Date of Award

Spring 1999

Document Type

Thesis - Restricted

Degree Name

Master of Science (MS)

Department

Mathematics, Statistics and Computer Science

First Advisor

Corliss, George

Second Advisor

Slattery, Michael

Third Advisor

Krenz, Gary

Abstract

The purpose of the research project was to develop a finite element program based on interval, rather than floating point arithmetic. Finite element programs are used by designers to model and refine a design on the computer before an expensive prototype is constructed. An interval based finite element program can be implemented using the global optimization program GlobSol, providing true global optimization of a design parameter of the finite element model. Global optimization of a finite element design parameter is highly desirable by industrial and design engineers since ordinarily, the finite element program can not tell the designer which model parameters need to be modified to achieve a desired design goal. With traditional finite element modeling programs, the engineer can never know for certain that some additional improvement could be achieved in the design by changing a different combination of model parameters. Global optimization applied to a finite element modeling program can assure the designer that no further stress reductions or performance improvements can be obtained from the design. The research project was conducted in two phases. In the first phase of the project, the MacNeal-Schwendler Corporation provided us with a manufacturing optimization problem of a NASA rocket nozzle. The nozzle is manufactured from hundreds of layer-wound carbon fiber ply patterns sandwiched together and baked under pressure to form a solid nozzle. The orientation of the carbon fiber ply pattern, the number of plys, and the ply thickness determine the mechanical working stress limit the nozzle can withstand while in operation. MacNeal-Schwendler provided us with the source code to their CPATCHES finite element software. Our intention was to rewrite sections of the code to operate in interval arithmetic. Once the code conversion was complete, the CPATCHES program could be executed with the GlobSol optimization program, returning the optimized values for the ply thickness, count, and orientation parameters. We discovered that the CPATCHES program would be too difficult to convert to interval arithmetic, since GlobSol required the objective function of the problem to be in a different form than that provided by CPATCHES. This new insight into the optimization problem led us to the second phase of the project. In the second phase of the research project, we broke new ground and developed our own interval finite element program based on the theory of the finite element method, but scaled to a more manageable size. Instead of applying our interval finite element program to a problem as large as the NASA rocket nozzle, we elected to model an axisymmetric elastic ring. We selected a ring as our model since in theory, the ring and the rocket nozzle share similar geometric features , and are subjected to exactly the same types of internal stresses. We defined the model geometry, material properties, and maximum stress constraints, and then asked GlobSol to find the optimized value of the internal pressure applied to the inside wall of the ring that would not exceed our desired material yield strength. GlobSol successfully returned the optimized internal pressure value that satisfied all model geometry, material, and maximum stress constraints. The results returned by GlobSol agreed completely with the empirical results calculated for our ring from the standard thick-wall pressure vessel equations. Our research into the implementation of interval mathematics to the finite element method broadens the scope of the practical use of global optimization in industrial design and manufacturing applications.

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