Using hierarchical finite element shape functions in material point method-based geotechnical analysis and simulation
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Publication Number
US-12099786-B1
Publication Date
2024-09-24
Expiration Date
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.
Core Innovation
The invention performs MPM-based geotechnical analysis and simulation using hierarchical (non-negative) finite element shape functions within a background mesh when material points partially cover elements. It receives material points that describe a continuum of soil, rock and/or groundwater and conducts the analysis by including the element at least in part based on those material points. The numerical integration applies hierarchical shape functions to the material points while producing a system matrix and right-hand side vector for the MPM analysis.
Hierarchical shape functions include at least one higher-order polynomial and lower-order polynomials. The invention avoids internal cancellations that can make the system matrices ill-conditioned when higher-order Lagrangian functions are used. The hierarchical shape functions may violate a partition-of-unity property and a Kronecker delta property, so the invention compensates by subtracting lower-order polynomial contributions from higher-order polynomial contributions during interpolation of state variables and during calculation of boundary conditions.
In the interpolation, the invention subtracts out contributions to the state variables of any lower-order polynomials from higher-order polynomials of the hierarchical shape functions to compensate for lack of a partition of unity property. For boundary conditions, it subtracts out contributions to a resulting field of any lower-order polynomials from higher-order polynomials of the hierarchical shape functions to compensate for loss of the Kronecker delta property. The invention includes displaying and/or storing results from the MPM-based geotechnical analysis and simulation.
Claims Coverage
The independent claims (a method, a computing device, and a non-transitory computing device readable medium) share a common set of inventive features: using numerical integration in MPM-based geotechnical analysis with hierarchical shape functions that include higher-order and lower-order polynomials, and compensating for loss of polynomial-related basis properties by subtracting lower-order contributions from higher-order contributions during interpolation of state variables and/or calculation of boundary conditions. Across these independent claims, the inventive features are implemented in software modules and/or stored as executable instructions, and the results are displayed and/or stored.
MPM-based geotechnical analysis using hierarchical shape-function integration
Receives material points covering at least a portion of an element of a background mesh describing a continuum of soil, rock and/or groundwater, conducts MPM-based geotechnical analysis by performing numerical integration over the material points to produce a system matrix and right-hand side vector, where the numerical integration applies hierarchical shape functions to the material points, and where the hierarchical shape functions include at least one higher-order polynomial and lower-order polynomials.
Compensation by subtracting lower-order from higher-order during state-variable interpolation
Interpolates one or more state variables for the material points to the background mesh, and performs the interpolating to subtract out contributions to the state variables of any lower-order polynomials from higher-order polynomials of the hierarchical shape functions to compensate for lack of a partition of unity property.
Compensation by subtracting lower-order from higher-order during boundary-condition calculation
Calculates one or more boundary conditions for the material points, and the calculating subtracts out contributions to a resulting field of any lower-order polynomials from higher-order polynomials of the hierarchical shape functions to compensate for loss of Kronecker delta property.
Geotechnical engineering software with mesher and solver modules using hierarchical shape functions
Stores geotechnical engineering software including a mesher module configured to generate a background mesh composed of elements formed from nodes, each element including material points describing a continuum of soil, rock and/or groundwater; and a solver module configured to use MPM to perform analysis and simulation of behavior by performing numerical integration on each element using hierarchical shape functions, where the analysis includes interpolating one or more state variables for the material points to the background mesh and/or calculating one or more boundary conditions for the material points.
Non-transitory medium instructions for hierarchical shape-function integration and compensation
Provides non-transitory computing device readable medium instructions that, when executed, receive material points covering at least a portion of an element of a background mesh describing a continuum of soil, rock and/or groundwater; conduct MPM-based geotechnical analysis by performing numerical integration over the material points to produce a system matrix and right-hand side vector using hierarchical shape functions including higher-order and lower-order polynomials; and calculate one or more boundary conditions by subtracting lower-order polynomial contributions from higher-order polynomial contributions to compensate for loss of Kronecker delta property.
Across the independent claims, the core claim coverage centers on applying hierarchical shape functions (including higher-order and lower-order polynomials) within MPM-based geotechnical numerical integration to form a system matrix and right-hand side vector, and compensating for lost polynomial basis properties by subtracting lower-order contributions from higher-order contributions during state-variable interpolation and/or boundary-condition calculation, with results displayed and/or stored.
Stated Advantages
Hierarchical shape functions avoid internal cancellations that can otherwise make system matrices ill-conditioned when higher-order Lagrangian functions are used.
Documented Applications
MPM-based geotechnical analysis and simulation of the behavior of a continuum of soil, rock and/or groundwater using material points and a background mesh.
Displaying and/or storing results of the MPM-based geotechnical analysis and simulation in a geotechnical engineering software implementation.
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