2,{Q}_3] \quad ,\quad \displaystyle Q_{i}^{ab} = \int _{\Omega ^e}\tau _i {\partial N^a \over \partial x_i} N^b d\Omega \quad \quad \hbox{no sum in }i\\ \\ \displaystyle \hat{C}= [\hat{C}_1,\hat{C}_2,\hat{C}_3] \quad ,\quad \displaystyle \hat{C}_1^{ab}=\hat{C}_2^{ab}=\hat{C}_3^{ab} = \int _{\Omega ^e} \rho ^2 N^a ({u}^T {\boldsymbol \nabla }N^b)d\Omega \\ \\ \displaystyle \hat {M}^{ab}_{ij}= \left(\int _{\Omega ^e} \tau _i N^a N^b d\Omega \right)\delta _{ij}\quad ,\quad \displaystyle {f}_i^a = \int _{\Omega ^e} N^a f_i d\Omega + \int _{\Gamma ^e} N^a t_i d\Gamma \end{array}</math> | + | | style="text-align: center;" | <math>\begin{array}{l} \displaystyle M_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a N^b d\Omega \right)\delta _{ij} \quad ,\quad A_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a ({\textbf u}^T {\nabla } N^b) d\Omega \right)\delta _{ij}\\ \\ |
+ | \displaystyle {K}_{ij}^{ab} = \left(\int _{\Omega ^e} \mu {\boldsymbol \nabla }^T N^a{\boldsymbol \nabla }N^b d\Omega \right)\delta _{ij} \quad ,\quad {\boldsymbol \nabla } = \left[{\partial \over \partial x_1},{\partial \over \partial x_2},{\partial \over \partial x_3}\right]^T\\ \\ | ||
+ | \displaystyle \hat{K}_{ij}^{ab} = \left({1\over 2} \int _{\Omega ^e} ({\textbf h}^T {\boldsymbol \nabla } N^a)(\rho {\textbf u}^T {\boldsymbol \nabla } N^b)d\Omega \right)\delta _{ij}\quad ,\quad {G}_{i}^{ab}=\int _{\Omega ^e} {\partial N^a \over \partial x_i}N^b d\Omega \\ \\ | ||
+ | \displaystyle {\textbf C}= \left[\begin{matrix}{\textbf C}_1\\ {\textbf C}_2\\ {\textbf C}_3\\\end{matrix}\right]\quad ,\quad {C}_{ij}^{ab}=\left({1\over 2} \int _{\Omega ^e} [{\textbf h}^T {\boldsymbol \nabla } N^a]N^bd\Omega \right)\delta _{ij}\\ \\ | ||
+ | \displaystyle \hat L^{ab}= \int _{\Omega ^e} ({\boldsymbol \nabla }^T N^a) [\tau ] {\boldsymbol \nabla } N^b d\Omega \quad ,\quad [\tau ]= \left[\begin{matrix}\tau _1 &0 &0 \\ 0 & \tau _2&0\\ 0&0& \tau _3\\\end{matrix}\right]\\ \\ | ||
+ | \displaystyle {\textbf Q}= [{\textbf Q}_1,{\textbf Q}_2,{\textbf Q}_3] \quad ,\quad \displaystyle Q_{i}^{ab} = \int _{\Omega ^e}\tau _i {\partial N^a \over \partial x_i} N^b d\Omega \quad \quad \hbox{no sum in }i\\ \\ | ||
+ | \displaystyle \hat{\textbf C}= [\hat{\textbf C}_1,\hat{\textbf C}_2,\hat{\textbf C}_3] \quad ,\quad \displaystyle \hat{C}_1^{ab}=\hat{C}_2^{ab}=\hat{C}_3^{ab} = \int _{\Omega ^e} \rho ^2 N^a ({\textbf u}^T {\boldsymbol \nabla }N^b)d\Omega \\ \\ | ||
+ | \displaystyle \hat {M}^{ab}_{ij}= \left(\int _{\Omega ^e} \tau _i N^a N^b d\Omega \right)\delta _{ij}\quad ,\quad \displaystyle {f}_i^a = \int _{\Omega ^e} N^a f_i d\Omega + \int _{\Gamma ^e} N^a t_i d\Gamma \end{array}</math> | ||
|} | |} | ||
|} | |} | ||
Line 1,993: | Line 1,974: | ||
It is understood that all the arrays are matrices (except <math display="inline">{f}</math>, which is a vector) whose components are obtained by grouping together the left indexes in the previous expressions (<math display="inline">a</math> and possibly <math display="inline">i</math>) and the right indexes (<math display="inline">b</math> and possibly <math display="inline">j</math>). | It is understood that all the arrays are matrices (except <math display="inline">{f}</math>, which is a vector) whose components are obtained by grouping together the left indexes in the previous expressions (<math display="inline">a</math> and possibly <math display="inline">i</math>) and the right indexes (<math display="inline">b</math> and possibly <math display="inline">j</math>). | ||
− | Note that the stabilization matrix <math display="inline">\hat{K}</math> in Eq.([[#eq- | + | Note that the stabilization matrix <math display="inline">\hat{K}</math> in Eq.([[#eq-A.1|A.1]]) adds additional orthotropic diffusivity terms of value <math display="inline">\rho \displaystyle{h_ku_l\over 2}</math>. |
Published in Comput. Methods Appl. Mech. Engrg. Vol. 195, pp. 6750-6777, 2006
doi: 10.1016/j.cma.2004.10.018
The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk forming processes. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal forming problems and mixing processes are shown.
Keywords: Bulk forming processes, stabilized finite element method, particle method, particle finite element method, mixing processes.
The development of efficient and robust numerical methods for analysis of bulk forming problems has been a subject of intensive research in recent years [1-7]. Many of these problems require the solution of incompressible fluid flow situations (such as in mould filling problems) whereas in other cases (such as forging, rolling, extrusion, etc.) the numerical method must be able to account for the quasi/fully incompressible behaviour induced by the large plastic deformation. The solution of these problems has motivated the development of the so called stabilized numerical methods overcoming the two main sources of instability in the analysis of incompressible continua, namely those originated by the high values of the convective terms in fluid flow situtions and those induced by the difficulty in satisfying the incompressibility condition.
Different approaches to solve both type of problems in the context of the finite element method (FEM) have been recently developed [8]. Traditionally, the underdiffusive character of the Galerkin FEM for high convection flows has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations [8,9].
A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum and using implicit and explicit algorithms ad hoc such as artificial compressibility schemes [10] and preconditioning techniques [11]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms in fluid flows are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods [8,9,12]. A general class of Galerkin FEM has been developed where the standard Galerkin variational form is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme. Among the many FEM of this kind we can name the Streamline Upwind Petrov Galerkin (SUPG) method [8,13-16], the Galerkin Least Square (GLS) method [8,17], the Taylor-Galerkin method [18], the Characteristic Galerkin method [19] and its variant the Characteristic Based Split (CBS) method [20,21], the pressure gradient operator method [22] and the Subgrid Scale (SS) method [23]. A good review of these methods can be found in [24]. Extensions of the CBS and SS methods to treat incompressible problems in solid mechanics are reported in [25,26,58] and [27-29], respectively.
