YT L = φT i,L φPi,L T +φP b,L TKbiK−1 ii , (38) one forms the interaction problem for the internal dofs of a component as YLT Kii −ω2Mii YRζ=0. (39) This reduced interaction problem is also unsymmetric and reveals left and right eigenvectors, namelyvL andvR. The recovery is accomplished over the left and right eigenvectors of the interaction problem. The mode set to be included into the basis is foundby ψL =YLvL, (40) ψR =YRvR. (41) The final form of the reduction and projection bases are given in a compact format as TL cb T = I −KbiK−1 ii 0 ψLT , (42) TR cb = I 0 −K−1 ii Kib ψR . (43) 4 Application and results 4.1 Computational framework and the model An application code is being developed by the first author to test the above mentioned ideas. The finite element system matrices are generated in the commercial program ANSYS using the implemented fluid-structure coupling scheme [15]. The resulting matrices are extracted from the binary result files along with the boundary condition information. Partitioning operations are accomplished in Gmsh [16] and the related interface and internal dof information is found from the partitioned mesh structure of the model. According to the partitioning information, component matrices are built from the binary element matrix files of ANSYS. A MATLAB [17] interface to SLEPc/PETSc [18, 19] is created to solve sparse eigenvalue problems. This interface is used to solve the eigenvalue problems that are encountered in the course of this research paper. For the test of ideas, a two-dimensional academic model is created. Namely, a structural beam is coupled to a closed fluid volume that is filled with water to simulate the strong coupling between the structure and the fluid. The structural properties of the beam are taken as follows: E = 71GPa, ν=0.33 and the density as 2800 kg/m3. The speed of sound in water is taken as 1500 m/s and the density of water is Umut Tabak and Daniel J. Rix 274
RkJQdWJsaXNoZXIy MTMzNzEzMQ==