Support-Graph Preconditioners for 2-Dimensional Trusses

Abstract: We use support theory, in particular the fretsaw extensions of Shklarski and Toledo, to design preconditioners for the stiffness matrices of 2-dimensional truss structures that are stiffly connected. Provided that all the lengths of the trusses are within constant factors of each other, that the angles at the corners of the triangles are bounded away from 0 and $\pi$, and that the elastic moduli and cross-sectional areas of all the truss elements are within constant factors of each other, our preconditioners allow us to solve linear equations in the stiffness matrices to accuracy $\epsilon$ in time $O (n^{5/4} (\log^{2}n \log \log n)^{3/4} \log (1/\epsilon))$.