In this paper a different class of stabilized FEM for quasi and fully incompressible fluid and solid materials applicable to a wide range of bulk forming problems is presented. The starting point is the modified governing differential equations of the continuous problem formulated via a finite calculus (FIC) approach [30,31]. The FIC method is based in invoking the classical balance (or equilibrium) laws in a domain of finite size. This introduces naturally additional terms in the differential equations of infinitesimal continuum mechanics which are a function of the balance domain dimensions. The new terms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via the standard Galerkin FEM. One of the main advantages of the FIC formulation versus other alternative approaches (such as mixed FEM, etc.) is that it allows to solve incompressible fluid problems using low order finite elements (such as linear triangles and tetrahedra) with equal order approximations for the velocity and pressure variables [32-35]. The FIC formulation has been successfully used for analysis of fully or quasi incompressible solids [36,37].
The layout of the paper is the following. In the next section the basic FIC equations for incompressible flow problems formulated in an Eulerian frame are presented. The finite element discretization is introduced and the resulting discretized equations are detailed. A fractional step scheme for the transient solution is presented.
The stabilized Eulerian formulation is extended to account for thermal effects and the transport of the free surface which are needed for mould filling processes.
The following sections outline the FIC formulation for analysis of quasi/fully incompressible solids using a Lagrangian description and linear triangles and tetrahedra. An explicit algorithm for integrating in time the equations of motion of elasto-plastic solids in large-strain problems involving frictional contact is described. Examples of application to a casting problem and some bulk forming processes are presented.
In the last part of the paper a Lagrangian formulation for fluid flow analysis is presented as a straightforward extension of the formulation for solid mechanics. The procedure, called the Particle Finite Element Method (PFEM) [38-40], treats the mesh nodes in the fluid and solid domains as dimensionless particles which can freely move, an even separate from the main fluid domain, representing, for instance, the effect of liquid drops. A finite element mesh connects the nodes defining the discretized domain where the governing equations are solved in the standard FEM fashion. The main advantage of the Lagrangian flow formulation is that the convective terms do not enter in the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes. The final examples show that the PFEM is a promising method to solve mould filling and casting problems, material mixing processes and many other bulk metal forming problems involving the interaction between solids and fluids which can be treated with the same formulation.
The FIC governing equations for a viscous incompressible fluid can be written in an Eulerian frame of reference as [30-34]
Momentum
|
(1) |
Mass balance
|
(2) |
where
|
Above is the analysis domain which can evolve with time, is the number of space dimensions ( for 3D problems), is the velocity along the ith global axis, is the (constant) density of the fluid, is the absolute pressure (defined positive in compression), are the body forces and are the deviatoric stresses related to the viscosity by the standard expression
|
(5) |
where is the Kronecker delta and the strain rates are
|
(6) |
The boundary conditions are written in the FIC approach as
|
(7) |
|
(8) |
and the initial condition is for .
Summation convention for repeated indexes in products and derivatives is used unless otherwise specified.
In Eqs.(7) and (8) and are surface tractions and prescribed velocities on the boundaries and , respectively, are the components of the unit normal vector to the boundary and are the total stresses given by .
Eqs. (1) and (2) are obtained by invoking the classical balance equations in fluid mechanics in a domain of finite size and retaining higher order terms [30]. The in above equations are characteristic lengths of the domain where the balance of momentum and mass is enforced. In Eq.(7) these lengths define the domain where equilibrium of boundary tractions is established. In the discretized problem the coincide with a typical element dimension, as described in Section 5. Note that by making in these equations the standard infinitesimal form of the fluid mechanics equations is recovered [8,9,24].
Eqs.(1)–(8) are the starting point for deriving stabilized FEM for solving the incompressible Navier-Stokes equations using equal order interpolation for the velocity and pressure variables [32-35]. Application of the FIC formulation to meshless analysis of fluid flow problems using the finite point method can be found in [42].
Remark. In most metal forming processes the viscosity is a non linear function of the strain rate and the yield stress of the material (which also depends on the temperature) [8,43,44]. This dependence adds another non linearity to the problem.
Remark. In Eqs.(1)–(7) and respectively denote the actual volume and the boundary of the domain where the governing equations are solved at each instant of the forming processes. This domain is considered here to be fixed in space (Eulerian approach). Moving free surfaces, such as in the case of mold filling processes, can be modelled by using standard level set and volume of fluid (VOF) techniques [47,48]. An alternative is to use an arbitrary Lagragian-Eulerian (ALE) description or even a fully Lagrangian description to follow the motion of the particles during the forming process. The Lagrangian description is typical in the case of solid mechanics problems (Section 5). A particular Lagrangian formulation for fluid flow problems involving large motion of the free surface, named the Particle Finite Element Method, is presented in Section 8.
From the momentum equations it can be obtained [32-34]
|
(9) |
where
|
(10) |
Substituting Eq.(9) into Eq.(2) and retaining the terms involving the derivatives of with respect to only, leads to the following expression for the stabilized mass balance equation
|
(11) |
with
|
(12) |
The 's in Eq.(12), when scaled by the density, are termed in the stabilization literature intrinsic time parameters.
The weighted residual form of the momentum and mass balance equations (Eqs.(1) and (11)) is written as
|
(13) |
|
(14) |
where and are arbitrary weighting functions representing virtual velocities and virtual pressure fields. Integrating by parts the terms leads to
|
(15a) |
|
(15b) |
We will neglect hereonwards the third integral in Eq.(15b) by assuming that is negligible on the boundaries. The deviatoric stresses and the pressure terms in the first integral of Eq.(15a) are integrated by parts in the usual manner. The resulting momentum and mass balance equations are
|
(16a) |
|
(16b) |
In the derivation of the viscous term in Eq.(16a) we have used the following identity (prior to the integration by parts)
|
(17) |
Eq.(17) is identically true for the exact incompressible limit .
The computation of the residual terms can be simplified if we introduce now the convective and pressure gradient projections and , respectively defined as
|
(18) |
We can express in Eqs.(16a) and (16b) in terms of and , respectively which then become additional variables. The system of integral equations is now augmented in the necessary number of equations by imposing that the residual vanishes (in average sense) for both forms given by Eqs.(18). This gives the final system of governing equation as:
|
(19) |
|
(20) |
|
(21) |
|
(22) |
with . In Eqs.(21) and (22) and are appropriate weighting functions and the and weights are introduced for convenience.
We note that accounting for the convective and pressure gradient projections enforces the consistency of the formulation as it ensures that the stabilization terms in Eqs.(19–22) have a residual form which vanishes for the “exact” solution. Neglecting these terms lowers the accuracy of the numerical solution and it makes the formulation more sensitive to the value of the stabilization parameters as shown in [34,36,37].
