No abstract

No abstract

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. K E Y W O R D S computational homogenization, data-driven, FFT-based methods, nonlinear elasticity 1 INTRODUCTION Multiscale techniques are important for man-made and natural materials; one such approach is homogenization. Roughly speaking, homogenization is a rigorous version of what is known as averaging. It is a powerful tool to study the heterogeneous materials and composites. Based on the knowledge of the microstructure of materials, the objective is to Abbreviations: HDMR, high-dimensional model representation; FFT, fast Fourier transform; FEM, finite element method; NN, neural network.

Fourier-based approaches are a well-established class of methods for the theoretical and computational characterization of microheterogeneous materials.Driven by the advent of computational homogenization techniques, Fourier schemes gained additional momentum over the past decade. In recent contributions, the interpretation of Green operators central to Fourier solvers as projections opened up a new perspective. Based on such a viewpoint, the present work addresses a multiscale framework for magneto-mechanically coupled materials at finite strains. The key ingredient for the solution of magneto-mechanic boundary value problems at the microscale is the construction of suitable operators in Fourier space that project vector fields onto either curl-free or divergence-free subspaces. The resulting linear system of equations is solved by a conjugate gradient method. In addition to that, we describe the computation of the consistent macroscopic tangent operator based on the same linear operators as the microscopic equilibrium with appropriately defined right-hand sides. We employ the framework for the simulation of representative two-scale boundary value problems and compare the results with pure finite element schemes.

We develop a general approach to the description of dispersive shock waves (DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin-Ono type dispersion term involving the Hilbert transform. Integrability of the governing equation is not a pre-requisite for the application of this method which represents a modification of the DSW fitting method previously developed for dispersive-hydrodynamic systems of Korteweg-de Vries (KdV) type (i.e. reducible to the KdV equation in the weakly nonlinear, long wave, unidirectional approximation). The developed method is applied to the Calogero-Sutherland dispersive hydrodynamics for which the classification of all solution types arising from the Riemann step problem is constructed and the key physical parameters (DSW edge speeds, lead soliton amplitude, intermediate shelf level) of all but one solution type are obtained in terms of the initial step data. The analytical results are shown to be in excellent agreement with results of direct numerical simulations.