Material point methods applied to one-dimensional shock waves and dual domain material point method with sub-points

Abstract

Using a simple one-dimensional shock problem as an example, the present paper investigates numerical properties of the original material point method (MPM), the generalized interpolation material point (GIMP) method, the convected particle domain interpolation (CPDI) method, and the dual domain material point (DDMP) method.For a weak isothermal shock of ideal gas, the MPM cannot be used with accuracy. With a small number of particles per cell, GIMP and CPDI produce reasonable results. However, as the number of particles increases the methods fail to converge and produce pressure spikes. The DDMP method behaves in an opposite way. With a small number of particles per cell, DDMP results are unsatisfactory. As the number of particles increases, the DDMP results converge to correct solutions, but the large number of particles needed for convergence makes the method very expensive to use in these types of shock wave problems in two- or three-dimensional cases.The cause for producing the unsatisfactory DDMP results is identified. A simple improvement to the method is introduced by using sub-points. With this improvement, the DDMP method produces high quality numerical solutions with a very small number of particles.Although in the present paper, the numerical examples are one-dimensional, all derivations are for multidimensional problems. With the technique of approximately tracking particle domains of CPDI, the extension of this sub-point method to multidimensional problems is straightforward. This new method preserves the conservation properties of the DDMP method, which conserves mass and momentum exactly and conserves energy to the second order in both spatial and temporal discretizations.