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 | 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 technique used 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|right]] | [[File:MPMTree.png|right]] | ||
The first decision is to chose χ<sub>p</sub>(x); the chart shows three options. The choice of the Dirac delta function (δ(x)) leads to classic MPM methods. The choice of a Heaviside function (''i.e.'', χ<sub>p</sub>(x) = 1 in Ω<sub>p</sub> and 0 elsewhere) leads to a family of Heaviside GIMP methods. Finally the choice of some other function (''e.g.'', a Gaussian centered on the particle) could lead to other MPM methods, but is currently an unexplored area. | |||
The first decision is to | |||
=== Delta Function GIMP Methods - Classic MPM === | |||
If χ<sub>p</sub>(x) = δ(x), the shape functions reduce to the the grid shape function (e.g., S<sub>ip</sub> = N<sub>i</sub>(x)), 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> Note that "Classic GIMP predates GIMP even though it descends from GIMP in the MPM "genealogy." Although Classic MPM is the most efficient method, it is rarely used because it is not very robust for large displacement problems. Whenever particle cross cell boundaries, they can cause noise that soon leads to unsatisfactory results. These cell-crossing artifacts can be eliminated (or greatly reduced) by using different χ<sub>p</sub>(x). | |||
=== Heaviside GIMP Methods === | |||
MPM results are greatly improved by choosing χ<sub>p</sub>(x) to be a Heaviside function. With this function, the shape function integrals reduce to integrals over the particle domain of the grid-based shape function, which are normalized to particle volume (see short). All current MPM methods descend from this selection. The specific methods that descend from the Heaviside function depend on how that integrate over the particle domain. | |||
==== uGIMP ==== | ==== uGIMP ==== | ||
The simplest approach is to assume the particle domain has the same shape as the initial domain and merely translates with particle motion. This method, which is called uGIMP (for undeformed GIMP), has the greatest efficiency because shape functions can be explicitly calculated for all possible particle locations relative to a regular grid. uGIMP works well from small to fairly large deformation and greatly reduces cell-crossing artifacts. Very large tension strains (>50%), however, can cause undeformed particle domains to become numerically separated, which can lead to non-physical fracture of the object. | |||
The simplest approach is to assume the particle domain has the same shape as the initial domain and merely translates with particle motion. This method, which is called uGIMP (for undeformed GIMP), has the greatest efficiency because shape functions can be explicitly calculated for all possible particle locations. uGIMP works well | |||
==== Finite GIMP ==== | ==== Finite GIMP ==== | ||
Revision as of 08:14, 6 September 2013
These command select the MPM method to use and control time step and total time for the simulation.
Theory: Shape Functions
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 technique used 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 chose χp(x); the chart shows three options. The choice of the Dirac delta function (δ(x)) leads to classic MPM methods. The choice of a Heaviside function (i.e., χp(x) = 1 in Ωp and 0 elsewhere) leads to a family of Heaviside GIMP methods. Finally the choice of some other function (e.g., a Gaussian centered on the particle) could lead to other MPM methods, but is currently an unexplored area.
Delta Function GIMP Methods - Classic MPM
If χp(x) = δ(x), the shape functions reduce to the the grid shape function (e.g., Sip = Ni(x)), which recovers "Classic MPM" or the original derivation of MPM.[2] Note that "Classic GIMP predates GIMP even though it descends from GIMP in the MPM "genealogy." Although Classic MPM is the most efficient method, it is rarely used because it is not very robust for large displacement problems. Whenever particle cross cell boundaries, they can cause noise that soon leads to unsatisfactory results. These cell-crossing artifacts can be eliminated (or greatly reduced) by using different χp(x).
Heaviside GIMP Methods
MPM results are greatly improved by choosing χp(x) to be a Heaviside function. With this function, the shape function integrals reduce to integrals over the particle domain of the grid-based shape function, which are normalized to particle volume (see short). All current MPM methods descend from this selection. The specific methods that descend from the Heaviside function depend on how that integrate over the particle domain.
uGIMP
The simplest approach is to assume the particle domain has the same shape as the initial domain and merely translates with particle motion. This method, which is called uGIMP (for undeformed GIMP), has the greatest efficiency because shape functions can be explicitly calculated for all possible particle locations relative to a regular grid. uGIMP works well from small to fairly large deformation and greatly reduces cell-crossing artifacts. Very large tension strains (>50%), however, can cause undeformed particle domains to become numerically separated, which can lead to non-physical fracture of the object.
Finite GIMP
In finite GIMP, the integral is over the deformed particle domain. Although this has been shown to give improved results in 1D (where it is practical[1]), it is too inefficient for arbitarary deformation of 2D and 3D particle domains. As a result, finite GIMP is not used.
CPDI
A useful improvement for large deformation is CPDI or the Convected Partical Domain Integration approach.[3] In CPDI, the integral over the deformed particle domain is make practical by approximating the integrand. In brief:
- The deformed particle domain is discretized by isoparametric finite elements. In practice, the discretization uses a single element, but it could use more.
- The Integrand for Sip and Gip are then approximate by expanding them in the finite element shape functions
- Integration over then shape functions gives the final results.
Two versions of CPDI have appeared. The first version finds the edges of the deformed domain using the deformation gradient on the particle. In this approach, an initial 2D rectangle deforms to a parallelogram and an initial 3D orthogonal box deforms to a parallelepiped.[3] The coordinates for the corners of the deformed domain and this a function of the initial domain shape and the particle's deformation gradient. The final shape functions depends on the finite element shape functions used for the particle domain. For example lCPDI using linear shape functions (4 nodes for 2D domains and 8 nodes for 3D domains) while qCPDI using quadratic shape function (8 nodes for 2D domains). The options for "other..." versions allow for alternative finite element discretizations of the particle domains.
In CPDI2, the corner locations are separately tracked during the analysis and their current locations are used to define the deformed particle domain. For example, a 2D domain deforms to an arbitrary quadrilateral and a 2D domain an arbitrary 8-cornered domain.
Theory:Stress and Strain Updates
Commands
References
- ↑ 1.0 1.1 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).
- ↑ 3.0 3.1 A. Sadeghirad, R. M. Brannon, and J. Burghardt, "A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations," Int. J. Numer. Meth. Engng (2011)