MPM Methods and Simulation Timing
These command select the MPM method to use and control time step and total time for the simulation.
Theory
Many tasks in MPM involve extrapolations from particles to the grid or from the grid to the particles. These extrapolations are controlled by "Shape Functions," and the various MPM methods correspond to various methods for evaluating the shape functions. In the most generalized description of MPM (called GIMP for General Interpolation Material Point[1]), the shape function (Sip) and shape function gradient (Gip) for the node i/particle p pair are:
[math]\displaystyle{ S_{ip} = {\int_{\Omega_p} \chi_p(\vec x)N_i(\vec x) dV\over \int_{\Omega_p} \chi_p(\vec x) dV}\qquad{\rm and}\qquad G_{ip} = {\int_{\Omega_p} \chi_p(\vec x)\nabla N_i(\vec x) dV\over \int_{\Omega_p} \chi_p(\vec x) dV} }[/math]
where Ωp is the domain for particle p, χp(x) is the particle basis shape function for particle p, and Ni(x) is grid shape function for node i. The various MPM methods depend on the choice of χp(x) and on the method for evaluating the shape function integrals. The following chart shows an MPM "genealogy" or shows how the various MPM method descend from GIMP.
References
- ↑ S. G. Bardenhagen, J. E. Guilkey, K. M. Roessig, J. U. Brackbill, W. M. Witzel, and J. C. Foster, "An Improved Contact Algorithm for the Material Point Method and Application to Stress Propagation in Granular Material," Computer Modeling in Engineering & Sciences, 2, 509-522 (2001).