TY - JOUR

T1 - A surrogate model for computational homogenization of elastostatics at finite strain using high-dimensional model representation-based neural network

AU - Nguyen-Thanh, Vien Minh

AU - Nguyen, Lu Trong Khiem

AU - Rabczuk, Timon

AU - Zhuang, Xiaoying

N1 - Funding Information: V.M.N.‐T. thanks the sponsorship from Sofja Kovalevskaja Prize from Alexander von Humboldt Foundation (Zhuang as PI). X.Z. would like to thank the Heisenberg‐Professorship from German Research Foundation (DFG). L.T.K.N. wishes to thank Felix Selim Göküzüm, Matthias Rambausek, and Marc‐André Keip for introducing him to the topic computational homogenization and also for fruitful discussions during his research stay at University of Stuttgart as well as the financial support from DFG for the Cluster of Excellence in Simulation Technology (EXC 310).

PY - 2020/10/1

Y1 - 2020/10/1

N2 - We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium. In this manner, the fixed-point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT-based method for homogenization of elasticity at finite strain.

AB - We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium. In this manner, the fixed-point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT-based method for homogenization of elasticity at finite strain.

KW - cs.CE

KW - math.NA

KW - nonlinear elasticity

KW - data-driven

KW - FFT-based methods

KW - computational homogenization

UR - http://www.scopus.com/inward/record.url?scp=85089399114&partnerID=8YFLogxK

U2 - 10.1002/nme.6493

DO - 10.1002/nme.6493

M3 - Article

VL - 121

SP - 4811

EP - 4842

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 21

ER -