We choose continuous linear interpolations of the velocities, the pressure, the convection projections and the pressure gradient projections over three node triangles (2D) and four node tetrahedra (3D) [8]. The linear interpolations are written as
|
(23) |
where the sum goes over the number of nodes of each element ( for triangles/tetrahedra), denotes nodal variables and are the linear shape functions [8].
Substituting the approximations (23) into Eqs.(19–22) and choosing the Galerking form with leads to following system of discretized equations
|
(24a) |
|
(24b) |
|
(24c) |
|
(24d) |
The form of the different matrices is given in the Appendix.
The solution in time of the system of Eqs.(24) can be written in general form as
|
(25a) |
|
(25b) |
|
(25c) |
|
(25d) |
where , are the velocities evaluated at time and the parameter . The direct monolitic solution of Eqs.(25) is possible using an adequate iterative scheme. However, we have found more convenient to use a fractional step method as described in the next section.
A fractional step scheme is derived by noting that the discretized momentum equation (25a) can be split into the two following equations
|
(26) |
|
(27) |
In above equation is a predicted value of the velocity at time and is a variable whose values of interest are zero and one. For (first order scheme) the splitting error is of order , whereas for (second order scheme) the error is of order [45].
Eqs.(26) and (27) are completed with the following three equations emanating from Eqs.(25b-d)
|
(28a) |
|
(28b) |
|
(28c) |
The value of obtained from Eq.(27) is substituted into Eq.(28a) to give
|
(29) |
The product can be approximated by a laplacian matrix, i.e.
|
(30) |
where are the element contributions to (see Appendix).
The steps of the fractional step scheme chosen here are:
Step 1
The fractional nodal velocities can be explicitely computed from Eq.(26) by
|
(31) |
where is the diagonal form of obtaining by lumping the row terms into the corresponding diagonal terms.
Step 2 Compute from Eq.(29) as
|
(32) |
Step 3 Compute explicitely from Eq.(28a) as
|
(33) |
Step 4 Compute explicitely from Eq.(28b) as
|
(34) |
Step 5 Compute explicitely from Eq.(28c) as
|
(35) |
where is the lumped form of . A standard diagonal lumping procedure based in summing up the terms of each row has been used.
Note that all steps can be solved explicitely except for the computation of the pressure in Eq.(32) which requires to invert the sum of two laplacian matrices. This can be effectively performed using an iterative solution scheme such as the conjugate gradient method.
Above algorithm has improved stabilization properties versus the standard segregation methods due to the introduction of the laplacian matrix in Eq.(32). This matrix emanates from the FIC formulation.
The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities in Eq.(31). The prescribed velocities at the boundary are applied when solving for in step 3. The prescribed pressures at the boundary are imposed by making equal to the prescribed pressure values when solving Eq.(32).
Many metal forming processes can be simulated with the assumption that the convective terms are negligible (Stokes flow) [8,43,44]. The Stokes FIC formulation can be readily obtained simply by neglecting the convective terms in the Navier-Stokes formulation presented earlier. This also implies neglecting the convective stabilization terms in the momentum equations and, consequently, the convective projection variables are not larger necessary. Also the intrinsic time parameters take now the simpler form (see Eq.(12)):
|
(36) |
We note again that in metal forming problems the viscosity will typically be a function of the strain rate and the yield stress of the material 8.
The resulting discretized system of equations can be written as (see Eqs.(28))
|
(37) |
The algorithm of previous section can now be implemented.
The steady-state form of Eqs.(37) can be expressed in matrix form as
|
(38) |
The system is symmetric and always positive definite and therefore leads to a non singular solution. This property holds for any interpolation function chosen for and , therefore overcoming the Babuŝka-Brezzi (BB) restrictions [8].
A reduced velocity-pressure formulation can be obtained by elliminating the variables from the last row of Eq.(38) [37].
The FIC formulation for Stokes flows is applicable for analysis of quasi/fully incompressible solids. An analogous formulation based on solid mechanics concepts is derived in Section 6.
The effect of temperature can be easily introduced by solving the equation for heat transport coupled to the fluid flow equations.
The equation of balance of heat is written in the FIC formulation as [30]
|
(39a) |
with
|
(39b) |
In above is the temperature, and are the specific heat and the thermal conductivity of the material, respectively, is the heat source and are the characteristic length distances which are typical of the FIC formulation [30].
Eq.(39a) is completed with the Dirichlet and Neuman boundary conditions for the heat problem. For details see [35].
The convective velocities in Eq.(39b) are provided by the solution of the fluid flow problem. As usual in metal forming processes, the heat source is a function of the mechanical work generated in the flow of the material during the forming process. The temperature field affects in turn the flow viscosity via its dependence with the yield stress which is very sensitive to the temperature changes. The solution of the heat transfer equation is therefore fully coupled with that of the fluid flow problem [30,46].
For the treatment of the free boundary we use a standard volume of fluid (VOF) technique, also known as pseudo-concentration method or level set technique [22,47,48]. In the VOF method the motion of the free boudary is followed by solving the following transport equation (written using the FIC formulation)
|
(40a) |
where
|
(40b) |
In above is an auxiliary variable which takes a value equal to one on the free surface and zero elsewhere. The solution of Eq.(40a) in time in the discretized fluid-air domain allows to track the free surface which is characterized by the contours of which have a unit value.
The underlined term in Eq.(40a) emerges from the FIC formulation [33]. Again, the 's in Eq.(40a) are characteristic length distances of the order of the element size. This term introduces the necessary stabilization in the transient solution of Eq.(40a).
The analysis of mould filling problems typically requires the solution of the coupled-thermal flow problem accounting for the transport of the free surface.
The simplest explicit algorithm is as follows:
A number of implicit versions of the algorithm are possible and they all involve an iteration loop within each time step until convergence for the flow variables, the temperature and the free surface position is found. For details see [47,48].
The evaluation of the stabilization parameters is one of the crucial issues in stabilized methods. Excellent results have been obtained in all problems solved using linear tetrahedra with the characteristic length vector defined by
|
(41a) |
where and and are the “streamline” and “cross wind” contributions given by
|
(41b) |
|
(41c) |
where are the vectors defining the element sides ( for tetrahedra).
As for the free surface equation the following value of the characteristic length vector in Eq.(40a) has been taken
|
(42a) |
The streamline parameter has been obtained by Eq.(41b) whereas the cross wind parameter has been computed by
|
(42b) |
The cross-wind terms in eqs.(41a) and (42a) account for the effect of the gradient of the solution in the stabilization parameters. This is a standard assumption in most “shock-capturing” stabilization procedures [8,24].
As usual the governing equations of solid mechanics are written in a Lagrangian reference frame. Following the arguments of Section 2 the equilibrium equations for a solid are written using the FIC technique as [37]
|
(43) |
where for the dynamic case
|
(44) |
In Eqs.(43) and (44) are the displacements, and are the stresses and the body forces, respectively and are characteristic length distances of an arbitrary prismatic domain where equilibrium of forces is considered.
