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

Enhancement of the material point method using B‐spline basis functions

“…As the artificial viscosity is increased, the oscillations in the computed compressive stresses can still be seen with slightly reduced amplitudes, and meanwhile, the wave fronts become less sharp due to increased viscosity dissipation. Recently, the GIMP noise of particle stresses has also been observed in the gas shock problem . The noise of the GIMP and CPDI calculations in and Figure is mainly due to the cell‐crossing error, namely, discontinuity or rapid change of the gradient of the shape function gradient in a single time step, which can be characterized by the particle Courant number as defined in .…”

“…Recently, the GIMP noise of particle stresses has also been observed in the gas shock problem . The noise of the GIMP and CPDI calculations in and Figure is mainly due to the cell‐crossing error, namely, discontinuity or rapid change of the gradient of the shape function gradient in a single time step, which can be characterized by the particle Courant number as defined in . Therefore, in the cases involving large deformation and/or high particle velocity, the GIMP and CPDI improvements in reducing cell‐crossing error via the introduction of finite particle domain could be reduced or even lost.…”

Shock waves simulated using the dual domain material point method combined with molecular dynamics

“…For the continuum scale calculation, we choose the dual domain material point (DDMP) method [15] enhanced with sub-points [16]. The DDMP method uses Lagrangian particles, also called material points, in addition to an Eulerian grid, where the computational cells and nodal points are material points in the centers of these regions.…”

“…To reduce the number of material points needed, while having sufficient numerical accuracy in the DDMP internal force calculation, the recently developed sub-point algorithm [16] is used. The algorithm exploits the difference between the stress variation length scale, which is L m , and the length scale of the shape function, which is the typical mesh size ∆x.…”

Extension of B-spline Material Point Method for unstructured triangular grids using Powell–Sabin splines

“…The DDMP method replaces the gradients of the piecewise-linear Lagrange basis functions in standard MPM by smoother ones. The DDMP method with sub-points [6] proposes an alternative manner for numerical integration within the DDMP algorithm.…”