Mathematical Sciences: Finite Element Methods for Problems in Solid Mechanics

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9403552 Falk The first area of focus of the project is the numerical and analytical study of two dimensional plate models, commonly used to study thin three dimensional elastic bodies. A recently proposed finite element approximation scheme for the Reissner-Mindlin plate model will be analyzed to determine whether it avoids the "locking" problem which causes most methods to give poor approximations for thin plates. A numerical study of this and other schemes proposed in the literature will also be done to compare their effectiveness. A related portion of the project is to study formally "higher order" plate models, providing both a systematic derivation and mathematically rigorous error estimates for the stresses and displacements. The aim is to determine when any of these models can be rigorously shown to give better approximations to the full three dimensional equations than the simplest biharmonic model, thus providing a justification of the models and a way of deciding which models are the most appropriate in various circumstances. The second area of focus of the project is the study of space-time finite element methods for the approximation of two problems in Mechanics. The first problem involves a system of nonlinear partial differential equations which models the planar motion of a class of inextensible elastic rods. The system has an energy which is conserved and the goal of the project is to develop a family of arbitrary order energy conserving finite element schemes for this problem, in order to obtain more accurate approximations with less computational effort than are possible with a finite difference method previously developed by the PI. The second problem is concerned with surface diffusion. Specifically, the goal is to rigorously establish the stability and convergence of a family of space-time finite element methods developed by the PI for the approximation of a system of partial differential equations that model the changes of shape in duced in an isotropic and homogeneous solid body of constant density under the influence of mass diffusion within the body. Providing a rigorous foundation for these methods gives confidence that the numerical approximations provide accurate predictions of the behavior of the model The first area of focus of the project is the numerical and analytical study of two dimensional plate models, commonly used by engineers to predict displacements and stresses of thin three dimensional elastic bodies when various forces are applied. The use of the computer to obtain approximate solutions is necessary for these problems, since except in very special cases, exact solutions are not known. Hence, one important aspect of this project is the analysis and comparison of computational algorithms. Since many different two dimensional plate models appear in the literature, a related portion of the project is to determine when any of these models can be rigorously shown to give better approximations to the full three dimensional equations than the simplest model currently used, thus providing a justification of the models and a way of deciding which models are the most appropriate in various circumstances. The second area of focus of the project is the development of efficient computational algorithms for the approximation of two problems in mechanics. The first problem involves a mathematical model for the planar motion of a class of inextensible elastic rods. The dynamics of inextensible rods is important for applications ranging from flexible space structures to the modelling of the dynamics of long chain molecules such as polymers or DNA. The second problem to be investigated is concerned with shape changes of a body driven by surface tension. The particular mathematical model to be studied models the changes of shape induced in an isotropic and homogeneous solid body of constant density under the influence of mass diffusion within the body. The aim is t o both develop new approximation schemes and provide a rigorous analysis of the errors. The latter is important in providing confidence that the numerical approximations obtained are giving accurate predictions of the behavior of the models.