Convergence Estimation for Non-Linear PDE Simulations
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Current simulation methodologies in CAD and CAE systems lack efficient convergence estimation techniques, leading to poor quality solutions or wastage of computational resources due to unrequired iterations, as they rely on user judgment and indirect inference methods rather than quantifying error convergence.
Innovation Solution
The proposed method involves generating a system of equations in computer memory, iteratively solving them until convergence is reached, using an experimentally determined constant, estimate of the minimum eigenvalue, and residual to determine the error, allowing users to specify a tolerance for convergence, thereby quantifying error and optimizing iterations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If iterative simulations are performed until full convergence, then solution accuracy is improved, but computational time and resources are wasted due to unrequired iterations
Solution Approach 1:
The patent implements a feedback mechanism by continuously monitoring the estimated error during iterative simulations and comparing it against a user-specified tolerance. When the error estimate falls below the tolerance threshold, the simulation automatically terminates. This feedback-driven approach prevents unnecessary iterations while ensuring solution accuracy meets user requirements, directly resolving the contradiction between solution accuracy and computational time.
Solution Approach 2:
The patent employs preliminary action by using an error estimator to predict the convergence behavior and solution accuracy before completing all iterations. The error estimator provides advance information about whether the solution has reached the desired accuracy, allowing the simulation to terminate early when convergence is achieved. This preliminary assessment prevents wasteful computation of unrequired iterations.
2Ease of operation
If user judgment and indirect inference methods are used for convergence estimation, then ease of operation is maintained, but measurement precision of convergence is insufficient
Solution Approach 1:
The patent introduces an error estimator as an intermediary tool that objectively quantifies convergence between the user's tolerance requirement and the actual solution. This intermediary provides a numerical error estimate that bridges the gap between subjective user judgment and objective convergence measurement, enabling both ease of operation through automated assessment and high measurement precision through quantitative error analysis.
Data Source
AI summary
A method, according to an embodiment, provides a simulation of a physical real-world system, by first generating a system of equations that includes a discrete representation of the real-world system being simulated. Next, the real-world system is simulated. In simulating the system, a user specified tolerance of a solution of the system of equations is obtained. Then, the system of equations is iteratively solved until a solution to the system of equations for a given iteration is within the user specified tolerance of the solution of the system of equations for approximately infinite iterations. In such an embodiment, the solution to the system of equations for the given iteration is determined to be within the user specified tolerance using an experimentally determined constant, an estimate of a minimum eigenvalue of the system of equations for the given iteration, and a residual of the system of equations for the given iteration.