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• Evolutionary methods for optimal shape design

  A. Fung, E. Pahl, G. Bugeda, E. Oñate

  (2005). Research Report, Nº PI270

  
  

  Abstract

  An optimization code called Optima has been developed at the International Centre for Numerical Methods in Engineering (CIMNE). This report validates two reputably robust Evolutionary Algorithms available in Optima and employs them on aerodynamic shape optimization problems. The two schemes, Differential Evolution Scheme 1 (DE1) and Evolution Strategy coupled with Covariance Matrix Adaptation (ES-CMA), were tested and verified on three standard parametric optimization objective functions through comparison against existing results. Analysis of the test data allowed trends to be established and from this, settings to enhance the performance of the algorithms were proposed and substantiated. The algorithms and their suggested settings were applied on an inverse and a direct constrained shape optimization problem involving NACA four digit airfoils. The inverse task involves the recovery of an airfoil profile through its pressure distribution combination of lift and drag coefficients. Simulation of the airfoil in turbulent flow was done using a Computational Fluid Dynamics (CFD) setup. Finally, the results are presented and the ES-CMA method, using the setting CMA-B proposed in this paper, was found to be most robust. Suggestions for improvements and further work in the optimization configuration and the problem simulation are also proposed.  

  Abstract
  An optimization code called Optima has been developed at the International Centre for Numerical Methods in Engineering (CIMNE). This report validates two reputably robust [...]

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• Topology optimization of plates

  B. Boroomand, A. Barekatein

  (2006). Research Report, Nº PI298

  
  

  Abstract

  In this report we propose a stabilization method for topology optimization of planes. The method can be classified in the category of continuation methods. The new continuation method is based on using continuous of design variables (DV) defined on a set meshes different from the one used for the finite element solution. The optimization procedure stars with using a coarse DV-mesh compared to finite element one. Once the convergence is obtained in the optimizations steps, a finer DV-mesh is nominated for further steps. With such a continuation method one can control the bounds of the gradients of the DV while simultaneously smooth the values in a more logical fashion, compared to what conventional filters perform. The DV-mesh refinement can be continued until the final mesh becomes similar to the finite element mesh. Depending on the formulation and elements used for the plate problems, e.g. with Kirchhoff or Mindlin-Reissner hypothesis, the refinement may further be continued so that the DV elements become smaller than the plate elements. Application of the method is shown over a wide range of plate problems. Linear and nonlinear plate behaviors formulated by Kirchhoff or Mindlin Reissner hypothesis, while using several forms of DV, are considered to show the performance of the proposed method. As one of the main DV, density is used in a power-law approach (or in an artificial material approach). This is performed in two forms, on in obtaining the topology of Thickness is also used as a realistic design variable in order to show the performance of the method in a rather well-posed optimization problem. We have also included results from a homogenization approach. Comparison in made with conventional element/nodal based approaches using filter. The results show excellent and robust performance of the proposed method. Due to the wide range of cases studied, some inserting side conclusions are also given in this report.  

  Abstract
  In this report we propose a stabilization method for topology optimization of planes. The method can be classified in the category of continuation methods. The new continuation [...]

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• Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous Galerkin time discretization

  N. Parés, P. Díez, A. Huerta

  (2006). Research Report, Nº PI297

  
  

  Abstract

  Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques.

  Abstract
  Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds [...]

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• An orthotropic mesh corrected crack model

  M. Cervera

  (2006). Research Report, Nº PI296

  
  

  Abstract

  This paper recovers the original spirit of the continuous crack approaches, where displacements jumps across the crack are smeared over the affected elements and the behaviour is established through a softening stress–(total) strain law, using standard finite element displacement interpolations and orthotropic localconstitutive models. The paper focuses on the problem of shear locking observed in the discrete problem when orthotropic models are used. The solution for this drawback is found in the form of a mesh corrected crack model where the structure of the inelastic strain tensor is linked to the geometry of the cracked element. The discrete model is formulated as a non-symmetric orthotropic local damage constitutive model, in which the softening modulus is regularized according to the material fracture energy and the element size. The resulting formulation is easily implemented in standard non-linear FE codes and suitable for engineering applications. Numerical examples show that the results obtained using this crack model do not suffer from dependence on the mesh directional alignment, comparing very favourably with those obtained using related standard isotropic or orthotropic damage models.

  Abstract
  This paper recovers the original spirit of the continuous crack approaches, where displacements jumps across the crack are smeared over the affected elements and the behaviour [...]

