Stabilization Equation for Numerical Convergence in Large Deformation Simulations
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Solution Overview
Problem
Existing computer-aided design (CAD) and computer-aided engineering (CAE) systems face numerical instabilities and divergence when simulating models with large displacements and deformations, leading to incomplete analysis and failure in achieving converged solutions.
Innovation Solution
A computer-implemented method is introduced that stabilizes the simulation by defining a computer-based model with artificial internal forces and artificial constitutive modeling based on a non-linear deformation gradient matrix, ensuring numerical stability and convergence in determining the physical behavior of real-world objects.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If existing simulation methods are used for models with large displacements and deformations, then the simulation can be performed, but numerical instabilities and divergence occur leading to failure in achieving converged solutions
Solution Approach 1:
The patent introduces a stabilization equation as an intermediary mathematical component that mediates between the standard finite element equations and the solution process. This stabilization equation, which incorporates a non-linear deformation gradient matrix, acts as a mediator to prevent numerical instabilities and ensure convergence without fundamentally altering the physical model being simulated.
Solution Approach 2:
The patent modifies the simulation parameters by incorporating a stabilization equation that changes the mathematical parameters of the finite element analysis. This stabilization equation adjusts the deformation gradient matrix parameters to prevent numerical instabilities, allowing the simulation to converge for large displacement and deformation cases where standard methods fail.
2Measurement precision
If standard finite element methods are applied to large deformation problems, then the model can be analyzed, but ill-conditioned stiffness operators result in inaccurate or incomplete results
Solution Approach 1:
The stabilization equation serves as a mathematical intermediary that corrects the ill-conditioned stiffness operators. By introducing this additional equation that incorporates the non-linear deformation gradient matrix, the patent mediates between the problematic stiffness operators and the solution process, ensuring both numerical stability and accurate results.
Solution Approach 2:
The patent creates a composite mathematical model by combining the standard finite element equations with the stabilization equation. This composite approach integrates multiple mathematical components (the original equilibrium equations plus the stabilization equation) to achieve both numerical stability and accuracy in large deformation simulations.
3Productivity
If simulations with large deformations are performed using existing methods, then the analysis can proceed, but divergence occurs preventing completion of the analysis
Solution Approach 1:
The stabilization equation acts as a mediator that prevents divergence and ensures simulation completion. By incorporating this additional mathematical constraint that involves the non-linear deformation gradient matrix, the patent ensures that the analysis can proceed to completion without encountering numerical instabilities that would otherwise cause divergence.
Data Source
AI summary
Embodiments determine physical behavior of real-world objects. Using a computer-based model representing a real-world object, embodiments add pseudo-constitutive modeling to the computer-based model to alleviate local numerical instabilities caused by ill-conditioned elemental stiffness operators within elements of the model. A computer-based model, representing a real-world object using a plurality of elements, is defined which indicates one or more materials represented by the elements. Equations describing physics-based behaviors of the one or more materials are defined. A stabilization equation that is a function of a non-linear deformation gradient matrix is defined. A simulation is performed of the real-world object, subject to a load, using the defined computer-based model, the defined equations describing physics-based behaviors, and the defined stabilization equation. Performing the simulation includes applying the stabilization equation to each of the plurality of elements. Results of performing the simulation indicate the physical behavior of the real-world object.