Integration efficiency for model reduction in micro-mechanical analyses

Micro-structural analyses are an important tool to understand material behavior on a macroscopic scale. The analysis of a microstructure is usually computationally very demanding and there are several reduced order modeling techniques available in literature to limit the computational costs of repetitive analyses of a single representative volume element. These techniques to speed up the integration at the micro-scale can be roughly divided into two classes; methods interpolating the integrand and cubature methods. The empirical interpolation method (high-performance reduced order modeling) and the empirical cubature method are assessed in terms of their accuracy in approximating the full-order result. A micro-structural volume element is therefore considered, subjected to four load-cases, including cyclic and path-dependent loading. The differences in approximating the micro- and macroscopic quantities of interest are highlighted, e.g. micro-fluctuations and stresses. Algorithmic speed-ups for both methods with respect to the full-order micro-structural model are quantified. The pros and cons of both classes are thereby clearly identified.

[1]  Theodore Kim,et al.  Optimizing cubature for efficient integration of subspace deformations , 2008, SIGGRAPH Asia '08.

[2]  Siep Weiland,et al.  Missing Point Estimation in Models Described by Proper Orthogonal Decomposition , 2004, IEEE Transactions on Automatic Control.

[3]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[4]  P. Steinmann,et al.  A numerical study of different projection-based model reduction techniques applied to computational homogenisation , 2017, Computational Mechanics.

[5]  Lawrence Sirovich,et al.  Karhunen–Loève procedure for gappy data , 1995 .

[6]  J. Schröder,et al.  Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains , 1999 .

[7]  Muruhan Rathinam,et al.  A New Look at Proper Orthogonal Decomposition , 2003, SIAM J. Numer. Anal..

[8]  R. Hill On constitutive macro-variables for heterogeneous solids at finite strain , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  G. Dunteman Principal Components Analysis , 1989 .

[10]  C. Miehe,et al.  Computational micro-to-macro transitions of discretized microstructures undergoing small strains , 2002 .

[11]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1979 .

[12]  C. Farhat,et al.  Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency , 2014 .

[13]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[14]  E. A. de Souza Neto,et al.  Computational methods for plasticity , 2008 .

[15]  Pierre Ladevèze,et al.  A PGD-based homogenization technique for the resolution of nonlinear multiscale problems , 2013 .

[16]  Kenjiro Terada,et al.  Nonlinear homogenization method for practical applications , 1995 .

[17]  J. A. López del Val,et al.  Principal Components Analysis , 2018, Applied Univariate, Bivariate, and Multivariate Statistics Using Python.

[18]  Pierre Suquet,et al.  Nonuniform transformation field analysis of elastic–viscoplastic composites , 2009 .

[19]  J. Michel,et al.  Effective properties of composite materials with periodic microstructure : a computational approach , 1999 .

[20]  Robert Davis Cook,et al.  Finite Element Modeling for Stress Analysis , 1995 .

[21]  J. Peraire,et al.  A ‘best points’ interpolation method for efficient approximation of parametrized functions , 2008 .

[22]  M. Caicedo,et al.  Dimensional hyper-reduction of nonlinear finite element models via empirical cubature , 2017 .

[23]  E. Schmidt Zur Theorie der linearen und nichtlinearen Integralgleichungen , 1907 .

[24]  Fpt Frank Baaijens,et al.  An approach to micro-macro modeling of heterogeneous materials , 2001 .

[25]  Michel Loève,et al.  Probability Theory I , 1977 .

[26]  J. Michel,et al.  Nonuniform transformation field analysis , 2003 .

[27]  van Jaw Hans Dommelen,et al.  Investigation of the effects of the microstructure on the sound absorption performance of polymer foams using a computational homogenization approach , 2017 .

[28]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[29]  H. Hotelling Analysis of a complex of statistical variables into principal components. , 1933 .

[30]  V. G. Kouznetsova,et al.  Multi-scale computational homogenization: Trends and challenges , 2010, J. Comput. Appl. Math..

[31]  Marc G. D. Geers,et al.  A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials , 2017, J. Comput. Phys..

[32]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[33]  P. M. Squet Local and Global Aspects in the Mathematical Theory of Plasticity , 1985 .

[34]  J. Renard,et al.  Etude de l'initiation de l'endommagement dans la matrice d'un matériau composite par une méthode d'homogénéisation , 1987 .

[35]  E. B. Andersen,et al.  Modern factor analysis , 1961 .

[36]  M. Loève Probability theory : foundations, random sequences , 1955 .

[37]  Hamid Zahrouni,et al.  Compressive failure of composites: A computational homogenization approach , 2015 .

[38]  A. Huespe,et al.  High-performance model reduction techniques in computational multiscale homogenization , 2014 .

[39]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[40]  K. Karhunen Zur Spektraltheorie stochastischer prozesse , 1946 .

[41]  G. Dvorak Transformation field analysis of inelastic composite materials , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[42]  V. Kouznetsova,et al.  Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme , 2002 .

[43]  W. Brekelmans,et al.  Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling , 1998 .

[44]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[45]  Julien Yvonnet,et al.  The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains , 2007, J. Comput. Phys..

[46]  N. Kikuchi,et al.  Preprocessing and postprocessing for materials based on the homogenization method with adaptive fini , 1990 .