Nonconforming mixed finite element methods for linear elasticity

Son-Young Yi, Purdue University

Abstract

We consider mixed finite element methods for linear elasticity based on the Hellinger-Reissner variational formulation (stress-displacement formulation). We have developed two mixed finite element methods using rectangular elements. First, we construct a nonconforming mixed finite element method for 2 dimensions. A modification has been found for boundary elements with a (possiblely) curved edge so that a domain with curved boundary can be treated. We obtain the convergence rates of [special characters omitted](h) and [special characters omitted](h2) for the stress and displacement, respectively. This element extends to 3 dimensions with the same convergence rate for both stress and displacement. We confirm our theoretical result numerically for the 2-dimensional version of the method. Then, we construct a simpler mixed finite element method. The convergence analysis involves an estimate of the consistency error term and makes use of the properties of the Adini-Clough-Melosh rectangle. We establish error estimates for the pure traction boundary problem and pure displacement boundary problem separately. We prove an optimal (suboptimal) rate of [special characters omitted](h2) ([special characters omitted]([special characters omitted])) for the stress in the pure traction (displacement) boundary problem and an optimal rate of [special characters omitted](h) for the displacement in L 2 norm in both problems. However, numerical experiments have yielded optimal-order convergence rates of [special characters omitted](h2) for the stress in both problems and [special characters omitted](h) for the displacement, as expected, and have shown a superconvergence rate of [special characters omitted](h2) for the displacement at the midpoint of each element in both problems. Other numerical experiments have shown the same convergence rates for large values of the Lamé constant λ.

Degree

Ph.D.

Advisors

Douglas, Purdue University.

Subject Area

Mathematics

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