Multiscale modeling using goal-oriented adaptivity and numerical homogenization. Part II: Algorithms for the Moore-Penrose pseudoinverse

This paper is the second in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal-oriented finite element mesh adaptivity. The first paper developed the mathematical formulation of local homogenization. This was done using an appropriate averaging of the Moore–Penrose pseudoinverse of the element stiffness matrix. We also presented numerical verification of this approach and other numerical results. In the current paper, we present four algorithms to compute the Moore–Penrose pseudoinverse that also exploit the sparsity. These are based on QR factorization, a priori knowledge of the null-space, a regularization based characterization, and lastly an iterative algorithm based on proper splittings of matrices. We analyze some of these algorithms in detail for homogenizing hexahedral elements in the molecular base model of densification in Step and Flash Imprint Lithography.

[1]  Carlos A. Felippa,et al.  The construction of free–free flexibility matrices as generalized stiffness inverses , 1998 .

[2]  Editors , 1986, Brain Research Bulletin.

[3]  Leszek Demkowicz,et al.  A fully automatic hp-adaptivity for elliptic PDEs in three dimensions , 2007 .

[4]  M. Berezhnyya,et al.  Continuum limit for three-dimensional mass-spring networks and discrete Korn ’ s inequality , 2005 .

[5]  I︠u︡. E. Boi︠a︡rint︠s︡ev,et al.  Methods of Solving Singular Systems of Ordinary Differential Equations , 1992 .

[6]  Maciej Paszyński,et al.  On the Modeling of Step-and-Flash Imprint Lithography using Molecular Statics Models , 2005 .

[7]  Guillaume Rateau,et al.  The Arlequin method as a flexible engineering design tool , 2005 .

[8]  P. Donato,et al.  An introduction to homogenization , 2000 .

[9]  William J. Dauksher,et al.  Effects of etch barrier densification on step and flash imprint lithography , 2005 .

[10]  Serge Prudhomme,et al.  On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors , 1999 .

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

[12]  Doron Levy,et al.  On Wavelet-Based Numerical Homogenization , 2005, Multiscale Model. Simul..

[13]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[14]  Ronald E. Miller,et al.  The Quasicontinuum Method: Overview, applications and current directions , 2002 .

[15]  K Schulten,et al.  VMD: visual molecular dynamics. , 1996, Journal of molecular graphics.

[16]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[17]  Todd C. Bailey,et al.  Characterization and modeling of volumetric and mechanical properties for step and flash imprint lithography photopolymers , 2001 .

[18]  Leszek Demkowicz,et al.  Multiscale modeling using goal-oriented adaptivity and numerical homogenization. Part I: Mathematical formulation and numerical results , 2012 .

[19]  Todd Arbogast,et al.  Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media , 2000 .

[20]  Maciej Paszyński,et al.  Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications , 2007 .

[21]  R. RannacherInstitut,et al.  Weighted a Posteriori Error Control in Fe Methods , 1995 .

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

[23]  T. Belytschko,et al.  A bridging domain method for coupling continua with molecular dynamics , 2004 .

[24]  A. Berman,et al.  Cones and Iterative Methods for Best Least Squares Solutions of Linear Systems , 1974 .

[25]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[26]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[27]  T. Coleman,et al.  Large-Scale Optimization with Applications : Part III: Molecular Structure and Optimization , 1997 .

[28]  A. N. Tikhonov,et al.  REGULARIZATION OF INCORRECTLY POSED PROBLEMS , 1963 .

[29]  A. Charnes,et al.  Contributions to the Theory of Generalized Inverses , 1963 .

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

[31]  Mike Miller,et al.  Step and flash imprint lithography template fabrication for emerging market applications , 2007, Photomask Japan.

[32]  M. Zuhair Nashed,et al.  Perturbations and Approximations for Generalized Inverses and Linear Operator Equations , 1976 .

[33]  Timothy A. Davis,et al.  Multifrontral multithreaded rank-revealing sparse QR factorization , 2009, Combinatorial Scientific Computing.

[34]  C. Grant Willson,et al.  Kinetic parameters for step and flash imprint lithography photopolymerization , 2006 .

[35]  Tongxing Lu,et al.  Solution of the matrix equation AX−XB=C , 2005, Computing.

[36]  V. Sima,et al.  Solving linear matrix equations with SLICOT , 2003, 2003 European Control Conference (ECC).

[37]  J. Tinsley Oden,et al.  On the application of the Arlequin method to the coupling of particle and continuum models , 2008 .

[38]  Bernard Choi,et al.  Step and flash imprint lithography: a new approach to high-resolution patterning , 1999, Advanced Lithography.

[39]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[40]  Qingling Zhang,et al.  Some geometric properties of Lyapunov equation and LTI system , 2001, Autom..

[41]  L. Demkowicz.  One and two dimensional elliptic and Maxwell problems , 2006 .

[42]  S. Spagnolo,et al.  Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore , 1967 .

[43]  Gene H. Golub,et al.  Matrix computations , 1983 .

[44]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[45]  Stephan Knapek,et al.  Matrix-Dependent Multigrid Homogenization for Diffusion Problems , 1998, SIAM J. Sci. Comput..

[46]  M. Z. Nashed,et al.  Annotated Bibliography on Generalized Inverses and Applications , 1976 .

[47]  Leszek F. Demkowicz,et al.  A Fully Automatic hp-Adaptivity , 2002, J. Sci. Comput..

[48]  Yi Wei,et al.  Mesoscale modeling for SFIL simulating polymerization kinetics and densification , 2004, SPIE Advanced Lithography.

[49]  J. Oden,et al.  Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: Part II: a computational environment for adaptive modeling of heterogeneous elastic solids , 2001 .

[50]  C. Grant Willson,et al.  Study of the kinetics of step and flash imprint lithography photopolymerization , 2005 .

[51]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[53]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[54]  J. J. Moré,et al.  ISSUES IN LARGE-SCALE GLOBAL MOLECULAR OPTIMIZATION , 1997 .

[55]  J. Tinsley Oden,et al.  Adaptive multiscale modeling of polymeric materials using goal-oriented error estimation, arlequin coupling, and goals algorithms , 2008 .

[56]  Richard H. Byrd,et al.  Global Optimization For Molecular Clusters Using A New Smoothing Approach , 2000, J. Glob. Optim..

[57]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[58]  Todd C. Bailey,et al.  Step and Flash Imprint Lithography: An Efficient Nanoscale Printing Technology , 2002 .

[59]  Timothy A. Davis,et al.  Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization , 2011, TOMS.

[60]  Leszek Demkowicz,et al.  Adaptive methods for problems in solid and fluid mechanics , 1986 .

[61]  U. Hornung.  Homogenization and porous media , 1996 .

[62]  E Weinan,et al.  Some Recent Progress in Multiscale Modeling , 2004 .

[63]  Lois Curfman McInnes,et al.  TAO users manual. , 2003 .

[64]  J. Ericksen.  The Cauchy and Born Hypotheses for Crystals. , 1983 .

[65]  Peter Wriggers,et al.  An Introduction to Computational Micromechanics , 2004 .

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