Equations (43) and (44) are completed with the boundary conditions on the displacements
|
(45) |
and the equilibrium of surface tractions
|
(46) |
In the above and are prescribed displacements and tractions over the boundaries and , respectively and are the components of the unit normal vector.
Note that for consistency with the fluid flow equations of previous section and differently from the tradition in solid mechanics, the compression pressure has been taken as positive.
For simplicity the treatment of the constitutive equations will be explained for the linear elastic model. The approach extends naturally to the non linear elasto-plastic/viscoplastic constitutive equations typical of metal forming problems.
As usual in quasi-incompressible problems the stresses are split into deviatoric and volumetric (pressure) parts
|
(47) |
where is the Kronnecker delta function. The linear elastic constitutive equations for the deviatoric stresses are written as
|
(48) |
where is the shear modulus,
|
(49) |
The constitutive equation for the pressure can be written for an arbitrary domain of finite size of volume as
|
(50) |
where is the bulk modulus of the material and is the average value of the pressure over domain .
The value of can be approximated as [37]
|
(51) |
where is the pressure at an arbitrary point within the domain and are characteristic lengths of such a domain.
The ratio can be expressed as
|
(52) |
Substituting Eqs.(51) and (52) into Eq.(50) and neglecting second order terms in gives the FIC constitutive equation for the pressure as
|
(53) |
Note that for the standard relationship between the pressure and the volumetric strain of the infinitesimal theory is found.
For an incompressible material and Eq.(53) yields
|
(54) |
Eq.(54) expresses the limit incompressible behaviour of the solid. This equation is identical to Eq.(2) for incompressible flow problems and there arises from the mass continuity condition [30,32].
By combining Eqs. (43), (44), (47), (48) and (53) a mixed displacement–pressure formulation can be written as
|
(55) |
|
(56) |
From the observation of Eq.(55) we can obtain after some algebra [37]
|
(57) |
Substituting Eq.(57) into (53) the mass balance equation can be written as
|
(58) |
In the derivation of Eq.(58) we have neglected the terms involving products for .
The pressure gradient projections are introduced now as
|
(59) |
The final system of governing equations is
|
(60) |
|
(61) |
|
(62) |
In Eqs.(60)–(62) we note the following:
The weighted residual form of the governing equations can be written as (after integration by parts of the relevant terms)
|
(63a) |
|
(63b) |
|
(63c) |
where again the are introduced in Eq.(63c) for symmetry reasons.
The finite element discretization of the displacements, the pressure and the pressure gradient projections is written by expressions identical to Eqs.(23) with the nodal variables now being a function of the time . Substituting the approximations into eqs.(63) and using the Galerkin form gives the following system of discretized equations
|
(64a) |
|
(64b) |
|
(64c) |
where is the nodal acceleration vector,
|
(65) |
is the mass matrix,
|
(66) |
is the internal nodal force vector and the rest of matrices and vectors are defined in the Appendix. Note that the expression of of eq.(66) is adequate for non linear analysis.
A four steps semi-explicit time integration algorithm can be derived as follows:
Step 1. Compute the nodal velocities from Eq.(64a)
|
(67a) |
Step 2. Compute the nodal displacements
|
(67b) |
Step 3. Compute the nodal pressures from Eq.(64b)
|
(67c) |
Step 4. Compute the nodal projected pressure gradients from Eq.(64c)
|
(67d) |
In above, all matrices are evaluated at , denotes a lumped diagonal matrix and
|
(68) |
where the stresses are obtained by consistent integration of the adequate (non linear) constitutive law.
Note that steps 1, 2 and 4 are fully explicit as a diagonal form of matrices and has been chosen. The solution of step 3 requires invariably the inverse of a Laplacian matrix. This can be an inexpensive process using an iterative equation solution method (e.g. a preconditioned conjugate gradient method).
For the full incompressible case and in all above equations.
The critical time step is taken as that of the standard explicit dynamic scheme (see Section 6.5 and [9,37]).
Fully explicit algorithm
A fully explicit four steps algorithm can be obtained by computing from step 3 in eq.(67c) as follows
|
(69) |
Note that the explicit algorithm is not applicable in the full incompressible limit as the solution of Eq.(69) breaks down for and . The explicit form can however be used with success in problems where quasi-incompressible regions exist adjacent to standard “compressible” zones. Examples of this kind are shown in the next section. In both cases the semi-explicit and fully explicit schemes gave identical results with important savings in both computer time and memory storage requirements obtained when using the explicit form.
The effect of temperature can be easily accounted for by solving the heat transfer equation formulated in a Lagrangian frame as
|
(70) |
As usual, the source is dependent of the mechanical work generated during the forming process.
Note that the convective terms do not enter into Eq.(70) This also eliminates the need to stabilize the numerical solution.
The coupled thermal-mechanical problem requires the computation of the temperature at each time step using a transient solution scheme [47,48,50].
Remark. Above formulation is similar to that developed by Chiumenti et al. [27,29] and Cervera et al. [28] for analysis of incompressible problems in solid mechanics using a sub-grid scale approach.
In solid mechanics applications it is usual to accept that all are identical and constant within each element and given by
|
(71) |
where is the element volume (or the element area for 2D problems). This expression for does not take into account the element distorsions along a particular direction during the deformation process.
The correct value of the shear modulus in the expression of is another sensitive issue as, obviously, for non linear problems the value of will differ from the elastic modulus. This fact has been identified by Cervera et al. [28] for non linear analysis of incompressible problems using linear triangles.
A useful alternative to compute for explicit non linear transient situations is to make use of the value of the speed of sound in an elastic solid, defined by
|
(72) |
where is the Young modulus. The stability condition for explicit dynamic computations is given by the Courant condition defined as [8]
|
(73) |
where is the critical time step for the element.
Accepting that for the incompressible case and using Eqs.(72) and (73) (assuming the identity in Eq.(73)) an alternative expression for the element intrinsic time parameter in terms of the critical time step can be found as
|
(74) |
Eq.(74) shows clearly that the intrinsic time parameter varies across the mesh as a function of the critical time step for each element.
A numerical simulation of an aluminium casting process is presented as a demonstration of the accuracy of the stabilized formulation. The computations are performed with the finite element code VULCAN where the stabilized FEM presented has been implemented [56].
The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Material behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mould has been modeled by a simpler thermo-elastic model [51]. Geometrical and material data were provided by the foundry RUFFINI. Figure 1 shows the finite element mesh used for the part and the cooling system. The full mesh, including the mould has 380.000 four node tetrahedra. The pouring temperature is 650C. Initial temperature for the mould is obtained through a thermal die-cycling simulation. Figure 2 shows the evolution of the mould temperature after 6 cycles. The cooling system has been kept at 20C. Filling evolution has been simulated as in a pressure die-casting process using the stabilized VOF technique described in [47,48]. Figure 3 shows different time steps of the simulation.
