Vol. 3, No. 8, 2008

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A study of penalty formulations used in the numerical approximation of a radially symmetric elasticity problem

Adair R. Aguiar, Roger L. Fosdick and Jesús A. G. Sánchez

Vol. 3 (2008), No. 8, 1403–1427

We consider a class of two-dimensional problems in classical linear elasticity for which material overlapping occurs in the absence of singularities. Of course, material overlapping is not physically realistic, and one possible way to prevent it uses a constrained minimization theory. In this theory, a minimization problem consists of minimizing the total potential energy of a linear elastic body subject to the constraint that the deformation field must be locally invertible. Here, we use an interior and an exterior penalty formulation of the minimization problem together with both a standard finite element method and classical nonlinear programming techniques to compute the minimizers. We compare both formulations by solving a plane problem numerically in the context of the constrained minimization theory. The problem has a closed-form solution, which is used to validate the numerical results. This solution is regular everywhere, including the boundary. In particular, we show numerical results which indicate that, for a fixed finite element mesh, the sequences of numerical solutions obtained with both the interior and the exterior penalty formulations converge to the same limit function as the penalization is enforced. This limit function yields an approximate deformation field to the plane problem that is locally invertible at all points in the domain. As the mesh is refined, this field converges to the exact solution of the plane problem.

aeolotropic elasticity, constrained minimization, penalty method, finite element method
Received: 21 January 2008
Revised: 15 May 2008
Accepted: 6 August 2008
Published: 1 October 2008
Adair R. Aguiar
Department of Structural Engineering
São Carlos School of Engineering
University of São Paulo
Av. Trabalhador Sãocarlense, 400
13566-590 São Carlos, SP
Roger L. Fosdick
Aerospace Engineering and Mechanics
University of Minnesota
107 Akerman Hall
110 Union St. SE
Minneapolis, MN 55455-0153
United States
Jesús A. G. Sánchez
Department of Structural Engineering
São Carlos School of Engineering
University of São Paulo
Av. Trabalhador Sãocarlense, 400
13566-590 São Carlos, SP