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• Possibilities of finite calculus in computational mechanics

  E. Oñate

  Int. J. Numer. Meth. Engng. (2004). Vol. 60 (1), pp. 255-281

  
  

  Abstract

  The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations. 

  Abstract
  The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a [...]

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• Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

  E. Oñate, J. Rojek, R. Taylor, O. Zienkiewicz

  Int. J. Numer. Meth. Engng. (2004). Vol. 59 (11), pp. 1473-1500

  
  

  Abstract

  Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented.

  Abstract
  Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach [...]

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• A general advancing front technique for filling space with arbitrary objects

  R. Löhner, E. Oñate

  Int. J. Numer. Meth. Engng. (2004). Vol. 61 (12), pp. 1977-1991

  
  

  Abstract

  An advancing front space‐filling technique for arbitrary objects has been developed. The input required consists of the specification of the desired mean point distance in space and an initial triangulation of the surface. One object at a time is removed from the active front, and, if possible, surrounded by admissible new objects. This operation is repeated until no active objects are left. Two techniques to obtain maximum packing are discussed: closest object placement (during generation) and move/enlarge (after generation). Different deposition or layering patterns can be achieved by selecting the order in which objects are eliminated from the active front. Timings show that for simple objects like spheres the scheme is considerably faster than volume mesh generators based on the advancing front technique, making it possible to generate large (> 106) yet optimal clouds of points in a matter of minutes on a PC. For more general objects, the performance may degrade depending on the complexity of the penetration checks. Several examples are included that demonstrate the capabilities of the technique. 

  Abstract
  An advancing front space‐filling technique for arbitrary objects has been developed. The input required consists of the specification of the desired mean point distance [...]

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• The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

  S. Idelsohn, E. Oñate, F. Del Pin

  Int. J. Numer. Meth. Engng. (2004). Vol. 61 (7), pp. 964-989

  
  

  Abstract

  Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. 

  Abstract
  Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal [...]

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• The meshless finite element method

  S. Idelsohn, E. Oñate, N. Calvo, F. Del Pin

  Int. J. Numer. Meth. Engng. (2003). Vol. 58 (6), pp. 893-912

  
  

  Abstract

  A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the finite element method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C0 continuity between elements, which allows the treatment of material discontinuities, and (c) ease of introduction of the boundary conditions.

  Abstract
  A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, [...]

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• A finite point method for compressible flow

  R. Löhner, C. Sacco, E. Oñate, S. Idelsohn

  Int. J. Numer. Meth. Engng. (2002). Vol. 53 (8), pp. 1765-1779

  
  

  Abstract

  A weighted least squares finite point method for compressible flow is formulated. Starting from a global cloud of points, local clouds are constructed using a Delaunay technique with a series of tests for the quality of the resulting approximations. The approximation factors for the gradient and the Laplacian of the resulting local clouds are used to derive an edge‐based solver that works with approximate Riemann solvers. The results obtained show accuracy comparable to equivalent mesh‐based finite volume or finite element techniques, making the present finite point method competitive.

  Abstract
  A weighted least squares finite point method for compressible flow is formulated. Starting from a global cloud of points, local clouds are constructed using a Delaunay technique [...]

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• A basic thin shell triangle with only translational DOFs for large strain plasticity

  F. Flores, E. Oñate

  Int. J. Numer. Meth. Engng. (2001). Vol. 51 (1), pp. 57-83

  
  

  Abstract

  A simple finite element triangle for thin shell analysis is presented. It has only nine translational degrees of freedom and is based on a total Lagrangian formulation. Large strain plasticity is considered using a logarithmic strain–stress pair. A plane stress isotropic behaviour with an additive decomposition of elastic and plastic strains is assumed. A hyperelastic law is considered for the elastic part while for the plastic part a von Mises yield function with non‐linear isotropic hardening is adopted. The element is an extension of a previous similar rotation‐free triangle element based upon an updated Lagrangian formulation with hypoelastic constitutive law. The element termed BST (for basic shell triangle) has been implemented in an explicit (hydro‐) code adequate to simulate sheet‐stamping processes and in an implicit static/dynamic code. Several examples are shown to assess the performance of the present formulation.

  Abstract
  A simple finite element triangle for thin shell analysis is presented. It has only nine translational degrees of freedom and is based on a total Lagrangian formulation. Large [...]

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• Rotation‐free triangular plate and shell elements

  E. Oñate, F. Zárate

  Int. J. Numer. Meth. Engng. (2000). Vol. 47 (1-3), pp. 557-603

  
  

  Abstract

  The paper describes how the finite element method and the finite volume method can be successfully combined to derive two new families of thin plate and shell triangles with translational degrees of freedom as the only nodal variables. The simplest elements of the two families based on combining a linear interpolation of displacements with cell centred and cell vertex finite volume schemes are presented in detail. Examples of the good performance of the new rotation‐free plate and shell triangles are given.

  Abstract
  The paper describes how the finite element method and the finite volume method can be successfully combined to derive two new families of thin plate and shell triangles with [...]

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• Finite element solution of free‐surface ship‐wave problems

  S. Idelsohn, E. Oñate, C. Sacco

  Int. J. Numer. Meth. Engng. (1999). Vol. 45 (5), pp. 503-528

  
  

  Abstract

  An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface.

  Abstract
  An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes [...]

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• Solving structural optimization problems with genetic algorithms and simulated annealing

  S. Botello, J. Marroquín, E. Oñate

  Int. J. Numer. Meth. Engng. (1999). Vol. 45 (8), pp. 1069-1084

  
  

  Abstract

  In this paper we study the performance of two stochastic search methods: Genetic Algorithms and Simulated Annealing, applied to the optimization of pin‐jointed steel bar structures. We show that it is possible to embed these two schemes into a single parametric family of algorithms, and that optimal performance (in a parallel machine) is obtained by a hybrid scheme. Examples of applications to the optimization of several real steel bar structures are presented.

  Abstract
  In this paper we study the performance of two stochastic search methods: Genetic Algorithms and Simulated Annealing, applied to the optimization of pin‐jointed steel bar [...]

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