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Computational Technology Reviews
Computational Technology Reviews
Volume 2, 2010
The Use of Symbolic Processing and Explicit Formulations in Computational Mechanics
S.E. McCaslin1, B.H. Dennis2, P.S. Shiakolas2, W.S. Chan2, S. Nomura2, B.P. Wang2 and K.L. Lawrence2
1University of Texas at Tyler, Tyler TX, United States of America
S.E. McCaslin, B.H. Dennis, P.S. Shiakolas, W.S. Chan, S. Nomura, B.P. Wang, K.L. Lawrence, "The Use of Symbolic Processing and Explicit Formulations in Computational Mechanics", Computational Technology Reviews, vol. 2, pp. 83-102, 2010. doi:10.4203/ctr.2.4
Keywords: symbolic algebra, mechanics, finite elements, optimization, multi-body dynamics.
This paper presents the recent work in the use of computer algebra systems (CAS) in (1) applied mechanics (2) finite element methods (3) optimization and sensitivity analysis and (4) multi-body dynamics and mixed system analysis. CAS evolved from the earlier days of LISP based systems (REDUCE, Macsyma) to the highly sophisticated, all-purpose systems of today (Mathematica, Maple).
In applied mechanics, CAS was used to derive permissible functions in the Galerkin method that exactly satisfy the boundary conditions and continuity conditions across different phases. Differentiation and integration of these permissible functions were also carried out by CAS. Using this approach, Green's functions for heterogeneous systems were derived. CAS was also used to manipulate and automate computations involving tensor (index) algebra. It was also employed to automatically derive complex-valued Airy's stress functions in two-dimensional elasticity for a plate that has a second phase. Recently, an elastic inclusion problem where a spherical inclusion is embedded in layers of surrounding matrices with different material properties was solved analytically; however a huge output formula resulted which no other method could have possibly derived.
In finite element applications, CAS was used to generate closed-form stiffness matrices for various types of elements. The results are typically exported in a format compatible with numerical programming languages such as FORTRAN or C. CAS was successfully used to derive the closed-form stiffness matrices for linear and quadratic, straight-sided tetrahedrons; four-noded plane elasticity elements; quadrilateral elements; p-version curve-sided tetrahedral elements; plane stress, plane strain and axisymmetric eight-noded elements; 3-, 6-, 10-, and 15-node straight-sided triangular elements and straight-sided fourth order tetrahedral elements. It was reported that a significant speed-up in computing stiffness matrices was possible compared with the conventional approach. Post processing procedures to reduce the length of expressions generated by CAS was also explored.
CAS has also been employed in optimization and sensitivity analysis. The major use of CAS was to symbolically differentiate the objective function. As with other applications, symbolic differentiation often leads to significant intermediate expression growth. However, it was reported that a new algorithm to symbolically create difficult to find derivatives could speed up the computation by 10^3 times. Thus it appears there is room for enhancing the computational efficiency for symbolic derivative computation.
In multi-body applications, CAS enables engineers to derive, in closed form, the governing equations of multiple rigid-flexible bodies in combination with electrical, hydraulic, etc. systems. Rapidly generated, error-free equation sets, fast response simulation, and the capability to incorporate results in control systems are some of the benefits that result from the application of symbolic processing in this area of engineering.
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