A TPAA-SBFEM based full-scale forward and inverse interval analysis for dynamic viscoelastic problems

Abstract

This paper aims to propose efficient algorithms for the full-scale forward and inverse interval analysis of dynamic viscoelastic problems. Underpinned by a reliable deterministic forward solution provided by the Temporally Piecewise Adaptive Algorithm and Scaled Boundary Finite Element Method (TPAA-SBFEM), the deterministic solutions of displacement/stress and their derivatives with respect to constitutive parameters can be acquired with high fidelity, and an efficient surrogate of deterministic forward solutions can be constructed with sufficient accuracy. Utilizing these deterministic solutions, derivatives and the surrogate as well, two sets algorithm are proposed to solve forward/inverse interval dynamic viscoelastic problems.In the forward interval analysis, Taylor series expansion (TSE) and interval arithmetic (IA) are combined to estimate the bounds of displacements and stresses when the interval scale of constitutive parameter is relatively small, and the acquirement of the first-order derivatives of displacements and stresses are stressed; when the interval scale is relatively large, the Particle Swarm Optimization (PSO) is employed to estimate the intervals of displacements and stresses by solving two deterministic optimization problems, and a Legendre Polynomial Chaos Expansion (LPCE) based surrogate is addressed to reduce the computational cost in the optimization process. In the inverse interval analysis, the interval estimation of constitutive parameters can be formulated by two deterministic optimization problems when intervals of displacements are partially known, which are solved by the Levenberg Marquardt method (LMM) when the interval scale is small, and by the combination of PSO with LPCE when the interval scale is lager.The accuracy and efficiency of the proposed approaches are demonstrated via numerical examples. Forward/inverse interval problems with stress singularity and heterogeneity are emphatically analyzed, and the impacts of interval scales, step sizes, noisy data, and so forth are taken into account.