Difference between revisions of "MPM Methods and Simulation Timing"
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<math>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> | <math>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 Ω<sub>p</sub> is the domain for particle p, χ<sub>p</sub>(x) is the particle basis shape function for particle p, and N<sub>i</sub>(x) is grid shape function for node i. The various MPM methods depend on the choice of χ<sub>p</sub>(x) and on the method for evaluating the shape function integrals. The | where Ω<sub>p</sub> is the domain for particle p, χ<sub>p</sub>(x) is the particle basis shape function for particle p, and N<sub>i</sub>(x) is grid shape function for node i. The various MPM methods depend on the choice of χ<sub>p</sub>(x) and on the method for evaluating the shape function integrals. The chart on the right shows an MPM "genealogy" or shows how the various MPM methods descend from GIMP. | ||
[[File:MPMTree.png| | [[File:MPMTree.png|right]] | ||
The first decision is to choose χ<sub>p</sub>(x). If this function is set to the Dirac delta function (&delta(x)), the shape function reduced to the grid shape function, which recovers "Classic MPM" or the original derivation of MPM.<ref name='SS'>D. Sulsky, Z. Chen, and H. L. Schreyer, "A Particle Method for History-Dependent Materials,&wuot; ''Comput. Methods Appl. Mech. Engrg.'', '''118''', 179-186 (1994).</ref> This original MPM method is not very robust for large displacement problems because particles crossing cell boundaries in the mesh can cause significant noise. The MPM results are greatly improved by choosing χ<sub>p</sub>(x) to be a Heaviside function or to be 1 in the particle domain and zero elsewhere. All current MPM style descend from this selection. In principle, one could chose other χ<sub>p</sub>(x) (e.g., a Gaussian function centered on the particle) and derive new MPM options, but this area is unexplored. | |||
Next is how to integrate | |||
== Commands == | |||
* [[MPMMethod Command]] | * [[MPMMethod Command]] | ||
Revision as of 16:44, 5 September 2013
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 chart on the right shows an MPM "genealogy" or shows how the various MPM methods descend from GIMP.
The first decision is to choose χp(x). If this function is set to the Dirac delta function (&delta(x)), the shape function reduced to the grid shape function, which recovers "Classic MPM" or the original derivation of MPM.[2] This original MPM method is not very robust for large displacement problems because particles crossing cell boundaries in the mesh can cause significant noise. The MPM results are greatly improved by choosing χp(x) to be a Heaviside function or to be 1 in the particle domain and zero elsewhere. All current MPM style descend from this selection. In principle, one could chose other χp(x) (e.g., a Gaussian function centered on the particle) and derive new MPM options, but this area is unexplored.
Next is how to integrate
Commands
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).
- ↑ D. Sulsky, Z. Chen, and H. L. Schreyer, "A Particle Method for History-Dependent Materials,&wuot; Comput. Methods Appl. Mech. Engrg., 118, 179-186 (1994).