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Figure 1: Finite element discretization of the aluminium casting. |
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Figure 2: Temperature die-cycling. |
The final temperature field obtained after the filling simulation is taken as the initial condition for the solidification and cooling analysis. Temperature and liquid-fraction distributions during solidification are shown in Figures 4 and 5, respectively. The heat transfer coefficient takes into account the air-gap resistance due to the casting shrinkage during the solidification process. Figure 6 shows volumetric and von Mises deviatoric stress distributions in a x-y section. The figures also show the air-gap between the part and the mould, responsible of a non-uniform heat flux at the contact interface.
Other examples of application of the stabilized formulation to casting problems can be found in [47-51].
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Figure 3: Filling evolution: pressure die-casting simulation. |
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Figure 4: Temperature evolution during cooling phase. |
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Figure 5: Liquid-fraction evolution during phase-change. |
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Figure 6: Aluminium casting. Stress-trace and von Mises deviatoric stress indicator during phase-change (plane xy). |
A cylinder 100 mm long with a radius of 100 mm is subjected to sidepressing between two plane dies. It is compressed to 100 mm. The material properties are the following: GPa, , kg/m, MPa, MPa, friction coefficient = 0.2. The die velocity is assumed to be 2 m/s. Initial set-up is shown in Figure 7. A quarter of a cylinder was discretized with tri-linear hexahedra and linear tetrahedra meshes.
Figure 8 shows the results obtained using the hexahedral mesh and a standard mixed formulation. The results show the distribution of the effective plastic strain and pressure on the deformed shape. The sensitivity of the FIC results with the expression of the intrinsic time parameter of Eq.(71) was studied by defining and solving the problem for different values of . The results obtained with the FIC method are shown in Figs. 9 and 10 for and , respectively. The alternative expression for of Eq.(74) has also been studied. The results for this case are shown in Fig. 11. Quite a good agreement can be seen between the FIC solutions and the reference solution with the best results for the effective plastic strain obtained for calculated according to Eq. (71) with . The results for the two alternative formulae for are similar, but those obtained using Eq. (71) seem to be slightly better. In any case, the results on the pressure distribution are quite insentitive to the value of . This is also confirmed in Fig. 12b, which displays the distribution of the pressure along the line defined in Fig. 12a. A small perturbation can be seen at the sharp edges of the deformed body.
All the calculations here have been carried out using fully explicit version of the algorithm which is more efficient than the semi-implicit one with giving practically the same results.
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Figure 7: Sidepressing of a cylinder: (a) initial tetrahedral mesh; (b) initial hexahedral mesh |
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Figure 8: Sidepressing of a cylinder, mixed formulation, hexahedral mesh: (a) effective plastic strain; (b) pressure distribution |
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Figure 9: Sidepressing of a cylinder, FIC algorithm (), tetrahedra, mesh of 22186 elements: (a) effective plastic strain; (b) pressure distribution |
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Figure 10: Sidepressing of a cylinder, FIC algorithm (), tetrahedra, mesh of 22186 elements: (a) effective plastic strain; (b) pressure distribution |
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Figure 11: Sidepressing of a cylinder, FIC algorithm ( calculated according to Eq. (74)), tetrahedra, mesh of 22186 elements: (a) effective plastic strain; (b) pressure distribution |
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Figure 12: a) Definition of the line for comparison of pressure distribution b) Pressure distribution along the line |
Backward extrusion of a cylinder made of steel 16MNCr5 has been analysed using an axi-symmetric formulation. This is a benchmark example of the finite element program for forming simulation MARC/Autoforge [54]. The tooling and billet geometry are given in Fig. 13a. Initial material dimensions are the following: length 30 mm and diameter 30 mm. The punch of diameter 20 mm has a prescribed stroke of 28 mm. Material properties are as follows: Young's modulus MPa, Poissson's coefficient , material density kg/m, yield stress MPa and hardening modulus MPa. Friction between the material and tools is defined by the Coulomb friction coefficient .
The simulation of the backward extrusion process was carried out with a particularization of the FIC formulation of Section 6 for axisymmetric solids. Regeneration of the mesh was performed when element distorsion was excessive. Figures 13b and 13c show the results in the form of the final deformed shape with the distribution of the effective plastic strain obtained using quadrilaterals with a mixed formulation, and using triangles and the FIC algorithm, respectively. The results are in a good agreement with the solution given in [54]. This example demonstrates the efficiency of the FIC algorithm for simulation of bulk forming processes. Different stages of the forming process are shown in Fig. 14.
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Figure 14: Backward extrusion – deformed shapes with effective plastic strain distribution at different stages of forming. Solution with triangles and the FIC algorithm |
The stabilized formulation was applied to model the manufacturing of a hose-clamp band of steel AISI 409L (Fig. 15). The initial set-up of the tooling and band is shown in Fig. 16. A series of grooves are forged in the band by the roll passing over the band placed on the toothed punch. The band thickness is 0.7 mm and its width 8 mm. Plane strain conditions have been assumed. Material properties are as follows: Young's modulus MPa, Poissson's coefficient , material density kg/m, the true stress–true strain relationship is given by the power Ludwik–Nadai equation MPa, friction between the material and tools is defined by the Coulomb friction coefficient .
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Figure 15: A hose clamp |
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Figure 16: A hose clamp – initial set-up |
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Figure 17: A hose clamp – deformed shapes at different stages of forming with distribution of effective plastic strain |
Simulation was carried out using linear triangular elements and the FIC formulation. As in the previous example the meshes where regenerated when element distorsion was excessive. The purpose of the simulation was to check if the expected groove depth and tooth height in the band were obtained. Figures 17a and 17b show the results in the form of the deformed shape with the distribution of the effective plastic strain at different stages of forming. The results are in good agreement with the real process. Figure 18 shows a detail of the deformed shape with finite element discretization and distribution of effective plastic strain. In this figure the obtained dimensions are compared with the required ones shown in brackets. Effects of elastic springback are also clearly seen.
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Figure 18: Detail of the deformed shape with finite element discretization and distribution of effective plastic strain |
Remark. The examples presented show that the FIC/FEM formulation is an effective procedure for solving bulk metal forming problems involving full or quasi-incompressible situations. The key advantage of the FIC approach versus more standard mixed FEM formulations is that it provides a natural theoretical framework for equal order finite element interpolations for the velocity and pressure variables, both in the context of implicit and explicit solution schemes. We note the simplicity and effectiveness of the full explicit algorithm as demonstrated in the examples presented. The FIC formulation reproduces also the best feature of the so called stabilized FEM method for incompressible problems, such as the CBS scheme [20,21,25,26,58], the pressure gradient operator method [22] and the subgrid scale method [23] among others.
