MPM Methods and Simulation Timing

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), the shape function (S_ip) and shape function gradient (G_ip) 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 Ω_p is the domain for particle p, χ_p(x) is the particle basis shape function for particle p, and N_i(x) is grid shape function for node i. The various MPM methods depend on the choice of χ_p(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

Classic MPM

The first decision is to choose χ_p(x). If this function is set to the Dirac delta function (δ(x)), the shape function reduced to the grid shape function, which recovers "Classic MPM" or the original derivation of MPM. This original MPM method is not very robust for large displacement problems because particles crossing cell boundaries in the mesh can cause significant noise.

uGIMP

MPM results are greatly improved by choosing χ_p(x) to be a Heaviside function or to be 1 in the particle domain and zero elsewhere. All current MPM styles descend from this selection. In principle, one could chose other χ_p(x) (e.g., a Gaussian function centered on the particle) and derive new MPM options, but this area is unexplored. The use of the Heaviside function leads to simplification of both S_ip and G_ip, but the integrals still need to be evaluated.

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 for small and modest deformation. Very large tension strains (>50%) can cause particles to numerically separate because the undeformed domains no longer interact with each other.

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), 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. In CPDI, the integral over the deformed particle domain is make practical by approximating the integrand. In brief:

1. The deformed particle domain is discretized by isoparametric finite elements. In practice, the discretization uses a single element, but it could use more.
2. The Integrand for S_ip and G_ip are then approximate by expanding them in the finite element shape functions
3. 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. 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.

Commands

• MPMMethod Command
• PtsPerElement Command
• TimeStep Command
• MaximumTime Command
• DefGradTerms Command