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.
A numerical study of branching and stability of solutions to three-dimensional martensitic phase transformations using gradient-regularized, non-convex, finite strain elasticity
A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations
Centre de Recerca Matemàtica（数学研究中心）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；Department of Mathematics, The Pennsylvania State University（宾夕法尼亚州立大学数学系）
Numerical analysis of a locking-free primal hybrid method for linear elasticity with $H(\mathrm{div})$-conforming stress recovery
Universidade Estadual de Campinas (Unicamp), Center for Energy and Petroleum Studies（坎皮纳斯州立大学（Unicamp），能源与石油研究中心）；Department of Applied Mathematics, Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学，数学、统计与计算机科学研究所，应用数学系）
A Variational Bayesian Inference Theory of Elasticity and Its Mixed Probabilistic Finite Element Method for Inverse Deformation Solutions in Any Dimension
Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University（越南国立大学科学大学数学与计算机科学学院）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
