Analysis of numerical methods for transient viscoelastic waves
Date
2020
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Publisher
University of Delaware
Abstract
Viscoelasticity has been used to model the deformation and stress displayed by many types of solids including biological materials, polymers and metals. Viscoelastic materials exhibit a memory effect, where the stress at the current time depends on the previous history of the strain. This relation is identified by a convolution form, and has been expressed via different models: Maxwell, Voigt, Zener, and their fractional versions. We offer a unified discretization scheme based on the Finite Element Method (FEM) in space, coupled with a Convolution Quadrature (CQ) rule in time. Our method is applicable to general viscoelastic models including classical and fractional ones. We establish the stability and optimal convergence of our fully-discrete scheme, and demonstrate numerical examples supporting our analysis. Later we also analyze another semidiscrete scheme by utilizing semigroup theory, and discuss its approximation properties. ☐ This work contains the study of the Trapezoidal Rule based Convolution Quadrature (TRCQ) method. Like other CQ methods, the TRCQ method offers an approximation of convolution forms, and can be used to discretize homogeneous evolution equations. Here we close an important gap in the literature by showing the second order convergence of the TRCQ method with polynomially bounded error terms in time. This result also complements our discretization algorithm for viscoelastic waves.
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Keywords
Convolution quadrature, Finite element method, Numerical analysis, Partial Differential equations, Semigroup Theory, Viscoelasticity