Fixed-Step Solver Error Energy Minimization
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Solution Overview
Problem
Fixed-step solvers in computational simulation of physical systems often overshot critical events, leading to inaccurate simulation results due to their inability to adjust time steps, resulting in erroneous or impractical longer simulation times.
Innovation Solution
A method that minimizes the energy error between the initial and final states of a physical system by using a critical event energy minimizer, allowing fixed-step solvers to maintain accuracy and efficiency even when large time steps are used, and enabling the simulation to correct errors at the time of critical event detection.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If fixed-step solvers are used to simulate physical systems, then simulation efficiency and predictable runtime are improved, but accuracy at critical events deteriorates due to overshot detection
Solution Approach 1:
The patent applies preliminary action by projecting the state onto the new constraint manifold before the actual critical event occurs. The projection is performed at the time sample when the event is detected, but it anticipates the need for accurate transition by pre-computing the projected state that would result from an ideal instantaneous constraint change. This allows the fixed-step solver to maintain accuracy without changing the time step.
2Measurement precision
If time step is decreased to improve accuracy at critical events, then measurement precision is improved, but simulation time increases
Solution Approach 1:
The patent changes the parameter of state representation by introducing a projected state that lies on the new constraint manifold. Instead of changing the time step parameter, the invention transforms the state variable to account for the missed critical event. This parameter change allows the solver to maintain accuracy with the existing fixed time step, avoiding the need to decrease the time step and thereby avoiding increased simulation time.
3Measurement precision
If variable-step solvers are used to detect critical events precisely, then measurement precision is improved, but productivity deteriorates due to unpredictable runtime
Solution Approach 1:
The patent introduces an intermediary projection operation that mediates between the fixed-time sampling and the need for accurate critical event handling. The projection acts as a bridge that translates the discrete, fixed-step time samples into accurate representations of the continuous physical system state at critical events. This intermediary allows the system to maintain both fixed-step efficiency and event-detection accuracy without needing variable-step complexity.
Data Source
AI summary
The present invention improves in one aspect the accuracy, efficiency, and robustness of fixed-step solvers when simulating a physical system that does not change in energy at the critical event. The present invention uses the energy of the error as the figure of merit in finding the first state of the system under the new constraints. The present invention enables a fixed-step solver to be more robust even when the critical event falls between two time samples. The present invention may also be applied to variable-step solvers. Moreover, the present invention may also be used to simulate a physical system when the energy changes at the critical event in the physical system.