Variational Crimes in Meshfree Methods

Computational and Applied Mathematics Colloquium

Meeting Details

For more information about this meeting, contact Kendra Stauffer, Ludmil Zikatanov, Jinchao Xu, Wenrui Hao.

Speaker: Mike Hillman, Penn State University

Abstract: Meshfree methods provide a methodology to solve boundary value problems without relying on isoparametric finite element formulations. This relaxes the strong tie between the quality of the discretization and the quality of the numerical solution, and relieves mesh entanglement and other issues that are problematic in extremely large deformation solid mechanics problems. The unique properties of meshfree methods allow effective simulation of extreme events such as natural and man-made disasters, which often include fragmentation, complex contact, material flow, and severe material damage. Implementation of Galerkin methods inevitably involves certain abuses of the Galerkin formulation, abuses which Gilbert Strang termed variational crimes [1]. In this talk, I will discuss two non-trivial crimes committed in the commonplace employment of meshfree methods. First, the Galerkin method relies on construction of proper approximation spaces for the test and trial functions in the weak formulation. For meshfree methods, it is first shown that it is difficult to construct such spaces, and the inability leads to sub-optimal convergence rates. To rectify this situation, two new weak forms are introduced that relax the requirements on the test and trial functions. It is demonstrated that using these formulations, optimal convergence rates can be restored. The employment of quadrature in the weak form can lead to loss of Galerkin orthogonality according to Strang’s first lemma [1], and suboptimal convergence of the solution may also be encountered unless computationally intensive high order quadrature is employed. To circumvent this issue, a framework of variationally consistent integration is introduced [2], which restores Galerkin orthogonality and can yield optimal convergence using low-order quadrature. A naturally stabilized integration [3] method is also introduced to remedy rank deficiency which is another difficult issue in low order quadrature. Simulation of extreme events such as blast loads on structures, penetration events, and landslides, will be presented to demonstrate the effectiveness of the proposed meshfree formulation. [1] G.J. Strang, Gilbert and Fix, An analysis of the finite element method, Prentice-hall, Englewood Cliffs, NJ, 1973. [2] J.-S. Chen, M. Hillman, M. Rüter, "An arbitrary order variationally consistent integration for Galerkin meshfree methods," Int. J. Numer. Methods Eng. (95) 5, 387–418, 2013. [3] M. Hillman, J.-S. Chen, "An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics," Int. J. Numer. Methods Eng. (107) 603–630, 2016.


Room Reservation Information

Room Number: 114 McAllister

Date: 09/11/2017

Time: 2:30pm - 3:30pm