Inverse Mass Matrix via the Method of Localized Lagrange Multipliers

Abstract

Abstract. An efficient method for generating the mass matrix inverse is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces, and constructs the free-free inverse mass matrix employing the standard FEM procedure. The various boundary conditions are realized by the the method of localized Lagrange multipliers. Numerical experiments with the proposed inverse mass matrix method are carried out to validate the effectiveness proposed technique when applied to vibration analysis of bars and beams. A perfect agreement is found between the exact inverse of the mass matrix and its direct inverse computed through biorthogonal basis functions.

Large‐step explicit time integration via mass matrix tailoring

“…To derive the partitioned equations of motion of a linear structural dynamical system, we use the variational formulation proposed by Park and Felippa where the problem is treated like if all bodies were entirely free. Then, the total virtual work of the complete system is obtained by summing up the contributions of each substructure, plus the contribution of the interface constraints via the method of localized Lagrange multipliers (LLM): terms corresponding to the virtual work of the free‐floating substructures and localized interface constraints, respectively.…”

“…[2][3][4][5][6] To alleviate the stepsize limitations imposed by the mesh frequencies whose response components contribute very little when low modes dominate the transient responses, various mass matrix tailoring have been introduced by altering the mass matrices to reduce/filter out the high frequencies of the dynamical system without affecting the low-mid frequencies. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] Most of the existing methods cited above require either replacing existing elements by tailored elements and/or adopt element component-dependent time stepping procedures, leading to either elemental and/or global approaches, depending on how the modification of the mass matrix is made.…”

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

“…Different local and global dual interpolation functions of this kind, originally designed and tested for mortar methods, have been recently proposed in the literature [30][31][32] for NURBS and B-spline interpolations. Specifically for the construction of RMMs, biorthogonal basis functions have been proposed under the framework of FEM 8,9 and also IGA. 21 The different options available for the construction of dual bases are discussed next.…”

“…Remark 2. Expression (30), with the element projection matrix A e equal to the lumped element mass matrix, was originally proposed by González et al 9 for the construction of RMMs in the context of FEM. The same projection matrix was later employed by Schaeuble et al 21 to construct variationally consistent masses and reciprocal masses for IGA, eliminating the density inverse from its definition by introducing a momentum velocity field instead of a pure momentum field.…”

“…The preceding complications in generating biorthogonal basis functions for the momentum vector in treating the boundary conditions were alleviated in the work of González et al 9 for FEM by introducing the method of localized Lagrange multipliers (LLMs). [10][11][12][13] In that work, the system is considered completely free floating such that no special treatment is required for generating the biorthogonal interpolation of the momentum vector.…”