A WENO finite-difference scheme for a new class of Hamilton–Jacobi equations in nonlinear solid mechanics

Abstract

This paper puts forth a high-order weighted essentially non-oscillatory (WENO) finite-difference scheme to numerically generate the viscosity solution of a new class of Hamilton–Jacobi (HJ) equations that has recently emerged in nonlinear solid mechanics. The solution W of the prototypical version of the HJ equations considered here corresponds physically to the homogenized free energy that describes the macroscopic magneto-electro-elastic response of a general class of two-phase particulate composite materials under arbitrary quasi-static finite deformations, electric fields, and magnetic fields in N=2,3 dimensions. An important mathematical implication of its physical meaning is that W – although it may exhibit steep gradients – is expected to be at least twice continuously differentiable. This is in contrast to the viscosity solutions of the majority of HJ equations that have appeared in other scientific disciplines, which are merely Lipschitz continuous. Three other defining mathematical features that differentiate this new class of HJ equations from most of the existing HJ equations in the literature are that: (i) their “space” variables are defined over non-periodic unbounded or semi-unbounded domains, (ii) their Hamiltonians depend explicitly on all variables, namely, on the “space” and “time” variables, the “space” derivatives of W, and on the function W itself, and (iii) in general, their integration in “time” needs to be carried out over very long “times”. The proposed WENO scheme addresses all these features by incorporating a high-order accurate treatment of the boundaries of the domains of computation and by employing a high-order accurate explicit Runge–Kutta “time” integration that remains stable over very large “time” integration ranges. The accuracy and convergence properties of the proposed scheme are demonstrated by direct comparison with a simple explicit solution W available for the case when the general HJ equation is specialized to model the elastic response of isotropic porous Gaussian elastomers. Finally, for showcasing purposes, the scheme is deployed to probe the magneto-elastic response of a novel class of magnetorheological elastomers filled with ferrofluid inclusions.