A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

A reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced‐order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high‐dimensional operations. In this proposed proper orthogonal decomposition–energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced‐order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non‐parametric settings. Next, a classical offline–online training approach is performed to build a parametric hyper reduced‐order macroscale model, which completes the construction of a fully hyper reduced‐order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced‐order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable. Copyright © 2017 John Wiley & Sons, Ltd.

[1]  Charbel Farhat,et al.  Accelerated mesh sampling for the hyper reduction of nonlinear computational models , 2017 .

[2]  Jaroslaw Knap,et al.  A computational framework for scale‐bridging in multi‐scale simulations , 2016 .

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

[4]  Bernard Haasdonk,et al.  A POD-EIM reduced two-scale model for crystal growth , 2015, Adv. Comput. Math..

[5]  Mgd Marc Geers,et al.  Thermo-mechanical analyses of heterogeneous materials with a strongly anisotropic phase: the case of cast iron , 2015 .

[6]  C. Farhat,et al.  Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models , 2015 .

[7]  Shun Zhang,et al.  Reduced Basis Multiscale Finite Element Methods for Elliptic Problems , 2015, Multiscale Model. Simul..

[8]  Charbel Farhat,et al.  Projection‐based model reduction for contact problems , 2015, 1503.01000.

[9]  Mario Ohlberger,et al.  Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment , 2015, SIAM J. Sci. Comput..

[10]  Assyr Abdulle,et al.  An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems , 2014 .

[11]  Charbel Farhat,et al.  Progressive construction of a parametric reduced‐order model for PDE‐constrained optimization , 2014, ArXiv.

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

[13]  A. Abdulle,et al.  Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems , 2014 .

[14]  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 .

[15]  Assyr Abdulle,et al.  Adaptive reduced basis finite element heterogeneous multiscale method , 2013 .

[16]  Gaffar Gailani,et al.  Advances in assessment of bone porosity, permeability and interstitial fluid flow. , 2013, Journal of biomechanics.

[17]  Charbel Farhat,et al.  The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows , 2012, J. Comput. Phys..

[18]  Assyr Abdulle,et al.  Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems , 2012, J. Comput. Phys..

[19]  C. Farhat,et al.  A low‐cost, goal‐oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems , 2011 .

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

[21]  Ralph Müller,et al.  A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures , 2008 .

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

[23]  J. Fish,et al.  N-Scale Model Reduction Theory , 2009 .

[24]  Ngoc Cuong Nguyen,et al.  A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales , 2008, J. Comput. Phys..

[25]  Jacob Fish,et al.  Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading , 2008 .

[26]  J. Peraire,et al.  An efficient reduced‐order modeling approach for non‐linear parametrized partial differential equations , 2008 .

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

[28]  Ralph Müller,et al.  A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures , 2008 .

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

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

[31]  Hamid Zahrouni,et al.  A model reduction method for the post-buckling analysis of cellular microstructures , 2007 .

[32]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[33]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[34]  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..

[35]  Jacob K. White,et al.  Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations , 2006 .

[36]  Yvon Maday,et al.  A reduced basis element method for the steady stokes problem , 2006 .

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

[38]  Gianluigi Rozza.  Shape design by optimal flow control and reduced basis techniques , 2005 .

[39]  Mark F. Adams,et al.  Ultrascalable Implicit Finite Element Analyses in Solid Mechanics with over a Half a Billion Degrees of Freedom , 2004, Proceedings of the ACM/IEEE SC2004 Conference.

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

[41]  Charbel Farhat,et al.  Aeroelastic Dynamic Analysis of a Full F-16 Configuration for Various Flight Conditions , 2003 .

[42]  Gregory W. Brown,et al.  Application of a three-field nonlinear fluid–structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter , 2003 .

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

[44]  W. Brekelmans,et al.  Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling , 2000 .

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

[46]  J. Schröder,et al.  Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials , 1999 .

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

[48]  Hans Werner Meuer,et al.  Top500 Supercomputer Sites , 1997 .

[49]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

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

[51]  L. Sirovich.  Turbulence and the dynamics of coherent structures. II. Symmetries and transformations , 1987 .

[52]  Rajesh Sharma,et al.  Asymptotic analysis , 1986 .

[53]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[54]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .

[55]  Rodney Hill,et al.  Continuum micro-mechanics of elastoplastic polycrystals , 1965 .

[56]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[57]  Geoffrey Ingram Taylor,et al.  The use of flat-ended projectiles for determining dynamic yield stress I. Theoretical considerations , 1948, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.