Hyper-dual number-based numerical differentiation of eigensystems

Abstract

Considering application to nonlinear material models, this study proposes a novel numerical method for computing arbitrary-order derivatives of eigenvalues and eigenvectors (eigensystems) for second-order tensors (matrices) based on hyper-dual numbers (HDNs). This study introduces two numerical differentiation approaches for eigensystems: one obtains derivatives of both eigenvalues and eigenvectors by inductively solving linear equations derived from an HDN formulation of eigensystem, and the other specializes in solving derivatives of a function represented by eigenvalues, such as the Helmholtz free energy function in the Ogden model, a representative example of the eigenvalue-based hyperelastic material models. Because these approaches differ considerably in applicability and computational cost, this study proposes a numerical differentiation method that uses both complementarily. The proposed method can accurately compute higher-order derivatives even if the second-order tensors have multiple solutions for eigenvalues. Moreover, applying the proposed method to an incremental variational formulation (IVF) leads to robust and accurate computations for the Ogden material model. These results imply that IVF facilitates the implementation of several constitutive models in a unified manner, and the proposed method broadens the scope of these models, including eigensystem expressions.