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

Closed-form matrix expressions obtained through exact symbolic integration for use in finite element analysis allow for exact representation of stiffness matrices, equivalent nodal loads, error estimators, etc., whereas the more commonly used numerical integration, involving techniques such as Gaussian quadrature and cubature, are inherently approximate in nature [1]. Past research, specifically in the area of straight-sided triangular and tetrahedral elements, has demonstrated the efficiency of the closed-form method when compared to its numerically integrated counterpart.

Schuetze et al. [2] showed that the closed-form straight-sided isoparametric tetrahedral elements can successfully be combined with compatible numerically integrated curve-sided elements to provide efficient solutions to problems where the geometries require elements of both types, especially where straight-sided elements (generally found in the interior of the model domain) outnumber the curve-sided elements.

In previous work, Shiakolas et al. [3] developed closed-form expressions for the isoparametric constant strain, linear strain, and quadratic strain straight-sided tetrahedral elements. In this paper, his work is extended to fourth-order isoparametric elements with the development and testing of the closed-form stiffness matrix.

The closed form stiffness matrices for all p-levels through the fourth order are verified through solution of a standard set of test problems: cantilever beam subject an axial load, cantilever beam subject to a bending, and a plate with a hole subject to a tensile load. These test problems also serve a secondary purpose: to verify that for a given mesh, increasing the p-level produces results that approach expected values for strain energy, stress, and displacement.

The main objective of this research is to demonstrate the potential computational performance of the closed-form implementation, and to this end performance tests were performed. These tests were based on the number floating point operations (flops) required to implement a single stiffness matrix in closed-form versus the number of flops required to calculate the same stiffness matrix using numerical approximation.

The results of these tests confirmed earlier findings showing a speed up ratio (based on flops) of 15 for p-level 2 and 33 for p-level 3. New results for the fourth order isoparametric element show a speed up ratio of 45.

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

K.T. Schuetze, P.S. Shiakolas, S.N. Muthukrishnan, R.V. Nambiar, K.L. Lawrence, "A Study of Adaptively Remeshed Finite Element Problems using Higher Order Tetrahedra", Computers & Structures, 54, 279-288, 1995. doi:10.1016/0045-7949(94)00320-3

3

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

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