A fast hierarchical algorithm for 3-D capacitance extraction

Abstract

We presen t a new algorithm for computing the capacitance of three-dimensional perfect electrical conductors of complex structures. The new algorithm is signi cantly faster and uses much less memory than previous best algorithms, and is kernel independent.The new algorithm is based on a hierarchical algorithm for the n-body problem, and is an acceleration of the boundaryelement method for solving the integral equation associated with the capacitance extraction problem. The algorithm rst adaptively subdivides the conductor surfaces into panels according to an estimation of the potential coe cients and a user-supplied error bound. The algorithm stores the poten tial coe cient matrix in a hierarchical data structure of size On, although the matrix is size n 2 if expanded explicitly , wheren is the n umber of panels. The hierarchical data structure allows us to multiply the coe cient matrix with an y vector in On time. Finally, w e use a generalized minimal residual algorithm to solve m linear systems each of size n n in Omn time, where m is the n umberof conductors.The new algorithm is implemented and the performance is compared with previous best algorithms. F or the k k bus example, our algorithm is 100 to 40 times faster than F astCap, and uses 1=100 to 1=60 of the memory used by F astCap. The results computed by the new algorithm are within 2.7 from that computed by F astCap.

A direct integral-equation solver of linear complexity for large-scale 3D capacitance and impedance extraction

“…O n k . If we convert a full matrix block to an admissible block, the best approximation of ( 2,2) t r full G for the admissible block is G . The only approximation in this case is from the collect operation done in step (c), the accuracy of which is controllable.…”

“…8(c), we plot the inverse time and the total CPU time of the proposed direct solver with respect to N. Clearly a linear complexity can be observed. For comparison, the solution time of the HiCap [2] is also plotted. The advantage of the proposed direct solver is clearly demonstrated even though HiCap only calculated the results for m right hand sides with m being the number of conductors, whereas the proposed solver obtained the entire inverse, i.e., the results for N right hand sides.…”

“…When an iterative solver is used, the memory requirement remains the same, and the time complexity is O(N it N 2 ), where N it denotes the total number of iterations required to reach convergence. In state-of-the-art IE-based capacitance solvers, Fast Multipole Method (FMM) and hierarchical algorithms [1][2][3] were used to perform a matrix-vector multiplication in O(N) complexity, thereby significantly reducing the complexity of iterative solvers. In the limited work reported on the direct IE solutions for capacitance extraction [4], no linear complexity has been achieved.…”

“…Both Storage requirements and matrix-vector multiplications using -matrices are of complexity O(Nlog D N). 2 -matrices, which are a specialized subclass of hierarchical matrices, were later introduced in [6]. It was shown that the storage requirements and matrix-vector products are of complexity O(N) for 2 -based representation of both quasi-static and electrodynamic problems [7][8].…”

Large-scale capacitance calculation

“…The required matrix-vector products are accelerated with various compression schemes. Examples of previous work following this strategy include versions of FastCap [5,6], IES 3 [3], and the method of Shi et al [8].…”