The Lagrangian formulation is an excellent procedure for treating bulk forming processes involving the interaction of fluids and solids using a unified formulation. An important advantage of the Lagrangian formulation is that both the motions of the solid and the fluid are defined in the same frame of reference and modelled with the some governing equations.
The Lagrangian fluid flow equations can be simply obtained by noting that the velocity of the mesh nodes and that of the fluid particles are the same. Hence the the convective terms vanish in the momentum equations, while the rest of the fluid flow equations remain unchanged. The resulting governing equations have an identical form as those of the Stokes flow problem, with the motion of the flow particles being referred now to a Lagrangian coordinate frame.
The FEM algorithms for solving the Lagrangian flow equations are very similar to those presented for incompressible solids in a previous section. Here we present a particular class of Lagrangian formulation called the particle finite element method (PFEM) [38-41]. The PFEM treats the mesh nodes in the fluid and solid domains as dimensionless particles which can freely move an even separate from the main fluid domain representing, for instance, the effect of fluid drops. A finite element mesh connects the nodes defining the discretized domain where the governing equations are solved in the standard FEM fashion.
The quality of the numerical solution depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur.
A typical solution with the PFEM involves the following steps.
The iterative solution scheme chosen is a particularization of the fractional step algorithm of Section 3.1 for . In summary the solution steps are the following.
Step 3.1. Compute the predicted velocities (viz, Eq.(31))
|
(75) |
Step 3.2. Compute from Eq.(32) for .
Step 3.3. Compute explicitely from Eq.(33) with .
Step 3.4. Compute explicitely from Eq.(35).
Step 3.5. Solve for the motion of the solid. This can be performed by integrating the dynamic equations of motion in the solid. Here the algorithm of Section 6 can be used, as it applies to both “compressible” and incompressible materials.
Step 3.6. Estimate a new position of the mesh nodes in terms of the time increment size as
|
(76) |
It is important to note that all matrices in Steps 3.1–3.5 are evaluated at the last predicted configuration .
In steps 3.1–3.6 superindex denotes the iteration within each time step.
Step 3.7. Check the convergence of the velocity and pressure fields in the fluid and the displacements, strains and stresses in the solid. If convergence is achieved froze the final position of the mesh nodes and move to the next time increment (step 4), otherwise return to step 3.1 for the next iteration.
Above algorithm can be found to be analogous to the standard updated lagrangian scheme typically used in non linear solid mechanics problems [8,57].
Despite the motion of the nodes within the iterative process, in general there is no need to regenerate the mesh at each iteration. In the examples presented in the paper the mesh in the fluid domain has been regenerated at each time step. A cheaper alternative is to generate a new mesh only after a prescribed number of converged time steps, or when the nodal displacements induce significant geometrical distorsions in some elements.
The boundary conditions are applied as described in Section 3.1.
In the examples presented in the paper the time increment size has been chosen as
|
(77) |
where is the distance between node and the closest node in the mesh.
The condition of prescribed velocities or pressures at the solid boundaries in the PFEM are applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. In some problems it is useful to define a layer of nodes adjacent to the external boundary in the fluid where the condition of prescribed velocity is imposed. These nodes typically remain fixed during the solution process. Contact between liquid particles and the solid boundaries is accounted for by the incompressibility condition which naturally prevents the liquid nodes to penetrate into the solid boundaries [38-41].
One of the key points for the success of the PFEM is the fast regeneration of a finite element mesh at every time step on the basis of the position of the nodes in the space domain. In our work the mesh is typically generated using the so called extended Delaunay tesselation (EDT) [38,52]. The EDT allows one to generate non standard meshes combining elements of arbitrary polyhedrical shapes (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrary polyhedra in 3D) in a computing time of order , where is the total number of nodes in the mesh. The continuous shape functions of the elements are obtained using the so called meshless finite element interpolation (MFEM) [53].
The PFEM requires the correct definition of the boundary domain. Sometimes, boundary nodes are clearly distinguished from internal nodes. In other cases, the total set of nodes is the only information available and the algorithm must recognize the boundary nodes.
Considering that the nodes follow a variable distribution, where is the minimum distance between two nodes, the following criterion has been used. For each two nodes (three nodes in 3D) a circle (a sphere in 3D) of radius equal to containing the nodes is plotted. All nodes laying on a circle (sphere) with a radius greater than , not containing any other node in the interior are considered as boundary nodes. In practice, is a parameter close to, but greater than one. This criterion coincides with the Alpha Shape concept [39,55].
The method also allows to identify isolated fluid particles (nodes) outside the main fluid domain. These particles are treated as part of the external boundary where the pressure is fixed to the atmospheric value. We note that the “flying particles” are in fact “points” with no mass which motion is followed by integrating the dynamic equations of continuum mechanics for known values of the mass, the initial velocity and acceleration, gravity body forces and a prescribed zero (atmospheric) pressure.
Thermal-mechanical problems can be effectively treated with the PFEM. This requires solving for the temperature field at each time step, accounting for the couplings induced by the mechanical equations. The effect of temperature in the mechanical problem is introduced via the constitutive equation in the usual manner.
The form of the heat transfer equation is identical to that of Eq.(70). Recall that the Lagrangian formulation eliminates the convective term in the heat transfer equation and hence the FIC stabilization is here not needed [41].
The key advantage of the PFEM versus conventional particle method is that it retains the best feature of the FEM to solve problems in continuum mechanics using a variational approach. A standard finite element mesh is used to discretize in space the problem variables and for solving the governing equations precisely as in the standard FEM. The combination of the lagrangian FEM with the Alpha-Shape approach for identification of the domain boundary at each time step and the frequent regeneration of the finite element mesh are the distinct features of the PFEM.
The sloshing example shown in Figure 19 ilustrates the ability of the PFEM to simulate the flow of liquids within closed cavities with large motions of the free surface. This feature of the PFEM is essential to model the filling of moulds as shown in the next examples. More applications of the PFEM to sloshing problemas are reported in [39].
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Figure 19: PFEM results for a large amplitude sloshing problem |
Figure 20 shows the results of the filling of a 2D mould cavity with a powder material using the PFEM. Note that the essential feature of the filling process are well reproduced. The finite element mesh used for the computation in a certain instant is shown in Figure 20b. This ilustrates the fact that the PFEM is, in fact, a blending of particle and finite element procedure. Other applications of the PFEM to powder compaction problems are reported in [57].
Figure 21 shows the different stages of the 3D simulation of the filling of a mould with a casted metal using the PFEM. The pictures show the evolution of the casting during the filling process. The metal flows through the vertical mould first and then fills the lower and upper lateral ducts, as expected. The gray colour intensities denote the different temperatures in the melt with a whiler colour indicating the cooler region. Note that the metal cools down as it progresses within the mould.
The example in Figure 22 shows the simulation of the casting of a mechanical part using the PFEM. Note that the mesh discretizing the casted region is progressively generated as the mould is filled. This is a distinct feature of the PFEM which distinguishes this method from more standard VOF techniques using Eulerian and ALE approaches.
