Residual error based adaptive mesh refinement with the non-intrusive patch algorithm

This paper deals with the introduction of mesh refinement techniques within the non-intrusive patch process. For this, an ad hoc residual based explicit error estimator is built, which is adapted to a multi-scale solution, associated with those non-intrusive mesh refinement technique. Moreover, to reduce the global cost of the process, one introduces an estimate of the convergence error of the non-intrusive algorithm, which allows to reduce the number of iterations. This method is discussed and illustrated in various numerical examples.

[1]  L. Gallimard.  A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress , 2009 .

[2]  Olivier Pironneau,et al.  Numerical zoom for advection diffusion problems with localized multiscales , 2011 .

[3]  Pierre Ladevèze,et al.  Strict and practical bounds through a non-intrusive and goal-oriented error estimation method for linear viscoelasticity problems , 2009 .

[4]  Anthony Gravouil,et al.  Isogeometric local h-refinement strategy based on multigrids , 2015 .

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

[6]  Pedro Díez,et al.  Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method , 2009 .

[7]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[8]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[9]  Pierre Alliez,et al.  Anisotropic polygonal remeshing , 2003, ACM Trans. Graph..

[10]  Mathilde Chevreuil,et al.  A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties , 2013 .

[11]  Carsten Carstensen,et al.  Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM , 2001 .

[12]  Pedro Díez,et al.  Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates , 2003 .

[13]  C. Carstensen,et al.  Constants in Clément-interpolation error and residual based a posteriori estimates in finite element methods , 2000 .

[14]  S.,et al.  " Goal-Oriented Error Estimation and Adaptivity for the Finite Element Method , 1999 .

[15]  K. Bathe,et al.  Review: A posteriori error estimation techniques in practical finite element analysis , 2005 .

[16]  Pierre Gosselet,et al.  A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods , 2014 .

[17]  Thomas Grätsch,et al.  Review: A posteriori error estimation techniques in practical finite element analysis , 2005 .

[18]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

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

[20]  Leszek Demkowicz,et al.  Adaptive finite elements for flow problems with moving boundaries. part I: Variational principles and a posteriori estimates , 1984 .

[21]  Steven E. Benzley,et al.  Localized coarsening of conforming all-hexahedral meshes , 2010, Engineering with Computers.

[22]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[23]  Ivo Babuška,et al.  A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis , 1983 .

[24]  Harold Liebowitz,et al.  Hierarchical mesh adaptation of 2D quadrilateral elements , 1995 .

[25]  Roland Glowinski,et al.  Finite element approximation of multi-scale elliptic problems using patches of elements , 2005, Numerische Mathematik.

[26]  Rolf Rannacher,et al.  A posteriori error control in finite element methods via duality techniques: Application to perfect plasticity , 1998 .

[27]  Anthony T. Patera,et al.  A hierarchical duality approach to bounds for the outputs of partial differential equations , 1998 .

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

[29]  Julien Réthoré,et al.  Direct estimation of generalized stress intensity factors using a multigrid XFEM , 2011 .

[30]  Pedro Díez,et al.  Accurate upper and lower error bounds by solving flux-free local problems in “stars” , 2004 .

[31]  Francesca Rapetti,et al.  Basics and some applications of the mortar element method , 2005 .

[32]  Olivier Allix,et al.  Nonintrusive coupling of 3D and 2D laminated composite models based on finite element 3D recovery , 2014, 1501.01933.

[33]  Jacques Rappaz,et al.  Multiscale algorithm with patches of finite elements , 2007 .

[34]  Paul T. Bauman,et al.  An adaptive strategy for the control of modeling error in two-dimensional atomic-to-continuum coupling simulations , 2009 .

[35]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[36]  Gunar Matthies,et al.  The Inf-Sup Condition for the Mapped Qk−Pk−1disc Element in Arbitrary Space Dimensions , 2002, Computing.

[37]  Mickaël Duval,et al.  Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition , 2016 .

[38]  J. Tinsley Oden,et al.  Adaptive multiscale modeling of polymeric materials with Arlequin coupling and Goals algorithms , 2009 .

[39]  Pierre Ladevèze,et al.  Local error estimators for finite element linear analysis , 1999 .

[40]  Yvon Maday,et al.  Mesh adaptivity in finite elements using the mortar method , 2000 .

[41]  Pierre Gosselet,et al.  Fast estimation of discretization error for FE problems solved by domain decomposition , 2010 .

[42]  Robin Bouclier,et al.  Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture , 2016 .

[43]  Ernst Rank,et al.  Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries , 2014 .

[44]  Vincent Heuveline,et al.  H1-interpolation on quadrilateral and hexahedral meshes with hanging nodes , 2007, Computing.

[45]  R. Verfürth.  A review of a posteriori error estimation techniques for elasticity problems , 1999 .

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

[47]  Johann Rannou,et al.  A local multigrid X‐FEM strategy for 3‐D crack propagation , 2009 .

[48]  P. Ladevèze.  Constitutive Relation Error Estimators and Adaptivity in Structural Engineering , 2005 .

[49]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

[50]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[51]  Jeong-Ho Kim,et al.  A new generalized finite element method for two‐scale simulations of propagating cohesive fractures in 3‐D , 2015 .

[52]  J. Oden,et al.  A unified approach to a posteriori error estimation using element residual methods , 1993 .

[53]  Martin J. Gander,et al.  Schwarz Methods over the Course of Time , 2008 .

[54]  I. Babuska,et al.  A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator , 1987 .

[55]  E. Stein,et al.  An explicit residual‐type error estimator for Q1 ‐quadrilateral extended finite element method in two‐dimensional linear elastic fracture mechanics , 2012 .

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

[57]  Michel Fortin,et al.  A fully optimal anisotropic mesh adaptation method based on a hierarchical error estimator , 2012 .

[58]  O. Allix,et al.  Non-intrusive and exact global/local techniques for structural problems with local plasticity , 2009 .

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