Element-by-Element Solution Procedures for Nonlinear Transient Heat Conduction Analysis

Author: Winget, James Michael

Year: 1984

Degree: Dissertation (Ph.D.)

Advisor: Knowles, James K.

Committee Members: Knowles, James K.; Caughey, Thomas Kirk; Hall, John F.; Hughes, Thomas J. R.; Sabersky, Rolf H.

Option: Applied Mechanics

DOI: 10.7907/G7VB-EV65

Abstract

Despite continuing advancements in computer technology, there are many problems of engineering interest that exceed the combined capabilities of today's numerical algorithms and computational hardware. The resources required by traditional finite element algorithms tend to grow geometrically as the "problem size" is increased. Thus, for the forseeable future, there will be problems of interest which cannot be adequately modeled using currently available algorithms. For this reason, we have undertaken the development of algorithms whose resource needs grow only linearly with problem size. In addition, these new algorithms will fully exploit the "parallel-processing" capability available in the new generation of multi-processor computers.

The approach taken in the element-by-element solution procedures is to approximate the global implicit operator by a product of lower order operators. This type of "product" approximation originated with ADI techniques and was further refined into the "method of fractional steps." The current effort involves the use of a more natural operator split for finite element analysis based on "element operators." This choice of operator splitting based on element operators has several advantages. First, it fits easily within the architecture of current FE programs. Second, it allows the development of "parallel" algorithms. Finally, the computational expense varies only linearly with the number of elements.

The particular problems considered arise from nonlinear transient heat conduction. The nonlinearity enters through both material temperature dependence and radiation boundary conditions. The latter condition typically introduces a "stiff" component in the resultant matrix ODE's which precludes the use of explicit solution techniques. Implicit solution techniques can be prohibitively expensive. Instead, the matrix equations are solved by combining a modified Newton-Raphson iteration scheme with an element-by-element preconditioned conjugate gradient subiteration procedure. The resultant procedure has proven to be both accurate and reliable in the solution of medium-size problems in this class.

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