# Fast Methods for Variable-Coefficient Peridynamic and Non-Local Diffusion Models

12-14-2015

## Document Type

Open Access Dissertation

Mathematics

Hong Wang

## Abstract

In previous studies, scientists developed the classical solid mechanic theory. The model has been widely used in scientific research and practical production. The main assumption of the classical theory of solid mechanics is that all internal forces act through zero distance. Because of this assumption, the mathematical model always leads to partial differential equations, which meet with problems when describing the spontaneous formation of discontinuities and other singularities. A peridynamic model was proposed as a reformation of solid mechanics [40, 41, 43, 44, 45], which leads to a non-local framework that does not explicitly involve the notion of deformation gradients, and so provides a more accurate description of the problems[5, 15, 27]. With a steady state, the two dimensional non-local diffusion model has the same properties as the peridynamic model[13, 14]. Hence, we can consider it as a scalarvalued version of a peridynamic model. Our discussion will focus on the one dimensional peridynamic model and two dimensional non-local diffusion model. Starting in the 1970s, scientists began to focus on the research of numerical simulation of integral or boundary integral equations by collocation methods and Galerkin finite element methods[7, 24, 25, 47]. After the peridynamic model was developed, an enormous amount of research effort went to the peridynamic and its numerical simulation[ 17, 18, 42]. It was found that there are close relations between the peridynamic model[22], non-local diffusion model and fractional partial differential equations [12, 13, 19, 28, 29, 30, 31, 32, 33, 35, 37]. In contrast to those for classical elasticity models of solid mechanics and integer-order partial differential equations, numerical methods for peridynamic models, like those for space-fractional partial differential equations [19, 28, 29, 30, 32, 33], usually generate dense or full stiffness matrices for which widely used direct solvers typically require O(N3) operations and O(N2) memory storage where N refers to the number of unknowns. A simplified peridynamic model was proposed to reduce the computational cost and memory requirement of the corresponding numerical methods, in which the horizon of the material in the peridynamic model was assumed to be =O(N−1) [10]. The advantage of the simplified model was that it reduced the computational cost and memory requirement to O(N), but at the cost of a reduced convergence rate of their numerical approximation. Furthermore, it is not clear from the physical relevance that the material property (the radius of the horizon) can be assumed to be of the same order as the numerical mesh size. In previous research, we developed a fast numerical method for the constantcoefficient one dimensional peridynamic model and two dimensional non-local diffusion model[50, 49, 46], which reduced the memory requirement from O(N2) to O(N), and the computational cost from O(N3) to O(NlogN). These works relied heavily on the Toeplitz-like structure of the stiffness matrix like fractional partial differential equations[48, 53, 51, 52]. After Mengesha and Du developed the variable-coefficient peridynamic model[34], we found that the stiffness matrix was no longer in Toeplitz structure, and our fast method could not be applied. Moreover, the variable coefficient also increased the computational work of evaluating the entries of the stiffness matrix. Since our fast method significantly reduced the computational complexity and memory requirement of the constant coefficient models, we hope to reform the stiffness matrix in order to explore useful structure and improve the efficiency of the numerical simulation. In this thesis, we will briefly introduce the tools we will use in developing the fast method in Chapter 1, and show the general idea of our fast method by implementing it on a one dimensional peridynamic model in Chapter two. In Chapter three, we will derive the fast method for a special case of a one dimensional peridynamic model by collocation numerical simulation. Finally, we will extend the fast method to a two dimensional non-local diffusion model in Chapter 4.