Methods for selective scaling of strain and inertia for explicit dynamics with tetrahedral finite elements
Final Report Abstract
The goal of the project is the development of advanced tetrahedral finite elements and supporting finite element technology in the context of explicit dynamics. These developments extend current applications of reciprocal mass matrices, which are sparse matrices for direct computation of the acceleration vector from the force vector. Four main contributions include a novel reciprocal mass matrix formulation for the elements with Allman’s rotations, a sharp estimator for feasible time step size for the elements with Allman’s rotations, a method for heuristic customization for low numerical dispersion and improved understanding of reflectiontransmission error for reciprocal mass matrices. The well-established construction of reciprocal mass matrix for solid elements fails for Allmantype interpolation. The proposed construction splits displacement and rotary DOF. The vertex displacements are assigned with the standard variationally constructed reciprocal mass matrix and the vertex rotations are assigned with a specially constructed diagonal inertia. Furthermore, the Gershgorin-based nodal estimates for the feasible time step are inefficient in the presence of mixed dimensions DOF at each node. A special similarity transformation and Ostrowski’s circle theorem are employed for the novel nodal estimator. The proposed estimate proved to be sharp for various tests even for highly distorted meshes. A combination of the proposed reciprocal mass matrix and the time step estimator provide a moderate speed-up of 25% w.r.t. simulation with the diagonal mass matrix. Customization for low numerical dispersion is not trivial for large representative patches because a determinant of the representative dynamic stiffness matrix is a bottleneck for such a symbolic manipulation. An alternative symbolic-numeric approach is proposed that avoids a symbolic expansion of the determinant. It reformulates the customization process as a rank minimization problem for the representative dynamic stiffness matrix evaluated at discrete pairs of wave-vectors and wave frequencies. The latter problem is solved numerically via log-det heuristic. The final aspect of the project is the analysis of reflection-transmission error for heterogeneous domains. Three possible formulations are compared: coupling of subdomains using the method of localized Lagrange multipliers and using one domain with two versions of the biorthogonal bases employed for the construction of the reciprocal mass matrices. New construction of the biorthogonal basis and the method of localized Lagrange multipliers perform similarly well.
Publications
- Time step estimates for reciprocal mass matrices using Ostrowski’s bounds, Proceedings of 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM-PDYN2019), Crete, Greece, June 21–23, 2019, Volume 1, pp. 774–785
Tkachuk, A.; Kolman, R.; González, A.; Bischoff, M.; Kopačka, J.
(See online at https://doi.org/10.7712/120119.6956.18956) - Customization of reciprocal mass matrices via log-det heuristic. IJNME, available online, 2020:121:4; 690–711
A. Tkachuk
(See online at https://doi.org/10.1002/nme.6240) - Reciprocal mass matrices and a feasible time step estimator for finite elements with Allman’s rotations. International Journal for Numerical Methods in Engineering 2020, 122(10)
A. Tkachuk
(See online at https://doi.org/10.1002/nme.6583)
