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• A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

  P. Nadukandi, E. Oñate, J. García-Espinosa

  Int. J. Numer. Meth. Engng. (2012). Vol. 89 (11), pp. 1367-1391

  
  

  Abstract

  A new Petrov–Galerkin (PG) method involving two parameters, namely α1 and α2, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is, α1 = α2 = α; and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86:18–46) for the Helmholtz equation if the parameters are distinct, that is, α1 ≠ α2. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by α1) and the mass terms (specified by α2) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18–46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions.

  Abstract
  A new Petrov–Galerkin (PG) method involving two parameters, namely α1 and α2, is presented, which yields the following schemes on rectangular [...]

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• A four‐noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory

  A. Eijo, E. Oñate, S. Oller

  Int. J. Numer. Meth. Engng. (2013). Vol. 95 (8), pp. 631-660

  
  

  Abstract

  A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on the refined zigzag theory is presented. The element has seven kinematic variables per node. Shear locking is avoided by introducing an assumed linear shear strain field. The performance of the element is studied in several examples where the reference solution is the 3D finite element analysis using 20‐noded hexahedral elements.

  Abstract
  A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on [...]

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• Improving mass conservation in simulation of incompressible flows

  P. Ryzhakov, E. Oñate, R. Rossi, S. Idelsohn

  Int. J. Numer. Meth. Engng. (2012). Vol. 90 (12), pp. 1435-1451

  
  

  Abstract

  We propose a technique for improving mass‐conservation features of fractional step schemes applied to incompressible flows. The method is illustrated by using a Lagrangian fluid formulation, where the mass loss effects are particularly apparent. However, the methodology is general and could be used for fixed grid approaches. The idea consists in reflecting the incompressibility condition already in the intermediate velocity. This is achieved by predicting the end‐of‐step pressure and using this prediction in the fractional momentum equation. The resulting intermediate velocity field is thus much closer to the final incompressible one than that of the standard fractional step scheme. In turn, the predicted pressure can be used as the boundary condition necessary for the solution of the pressure Poisson equation in case a continuous Laplacian matrix is employed. Using this approximation of the end‐of‐step incompressible pressure as the essential boundary condition considerably improves the conservation of mass, specially for the free surface flows of fluids with low viscosity. The pressure prediction does not require the resolution of any additional equations system. The efficiency of the method is shown in two examples. The first one shows the performance of the method with respect to mass conservation. The second one tests the method in a challenging fluid–structure interaction benchmark, which can be naturally resolved by using the presented approach.

  Abstract
  We propose a technique for improving mass‐conservation features of fractional step schemes applied to incompressible flows. The method is illustrated by using a Lagrangian [...]

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• Combined Eulerian–PFEM approach for analysis of polymers in fire situations

  J. Marti, P. Ryzhakov, S. Idelsohn, E. Oñate

  Int. J. Numer. Meth. Engng. (2012). Vol. 92 (9), pp. 782-801

  
  

  Abstract

  In this paper, we present a computational algorithm for solving an important practical problem, namely, the thermoplastic polymer melting under fire conditions. We propose here a technique that aims at minimizing the computational cost. This is basically achieved by using the immersed boundary‐like approach, combining the particle finite element method for the polymer with an Eulerian formulation for the ambience. The polymer and ambience domains interact over the interface boundary. The boundary is explicitly defined by the position of the Lagrangian domain (polymer) within the background Eulerian mesh (ambience). This allows to solve the energy equation for both subdomains on the Eulerian mesh with different thermal properties. Radiative transport equation is exclusively considered for the ambience, and the heat exchange at the interface is modeled by calculating the radiant heat flux and imposing it as a natural boundary condition.

  Abstract
  In this paper, we present a computational algorithm for solving an important practical problem, namely, the thermoplastic polymer melting under fire conditions. We propose [...]

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• On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

  J. Ródenas, G. Bugeda, J. Albelda, E. Oñate

  Int. J. Numer. Meth. Engng. (2011). Vol. 87 (11), pp. 1105-1126

  
  

  Abstract

  This work analyzes the influence of the discretization error associated with the finite element (FE) analyses of each design configuration proposed by the structural shape optimization algorithms over the behavior of the algorithm. The paper clearly shows that if FE analyses are not accurate enough, the final solution provided by the optimization algorithm will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper proposes the combination of two strategies to reduce the computational cost related to the use of mesh adaptivity in evolutionary optimization algorithms: (a) the use of an algorithm for the mesh generation by projection of the discretization error, which reduces the computational cost associated with the adaptive FE analysis of each geometrical configuration and (b) the successive increase of the required accuracy of the FE analyses in order to obtain a considerable reduction of the computational cost in the early stages of the optimization process.

  Abstract
  This work analyzes the influence of the discretization error associated with the finite element (FE) analyses of each design configuration proposed by the structural shape [...]

