An Efficient Particle Subdomain Quadrature Scheme for the Material Point Method
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ISSN 1860-2134
An Efficient Particle Subdomain Quadrature Scheme for the Material Point Method Zheng Sun1
Xiaomin Zhou1
1
( School of Civil and Surveying & Mapping Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China)
Received 20 May 2020; revision received 7 August 2020; Accepted 24 August 2020 c The Chinese Society of Theoretical and Applied Mechanics 2020
ABSTRACT The material point method (MPM) has been proved to be an effective numerical method for large deformation problems. However, the MPM suffers from the cell crossing error as that the material particles are used to represent the deformed material and to perform the particle quadrature. In this paper, an efficient subdomain quadrature material point method (sqMPM) is proposed to eliminate the cell crossing error efficiently. The particle domain is approximated to be the line segment, rectangle, and cuboid for the one-, two-, and three-dimensional problems, respectively, which are divided into several different subdomains based on the topological relationship between the particle domain and background grid. A single Gauss quadrature point is placed at the center of each subdomain and used for the information mapping. The material quantities of each Gauss quadrature point are determined by the corresponding material particle and the subdomain volume without the cumbersome reconstruction algorithm. Numerical examples for one-, two-, and three-dimensional large deformation problems demonstrate the effectiveness and highly enhanced convergence and efficiency of the proposed sqMPM.
KEY WORDS Material point method, Cell crossing error, Subdomain quadrature, Large deformation
1. Introduction The material point method (MPM) [1, 2] is a particle-based meshless method utilizing a set of Lagrangian material particles to discretize the problem domain and an Eulerian background grid to calculate material derivatives and solve equations of motion. The MPM has been proved to be an extremely successful method for simulating complicated engineering problems such as extreme deformation [3–5], fluid–structure interaction [6, 7], landslides [8, 9], multi-physics [10], multi-scale [11, 12], multi-phase [13], and other problems [14–16]. In the standard MPM, the Lagrangian material particles are not only used to represent the deformed material but also employed as quadrature points to perform the particle quadrature. In simulations, however, material particle position is changing and seemingly “random” and hence reduces the accuracy of particle quadrature, especially when material particles cross the background grid as named cell crossing error [17]. The generalized interpolation material point method (GIMP) [17], the convected particle domain interpolation method (CPDI) [18, 19], the dual domain material point method (DDMP) [20], and the B-spline material point method (BSMPM) [21–23] have been proposed to reduce this cell
Corresponding author. E-mail: [email protected]
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Fig. 1. Schematic illustrations for a one-dim
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