A Dimensional Reduction Algorithm and Software for Acyclically Dependent Constraints

Abstract

For discrete equations of motion with acyclic equality constraints and within the context of the null-space method, an original Algorithm is introduced. By first permuting and then topologically ordering the degrees-of-freedom in the constraint gradient matrix, the saddle point problem can be solved with a sparse triangular system for the constraint equations. In this work, we show that saddle problems resulting from constrained (nonlinear) mechanical problems can always be set in this form, with constraint pivots being selected a priori. Given n discrete motion equations and m equality constraints, the original square sparse (n+m)2 ( n + m ) 2 system is replaced by a sparse system (n−m)2 ( n − m ) 2 and a sparse triangular solve with m2 coefficients and n – m right-hand sides. This triangular solve, which involves three sparse matrices (in existing literature only two of the three matrices are sparse), is here discussed in detail. Seven sparse operations are addressed (five standard and two nonstandard) in addition to some specific ad-hoc operations. Algorithms, source code and examples are presented in this work.