Design Parameter Space Visualization via Quantifier Elimination
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Solution Overview
Problem
In multi-objective optimization design, existing techniques struggle to effectively visualize the relationship between design parameters and costs, particularly when dealing with multiple routes in the design parameter space and branches or mergers, and fail to grasp how design parameter values change with cost variations or identify optimal conditions for better cost values.
Innovation Solution
A display processing method is developed, involving generating a constraint equation from cost function data and using quantifier elimination to obtain a logical expression, which is then used to determine true or false values for points in the design parameter space, allowing for the display of routes corresponding to designated routes in the cost space.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If quantifier elimination method is used to obtain logical expression of solution, then the ability to handle arbitrary routes and branches in design parameter space is improved, but the computational complexity and processing time increase
Solution Approach 1:
The patent pre-calculates and stores the logical expression of the solution using quantifier elimination before actual route queries. By performing the computationally intensive quantifier elimination in advance and caching the results, the system avoids repeating heavy computations during interactive design exploration, thus resolving the contradiction between handling complex arbitrary routes and maintaining computational efficiency.
2Device complexity
If the search in design parameter space is limited to perpendicular bisector, then the computational processing is simplified, but the ability to deal with multiple routes and branches is lost
Solution Approach 1:
Instead of limiting the search to perpendicular bisectors and trying to handle complexity, the patent inverts the approach by using quantifier elimination to directly compute the logical expression that defines all valid points in the design parameter space. This allows arbitrary routes and branches to be naturally captured without imposing geometric constraints, resolving the contradiction between computational simplicity and route handling capability.
3Area of stationary object
If gridlike arrangement of points is used in design parameter space, then the coverage of search space is improved, but the ability to trace specific routes corresponding to cost space routes is lost
Solution Approach 1:
The patent uses the logical expression of the solution as a feedback mechanism to evaluate whether each grid point satisfies the route conditions. By substituting grid point coordinates into the logical expression and checking truth values, the system identifies which grid points lie on the route corresponding to the designated cost space route, thus maintaining both comprehensive search space coverage and accurate route correspondence.
Data Source
AI summary
This method includes: generating a constraint equation from data of an approximate expression of a cost function representing a relationship between a plurality of design parameters and a cost, data of a route in a cost space and data of a search range in a design parameter space; obtaining a logical expression of a solution for the constraint equation from a quantifier elimination processing unit that carries out a processing according to a quantifier elimination method; substituting coordinates of each of a plurality of points within the search range in the design parameter space into the logical expression of the solution to determine, for each of the plurality of points, true or false of the logical expression of the solution; and displaying the design parameter space in which a display object including a first point for which true is determined is disposed at the first point.