CSE 245 ComputerAided Circuit Simulation and Verification Multipole

CSE 245: Computer-Aided Circuit Simulation and Verification Multipole method based on thesis of Leslie Greengrad, 1987 Chung-Kuan Cheng

Outlines • Strategy • A 2 Dimensional Example – Source grouping – Destination grouping • A 3 Dimensional Example • Remark

Strategy: Definition Dipole: • a pair of equal and opposite electric charges or magnetic poles of opposite sign separated especially by a small distance Multipole: • having or involving several or many poles

Strategy: The applications • System of huge variables • Simulation of physical systems – EM wave, stellar clusters, protein folding, acoustics, turbulence • Learning theory – Kernel methods, support vector machines • Graphics and vision – Light scattering

Strategy • Derivation of Potential vs. Field • Special case of matrix vector multiplication – Si= sumj f(zi, zj) mj • Multipole: two reference systems based on the local regions of obj i and obj j. • Complexity O(n 2) => O(n logn)

Strategy of Multipole Method

A 2 Dimensional Example

Multipole: Source Grouping

Separation of the region

Source Grouping

Definition of Regions

Destination Grouping

Multipole Algorithm

Regions of Algorithm

Boundary Condition

A 3 Dimensional Example

3 Dimension Space

3 Dimension: Polar Coordinate System

Legendre Polynomials

Legendre Polynomials

Source Grouping

Destination Grouping

Destination Grouping

Remark • Format of the matrix vector multiplication • Dimensions – 2 D Case: Complex space – 3 D Case: Polar coordinate system • Data structure: hierarchy of multiple level regions • Error estimation • Boundary conditions • Sensitivity derivation