A direct solution to linear constraints in the finite element analysis and its application illustrations

Abstract

Purpose

This article aims to present a direct solution to handle linear constraints in finite element (FE) analysis without penalties or the Lagrange multipliers introduced.

Design/methodology/approach

First, the system of linear equations corresponding to the linear constraints is solved for the leading variables in terms of the free variables and the constants. Then, the reduced system of equilibrium equations with respect to the free variables is derived from the finite-dimensional virtual work equation. Finally, the algorithm is designed.

Findings

The proposed procedure is promising in three typical cases: (1) to enforce displacement constraints in any direction; (2) to implement local refinements by allowing hanging nodes from element subdivision and (3) to treat non-matching grids of distinct parts of the problem domain. The procedure is general and suitable for 3D non-linear analyses.

Research limitations/implications

The algorithm is fitted only to the Galerkin-based numerical methods.

Originality/value

The proposed procedure does not need Lagrange multipliers or penalties. The tangential stiffness matrix of the reduced system of equilibrium equations reserves positive definiteness and symmetry. Besides, many contemporary Galerkin-based numerical methods need to tackle the enforcement of the essential conditions, whose weak forms reduce to linear constraints. As a result, the proposed procedure is quite promising.

Citation

Zhang, N., Zheng, H., Yuan, C. and Wu, W. (2023), "A direct solution to linear constraints in the finite element analysis and its application illustrations", Engineering Computations, Vol. 40 No. 9/10, pp. 2328-2347. https://doi.org/10.1108/EC-06-2022-0400