The nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions
exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum extent of nonlocal
interactions. A multiscale finite element implementation in one dimension and two dimensions of the nonlocal models is developed by taking
advantage of the proven fact that, for smooth solutions, the nonlocal models reduce, as the horizon parameter tends to zero, to well-known
local partial differential equations models. The implementation features adaptive abrupt mesh refinement based on the detection of defects
and resulting in an abrupt transition between refined elements that contain defects and unrefined elements that do not do so. Additional
difficulties encountered in the implementation that are overcome are the design of accurate quadrature rules for stiffness matrix
construction that are valid for any combination of the grid size and horizon parameter. As a result, the methodology developed can attain
optimal accuracy at very modest additional costs relative to situations for which the solution is smooth. Portions of the methodology can
also be used for the optimal approximation, by piecewise linear polynomials, of given functions containing discontinuities. Several
numerical examples are provided to illustrate the efficacy of the multiscale methodology. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the
requirements for the degree of Doctor of Philosophy. / Fall Semester 2015. / December 02, 2015. / anomalous diffusion, discontinuous displacements, finite element methods, multiscale methods, nonlocal models,
peridynamics / Includes bibliographical references. / Max Gunzburger, Professor Directing Dissertation; Xiaoming Wang, University Representative; John
Burkardt, Committee Member; Janet Peterson, Committee Member; Xiaoqiang Wang, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_360485 |
| Contributors | Xu, Feifei (authoraut), Gunzburger, Max D. (professor directing dissertation), Wang, Xiaoming (university representative), Burkardt, John V. (committee member), Wang, Xiaoqiang (committee member), Florida State University (degree granting institution), College of Arts and Sciences (degree granting college), Department of Scientific Computing (degree granting department) |
| Publisher | Florida State University, Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text |
| Format | 1 online resource (115 pages), computer, application/pdf |
| Rights | This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them. |
Page generated in 0.0105 seconds