Application of the Q-Less QR Factorization to Resolve Sparse Linear Over-Constraints

Over-constraint issues frequently arise in complex models with multiple constraints, such as those used in multibody systems and contact analyses. When over-constraints exist, commonly used constraint-handling methods, such as the transformation method, the Lagrange multiplier method, and the penalty method, can produce small or zero pivot values in the system matrix. This may result in non-convergence, slow convergence, or incorrect solutions. Furthermore, in the transformation method, in which the slave degrees of freedom (DOFs) corresponding to each constraint equation are eliminated, most finite element software restricts the reuse of slave DOFs that have already been referenced in other constraint equations. This makes an already complex modeling process even more challenging. This paper first summarizes a systematic algorithm proposed by Cho for detecting and resolving over-constraints in linear constraint equations, highlighting its efficiency in less complex scenarios but also its limitations for large-scale models due to excessive computation time. To address these limitations, this paper introduces a new algorithm that leverages the sparsity of the constraint matrix and applies sparse QR factorization without generating a dense Q matrix. The effectiveness of this approach in handling complex, large-scale finite element (FE) models is demonstrated, confirming its applicability to complex engineering problems.