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Figure 21: Filling of a 3D mould with a casted metal using the PFEM. Colours indicate the temperature values |
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Figure 22: Simulation of a casting process using the PFEM |
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Figure 22: (cont.) Cut through the mesh |
The PFEM is particularly suited for analyzing problems involving mixing of two fluids or that of a collection of particles in a fluid. This is very useful to model a wide class of material forming processes [5,7].
A first example of this kind is shown in Figure 23. The problem represents the mixing of two fluids of different density. The PFEM allows to track the position of the fluid particles of lower density which mix within the second fluid.
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Figure 23: Analysis of the mixing of two fluids of different density with the PFEM |
Figure 24 shows a last example of application of the PFEM to the mixing of a collection of particles within a container filled with a fluid of a higher density. Initially the particles are thrown into the container and mix within the fluid as shown. As time evolves the particles move up naturally towards the surface of the fluid due to their lower density.
The last two examples clearly show the possibilities of the PFEM for analysis of material mixing situations. Extensions of the approach accounting for thermal-coupled effects and chemical reactive situations are straightforward [41].
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Figure 24: Mixing of particles in a fluid. Evolution of the particles during the mixing process |
The finite calculus form of the balance equations in mechanics is a good starting point for deriving stabilized finite element methods for solving a variety of bulk metal forming problems involving fluid flow and large deformation of solids using Euler and Lagrangian descriptions. Free surface effects and coupled thermal situations can be accounted for in a straightforward manner within the general algorithm. A fractional step scheme with intrinsic stabilization properties has been described. Applications of the Eulerian and Lagrangian methods presented range from mould filling, solidification and cooling situations in casting processes to forging and extrussion problems, among many others. The Lagrangian PFEM formulation is very effective for the analysis of bulk metal forming problems and material design processes involving filling of moulds, material mixing situations and complex fluid-structure interactions.
The authors thank the company QUANTECH ATZ, SA (www.quantech.es) for providing the code VULCAN for solution of the casting problem presented. Thanks are also given to MIKALOR SA for providing data for the hose clamp example.
[1] J.L. Chenot and E. Oñate, Modelling of Metal Forming Processes, Kluwer Academic, Dordrecht, 1988.
[2] J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (Eds), Numerical Analysis of Forming Processes, Wiley, Chichester, 1984.
[3] J. Huetnik and F.P.T. Baaijens (Eds), Simulation of Materials Processing: Theory, Methods and Applications, NUMIFORM 98, Enschede, Países Bajos, A.A. Balkema, Rotterdam, 22–5 Jun. 1998.
[4] M. Pietrzyk, J. Kusiak, J. Majta, P. Hartley and I. Pillinger, Metal Forming 2000, A.A. Balkema, Rotterdam, 2000.
[5] K. Mori, Simulation of Material Processing: Theory, Methods and Applications, A.A. Balkema, Lisse, 2001.
[6] D.R.J. Owen, E. Oñate and B. Suarez (Eds.), Computational Plasticity VII. Fundamentals and Applications, CIMNE, Barcelona 2003.
[7] S. Ghosh, J.C. Castro and J.K. Lee, ``Material Processing and Design: Simulation and ApplicatOñate,ions, NUMIFORM 2004, American Institute of Physics, 2004.
[8] O.C. Zienkiewicz and R.C. Taylor,The finite element method, 5th Edition, 3 Volumes, Butterworth–Heinemann, 2000.
[9] C. Hirsch, Numerical computation of internal and external flow, J. Wiley, Vol. 1 1988, Vol. 2, 1990.
[10] J. Peraire, K. Morgan and J. Peiro, “The simulation of 3D incompressible flows using unstructured grids”, In Frontiers of Computational Fluid Dynamics, Caughey DA and Hafez MM. (eds), Chapter 16, J. Wiley, 1994.
[11] M. Storti, N. Nigro and S.R. Idelsohn, “Steady state incompressible flows using explicit schemes with an optimal local preconditioning”, Computer Methods in Applied Mechanics and Engineering, 124, 231–252, 1995.
[12] J.C. Heinrich, P.S. Hayakorn and O.C. Zienkiewicz, “An upwind finite element scheme for two dimensional convective transport equations”, Int. J. Num. Meth. Engng., 11, 131–143, 1977.
[13] A. Brooks and T.J.R. Hughes, “Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”, Comput. Methods Appl. Mech. Engrg, 32, 199–259, 1982.
[14] T.J.R. Hughes and M. Mallet, “A new finite element formulations for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems”, Comput Methods Appl. Mech. Engrg., 58, pp. 305–328, 1986.
[15] M.A. Cruchaga and E. Oñate, “A finite element formulation for incompressible flow problems using a generalized streamline operator”, Comput. Methods in Appl. Mech. Engrg., 143, 49–67, 1997.
[16] M.A. Cruchaga and E. Oñate, “A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces”, Comput. Methods in Appl. Mech. Engrg., 173, 241–255, 1999.
[17] T.J.R. Hughes, L.P. Franca and G.M. Hulbert, “A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations”, Comput. Methods Appl. Mech. Engrg., 73, pp. 173–189, 1989.
[18] J. Donea, “A Taylor-Galerkin method for convective transport problems”, Int. J. Num. Meth. Engng., 20, 101–119, 1984.
[19] R. Löhner, K. Morgan and O.C. Zienkiewicz, “The solution of non-linear hyperbolic equation systems by the finite element method”, Int. J. Num. Meth. in Fluids, 4, 1043, 1984.
[20] R. Codina, M. Vazquez and O.C. Zienkiewicz, “A general algorithm for compressible and incompressible flow - Part III. The semi-implicit form” Int. J. Num. Meth. in Fluids, 27, 13–32, 1998.
[21] R. Codina and O.C. Zienkiewicz, “CBS versus GLS stabilization of the incompressible Navier-Stokes equations and the role of the time step as stabilization parameter”, Communications in Numerical Methods in Engineering, 18 (2), 99–112, 2002.
[22] R. Codina and O. Soto, “A numerical model to track two-fluid interfaces based on a stabilized finite element method and the level set technique”, Int. J. Num. Meth. in Fluids, 4, 293–301, (2002).
[23] R. Codina, “Stabilized finite element approximation of transient incompressible flows using orthogonal subscales”, Comput. Methods Appl. Mech. Engrg., 191, 4295–4321, 2002.
[24] J. Donea and A. Huerta, "Finite element method for flow problems", J. Wiley, 2003.
[25] O.C. Zienkiewicz, J. Rojek, R.L. Taylor and M. Pastor, “Triangles and tetrahedra in explicity dynamic codes for solids”, Int. J. Num. Meth. Engng., 43, 565–583, 1998.
[26] J. Rojek, O.C. Zienkiewicz, E. Oñate and E. Postek, “Advances in FE explicit formulations for simulation of metal forming processes”, J. of Material Processing Technol., 119, 41–47, 2001.
