Doctoral Dissertations

Date of Award

8-1996

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Engineering Science

Major Professor

Raymond D. Krieg

Committee Members

J. A. M. Boulet, Thomas G. Carley, Xiaobing Feng, Christopher D. Pionke

Abstract

The finite element method is used by today's engineering analyst to solve a wide variety of problems. These analyses can be computationally expensive, especially if nonlinearities are involved. This work presents a multigrid method to reduce the computational cost of performing large strain, nonlinear, quasi-static, structural finite element analyses. The multigrid method uses several grids in conjunction with existing iterative solvers to produce a very efficient solution technique. An added benefit of the multigrid method is the ability to obtain the solution to problems which were previously unsolvable. Multigrid methods are first presented for the single load step problem (i.e., one that moves the solution from the initial to the final configuration in one step) and finally for the multiple load step problem (i.e., one that breaks the loading into many small increments so that the proper loading path is followed). The multigrid methods as implemented here can be used with any existing code and are capable of handling a variety of nonlinearities including material and geometric nonlinearities as well as those which arise from frictional contacts. The implementation here also allows arbitrary grids to be used where the grids need not have any common nodes. All of the algorithms are implemented into existing large finite element codes using both conjugate gradient and dynamic relaxation solvers. Example problems and results are given to illustrate the various capabilities of the multigrid methods. Two- and three-dimensional examples include beam bending, torsion of square cross section rods, compression of a plate against the side of a cylinder and a forging example. The single load step multigrid method reduces the computational cost by at least a factor of two and as much as a factor of twenty-five. For every example problem, the single step multigrid method reduces the computational cost of the analysis. The multiple load step multigrid methods reduced the computational cost of some of the load steps but not others. This is a result of the effectiveness of current algorithms for multiple load step problems. A procedure is outlined for implementing the algorithms into existing finite element codes and recommendations are made for its use. Further areas of study are also suggested.

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