Parallelization of Closed-Form Stiffness Matrix Generation for Tetrahedral Finite Elements

Abstract

This paper expands on previous research into the use of parallelization tools for closed-form solutions with a focus on applications in finite element analysis (McCaslin SE (2010) “Parallelization of shape function generation for hierarchical tetrahedral elements.” In: Sobh T, Elleithy K (eds.) Innovations in computer science and software engineering, pp 409–414). Recent advances in parallelization tools for computer algebra systems allow users to easily implement parallelization, using built-in tools. Previous work used the generation of higher-order shape functions for tetrahedral elements through p-level 20 as a simple test case to demonstrate the potential for increased computational efficiency (McCaslin SE (2010) “Parallelization of shape function generation for hierarchical tetrahedral elements.” In: Sobh T, Elleithy K (eds.) Innovations in computer science and software engineering, pp 409–414). This research generates closed-form upper triangular stiffness matrices (not merely the shape functions) for straight-sided hierarchical tetrahedral elements (Shiakolas PS, Lawrence KL, Nambiar RV (1994) Closed-form expressions for the linear and quadratic strain tetrahedral finite elements. Comput Struct 50:743–747; McCaslin SE, Shiakolas PS, Dennis, BH, Lawrence KL (2009) “Closed-form matrices for higher order tetrahedral elements.” In: Proceedings of the 12th international conference on civil, structural and environmental engineering computing, Madeira, Portugal, September 2009) through the 9th order on a dual-core desktop by exploiting available parallelization tools. A speed-up ratio of over 2,000 and reduction in required memory of up to 83 % was achieved.