Title:
Geometric discretization schemes and differential complexes for elasticity

dc.contributor.advisor Yavari, Arash
dc.contributor.author Angoshtari, Arzhang
dc.contributor.committeeMember Desroches, Reginald
dc.contributor.committeeMember Gangbo, Wilfrid
dc.contributor.committeeMember Garmestani, Hamid
dc.contributor.committeeMember Thadhani, Naresh
dc.contributor.department Civil and Environmental Engineering
dc.date.accessioned 2013-09-20T13:24:41Z
dc.date.available 2013-09-20T13:24:41Z
dc.date.created 2013-08
dc.date.issued 2013-05-15
dc.date.submitted August 2013
dc.date.updated 2013-09-20T13:24:41Z
dc.description.abstract In this research, we study two different geometric approaches, namely, the discrete exterior calculus and differential complexes, for developing numerical schemes for linear and nonlinear elasticity. Using some ideas from discrete exterior calculus (DEC), we present a geometric discretization scheme for incompressible linearized elasticity. After characterizing the configuration manifold of volume- preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible. On the other hand, it has been observed that the linear elastostatics complex can be used to find very efficient numerical schemes. We use some geometric techniques to obtain differential complexes for nonlinear elastostatics. In particular, by introducing stress functions for the Cauchy and the second Piola-Kirchhoff stress tensors, we show that 2D and 3D nonlinear elastostatics admit separate kinematic and kinetic complexes. We show that stress functions corresponding to the first Piola-Kirchhoff stress tensor allow us to write a complex for 3D nonlinear elastostatics that similar to the complex of 3D linear elastostatics contains both the kinematics an kinetics of motion. We study linear and nonlinear compatibility equations for curved ambient spaces and motions of surfaces in R3. We also study the relationship between the linear elastostatics complex and the de Rham complex. The geometric approach presented in this research is crucial for understanding connections between linear and nonlinear elastostatics and the Hodge Laplacian, which can enable one to convert numerical schemes of the Hodge Laplacian to those for linear and possibly nonlinear elastostatics.
dc.description.degree Ph.D.
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/49026
dc.language.iso en_US
dc.publisher Georgia Institute of Technology
dc.subject Geometric numerical schemes
dc.subject Elasticity complex
dc.subject Nonlinear stress functions
dc.subject.lcsh Elasticity
dc.subject.lcsh Hodge theory
dc.subject.lcsh Differential equations, Partial.
dc.subject.lcsh Laplacian operator
dc.title Geometric discretization schemes and differential complexes for elasticity
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Yavari, Arash
local.contributor.corporatename School of Civil and Environmental Engineering
local.contributor.corporatename College of Engineering
relation.isAdvisorOfPublication 8c256ff1-f314-407a-ad6f-1c3ae65f35b0
relation.isOrgUnitOfPublication 88639fad-d3ae-4867-9e7a-7c9e6d2ecc7c
relation.isOrgUnitOfPublication 7c022d60-21d5-497c-b552-95e489a06569
thesis.degree.level Doctoral
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