The construction of free–free flexibility matrices for multilevel structural analysis

Abstract

This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations at the substructure level. Direct or indirect computation of flexibilities as “influence coefficients” has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements. With the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This generalized inverse preserves the stiffness spectrum. The definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free–free flexibility has proven useful in special application areas of finite element structural analysis, notably massively parallel processing, model reduction and damage localization. It can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition. If substructures contain thousands of d.o.f., exploitation of the stiffness sparseness is important. For that case this paper presents a computation procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full free–free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness.

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