A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics.

We propose in this paper a reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No a priori knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture.

[1]  P. Tallec.  Domain decomposition methods in computational mechanics , 1994 .

[2]  P. Gosselet,et al.  Non-overlapping domain decomposition methods in structural mechanics , 2006, 1208.4209.

[3]  Francisco Chinesta,et al.  Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models , 2010 .

[4]  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.

[5]  Olivier Allix,et al.  On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework , 2010, ArXiv.

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

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

[8]  Olivier Allix,et al.  A relocalization technique for the multiscale computation of delamination in composite structures , 2011, 1109.4799.

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

[10]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[11]  D. Sorensen,et al.  Approximation of large-scale dynamical systems: an overview , 2004 .

[12]  C. Farhat,et al.  Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity , 2008 .

[13]  Anthony Gravouil,et al.  A global model reduction approach for 3D fatigue crack growth with confined plasticity , 2011 .

[14]  D. Ryckelynck,et al.  A priori hyperreduction method: an adaptive approach , 2005 .

[15]  Stefanie Reese,et al.  A simulation strategy for life time calculations of large, partially damaged structures , 2006 .

[16]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

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

[18]  David Dureisseix,et al.  A micro / macro approach for parallel computing of heterogeneous structures , 2000 .

[19]  D. Rixen.  A dual Craig-Bampton method for dynamic substructuring , 2004 .

[20]  P. Ladevèze,et al.  The LATIN multiscale computational method and the Proper Generalized Decomposition , 2010 .

[21]  David Ryckelynck,et al.  A robust adaptive model reduction method for damage simulations , 2011 .

[22]  K. Kunisch,et al.  Optimal snapshot location for computing POD basis functions , 2010 .

[23]  Jacob K. White,et al.  A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[24]  A. Nouy.  A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations , 2010 .

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

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

[27]  Zdeněk P. Bažant,et al.  Mechanics of solid materials , 1992 .

[28]  M. J. Wittbrodt,et al.  A critique of mode acceleration and modal truncation augmentation methods for modal response analysis , 1997 .

[29]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[30]  J. Mandel.  Balancing domain decomposition , 1993 .

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

[32]  J. Chaboche,et al.  Mechanics of Solid Materials , 1990 .

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

[34]  Olivier Allix,et al.  A three-scale domain decomposition method for the 3D analysis of debonding in laminates , 2009, 1109.6111.

[35]  Walter C. Hurty,et al.  Vibrations of Structural Systems by Component Mode Synthesis , 1960 .

[36]  Angelo Iollo,et al.  Iterative Methods for Model Reduction by Domain Decomposition , 2007, 0712.0691.

[37]  Daniel Rixen,et al.  Domain decomposition techniques for the efficient modeling of brittle heterogeneous materials , 2011 .

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

[39]  Karl Pearson F.R.S..  LIII. On lines and planes of closest fit to systems of points in space , 1901 .

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

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

[42]  Piotr Breitkopf,et al.  Model reduction by CPOD and Kriging , 2010 .

[43]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

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

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

[46]  Wojtek J. Krzanowski,et al.  Cross-Validation in Principal Component Analysis , 1987 .

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

[48]  H. Abdi,et al.  Principal component analysis , 2010 .

[49]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[50]  Damijan Markovic,et al.  Reduction of substructural interface degrees of freedom in flexibility‐based component mode synthesis , 2007 .

[51]  Yong-hwa Park,et al.  Partitioned Component Mode Synthesis via a Flexibility Approach , 2004 .

[52]  David Ryckelynck,et al.  Multi-level A Priori Hyper-Reduction of mechanical models involving internal variables , 2017 .

[53]  Yong P. Chen,et al.  A Quadratic Method for Nonlinear Model Order Reduction , 2000 .

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

[55]  Laurent Champaney,et al.  A multiscale extended finite element method for crack propagation , 2008 .

[56]  Siamak Niroomandi,et al.  Real-time deformable models of non-linear tissues by model reduction techniques , 2008, Comput. Methods Programs Biomed..

[57]  Manuel Laso,et al.  MESHLESS STOCHASTIC SIMULATION OF MICRO-MACROKINETIC THEORY MODELS , 2011 .

[58]  Juan J. Alonso,et al.  Dynamic Domain Decomposition and Error Correction for Reduced Order Models , 2003 .

[59]  P. Astrid,et al.  Reduction of process simulation models : a proper orthogonal decomposition approach , 2004 .

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

[61]  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..

[62]  Dan Givoli,et al.  Optimal modal reduction of vibrating substructures , 2003 .

[63]  Frédéric Feyel,et al.  High‐performance parallel simulation of structure degradation using non‐local damage models , 2007 .

[64]  Elías Cueto,et al.  Coupling finite elements and proper generalized decompositions , 2011 .

[65]  J. Borggaard,et al.  A Domain Decomposition Approach to POD , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[66]  Rakesh K. Kapania,et al.  Global/local analysis of composite plates with cracks , 1998 .