Spatial Interpolation for Circuit Simulation Process Variables
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Conventional methods for distributing process variables in circuit simulation tools fail to continuously distribute variables across a chip, leading to discontinuities and inefficiencies due to their grid-based approaches, which do not maintain local spatial correlation effectively.
Innovation Solution
A method using equilateral triangles to interpolate process variables, where numerical values are assigned at vertices and distributed across test points with correction factors to ensure continuous and statistically correct spatial interpolation, maintaining global distribution and local spatial correlation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If rectangular grid methods are used for spatial correlation, then computational simplicity is improved, but manufacturing precision deteriorates due to discontinuities at grid boundaries
Solution Approach 1:
The chip area is divided into multiple overlapping triangular elements rather than non-overlapping rectangular grids. This segmentation allows continuous spatial correlation across element boundaries while maintaining computational efficiency through the use of simple linear interpolation formulas within each triangle.
Solution Approach 2:
The invention uses triangular elements with curved (hyperbolic) interpolation surfaces instead of flat rectangular grids. The hyperbolic interpolation provides continuous spatial correlation across the entire chip area, eliminating the discontinuities that occur at rectangular grid boundaries while maintaining computational simplicity.
2Manufacturing precision
If hexagonal grid approach is used, then manufacturing precision is improved through higher packing density, but device complexity increases
Solution Approach 1:
The invention applies different interpolation methods to different regions: linear interpolation is used within each triangular element for simplicity, while hyperbolic interpolation is applied across element boundaries to ensure continuous spatial correlation. This local differentiation optimizes both accuracy and computational efficiency.
3Ease of operation
If conventional grid-based methods are used, then ease of operation is improved, but reliability deteriorates due to inability to maintain continuous spatial correlation
Solution Approach 1:
The invention pre-divides the chip area into triangular elements and pre-calculates the interpolation coefficients for each element based on process variable measurements at vertices. This preliminary preparation allows continuous and accurate spatial correlation to be maintained during actual circuit simulation without adding computational burden at runtime.
Data Source
AI summary
Methods for distributing a random variable by spatial interpolation with statistical corrections. The method includes assigning a numerical value of the random variable at each vertex of an array of equilateral triangles formed in a planar coordinate frame and defining a plurality of test points at respective spatial locations in the planar coordinate frame that are bounded by the array of equilateral triangles. A numerical value of the random variable is distributed at each of the test points by spatial interpolation from one or more of the numerical values of the random variable assigned at each vertex of the array of equilateral triangles. The method further includes adjusting the numerical value of the random variable distributed at each of the test points with a respective correction factor.