Automated multi-level sub-structuring for fluid-solid interaction problems. (Q2889394)

The Automated Multi-Level Sub-structuring (AMLS) method is a multilevel extension of the Component Mode Synthesis (CMS) method. Recent studies show that for symmetric eigenvalues problems that arise in very large finite element models, AMLS is considerably faster then Lanczos-type approaches.NEWLINENEWLINEIn the paper authors consider free vibrations of a linear elastic structure which interacts with homogeneous, inviscid and compressible fluid. They assume that the structure and fluid occupy bounded disjoint domains with a common interface. On the remaining parts of the boundaries there are prescribed Dirichlet and Neumann boundary conditions. The finite element discretization of physical behaviour of the system leads to an unsymmetric matrix eigenvalue problem in which stiffness and mass matrices of the structure and mass matrix of the fluid are symmetric positive definite, stiffness matrix of the fluid is symmetric and positive semidefinite. The structure and fluid form a strongly coupled system.NEWLINENEWLINEAuthors propose a version of AMLS method for the unsymmetric eigenvalue problem that incorporates the coupling term into the reduction process. They formulated the unsymmetric eigenvalue problem as a symmetric one of doubled dimension with desired eigenvalues located at neither end of the spectrum.NEWLINENEWLINESimilarly as the standard AMLS, also the modified AMLS algorithm consists of two steps on each sub-structure. The first step is to transform the current approximating pencil by symmetric block Gauss elimination to an equivalent one by eliminating all off-diagonal blocks corresponding to the current sub-structure. The second step (usually parallelized) requires to solve the sub-structure eigenvalue problem. Details are given in three algorithms.NEWLINENEWLINEThe modified AMLS method for fluid-solid unsymmetric system requires essentially the same cost as AMLS for a symmetric problem of the same dimension. For strongly coupled problem this improves the approximation properties substantially.NEWLINENEWLINEThe AMLS method on two levels with a prescribed cut-off frequency on the sub-structures and no eigenvalue truncation on the interface is mathematically equivalent to a version of CMS algorithm. An a priori CMS-bound for fluid-structure interaction problems can be generalized to AMLS by recursive application. The authors proved the a priori bound for modified AMLS method. It overestimates the errors by orders of magnitude but it cannot be improved without further assumptions.