Efficient finite element methodology based on cartesian grids: application to structural shape optimization

This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain’s geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

[1]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[2]  Philippe Angot,et al.  Fictitious domain methods to solve convection-diffusion problems with general boundary conditions , 2005 .

[3]  Patrick Patrick Anderson,et al.  A combined fictitious domain/adaptive meshing method for fluid–structure interaction in heart valves , 2004 .

[4]  Lucy T. Zhang,et al.  Immersed finite element method , 2004 .

[5]  Marc Duflot,et al.  Derivative recovery and a posteriori error estimate for extended finite elements , 2007 .

[6]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[7]  O. C. Zienkiewicz,et al.  Superconvergence and the superconvergent patch recovery , 1995 .

[8]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[9]  Nils-Erik Wiberg,et al.  Enhanced Superconvergent Patch Recovery incorporating equilibrium and boundary conditions , 1994 .

[10]  Grégory Legrain,et al.  Image-based computational homogenization and localization: comparison between X-FEM/levelset and voxel-based approaches , 2012, Computational Mechanics.

[11]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[12]  Ted Belytschko,et al.  Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements , 1994 .

[13]  Farhang Daneshmand,et al.  Static and dynamic analysis of 2D and 3D elastic solids using the modified FGFEM , 2009 .

[14]  R. Löhner,et al.  Adaptive embedded unstructured grid methods , 2004 .

[15]  Mark S. Shephard,et al.  An algorithm for multipoint constraints in finite element analysis , 1979 .

[16]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[17]  Valentino Pediroda,et al.  Fictitious Domain approach with hp-finite element approximation for incompressible fluid flow , 2009, J. Comput. Phys..

[18]  Daniel Rixen,et al.  Incorporation of linear multipoint constraints in substructure based iterative solvers. Part 1: a numerically scalable algorithm , 1998 .

[19]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[20]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .

[21]  S. Bordas,et al.  A posteriori error estimation for extended finite elements by an extended global recovery , 2008 .

[22]  Gabriel Bugeda,et al.  On the need for the use of error‐controlled finite element analyses in structural shape optimization processes , 2011 .

[23]  I. Babuska,et al.  Rairo Modélisation Mathématique Et Analyse Numérique the H-p Version of the Finite Element Method with Quasiuniform Meshes (*) , 2009 .

[24]  Pedro Díez,et al.  Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error , 2007 .

[25]  Raino A. E. Mäkinen,et al.  A moving mesh fictitious domain approach for shape optimization problems , 2000 .

[26]  Ted Belytschko,et al.  Modelling crack growth by level sets in the extended finite element method , 2001 .

[27]  R. Storn,et al.  Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces , 2004 .

[28]  Juan José Ródenas,et al.  Error estimation and error bounding in energy norm based on a displacement recovery technique , 2012 .

[29]  J. Prévost,et al.  Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation , 2003 .

[30]  A. Diaz,et al.  Solving three-dimensional layout optimization problems using fixed scale wavelets , 2000 .

[31]  Javier Oliver,et al.  CRITERIA TO ACHIEVE NEARLY OPTIMAL MESHES IN THEh-ADAPTIVE FINITE ELEMENT METHOD , 1996 .

[32]  Osvaldo M. Querin,et al.  Topology design for multiple loading conditions of continuum structures using isolines and isosurfaces , 2010 .

[33]  Ivo Babuška,et al.  Assessment of the cost and accuracy of the generalized FEM , 2007 .

[34]  A. Jahangirian,et al.  Adaptive unstructured grid generation for engineering computation of aerodynamic flows , 2008, Math. Comput. Simul..

[35]  I. Babuska,et al.  The generalized finite element method , 2001 .

[36]  Martin Rumpf,et al.  Finite Element Simulation of Bone Microstructures , 2007 .

[37]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[38]  O. C. Zienkiewicz,et al.  Superconvergent patch recovery techniques – some further tests , 1993 .

[39]  Joseph E. Bishop,et al.  Rapid stress analysis of geometrically complex domains using implicit meshing , 2003 .

[40]  Juan José Ródenas,et al.  Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR‐C technique , 2007 .

[41]  M. Berger,et al.  An Adaptive Version of the Immersed Boundary Method , 1999 .

[42]  Charbel Farhat,et al.  A fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems. Part 2: Neumann boundary conditions , 2002 .

[43]  Stéphane Bordas,et al.  Strain smoothing in FEM and XFEM , 2010 .

[44]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[45]  Stéphane Bordas,et al.  Finite element analysis on implicitly defined domains: An accurate representation based on arbitrary parametric surfaces , 2011 .

[46]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[47]  Kyung K. Choi,et al.  Remesh-free shape optimization using the wavelet-Galerkin method , 2004 .

[48]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[49]  Juan José Ródenas,et al.  Report: Error estimation of recovered solution in FE analysis , 2012, ArXiv.

[51]  Grant P. Steven,et al.  Fixed grid finite elements in elasticity problems , 1999 .

[52]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[53]  Grégory Legrain,et al.  High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation , 2012 .

[54]  E. Maunder,et al.  Effective error sttimation from continous, boundary admissible estimated stress fields , 1996 .

[55]  W. Shyy,et al.  Regular Article: An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries , 1999 .

[56]  Juan José Ródenas,et al.  A recovery‐type error estimator for the extended finite element method based on singular+smooth stress field splitting , 2008 .

[57]  D. Greaves,et al.  Using hierarchical Cartesian grids with multigrid acceleration , 2007 .

[58]  Peter Hansbo,et al.  Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method , 2010 .

[59]  Bhushan Lal Karihaloo,et al.  Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery , 2006 .

[60]  O. González-Estrada,et al.  Accurate recovery-based upper error bounds for the extended finite element framework , 2010 .

[61]  Yuri A. Kuznetsov,et al.  Fictitious Domain Methods For The Numerical Solution Of Three-Dimensional Acoustic Scattering Proble , 1999 .

[62]  S. Bordas,et al.  A simple error estimator for extended finite elements , 2007 .