A homogenization method for nonlinear inhomogeneous elastic materials

Background Fast simulation techniques are strongly favored in computer graphics,especially for the nonlinear inhomogeneous elastic materials.The homogenization theory is a perfect match to simulate inhomogeneous deformable objects with its coarse discretization,as it reveals how to extract information at a fine scale and to perform efficient computation with much less DOF.The existing homogenization method is not applicable for ubiquitous nonlinear materials with the limited input deformation displacements.Methods In this paper,we have proposed a homogenization method for the efficient simulation of nonlinear inhomogeneous elastic materials.Our approach allows for a faithful approximation of fine,heterogeneous nonlinear materials with very coarse discretization.Modal analysis provides the basis of a linear deformation space and modal derivatives extend the space to a nonlinear regime;based on this,we exploited modal derivatives as the input characteristic deformations for homogenization.We also present a simple elastic material model that is nonlinear and anisotropic to represent the homogenized materials.The nonlinearity of material deformations can be represented properly with this model.The material properties for the coarsened model were solved via a constrained optimization that minimizes the weighted sum of the strain energy deviations for all input deformation modes.An arbitrary number of bases can be used as inputs for homogenization,and greater weights are placed on the more important low-frequency modes.Results Based on the experimental results,this study illustrates that the homogenized material properties obtained from our method approximate the original nonlinear material behavior much better than the existing homogenization method with linear displacements,and saves orders of magnitude of computational time.Conclusions The proposed homogenization method for nonlinear inhomogeneous elastic materials is capable of capturing the nonlinear dynamics of the original dynamical system well.

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin(DG)method for the solution of nonlinear hyperbolic systems of partial differential equations(PDE)on unstructured polygonal Voronoi meshes.Rather than using classical polynomials of degree N inside each element,in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element,using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element(AFE)basis for the DG method on general polygons,since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles.The basis functions on each sub-triangle are defined,as usual,on a universal reference element,hence allowing to compute universal mass,flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only.Consequently,the construction of an efficient quadrature-free algorithm is possible,despite the unstructured nature of the computational grid.High order of accuracy in time is achieved thanks to the ADER approach,making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations.The numerical results have been checked with reference solutions available in literature and also systematically compared,in terms of computational efficiency and accuracy,with those obtained by the corresponding modal DG version of the scheme.