Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization

In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element. We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model can be built.

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

[2]  Danny C. Sorensen,et al.  Discrete Empirical Interpolation for nonlinear model reduction , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[3]  M. Stein.  Large sample properties of simulations using latin hypercube sampling , 1987 .

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

[5]  Somnath Ghosh,et al.  Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method , 2011 .

[6]  Juan José Ródenas,et al.  Certification of projection‐based reduced order modelling in computational homogenisation by the constitutive relation error , 2013, ArXiv.

[7]  Juan J. Alonso,et al.  Investigation of non-linear projection for POD based reduced order models for Aerodynamics , 2001 .

[8]  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 .

[9]  Mario Aachen,et al.  Micromechanics Overall Properties Of Heterogeneous Materials , 2016 .

[10]  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..

[11]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[12]  V. Buryachenko.  Micromehcanics of Heterogenous Materials , 2007 .

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

[14]  P. Breitkopf,et al.  A reduced multiscale model for nonlinear structural topology optimization , 2014 .

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

[16]  P Kerfriden,et al.  A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics. , 2012, Computer methods in applied mechanics and engineering.

[17]  Olivier Goury,et al.  Computational time savings in multiscale fracturemechanics using model order reduction , 2015 .

[18]  Bhushan Lal Karihaloo,et al.  Improved Lattice Model for Concrete Fracture , 2002 .

[19]  S. Forest,et al.  Asymptotic analysis of heterogeneous Cosserat media , 2001 .

[20]  L. Sirovich.  TURBULENCE AND THE DYNAMICS OF COHERENT STRUCTURES PART I : COHERENT STRUCTURES , 2016 .

[21]  Julien Yvonnet,et al.  Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction , 2008 .

[22]  Thomas Böhlke,et al.  Three‐dimensional finite element implementation of the nonuniform transformation field analysis , 2010 .

[23]  S. Nemat-Nasser,et al.  Micromechanics: Overall Properties of Heterogeneous Materials , 1993 .

[24]  Marcus Meyer,et al.  Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods , 2003 .

[25]  Qiqi Wang,et al.  Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems , 2010, SIAM J. Sci. Comput..

[26]  Jacob Fish,et al.  Hybrid impotent–incompatible eigenstrain based homogenization , 2013 .

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

[28]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[29]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

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

[31]  Danny C. Sorensen,et al.  A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems , 2014, SIAM J. Sci. Comput..

[32]  Karen Willcox,et al.  Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space , 2008, SIAM J. Sci. Comput..

[33]  S. Reese,et al.  Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive sub-structuring , 2014 .

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

[35]  Jacob Fish,et al.  Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials , 2007 .

[36]  J. Willis.  Bounds and self-consistent estimates for the overall properties of anisotropic composites , 1977 .

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

[38]  Timon Rabczuk,et al.  STATISTICAL EXTRACTION OF PROCESS ZONES AND REPRESENTATIVE SUBSPACES IN FRACTURE OF RANDOM COMPOSITES , 2012, 1203.2487.

[39]  Mark S. Shephard,et al.  Computational plasticity for composite structures based on mathematical homogenization: Theory and practice , 1997 .

[40]  P Kerfriden,et al.  Bridging Proper Orthogonal Decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. , 2011, Computer methods in applied mechanics and engineering.

[41]  Peter Wriggers,et al.  A description of macroscopic damage through microstructural relaxation , 1998 .

[42]  David Ryckelynck.  Hyper‐reduction of mechanical models involving internal variables , 2009 .

[43]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[44]  SteinMichael.  Large sample properties of simulations using latin hypercube sampling , 1987 .

[45]  P. Sagaut,et al.  Towards an adaptive POD/SVD surrogate model for aeronautic design , 2011 .

[46]  Donald R. Jones,et al.  A Taxonomy of Global Optimization Methods Based on Response Surfaces , 2001, J. Glob. Optim..

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

[48]  David Amsallem,et al.  An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models , 2015 .

[49]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[50]  V. Buryachenko.  Micromechanics of Heterogeneous Materials , 2007 .

[51]  Kevin Carlberg,et al.  The ROMES method for statistical modeling of reduced-order-model error , 2014, SIAM/ASA J. Uncertain. Quantification.

[52]  C. Farhat,et al.  Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations , 2011 .

[53]  Gianluigi Rozza,et al.  Certified reduced basis approximation for parametrized partial differential equations and applications , 2011 .

[54]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[55]  Peter Wriggers,et al.  Introduction to Computational Micromechanics (Lecture Notes in Applied and Computational Mechanics) , 2004 .

[56]  Jacob Fish,et al.  Higher-Order Homogenization of Initial/Boundary-Value Problem , 2001 .

[57]  Stefan Volkwein,et al.  Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics , 2002, SIAM J. Numer. Anal..

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

[59]  G. Milton.  The Theory of Composites , 2002 .

[60]  Christian Miehe,et al.  Strain‐driven homogenization of inelastic microstructures and composites based on an incremental variational formulation , 2002 .

[61]  E. S. Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .