CPDI shape functions for the Material Point Method

In a conventional MPM formulation, the shape functions on the grid are the same as in a traditional FEM solution. In the CPDI, the shape functions on the grid are replaced by alternative (and still linearly complete*) shape functions, given by piecewise linear interpolations of the traditional FEM shape functions to the boundaries of the particles.  This change provides FEM-level accuracy in moderately deforming regions while retaining the attractive feature of MPM that particles can move arbitrarily relative to one another in massively deforming regions (provided, of course, that the deformation is updated in a manner compatible with the constitutive model).

In the images below, the shaded regions are the traditional FEM “tent” linear shape functions in 1-D, and the solid lines are the CPDI interpolated shape functions, which clearly change based on particle position relative to the grid.  Both the traditional FEM tent functions and these new CPDI functions are linearly complete (i.e., they can exactly fit any affine function). The tremendous advantage of CPDI is that the basis functions are extraordinarily simple over a particle domain, thus facilitating exact and efficient evaluation of integrals over particle domains.

The greatest advantage of the CPDI alternative shape functions is that they allow an MPM formulation to retain particle-to-particle coupling regardless of the amount of particle stretch:

CPDI’s support for arbitrary stretch allows this method to avoid the so-called extension instability that results in apparent fracture when particles separate by more than one grid cell.  With CPDI, fracture is returned to being a constitutive feature, not a numerical artifact. For further information, refer to our full publication.

These images show CPDI’s adaptive grid basis functions (equal to coefficients of the CPDI nodal values in the field descriptions used in the weak form). To see similar animations of the particle basis functions (equal to coefficient of particle values in the same field), see https://csmbrannon.net/2013/09/29/particle-basis-function-in-the-cpdi-method/. That post also illustrates performance of the method to describe constant and linear fields for a variety of grid-particle morphologies.

* Strictly speaking, linear completeness is ensured exactly in 1-D, but only asymptotically for 2-D and 3-D.  To be linearly complete, the approximate basis functions need to provide a partition of unity and be able to exactly represent a linear field, which means that the particle characteristic domains (convecting parallelograms in a basic CPDI formulation) must have no gaps and no overlaps.  The CPDI parallelograms provide only a first-order paving, with tiny gaps and tiny overlaps at corners of the parallelograms. Thus, partition of unity is satisfied only approximately. The error is extremely small because CPDI phrases all integrals in terms of averages over the particle domains. In comparison to the total size of the particle domain, the topological measure of the gap/overlap regions is very small and it rapidly goes to zero with refinement of the discretization, thus providing a superior rate of convergence despite small violations of partition of unity.