 |
Center for Nonlinear Studies |
Like many other particle methods, the material point method (MPM) [1][2] gains computational efficiency by solving the governing equations on a background grid that can be reset or refined at the end of each time step, while field data are stored at moving Lagrangian particles. For such methods, spurious non-monotonic variation in stress can occur at material interfaces passing through the interior of a grid cell. This problem is caused by low-order shape functions being incapable of describing the jump in strain gradients needed to allow the compliant materials within the cell to deform more than stiff materials. The recently developed convected particle domain interpolation (CPDI) method [3], which allows integrals in the generalized interpolation material point (GIMP) method [4] to be evaluated on continuously deforming domains, is enhanced to describe fields over particle domains based on values of those fields mapped to the particle domain’s corners. In the original CPDI method, now called CPDI1, the particle domains were approximated to be parallelograms (or parallelepipeds in 3D). An enhancement, called CPDI2, describes particle domains by quadrilaterals (hexahedra in 3D). Advantages of CPDI2 over CPDI1 include (1) more accurately tracking particle domains with very little computational overhead in comparison to CPDI1, (2) removing gaps/overlapping between particle domains, and (3) giving more flexibility in choosing particle domains shape in the initial configuration. An additional advantage is that the nodal degrees of freedom already defined on the background grid can be supplemented with the corner values of the fields at the particles that are known to be near a material interface. This approach provides enrichment capable of properly describing weak discontinuities in the displacement field (i.e., strong discontinuities in strain) across a material interface that passes through the interior of a grid cell. Effectiveness of this approach is, at first, trivially demonstrated in a simplistic 1-D context, where spurious stress spikes and dips caused by incorrect partitioning of deformation within a cell are eliminated through CPDI2 enrichment, thus giving results comparable to a traditional Lagrangian finite-element simulation that has a node at the material interface. Recognizing that particle methods are adopted in situations for which traditional Lagrangian finite elements or Eulerian finite difference methods are unsatisfactory (e.g., massive deformations of history-dependent materials arranged in complicated geometries), improvements in accuracy of CPDI2 over CPDI1, and even more dramatic improvements in comparison to legacy methods of evaluating non-advecting GIMP integrals, are presented with and without material interfaces in cell boundaries.
Host: Matt Lewis, Ph.D. Deputy Group Leader, W-13 Advanced Engineering Analysis Weapons System Engineering Division