[Paper Review] Consistent mass matrix of ten nodes tetrahedral element based on analytical integration

Eli Hanukah|arXiv (Cornell University)|Nov 5, 2014

Computational Geometry and Mesh Generation12 references3 citations

TL;DR

This paper presents closed-form analytical expressions for the consistent mass matrix of a 10-node tetrahedral finite element, using exact integration for straight-sided elements and semi-analytical approximations for curved-sided elements. The proposed constant (CM), linear (LM), and quadratic (QM) metric approximations yield highly accurate, computationally efficient mass matrices that outperform standard Gauss quadrature schemes in both accuracy and efficiency.

ABSTRACT

Currently, components of consistent mass matrix are computed using various numerical integration schemes, each one alters in number of integration (Gauss) points, requires different amount of computations and possess different level of accuracy. We discuss the closed-form mass matrix based on analytical integration. Curved-sided and straight-sided elements are considered. For a straight-sided element we derive an exact analytical easy to implement consistent mass matrix. For a curved-sided element an exact analytical mass matrix is derived, however it is rather lengthy, hence approximations are proposed. Three systematic approximations to the metric (jacobian determinant) are suggested; constant metric (CM), linearly varying metric (LM) and quadratic metric (QM). CM requires evaluation of the metric at the centroid, LM requires metric evaluations at the four corner nodes and QM uses metric values at all the ten nodes. Analytical integration together with approximated metric models yields closed-form semi-analytical mass matrices. The accuracy of the schemes is studied numerically using randomly generated coarse mesh. Our findings reveal significant superiority in accuracy and computations over equivalent schemes. An important implication of this study is that based on the results, it is superior to use our CM, LM and QM semi-analytical mass matrices over mass matrices based on numerical integration schemes which involve four, five and fifteen point Gauss quadrature. For a straight-sided element, CM, LM and QM admit an exact consistent mass matrix.

Motivation & Objective

• To develop exact and efficient consistent mass matrices for 10-node tetrahedral elements used in finite element analysis.
• To overcome the computational cost and accuracy limitations of traditional numerical integration schemes like Gauss quadrature.
• To provide closed-form solutions for straight-sided elements and practical semi-analytical approximations for curved-sided elements.
• To evaluate and compare the accuracy and computational efficiency of different metric approximation models (CM, LM, QM) against standard numerical integration methods.

Proposed method

• Derives an exact analytical consistent mass matrix for straight-sided 10-node tetrahedral elements using analytical integration over the element domain.
• Proposes three approximations to the Jacobian determinant (metric): constant at centroid (CM), linear variation from corner nodes (LM), and quadratic variation from all ten nodes (QM).
• Combines analytical integration with the approximated metric models to produce closed-form semi-analytical mass matrices.
• Employs numerical experiments on randomly generated coarse meshes to validate accuracy and computational performance.
• Uses Gauss quadrature with 4, 5, and 15 points as benchmarks for comparison.
• Applies the method to both curved-sided and straight-sided elements, with exact results only for the latter.

Experimental results

Research questions

• RQ1 Can an exact analytical consistent mass matrix be derived for straight-sided 10-node tetrahedral elements using analytical integration?
• RQ2 How can accurate and efficient mass matrices be constructed for curved-sided 10-node tetrahedral elements where exact integration is impractical?
• RQ3 How do the CM, LM, and QM metric approximations compare in accuracy and computational cost to standard Gauss quadrature schemes?
• RQ4 Does the proposed semi-analytical approach yield superior accuracy and efficiency compared to numerical integration with 4, 5, or 15 Gauss points?

Key findings

• For straight-sided 10-node tetrahedral elements, the CM, LM, and QM approximations yield an exact consistent mass matrix due to the polynomial nature of the shape functions and geometry.
• The CM, LM, and QM semi-analytical mass matrices achieve significantly higher accuracy than equivalent numerical integration schemes with 4, 5, or 15 Gauss points.
• The proposed method reduces computational cost while maintaining or improving accuracy, especially in coarse mesh configurations.
• The CM approximation requires only one metric evaluation (at the centroid), the LM requires four evaluations (at corner nodes), and the QM uses all ten node values, with increasing accuracy and complexity.
• Numerical results demonstrate that the semi-analytical approach outperforms standard Gauss quadrature in both accuracy and efficiency for the tested mesh configurations.
• The study concludes that using CM, LM, or QM semi-analytical mass matrices is superior to using numerical integration schemes for 10-node tetrahedral elements.