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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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• An adaptive finite point method for the shallow water equations

  E. Ortega, E. Oñate, S. Idelsohn, C. Buachart

  Int. J. Numer. Meth. Engng. (2011). Vol. 88 (2), pp. 180-204

  
  

  Abstract

  An adaptive Finite Point Method (FPM) for solving shallow water problems is presented. The numerical methodology we propose, which is based on weighted‐least squares approximations on clouds of points, adopts an upwind‐biased discretization for dealing with the convective terms in the governing equations. The viscous and source terms are discretized in a pointwise manner and the semi‐discrete equations are integrated explicitly in time by means of a multi‐stage scheme. Moreover, with the aim of exploiting meshless capabilities, an adaptive h‐refinement technique is coupled to the described flow solver. The success of this approach in solving typical shallow water flows is illustrated by means of several numerical examples and special emphasis is placed on the adaptive technique performance. This has been assessed by carrying out a numerical simulation of the 26th December 2004 Indian Ocean tsunami with highly encouraging results. Overall, the adaptive FPM is presented as an accurate enough, cost‐effective tool for solving practical shallow water problems. Copyright © 2011 John Wiley & Sons, Ltd.

  Abstract
  An adaptive Finite Point Method (FPM) for solving shallow water problems is presented. The numerical methodology we propose, which is based on weighted‐least squares approximations [...]

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• Fluid–structure interaction problems with strong added‐mass effect

  S. Idelsohn, F. Del Pin, R. Rossi, E. Oñate

  Int. J. Numer. Meth. Engng. (2009). Vol. 80, (10), pp. 1261-1294

  
  

  Abstract

  In this paper, the so‐called added‐mass effect is investigated from a different point of view of previous publications. The monolithic fluid–structure problem is partitioned using a static condensation of the velocity terms. Following this procedure the classical stabilized projection method for incompressible fluid flows is introduced. The procedure allows obtaining a new pressure segregated scheme for fluid–structure interaction problems, which has good convergent characteristics even for biomechanical application, where the added‐mass effect is strong. The procedure reveals its power when it is shown that the same projection technique must be implemented in staggered FSI methods. Copyright © 2009 John Wiley & Sons, Ltd.

  Abstract
  In this paper, the so‐called added‐mass effect is investigated from a different point of view of previous publications. The monolithic fluid–structure problem is [...]

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• An advancing front technique for filling space with arbitrary separated objects

  R. Löhner, E. Oñate

  Int. J. Numer. Meth. Engng. (2009). Vol. 78 (13), pp. 1618-1630

  
  

  Abstract

  An advancing front technique for filling space with arbitrary, separated objects has been developed. The input required consists of the specification of the desired object type, the mean object size, the distance between objects in space, as well as an initial triangulation of the surface. The objects are assumed to be described by a coarse mesh of tetrahedra. One face at a time is removed from the active front, and, if possible, surrounded by admissible new objects. This operation is repeated until no active faces are left. Two techniques to obtain maximum packing are discussed: closest object placement (during generation) and move/enlarge (after generation). Several examples are included that demonstrate the capabilities of the technique. Copyright © 2009 John Wiley & Sons, Ltd.

  Abstract
  An advancing front technique for filling space with arbitrary, separated objects has been developed. The input required consists of the specification of the desired object [...]

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