Imposing essential boundary conditions in mesh-free methods

Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations.

[1]  Pedro Díez,et al.  Convergence of finite elements enriched with mesh-less methods , 2003, Numerische Mathematik.

[2]  Roland Becker,et al.  Mesh adaptation for Dirichlet flow control via Nitsche's method , 2002 .

[3]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[4]  Wing Kam Liu,et al.  Implementation of boundary conditions for meshless methods , 1998 .

[5]  Juhani Pitkäranta,et al.  Boundary subspaces for the finite element method with Lagrange multipliers , 1979 .

[6]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[7]  Michael Griebel,et al.  A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs , 2000, SIAM J. Sci. Comput..

[8]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[9]  Rolf Stenberg,et al.  On weakly imposed boundary conditions for second order problems , 1995 .

[10]  Gregory J. Wagner,et al.  Hierarchical enrichment for bridging scales and mesh-free boundary conditions , 2001 .

[11]  I. Babuska.  The Finite Element Method with Penalty , 1973 .

[12]  Weimin Han,et al.  A reproducing kernel method with nodal interpolation property , 2003 .

[13]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[14]  Ted Belytschko,et al.  A coupled finite element-element-free Galerkin method , 1995 .

[15]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[16]  P. Hansbo,et al.  CHALMERS FINITE ELEMENT CENTER Preprint 2000-06 Discontinuous Galerkin Methods for Incompressible and Nearly Incompressible Elasticity by Nitsche ’ s Method , 2007 .

[17]  Sivakumar Kulasegaram,et al.  Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations , 2000 .

[18]  Wing Kam Liu,et al.  Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures , 1996 .

[19]  Editors , 1986, Brain Research Bulletin.

[20]  Wing Kam Liu,et al.  Admissible approximations for essential boundary conditions in the reproducing kernel particle method , 1996 .

[21]  I. I. Bakelʹman,et al.  Geometric Analysis and Nonlinear Partial Differential Equations , 1993 .

[22]  Ted Belytschko,et al.  Overview and applications of the reproducing Kernel Particle methods , 1996 .

[23]  I. Babuska,et al.  Meshless and Generalized Finite Element Methods: A Survey of Some Major Results , 2003 .

[24]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[25]  Antonio Huerta,et al.  Coupling Finite Elements and Particles for Adaptivity: An Application to Consistently Stabilized Convection-Diffusion , 2003 .

[26]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[27]  Michael Griebel,et al.  A Particle-Partition of Unity Method Part V: Boundary Conditions , 2003 .

[28]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[29]  Antonio Huerta,et al.  Enrichment and coupling of the finite element and meshless methods , 2000 .

[30]  Wing Kam Liu,et al.  A comparison of two formulations to blend finite elements and mesh-free methods , 2004 .

[31]  A. Huerta,et al.  Enrichissement des interpolations d'?l?ments finis en utilisant des m?thodes sans maillage , 2002 .

[32]  Gregory J. Wagner,et al.  Application of essential boundary conditions in mesh-free methods: a corrected collocation method , 2000 .