Real-World Object Simulation with Basis-Specific Quadrature Weights
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Solution Overview
Problem
Existing simulation methods for predicting the shape of real-world objects under mechanical force application are computationally expensive due to the high number of quadrature points required for accuracy, which exceeds the available computational resources and prolongs simulation time.
Innovation Solution
A modified weighted quadrature method that reduces the number of quadrature points by pre-calculating specific quadrature weights for individual basis functions, allowing simultaneous integration over entire macro-elements, thereby reducing computational complexity and time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional quadrature methods are used with sufficient quadrature points to ensure accuracy, then simulation accuracy is maintained, but computational time and resource consumption increase significantly
Solution Approach 1:
The patent pre-calculates and stores quadrature weights for individual basis functions before the simulation runs. This preliminary preparation allows the simulation to use these pre-computed weights directly during execution, eliminating the need to calculate quadrature weights on-the-fly and significantly reducing computational time while maintaining accuracy
Solution Approach 2:
The patent segments the quadrature calculation process by computing weights for each individual basis function separately and storing them in lookup tables. This segmentation allows the simulation to efficiently retrieve and apply only the specific weights needed for each basis function, rather than computing all quadrature weights globally, thus reducing overall computational burden
2Productivity
If the number of quadrature points is reduced to decrease computational complexity, then simulation time decreases, but simulation accuracy deteriorates
Solution Approach 1:
The patent changes the parameter representation by using pre-computed quadrature weights associated with each basis function. This parameter transformation allows the simulation to achieve accurate results with fewer quadrature points because each basis function's contribution is precisely captured through its dedicated pre-calculated weights, maintaining accuracy while improving efficiency
3Manufacturing precision
If traditional quadrature methods are applied to each element separately, then local accuracy is ensured, but overall computational complexity increases
Solution Approach 1:
The patent merges the quadrature weight calculations by pre-computing weights for individual basis functions across the entire model and storing them in unified lookup tables. This merging approach eliminates redundant calculations that would occur if each element computed its own quadrature weights separately, reducing overall computational complexity while maintaining local accuracy through the precise basis-function-specific weights
Data Source
Figure 1(A)~1(C)
Figure 2
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AI summary
A simulation computer predicts the shape (200) of a real-world object as a result of hypothetical force application to the real-world object. The computer calculates individual integrals with a trial function and a test function by applying quadrature rules with quadrature weights (W_i_r) that are specific to the individual basis function and that are specific to the parameter set that corresponds to interpolation points of the individual basis function. The weights (220, 230) are pre-calculated and fit to exactness conditions.