Date of Award

12-2018

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

Committee Member

Dr. Timo Heister, Committee Chair

Committee Member

Dr. Leo Rebholz

Committee Member

Dr. Hyesuk Lee

Abstract

The elasticity equations describe how an elastic material moves under a force. An elastic material is one that returns to its original shape after the force is lifted. Modeling elasticity is useful in manufacturing applications such as suspension cables and nail bending, and biological applications such as weight on bones and tendons [1]. In this thesis we study the modeling of nearly incompressible linearly elastic materials. A nearly incompressible material is one that does not change much under pressure. Linearly elastic materials exhibit small deformations under a force. Standard finite element methods do not work well on nearly incompressible materials when using the simplest form of the linear elasticity equations (the pure displacement form). Instead, they exhibit locking; in other words, they produce excessively small displacements on a coarse mesh. We examine several methods that fix this problem. One method is to use a different form of the linear elasticity equations that more closely resembles a Stokes’ formulation (the displacement-pressure formulation). This approach has theoretical support, but is somewhat computationally expensive. Another method is to use reduced integration for part of our equation. This approach has less theoretical support, but with the correct setups, this method is both cheaper and more accurate than the standard method, and solves the locking problem.

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