Closed-form stiffness matrices for higher order tetrahedral finite elements

Closed-form expressions for straight-sided isoparametric tetrahedral finite element stiffness matrices up to p-level 3 have been shown to offer significant time savings when compared to numerical integration. In this work, the development of closed-form stiffness matrices is extended to the next level of approximation, the subparametric p-level 4 element. The resulting stiffness matrices are verified, and computational efficiency was tested by comparing floating-point operations required for closed-form and numerically integrated solutions. Results showed that closed-form elements still provide time savings in stiffness matrix evaluation for all p-levels tested, with up to 61× speed gain for p-level 4 depending on the compiler used. It was also found that the compilers used could better optimize the code with closed form generated expressions for efficient execution when compared to the code for numerical integration.