Comparison and unification of material-point and optimal transportation meshfree methods

Abstract

Both the material-point method (MPM) and optimal transportation meshfree (OTM) method have been developed to efficiently solve partial differential equations that are based on the conservation laws from continuum mechanics. However, the methods are derived in a different fashion and have been studied independently of one another. In this paper, we provide a direct step-by-step comparison of the MPM and OTM algorithms. Based on this comparison, we derive the conditions, under which the two approaches can be related to each other, thereby bridging the gap between the MPM and OTM communities. In addition, we introduce a novel unified approach that combines the design principles from B-spline MPM and the OTM method. The proposed approach does not contain user-defined parameters and can decrease the costs of the standard OTM method. Moreover, it allows for the use of a consistent mass matrix without stability issues that are typically encountered in MPM computations.

The Local Maximum-Entropy Material Point Method

“…The simulations presented in [165] of MPM within LME show considerably more accurate stress approximations than traditional MPM schemes. However, how the regularization parameter β affects to the accuracy and stability of the solution is not assessed deeply in that research.…”

“…Solving this issue is the main goal of the employment of the LME shape functions. Local Maximum-Entropy (LME) approximation schemes were first introduced by Arroyo and Ortiz [8] and has been recently tested under MPM framework by Wobbes et al [165].…”

“…The latter methodology has been proven to have a good performance under the dynamic regime being worth mentioning the contributions of Li et al [89] and Navas et al [115,116] in the explicit regime and Navas et al [114,117,120] and Wriggers and coworkers [44] with implicit schemes. More recently, under MPM framework, the work made by Wobbes et al [165]. The proposed research delves into the benefits of the regularization parameter, β, and the analogy of the different shape functions derived by the tuning of this parameter and the traditional MPM ones.…”