Open Access

ARTICLE

On the (Meshless Local Petrov-Galerkin) MLPG-Eshelby Method in Computational Finite Deformation Solid Mechanics - Part II

by Z. D. Han¹, S. N. Atluri²

1 Livermore Software Technology Corporation, Livermore, CA, 94551, USA.
2 International Collaboratory for Fundamental Studies in Engineering and the Sciences, 4131 Engineering Gateway, University of California, Irvine, Irvine, CA, 92697, USA.
3 Fellow & Eminent Scholar, Texas Institute for Advanced Study, TAMU-3474, College Station, TX77843, USA.

Computer Modeling in Engineering & Sciences 2014, 97(3), 199-237. https://doi.org/10.3970/cmes.2014.097.199

Abstract

This paper presents a new method for the computational mechanics of large strain deformations of solids, as a fundamental departure from the currently popular finite element methods (FEM). The currently widely popular primal FEM: (1) uses element-based interpolations for displacements as the trial functions, and element-based interpolations of displacement-like quantities as the test functions; (2) uses the same type and class of trial & test functions, leading to a Galerkin approach; (3) uses the trial and test functions which are most often continuous at the inter-element boundaries; (4) leads to sparsely populated symmetric tangent stiffness matrices; (5) computes piecewise-linear predictor solutions based on the global weak-forms of the Newtonian Momentum Balance Laws for a Lagrangean Stress tensor, such as the symmetric Second Piola-Kirchhoff Stress tensor S [≡ JF⁻¹·σ·F^−t, where σ is the Cauchy Stress tensor and F the deformation gradient] in the initial or any other known reference configuration; and (6) computes a corrector solution, using Newton-Raphson or other Jacobian-inversion-free iterations, based on the global weak-forms of the Newtonian Momentum Balance Laws for the symmetric Cauchy Stress tensor σ in the current configuration. In a radical departure, the present approach blends the Energy-Conservation Laws of Noether and Eshelby, and the Meshless Local Petrov Galerkin (MLPG) Methods of Atluri, and is designated herein as the MLPG-Eshelby Method. In the MLPG-Eshelby Method, we: (1) use meshless node-based functions δX, for configurational changes of the undeformed configuration, as the trial functions; (2) meshless node-based functions δx, for configurational changes of the deformed configuration, as the test functions; (3) the trial functions δX and the test functions δx are necessarily different and belong to different classes of functions, thus naturally leading to a Petrov-Galerkin approach; (4) leads to sparsely populated unsymmetric tangent stiffness matrices; (5) the trial functions δX, as well as the test functions δx, may either be continuous or be discontinuous in their respective configurations; (6) generate piecewise-linear predictor solutions based on the local weak-forms of the Noether/Eshelby Energy Conservation Laws for the Lagrangean unsymmetric Eshelby Stress tensor T in the undeformed configuration [T = WI − P·F; where P = JF^-1·σ is the first Piola-Kirchhoff Stress tensor, and W is the stress-work density per unit initial volume of the solid] and (7) generate corrector solutions, based on Newton-Raphson or Jacobian-inversion-free iterations, using the local weak-forms of the Noether/Eshelby Energy Conservation Laws in the current configuration, for a newly introduced Eulerean symmetric Stress tensor S^˜ [which is the counter part of T] in the current configuration [S^˜ = (W/J)I − σ, often called by chemists as the Chemical Potential Tensor]. It is shown in the present paper that the present MLPG-Eshelby Method, based on the meshless local weak-forms of the Noether/Eshelby Energy Conservation Laws, converges much faster and leads to much better accuracies than the currently popular FEM based on the global weak-forms of the Newtonian Momentum Balance Laws. The present paper is limited to hyperelasticity, while large strains of inelastic solids will be considered in our forthcoming papers.

---