Closed-form Development Of A Family Of Higher Order Tetrahedral Elements Through The Fourth Order

This research is concerned with the development and implementation of a family of tetrahedral elements through the fourth order. The straight-sided tetrahedral elements are developed in closed-form. This work investigates the efficiency of closed-form implementation of stiffness matrices and error estimators compared to numerical implementation. An additional objective is the compaction of closed-form source-code files which require as little storage space as possible, a more pronounced requirement at high p-levels. For the straight-sided elements through p-level 4, the stiffness matrix, equivalent nodal load vectors, and error estimators (based on nodal averaging) are developed using closed-form equations obtained through the use of a computer algebra system. The stiffness matrix and error estimators are also implemented using numerical integration so that a timing comparison between the numerical and the closed-form approaches could be performed.
The curved-sided elements, including the stiffness matrix, equivalent nodal load vectors, and error estimators are also implemented using Gaussian quadrature only. A test conducted on a model of all curved-sided elements is used to verify that the elements are working correctly. Results indicate that the closed-form implementation solutions are comparable to the numerical solutions. For all p-levels the closed-form stiffness matrix is more efficient by a factor of at least 4 when compared with numerically integrated elements.