You are here

Numerical Methods for Deterministic and Stochastic Nonlocal Problem in Diffusion and Mechanics

Title: Numerical Methods for Deterministic and Stochastic Nonlocal Problem in Diffusion and Mechanics.
124 views
77 downloads
Name(s): Chen, Xi, 1980-, author
Gunzburger, Max, professor directing dissertation
Wang, Xiaoming, university representative
Peterson, Janet, committee member
Wang, Xiaoqiang, committee member
Ye, Ming, committee member
Burkardt, John, committee member
Department of Scientific Computing, degree granting department
Florida State University, degree granting institution
Type of Resource: text
Genre: text
Issuance: monographic
Date Issued: 2012
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
Physical Form: online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: In this dissertation, the recently developed peridynamic nonlocal continuum model for solid mechanics is extensively studied, specifically, the numerical methods for the deterministic and stochastic steady-state peridynamics models. In contrast to the classical partial differential equation models, peridynamic model is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to the fracture problems. This dissentation consists of three major parts. The first part focuses on the one-dimensional steady-state peridynamics model. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Optimal convergence rates for different continuous and discontinuous manufactured solutions are obtained. A strategy for identifying the discontinuities of the solution is developed and implemented. The convergence of peridynamics model to classical elasticity model is studied. Some relevant nonlocal problems are also considered. In the second part, we focus on the two-dimensional steady-state peridynamics model. Based on the numerical strategies and results from the one-dimensional peridynamics model, we developed and implemented the corresponding approaches for the two-dimensional case. Optimal convergence rates for different continuous and discontinuous manufactured solutions are obtained. In the third part, we study the stochastic peridynamics model. We focus on a version of peridynamics model whose forcing terms are described by a finite-dimensional random vector, which is often called the finite-dimensional noise assumption. Monte Carlo methods, stochastic collocation with full tensor product and sparse grid methods based on this stochastic peridynamics model are implemented and compared.
Identifier: FSU_migr_etd-4753 (IID)
Submitted Note: A Dissertation submitted to the Department of ScientifiC Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester, 2012.
Date of Defense: June 22, 2012.
Keywords: DISCONTINUOUS GALEKIN METHODS, FINITE ELEMENT METHODS, INTEGRAL DIFFERENTIAL EQUATIONS, NONLOCAL DIFFUSION PROBLEM, PERIDYNAMICS, STOCHASTIC
Bibliography Note: Includes bibliographical references.
Advisory Committee: Max Gunzburger, Professor Directing Dissertation; Xiaoming Wang, University Representative; Janet Peterson, Committee Member; Xiaoqiang Wang, Committee Member; Ming Ye, Committee Member; John Burkardt, Committee Member.
Subject(s): Numerical analysis
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-4753
Owner Institution: FSU