Mixed Strong Form Representation Particle Method for Solids and Structures

Document Type: Research Paper


Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE. Minneapolis, MN, 55455, USA


In this paper, a generalized particle system (GPS) method, a general method to describe multiple strong form representation based particle methods is described. Gradient, divergence, and Laplacian operators used in various strong form based particle method such as moving particle semi-implicit (MPS) method, smooth particle hydrodynamics (SPH), and peridynamics, can be described by the GPS method with proper selection of parameters. In addition, the application of mixed formulation representation to the GPS method is described. Based on Hu-Washizu principle and Hellinger-Reissner principle, the mixed form refers to the method solving multiple primary variables such as displacement, strain and stress, simultaneously in the FEM method; however for convenience in employing FEM with particle methods, a simple representation in construction only is shown. It is usually applied to finite element method (FEM) to overcome numerical errors including locking issues. While the locking issues do not arise in strong form based particle methods, the mixed form representation in construction only concept applied to GPS method can be the first step for fostering coupling of multi-domain problems, coupling mixed form FEM and mixed form representation GPS method; however it is to be noted that the standard GPS particle method and the mixed for representation construction GPS particle method are equivalent. Two dimensional simple bar and beam problems are presented and the results from mixed form GPS method is comparable to the mixed form FEM results.


Main Subjects

[1] S.N. Atluri, R.H. Gallagher, O.C. Zienkiewicz, Hybrid and Mixed Finite Element Methods. John Wiley Sons, 1983.

[2] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Butterworth-heinemann, 2005.

[3] K. Washizu, Variational Methods in Elasticity and Plasticity, Oxford; New York; Pergamon Press, 1982.

[4] E. Reissner, On a Variational Theorem in Elasticity, Studies in Applied Mathematics, 29(1-4), 1950, 90-95.

[5] T. Belytschko, Y.Y. Lu, L. Gu, Element Free Galerkin Method, International Journal for Numerical Methods in Engineering, 37, 1994, 229–256.

[6] Y.Y. Lu, T. Belytschko, L. Gu, A New Implementation of the Element Free Galerkin Method, Computer Methods in Applied Mechanics and Engineering, 113, 1994, 397–414.

[7] S.N. Atluri, T. Zhu, A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics, Computational Mechanics, 22, 1998, 117–127.

[8] S.N. Atluri, T. Zhu, A New Meshless Local Petrov-Galerkin (MLPG) Approach to Nonlinear Problems in Computer Modeling and Simulation, Computational Modeling and Simulation in Engineering, 3, 1998, 187–196.

[9] S. Koshizuka, Y. Oka, Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid, Nuclear Science and Engineering, 123, 1996, 421–434.

[10] R.A. Gingold, J.J. Monaghan, Smoothed Particle Hydrodynamics: Theory and Application to Non-spherical Stars, Monthly Notices of the Royal Astronomical Society, 181, 1977, 375–389.

[11] M. Shimada, D. Tae, T. Xue, R. Deokar, K.K. Tamma, Second order Accurate Particle-based Formulations: Explicit MPS-GS4-II Family of Algorithms for Incompressible Fluids with Free Surfaces, International Journal of Numerical Methods for Heat & Fluid Flow, 26(3/4), 2016, 897–915.

[12] X. Zhou, K.K. Tamma, Design, Analysis, and Synthesis of Generalized Single Step Single Solve and optimal Algorithms for Structural Dynamics, International Journal for Numerical Methods in Engineering, 59, 2004, 597-668.

[13] X. Zhou, K.K. Tamma, Algorithms by Design with Illustrations to Solid and Structural Mechanics/Dynamics, International Journal for Numerical Methods in Engineering, 66, 2006, 1738–1790.

[14] M. Shimada, K.K. Tamma, Explicit Time Integrators and Designs for First/Second Order Linear Transient Systems, Encyclopedia of Thermal Stresses, 3, 2013, 1524–1530.

[15] Y. Chikazawa, S. Koshizuka, Y. Oka, A Particle Method for Elastic and Visco-plastic Structures and Fluid-structure Interactions, Computational Mechanics, 27(2), 2001, 97–106.