Abstract: In one embodiment, a technique for numerical integration in material point method (MPM)-based geotechnical analysis and simulation is provided. Input terms for an element of a background mesh are received. The input terms including material points in the element that describe a continuum of soil, rock and/or groundwater. A set of constraints is created that defines an optimization problem. The set of constraints provide that numerical integration of the material points weighted by unknown integration weights equal exact integration for finite element shape functions. The optimization problem defined by the constraints is solved by an optimization algorithm to minimize numerical integration error for polynomials up to a given order to produce a set of integration weights. The set of integration weights is scaled to conserve the mass of the material points to produce optimized integration weights.

Abstract: In one embodiment, material points are received that cover at least a portion of an element of a background mesh that describes a continuum of soil, rock and/or groundwater. MPM-based geotechnical analysis and simulation is conducted at least in part by performing a numerical integration over the material points to produce a system matrix and right-hand side vector. The numerical integration applies hierarchical shape functions to the material points. The MPM-based geotechnical analysis and simulation also may subtract out contributions of any lower-order polynomials from higher-order polynomials of the hierarchical shape functions when interpolating one or more state variables for the material points to the background mesh. The MPM-based geotechnical analysis and simulation also may subtract out contributions any lower-order polynomials from higher-order polynomials of the hierarchical shape functions when calculating one or more boundary conditions for the material points.

Abstract: In an example embodiment, a process may calculate a fill ratio for each element in the background mesh. The process may identify partially filled elements of the background mesh based on the calculated fill ratios. The process may transform each partially filled element to an empty or filled element by identifying a node of the partially filled element and moving the identified node to a different location in the background mesh. The process may generate an updated background mesh, that includes only empty and filled elements, that may be utilized to perform one or more calculations for one or more time steps in a modeling/simulation environment to simulate a behavior of a physical material/object that may exhibit deformations.