# Semi-Analytical Approach to Element-Level Integration for the Solid Nonlinear Finite Element

## DOI:

https://doi.org/10.31224/3231## Keywords:

Finite Element Technology, Semi-Analytical Integration, Full Integration, Nonlinear Tangent Stiffness Matrix, Closed-Form, Element-Level Integration## Abstract

The Nonlinear Finite Element Method is a widely used numerical technique to solve engineering problems associated with nonlinear solid continua. Often, real-life engineering problems require immense computational resources. Thus, raising the computational efficiency of the method is a highly desirable goal. A nonlinear global system solver requires constant reevaluation of the element-level internal forces together with the nonlinear stiffness matrix. For this purpose, commercial software employs the standard numerical integration based on the quadrature method (e.g., Gauss points). Consequently, computational complexity increases linearly with the number of integration points. Therefore, the derivation of alternative, highly efficient integration approaches is essential, and this is the main goal of our study. Herein, we propose new element-level integration formulae of the internal forces and of the nonlinear tangent stiffness matrix. Our formulae admit the “full” order integration requirement. Moreover, we demonstrate analytically that the computational cost of the proposed schemes is roughly equivalent to One-Point quadrature, irrespective of the element type (e.g., tetrahedral, hexahedral, wedge, etc.,) and irrespective of the element order (e.g., 8-node brick, 20-node brick, etc.,). Code implementation of our formula follows a rather familiar, standard, manner. However, before code implementation, it requires sets of coefficients/special weights to be pre-computed or adopted from the literature. Undoubtedly, the proposed integrators require significantly more coefficients than the standard numerical integration. Recently, the Semi-Analytical approach has been employed to produce highly efficient case-specific integrators for the mass matrices. Here, we generalize and expand this approach to all element-level integrals. To this end, the integrands are decomposed into two multiplicative parts. The first part includes kinetic and kinematic functions which admit “full” integration criteria for one sampling point (e.g., the centroid). While the second part consists of mesh-independent and displacement-independent polynomial functions. We integrate the above polynomials analytically, to derive coefficient sets. Those coefficients are incorporated in the resulting scheme’s subroutine. In other words, we take advantage of the mathematical structure of the integrand to produce a highly efficient yet case-specific integration formula (e.g., our internal forces rule can’t be used for the mass of stiffness matrices and vice versa). Importantly, code implementation doesn’t involve either meta-programming or a computer algebra system for explicit (closed-form) code generation.

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Copyright (c) 2023 Eli Hanukah

This work is licensed under a Creative Commons Attribution 4.0 International License.