On the pure traction problem of linear elasticity: a regularized formulation and its robust approximation

TL;DR

This work addresses the pure traction (Neumann) elasticity problem, whose solutions are unique only up to infinitesimal rigid motions and are challenging to approximate numerically. By regularizing the energy with $V_\eta(v)=V(v)+\frac{\eta}{2}\|v\|^2_{[L^2(\Omega)]^d}$, the authors obtain a well-posed problem with a unique minimizer that converges to the minimal $L^2$-norm Neumann solution as $\eta\to0$, while preserving the sparsity of the stiffness matrix. They develop a finite element formulation for the regularized problem, prove convergence (and conditions under which it recovers the original Neumann solution), and provide an iterative solver to mitigate ill-conditioning as $\eta\to0$. The paper also extends the method to non-equilibrated loads via a predictor–corrector scheme, analyzes discretization effects on perturbed domains, and demonstrates robustness and accuracy through numerical examples including spheres, parallelepipeds, RVEs with periodic BCs, and a thermoelastic bunny geometry. Overall, the approach yields a simple, robust, and scalable pathway for solving pure traction problems without introducing extra DOFs or compromising sparsity, with practical impact for engineering simulations involving traction boundaries.

Abstract

The pure traction problem of elasticity appears frequently in engineering applications, and its complexity stems from the fact that its solution is unique only up to (infinitesimal) rigid body motions. When finite elements are employed to approximate this problem, one solution is typically singled out by applying carefully selected boundary conditions on the discrete model or by imposing global constraints on the deformation. However, neither of these strategies is both simple and computationally efficient. In this work, we propose a new approach to solving the pure traction problem that overcomes existing limitations. Our method builds on a regularized form of the problem whose solution is shown to be unique, converges to the original solution of minimal norm, and can be approximated with finite elements in a straightforward way, without additional degrees of freedom. Additionally, we analyze the situation in which the approximation of the solution domain renders the loading of the discretized problem non-equilibrated, making the problem ill-posed. In this case, we propose a regularized predictor--corrector finite element formulation that handles the incompatibilities of the loading, providing a solution that converges to that of the original Neumann problem as the mesh size and the regularizing parameter tend to zero. Numerical examples illustrate the effectiveness of the proposed approach for representative problems in mechanics where pure traction boundary conditions appear.

Figures (18)

• Figure 1: Simple mechanical problem.
• Figure 2: Finite element models for the elastic sphere problem. Number of elements, from top to bottom and left to right: $96\cdot8^j, j=0,\ldots,3$)
• Figure 3: $H^1$ error in the numerical solution of the elastic sphere problem. Top: solutions with nodal constraints; bottom: solutions obtained with the regularized method.
• Figure 4: Elastic sphere with a perturbed central load. Norm of the nodal reactions in the solution obtained with constrained degrees of freedom.
• Figure 5: Elastic sphere with a perturbed central load. Solution obtained with constrained degrees of freedom. Top: norm of mean displacement vs. mesh size; center: norm of displacement moment vs. mesh size; bottom: $H^1$ error in the solution.

Theorems & Definitions (27)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4: ciarlet1978tm
• Theorem 4.1
• Definition 5.1
• Lemma 5.2
• proof
• Lemma 5.3
• proof