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Keywords: expression growth, closed-form, finite element analysis, analytic, tetrahedral, isoparametric, stiffness.

Summary

The closed-form development of finite element matrix expressions allows for exact representation of stiffness matrices, equivalent nodal loads, error estimators, etc., whereas numerical integral evaluation using techniques such as Gaussian quadrature and cubature are inherently approximate in nature.

Most two- and three-dimensional finite element domains are composed of exterior and interior boundaries that may be straight or curved, however the interior is usually filled with finite elements that have straight edges. Thus it would seem to be appropriate to use numerical quadrature and cubature for curved elements on domain boundaries, and closed-form evaluation of compatible elements for interior and exterior straight edge elements, particularly in those cases where the straight edge elements greatly outnumber the curved elements. It has been shown that the time savings in the element formulation phase with this approach can be considerable [1,2,3].

As the order of the element increases, however, symbolic processing of the required integrals can lead to expressions of unmanageable size. Furthermore, the end use of these expression files necessitates that they be formatted and compiled using the programming language of choice.

This paper presents an algorithm based on an adaptive dictionary that allows expressions found in complex source code files to be reduced in length, resulting in smaller and more manageable source code as well as executable code files. The algorithm is tested on closed-form expressions resulting from the implementation of straight-sided, isoparametric tetrahedral finite elements through the fourth order and implemented for practical use using Fortran as the source code language.

Examples of the reduction method are given, followed by the results of compaction for the higher order elements of p-levels 3 and4, where individual source code file reductions of up to 78% are achieved. The relationship between p-level and overall storage requirements for the source code files is illustrated, along with the effects of compaction. For both p-level 3 and 4, compaction reduced the overall storage requirements by one third.

This algorithm can be applied to any source code file where (a) each expression exists on its own line and (b) where each expression has been grouped into terms that are contained inside a single set of parentheses. It is can easily be modified for use with any structured or object-oriented programming language.

References

1: C.K. Yew, J.T. Boyle, D. MacKenzie, "Closed Form Integration of Element Stiffness Matrices using a Computer Algebra System", Computers & Structures, 56, 529-539, 1995. doi:10.1016/0045-7949(94)00549-I
2: P.S. Shiakolas, K.L. Lawrence, R.V. Nambiar, "Closed-form Expressions for the Linear and Quadratic Strain Tetrahedral Finite Elements", Computers & Structures, 50, 743-747, 1994. doi:10.1016/0045-7949(94)90309-3
3: P.S. Shiakolas, K.L. Lawrence, R.V. Nambiar, "Closed-form Error Estimators for the Linear Strain and Quadratic Strain Tetrahedron Finite Elements", Computers & Structures, 47, 907-915, 1993. doi:10.1016/0045-7949(93)90295-O

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 91 PROCEEDINGS OF THE TWELFTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping, L.F. Costa Neves and R.C. Barros Paper 287 Control of Expression Growth in Symbolic Processing of Finite Element Stiffness Matrices S.E. McCaslin¹, B. Dennis², P. Shiakolas² and K.L. Lawrence² ¹Mechanical Engineering Department, University of Texas at Tyler, United States of America ²Mechanical and Aerospace Engineering Department, University of Texas at Arlington, United States of America doi:10.4203/ccp.91.287 purchase the full-text of this paper Full Bibliographic Reference for this paper S.E. McCaslin, B. Dennis, P. Shiakolas, K.L. Lawrence, "Control of Expression Growth in Symbolic Processing of Finite Element Stiffness Matrices", in B.H.V. Topping, L.F. Costa Neves, R.C. Barros, (Editors), "Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 287, 2009. doi:10.4203/ccp.91.287 Keywords: expression growth, closed-form, finite element analysis, analytic, tetrahedral, isoparametric, stiffness. Summary The closed-form development of finite element matrix expressions allows for exact representation of stiffness matrices, equivalent nodal loads, error estimators, etc., whereas numerical integral evaluation using techniques such as Gaussian quadrature and cubature are inherently approximate in nature. Most two- and three-dimensional finite element domains are composed of exterior and interior boundaries that may be straight or curved, however the interior is usually filled with finite elements that have straight edges. Thus it would seem to be appropriate to use numerical quadrature and cubature for curved elements on domain boundaries, and closed-form evaluation of compatible elements for interior and exterior straight edge elements, particularly in those cases where the straight edge elements greatly outnumber the curved elements. It has been shown that the time savings in the element formulation phase with this approach can be considerable [1,2,3]. As the order of the element increases, however, symbolic processing of the required integrals can lead to expressions of unmanageable size. Furthermore, the end use of these expression files necessitates that they be formatted and compiled using the programming language of choice. This paper presents an algorithm based on an adaptive dictionary that allows expressions found in complex source code files to be reduced in length, resulting in smaller and more manageable source code as well as executable code files. The algorithm is tested on closed-form expressions resulting from the implementation of straight-sided, isoparametric tetrahedral finite elements through the fourth order and implemented for practical use using Fortran as the source code language. Examples of the reduction method are given, followed by the results of compaction for the higher order elements of p-levels 3 and4, where individual source code file reductions of up to 78% are achieved. The relationship between p-level and overall storage requirements for the source code files is illustrated, along with the effects of compaction. For both p-level 3 and 4, compaction reduced the overall storage requirements by one third. This algorithm can be applied to any source code file where (a) each expression exists on its own line and (b) where each expression has been grouped into terms that are contained inside a single set of parentheses. It is can easily be modified for use with any structured or object-oriented programming language. References 1 C.K. Yew, J.T. Boyle, D. MacKenzie, "Closed Form Integration of Element Stiffness Matrices using a Computer Algebra System", Computers & Structures, 56, 529-539, 1995. doi:10.1016/0045-7949(94)00549-I 2 P.S. Shiakolas, K.L. Lawrence, R.V. Nambiar, "Closed-form Expressions for the Linear and Quadratic Strain Tetrahedral Finite Elements", Computers & Structures, 50, 743-747, 1994. doi:10.1016/0045-7949(94)90309-3 3 P.S. Shiakolas, K.L. Lawrence, R.V. Nambiar, "Closed-form Error Estimators for the Linear Strain and Quadratic Strain Tetrahedron Finite Elements", Computers & Structures, 47, 907-915, 1993. doi:10.1016/0045-7949(93)90295-O purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £140 +P&P)
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