A temporally piecewise adaptive multiscale scaled boundary finite element method to solve two-dimensional heterogeneous viscoelastic problems

Abstract

By combining the Multiscale Scale Boundary Finite Element Method (MsSBFEM) and the Temporally Piecewise Adaptive Algorithm (TPAA), a new numerical model is presented to reduce the solution scale of the two-dimensional heterogeneous viscoelastic problems. Utilizing TPAA, a spatially and temporally related problem is transformed into a series of recursive spatial problems, which are solved by MsSBFEM. The solution scale can be effectively reduced by recourse of a bridge between small-scale and large-scale via Scaled Boundary Finite Element Method (SBFEM) based numerical base functions, and the solution accuracy can be improved only by increasing nodes of coarse elements without increasing any new node inside. By virtue of singular, polygon and Quadtree elements, SBFEM renders the proposed algorithm more efficient and convenient to tackle with the stress singularity, and to generate SBFE mesh. TPAA provides a measure to secure the temporal computational accuracy via an adaptive process when the step size varies. Numerical examples are provided to elucidate the effectiveness of proposed approaches, and satisfactory results are achieved at both the large and small scales.