Mechanical Engineering Department seminar
Coefficients Determination for the Element Mass Matrix and Its Direct Inverse Following the J-Derivatives and the Optimization–Based Approaches
Speaker: Alex Zendel
Supervised by: Dr. Eli Hanukah
:Abstract
Many real-life engineering problems associated with solid mechanics are analyzed using the widely used Finite Element (FE) computational packages. A necessary step of this numerical procedure is the evaluation of the element-level quantities, among which are mass matrices. To this end, commercial programs adopt the Standard (ST) approach to numerical integration. Namely, a sufficiently accurate quadrature scheme (such as Gauss points) is employed. The higher the number of integration points, the higher the integration accuracy (integration order), yet the higher the cost (complexity). Thus, developing sufficiently accurate and highly efficient integration schemes is of great practical importance. Furthermore, efficient evaluation of the consistent element mass matrix inverse is crucial to transient explicit solvers.
Herein, we confine ourselves to the widely used 10-node tetrahedral element. We adopt the recently proposed J-Derivatives (JD) method to determine, for the first time, the necessary coefficients for a highly efficient and easy-to-implement integration scheme of the element mass matrix. In addition, we quantitatively evaluate the overall complexity of the specific integrator. According to our findings, the JD formula’s complexity is equivalent to 1-point ST numerical integration while sufficiently accurate for global convergence. This proves that the specific JD integration scheme is highly efficient and desirable for practical implementation.
In addition, we adopt the recently proposed Optimization-Based (OB) method to directly evaluate the inverse of the consistent element mass matrix. We determine for the first time all the necessary coefficients, resulting in a ready-to-use practical formula. To this end, the nonlinear optimization problem is solved by decomposing it into two linear least squares problems. Furthermore, the complexity of the resulting formula is evaluated.