Multiscale enrichment based on partition of unity for nonperiodic fields and nonlinear problems

We present a generalization of the Multiscale Enrichment based on Partition of Unity (MEPU) formulation originally reported in Fish and Yuan (Int J Numer Methods Eng 62:1341–1359, 2005) to account for boundary layers, nonperiodic fields and nonlinear systems. MEPU is aimed at extending the range of applicability of the mathematical homogenization theory to nonlinear nonperiodic systems with inseparable fine and coarse scales. Performance studies for both continuum and coarse grained discrete systems are conducted to validate the formulation.

[1]  Integral equations for the plane potential problem on domains with a corner or a slit , 1993 .

[2]  Jacob Fish,et al.  Discrete-to-continuum bridging based on multigrid principles , 2004 .

[3]  Ted Belytschko,et al.  The extended finite element method for rigid particles in Stokes flow , 2001 .

[4]  T. Belytschko,et al.  On the construction of blending elements for local partition of unity enriched finite elements , 2003 .

[5]  Jacob Fish,et al.  Multigrid method for periodic heterogeneous media Part 1: Convergence studies for one-dimensional case , 1995 .

[6]  Jacob Fish,et al.  A generalized space–time mathematical homogenization theory for bridging atomistic and continuum scales , 2006 .

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

[8]  Jacob Fish,et al.  Hierarchical composite grid method for global-local analysis of laminated composite shells , 1997 .

[9]  C. D. Mote.  Global‐local finite element , 1971 .

[10]  Jacob Fish,et al.  The s‐version of the finite element method for multilayer laminates , 1992 .

[11]  J. Fish,et al.  Multi-grid method for periodic heterogeneous media Part 2: Multiscale modeling and quality control in multidimensional case , 1995 .

[12]  Jacob Fish,et al.  Finite deformation plasticity for composite structures: Computational models and adaptive strategies , 1999 .

[13]  Jacob Fish,et al.  Multiscale finite element method for a locally nonperiodic heterogeneous medium , 1993 .

[14]  J. Fish,et al.  Microscale reduction error indicators and estimators for a periodic heterogeneous medium , 1994, Computational Mechanics.

[15]  J. N. Reddy,et al.  Variable Kinematic Modelling of Laminated Composite Plates , 1996 .

[16]  J. Fish,et al.  Hierarchical modelling of discontinuous fields , 1992 .

[17]  Jacob Fish,et al.  Micromechanical elastic cracktip stresses in a fibrous composite , 1993 .

[18]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[19]  W. B. VanderHeyden,et al.  Compatible Fluxes for van Leer Advection , 1998 .

[20]  M. Kanninen,et al.  A finite element calculation of stress intensity factors by a modified crack closure integral , 1977 .

[21]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

[22]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[23]  Jacob Fish,et al.  On adaptive multilevel superposition of finite element meshes for linear elastostatics , 1994 .

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

[25]  S. Scott Collis,et al.  The DG/VMS Method for Unified Turbulence Simulation , 2002 .

[26]  Naoki Takano,et al.  Multi-scale computational method for elastic bodies with global and local heterogeneity , 2000 .

[27]  Jacob Fish,et al.  Multiscale enrichment based on partition of unity , 2005 .

[28]  I. Babuska,et al.  Special finite element methods for a class of second order elliptic problems with rough coefficients , 1994 .

[29]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[30]  T. Strouboulis,et al.  The generalized finite element method: an example of its implementation and illustration of its performance , 2000 .

[31]  황재웅.  Efficient finite element analysis using mesh superposition technique , 2001 .

[32]  Ted Belytschko,et al.  Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM , 2004 .

[33]  Jacob Fish,et al.  Adaptive s-method for linear elastostatics , 1993 .

[34]  J. Fish.  The s-version of the finite element method , 1992 .