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• Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique

  M. Cervera, M. Chiumenti

  (2006). Research Report, Nº PI294

  
  

  Abstract

  This paper describes a procedure for the solution of problems involving tensile cracking using the so-called smeared crack approach, that is, standard finite elements with continuous displacement fields and a standard local constitutive model with strain-softening. An isotropic Rankine damage model is considered. The softening modulus is adjusted according to the material fracture energy and the element size. The resulting continuum and discrete mechanical problems are analyzed and the question of predicting correctly the direction of crack propagation is deemed as the main difficulty to be overcome in the discrete problem. It is proposed to use a crack tracking technique to attain the desired stability and convergence properties of the corresponding formulation. Numerical examples show that the resulting procedure is well-posed, stable and remarkably robust; the results obtained do not seem to suffer from spurious mesh-size or mesh-bias dependence.

  Abstract
  This paper describes a procedure for the solution of problems involving tensile cracking using the so-called smeared crack approach, that is, standard finite elements [...]

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• Modeling bed erosion in free surface flows by the particle finite element method

  E. Oñate, M. Celigueta, S. Idelsohn

  (2006). Research Report, Nº PI293

  
  

  Abstract

  We present a general formulation for modeling bed erosion in free surface flows using the particle finite element method (PFEM). The key feature of the PFEM is the use of an updated Lagrangian description to model the motion of nodes (particles) in domains containing fluid and solid subdomains. Nodes are viewed as material points (called particles) which can freely move and even separate from the fluid and solid subdomains representing, for instance, the effect of water drops or soil/rock particles. A mesh connects the nodes defining the discretized domain in the fluid and solid regions where the governing equations, expressed in an integral form, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. An incremental iterative scheme for the solution of the nonlinear transient coupled fluid-structure problem is described. The erosion mechanism is modeled by releasing the material adjacent to the bed surface according to the frictional work generated by the fluid shear stresses. The released bed material is subsequently transported by the fluid flow. Examples of application of the PFEM to solve a number of bed erosion problems involving large motions of the free surface and splashing of waves are presented.

  Abstract
  We present a general formulation for modeling bed erosion in free surface flows using the particle finite element method (PFEM). The key feature of the PFEM is the use of [...]

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• Modelo constitutivo para el comportamiento de tejidos biológicos blandos

  S. Oller

  (2006). Research Report, Nº PI292

  
  

  Abstract

  El objetivo de este trabajo es obtener una formulación constitutiva general que permita representar el comportamiento –crecimiento y decrecimiento natural / patológico, remodelación de absorción y de regeneración- de los tejidos biológicos blandos. Entiéndase por tejidos biológicos blandos a aquellos que conforman la piel y órganos del cuerpo humano y que pueden estás sometidos a acciones externas e internas de origen mecánico y metabólico. Estas acciones pueden producir grandes deformaciones transitorias y permanentes.

  Abstract
  El objetivo de este trabajo es obtener una formulación constitutiva general que permita representar el comportamiento –crecimiento y decrecimiento natural / patológico, [...]

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• A study on the lumped preconditioner and memory requirements of feti and related primal domain decompositions methods

  Y. Fragakis

  (2006). Research Report, Nº PI291

  
  

  Abstract

  In recent years, Domain Decomposition Methods (DDM) have emerged as advanced solvers in several areas of computational mechanics. In particular, during the last decade, in the area of solid and structural mechanics, they reached a considerable level of advancement and were shown to be more efficient than popular solvers, like advanced sparse direct solvers. The present contribution follows the lines of a series of recent publications by author on DDM. In the papers, the authors developed a unified theory of primal and dual methods and presented a family of DDM that were shown to be more efficient than previous methods. The present paper extends this work, presenting a new family of related DDM, thus enriching the theory of the relations between primal and dual methods. It also explores memory requirement issues, suggesting also a particularly memory efficient formulation.

  Abstract
  In recent years, Domain Decomposition Methods (DDM) have emerged as advanced solvers in several areas of computational mechanics. In particular, during the last decade, in [...]

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• Force and displacement duality in domain decomposition methods for solid and structural mechanics

  Y. Fragakis

  (2006). Research Report, Nº PI290

  
  

  Abstract

  In recent years, Domain Decomposition Methods (DDM) have emerged as advanced solvers in several of computational mechanics. In particular, during the last decade, in the area of solid and structural mechanics, they reached a considerable level of advancement and were shown to be more efficient than popular solvers, like advanced sparse direct solvers. The present paper explores the extent of application of the general concept of force-displacement duality in DDM. A general framework for the definition of DDM is set up and it is shown that if the definition of a DDM meets some requirements, then it can lead to one primal and one dual formulation. A number of DDM are included in this setting and particular implications for each one of them is researched.

  Abstract
  In recent years, Domain Decomposition Methods (DDM) have emerged as advanced solvers in several of computational mechanics. In particular, during the last decade, in the area [...]

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• Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscale

  R. Codina

  (2006). Research Report, Nº PI289

  
  

  Abstract

  In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf–sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. It is shown that this formulation is stable and optimally convergent for an adequate choice of the algorithmic parameters on which the method depends.

  Abstract
  In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility [...]

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