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• Consistent pressure Laplacian stabilization for incompressible continua via higher‐order finite calculus

  E. Oñate, S. Idelsohn, C. Felippa

  Int. J. Numer. Meth. Engng. (2011). Vol. 87 (1-5), pp. 171-195

  
  

  Abstract

  We present a stabilized numerical formulation for incompressible continua based on a higher‐order Finite Calculus (FIC) approach and the finite element method. The focus of the paper is on the derivation of a stabilized form for the mass balance (incompressibility) equation. The simpler form of the momentum equations neglecting the non‐linear convective terms, which is typical for incompressible solids, Stokes flows and Lagrangian flows is used for the sake of clarity. The discretized stabilized mass balance equation adds to the standard divergence of velocity term a pressure Laplacian and an additional boundary term. The boundary term is relevant for the accuracy of the numerical solution, especially for free surface flow problems. The Laplacian and boundary stabilization terms are multiplied by non‐linear parameters that have an extremely simple expression in terms of element sizes, the pressure and the discrete residuals of the incompressibility equation and the momentum equations, thus ensuring the consistency of the method. The stabilized formulation allows solving the incompressible problem iteratively using an equal‐order interpolation for the velocities (or displacements) and the pressure, which are the only unknowns. The use of additional pressure gradient projection variables, typical of many stabilized methods, is unnecessary. The formulation is particularly useful for heterogeneous incompressible materials with discontinuous material properties, as it allows computing all the stabilization matrices at the element level. Details of the finite element formulation are given. The good behaviour of the new pressure Laplacian stabilization (PLS) technique is shown in simple but demonstrative examples of application. A very accurate solution was obtained in all cases in 2–3 iterations. Copyright © 2010 John Wiley & Sons, Ltd.

  Abstract
  We present a stabilized numerical formulation for incompressible continua based on a higher‐order Finite Calculus (FIC) approach and the finite element method. The focus [...]

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• A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

  P. Nadukandi, E. Oñate, J. García-Espinosa

  Int. J. Numer. Meth. Engng. (2011). Vol. 86 (1), pp. 18-46

  
  

  Abstract

  We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f(x)+Δφ+ξ2φ=0. The scheme consists of taking the alpha‐interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha‐interpolation method (J. Comput. Appl. Math. 1982; 8(1):15–19) and in 2D making the choice α=0.5 we recover the generalized fourth‐order compact Padé approximation (J. Comput. Phys. 1995; 119:252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163:343–358) (therein using the parameter γ=2). We follow (SIAM Rev. 2000; 42(3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi‐stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((ξℓ)4), where ξ, ℓ represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L2 norm, the H1 semi‐norm and the l∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd.

  Abstract
  We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f(x)+Δφ+ξ2φ=0. The scheme consists of taking [...]

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• Finite volumes and finite elements: Two ‘good friends’

  E. Oñate, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1994). Vol. 37 (19), pp. 3323-3341

  
  

  Abstract

  In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. The paper shows that both procedures share a number of features, like mesh discretization and approximation. Moreover, it is shown that in many cases both techniques are completely equivalent. 

  Abstract
  In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. [...]

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• Melting and Spread of Polymers in Fire with the Particle Finite Element Method

  E. Oñate, R. Rossi, K. Butler, S. Idelsohn

  Int. J. Numer. Meth. Engng. (2010). Vol. 81 (8), pp. 1046-1072

  
  

  Abstract

  A new computational procedure for analysis of the melting and flame spread of polymers under fire conditions is presented. The method, termed Particle Finite Element Method (PFEM), combines concepts from particle-based techniques with those of the standard finite element method (FEM). The key feature of the PFEM is the use of an updated Lagrangian description to model the motion of nodes (particles) in the thermoplastic material. Nodes are viewed as material points which can freely move and even separate from the main analysis domain representing, for instance, the effect of melting and dripping of polymer particles. A mesh connects the nodes defining the discretized domain where the governing equations are solved as in the standard FEM. An incremental iterative scheme for the solution of the nonlinear transient coupled thermal-flow problem, including loss of mass by gasification, is used. Examples of the possibilities of the PFEM for the modelling and simulation of the melting and flame spread of polymers under different fire conditions are described. Numerical results are compared with experimental data provided by NIST.

  Abstract
  A new computational procedure for analysis of the melting and flame spread of polymers under fire conditions is presented. The method, termed Particle Finite Element Method [...]

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• Extended rotation-free plate and beam elements with shear deformation effects

  E. Oñate, F. Zárate

  Int. J. Numer. Meth. Engng. (2010). Vol. 83 (2), pp.196-227

  
  

  Abstract

  The paper describes a methodology for extending rotation-free plate and beam elements in order to accounting for transverse shear deformation effects. The ingredients for the element formulation are: a Hu-Washizu type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure we present an extension of the 3-noded rotation-free basic plate triangle (BPT) originally developed for thin plate analysis to accounting for shear deformation effects of relevance for thick plates and composite laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to computing the bending and shear deformation energies leading to a simple triangular plate element with 3 DOFs per node (termed BPT+ element). For the thin plate case the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A 2-noded rotation-free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems.

  Abstract
  The paper describes a methodology for extending rotation-free plate and beam elements in order to accounting for transverse shear deformation effects. The ingredients for [...]

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