[27] M. Chiumenti, Q. Valverde, C. Agelet de Saracibar and M. Cervera, “A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations”, Computer Methods in Applied Mechanics and Engineering, 191, (46), 5253–5264, (2002).
[28] M. Cervera, M. Chiumenti, Q. Valverde and C. Agelet de Saracibar, “Mixed linear/linear simplicial elements for incompressible elasticity and plasticity”, Comput. Meth. Appl. Mech. Engrg., 192 (49-50), 5249–5263, (2003).
[29] M. Chiumenti, Q. Valverde, C. Agelet de Saracibar and M. Cervera, “A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra”, Int. Journal of Plasticity, 20 (8-9), 1487–1504, (2004).
[30] E. Oñate, Derivation of stabilized equations for advective-diffusive transport and fluid flow problems, Comput. Meth. Appl. Mech. Engng., Vol. 151, pp. 233–267, (1998).
[31] E. Oñate, “Possibilities of finite calculus in computational mechanics”, Int. J. Num. Meth. Engng., 60(1), 255–281, 2004.
[32] E. Oñate, “A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation”, Comp. Meth. Appl. Mech. Engng., 182, 1–2, 355–370, 2000.
[33] E. Oñate and J. García, “A finite element method for fluid-structure interaction with surface waves using a finite calculus formulation”, Comput. Meth. Appl. Mech. Engrg., 191, 635–660, (2001).
[34] E. Oñate, J. García, S.R. Idelsohn and F. Del Pin. FIC formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian approaches, Submitted to Comput. Methods Appl. Mech. Engng., July 2004.
[35] E. Oñate, J. Arteaga, J, García and R. Flores, “Error estimation and mesh adaptivity in incompressible viscous flows using a residual power approach”, Submitted to Comput. Meth. Appl. Mech. Engrg., 2004.
[36] E. Oñate, R.L. Taylor, O.C. Zienkiewicz and J. Rojek, “A residual correction method based on finite calculus”, Engineering Computations, 20, (5/6), 629–658, 2003.
[37] E. Oñate, J. Rojek, R.L. Taylor and O.C. Zienkiewicz, “Finite calculus formulation for incompressible solids using linear triangles and tetrahedra”, Int. J. Num. Meth. Engng., 59, 1473–1500, 2004.
[38] S.R. Idelsohn, E. Oñate and F. Del Pin, “A lagrangian meshless finite element method applied to fluid-structure interaction problems”, Computer and Structures, 81, 655–671, 2003.
[39] E. Oñate, S.R. Idelsohn, F. Del Pin and R. Aubry, “The Particle finite element method. An overview”, Int. J. Comput. Meth., 1 (2), 267–307, 2004.
[40] S.R. Idelsohn, E. Oñate and F. Del Pin, “The Particle Finite Element Method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves”, Int. J. Num. Meth. Engng., (Submitted), 2004.
[41] R. Aubry, S.R. Idelsohn and E. Oñate, “Particle finite element method in fluid mechanics including thermal convection-diffusion”. Computer & Structures, submitted, 2004.
[42] E. Oñate, C. Sacco and S. Idelsohn, “A finite point method for incompressible flow problems”, Computing and Visualization in Science, 2, 67–75, 2000.
[43] O.C. Zienkiewicz, P.C. Jain y E. Oñate, Flow of solids during forming and extrusion: some aspects of numerical solutions, Int. J. Solids Struct., 14, 15–38, 1978.
[44] O.C. Zienkiewicz, E. Oñate and J.C. Heinrich, A general formulation for coupled thermal flow of metals using finite elements, Int. J. Num. Meth. Eng., 17, 1497–514, 1981.
[45] R. Codina, “Pressure stability in fractional step finite element method for incompressible flows”, J. Comput. Physics, 170, 112–140, 2001.
[46] E. Oñate and M. Manzan, A general procedure for deriving stabilized space-time finite element methods for advective-diffusive problems, Int. J. Num. Meth. Fluids, 31, 203–221, 1999.
[47] Codina, R., Schaefer, U., and Oñate, E., “Mould filling simulation using finite elements”, Int. J. Num. Meth. Engng., 4, 291–310, (1994).
[48] Codina, R., and Soto, O., “Finite element simulation of the filling of thin moulds”, Int. J. Num. Meth. Engng., 46, 1559–1573, (1999).
[49] C. Agelet de Saracibar, M. Cervera and M. Chiumenti, “ On the Formulation of coupled thermoplastic problems with phase-change”, International Journal of Plasticity, Vol. 15, 1–34, (1999).
[50] M. Cervera, C. Agelet de Saracibar and M. Chiumenti, “ Thermo-mechanical analysis of industrial solidification processes”, Int. Journal for Num. Methods in Engineering, Vol. 46, 1575-1591, (1999).
[51] C. Agelet de Saracibar, M. Cervera and M. Chiumenti, “On the constitutive modeling of coupled thermo-mechanical phase-change problems”, International Journal of Plasticity, Vol. 17, 1565-1622, (2001).
[52] N. Calvo, S.R. Idelsohn and E. Oñate, “The extended Delaunay tesselation”, Engineering Computations, 20 (5/6), 583–600, 2003.
[53] S.R. Idelsohn, E. Oñate, N. Calvo and F. del Pin, “The meshless finite element method”, Int. J. Num. Meth. Engng., Vol. 58, pp. 893- 912, 2003.
[54] MARC/Autoforge User's Guide 2.2. 1999.
[55] H. Edelsbrunner and E.P. Mucke, E.P., “Three dimensional alpha shapes”, ACM Trans. Graphics, 13, 43–72, 1999.
[56] VULCAN. Finite element system for casting simulation, Quantech ATZ, Barcelona, Spain, (2003), (www.quantech.es).
[57] J.C. Cante, J. Oliver and C. González, “Powder forming simulation with the particle finite element method”, Computational Mechanics, Z.H. Yao, M.W. Yuan and W.X. Zhong (Eds.), CD-Rom Proceedings of the WCCM VI, Beijing, September 5–10, 2004.
[58] J. Rojek, E. Oñate and R.L. Taylor, “CBS based stabilization in explicit solid mechanics”. Int. J. Num. Mech. Engng., 66(10), 1547-1568, 2006.
In Eqs.(24)
|
(A.1) |
If we denote the node indexes with superscripts , the space indexes with subscripts , the element contributions to the components of the arrays involved in Eqs.(24) and (A.1) are
|
|
(A.2) |
It is understood that all the arrays are matrices (except , which is a vector) whose components are obtained by grouping together the left indexes in the previous expressions ( and possibly ) and the right indexes ( and possibly ).
Note that the stabilization matrix in Eq.(A.1) adds additional orthotropic diffusivity terms of value .
Published on 01/01/2006
DOI: 10.1016/j.cma.2004.10.018
Licence: CC BY-NC-SA license
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