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78 Valério da Silva Almeida & João Batista de Paiva 0,0 -4,0x10 -2 Bending Moment M 2 (MN/m 2 ) -8,0x10 -2 -1,2x10 -1 -1,6x10 -1 -2,0x10 -1 Case a Case b Case c Case d -2,4x10 -1 -10 -5 0 5 10 Cross-sectional BB along the radier. Figure 8 - Bending moments M 2 along section BB 9x10 -2 Vertical Contact Pressures (MN/m 2 ) 6x10 -2 3x10 -2 Case a Case b Case c Case d 0 -10 -5 0 5 10 Cross-sectional AA along the radier. Figure 9 - Contact forces between soil and raft along section AA Table 4 Nominal forces (MN) in the columns of the building’s ground floor for the several cases of soil rigidity Column Case a Case b Case c Case d Indeformable soil A1 0.3627 0.3620 0.4849 0.4092 0.0220 A2 0.6247 0.0890 0.1183 0.1022 0.3188 A3 0.6643 0.6649 0.6593 0.6662 0.1509 D4 0.5139 0.5104 0.5211 0.5110 0.9588 8 CONCLUSION An analysis was made of nonhomogeneous soil-structure interaction by a BEM-FEM combination. The method of successive stiffness proposed by MAIER & NOVATI (1987) was extended to 3D problems, including the influence of the Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007
A mixed BEM-FEM formulation for layered soil-superstructure interaction 79 structure’s flexibility based on the shell finite element composed of DKT, BATOZ (1980), and Free Formulation elements, BERGAN & FELIPPA (1985). The method of successive stiffness offers two computational advantages over the standard boundary element method by zones (subregions), namely: (1) fewer computational operations, and (2) lower storage memory requirements for equations of the final soil system. It should also be pointed out that the influences of each stratum can be computed independently, then this technique can be used for distributed memory computers with the advantage of achieving high efficiency and loading balance naturally. The results obtained show good agreement with those found in literature, and the robustness of the responses remained unaltered even when very thin soil layers were considered. The implemented model provides an efficient numerical tool for solving real problems such as the soil-structure system. 8.1 Application of sparse techniques in the linear systems In the solution of the linear system generated by coupling the soil-shell-multistorey building – example 7.4 - the resulting matrix is neither symmetrical nor positively defined, but sparse with 26% of nonzero elements (see figure 10). Thus, it is imperative to use sparse techniques for the solution of the problem. In the present research three different types of methods for solving the final linear equations were used: i) the Fortran 90 code that uses IMSL routines AIRD et al. (1977); ii) the iterative method GMRES, SAAD & SCHULTZ (1986), optimized with a diagonal preconditioning (Jacobi); and iii) the Harwell sparse matrix package MA28, DUFF (1977). The first, which is based on the direct method does not take into account the sparsity pattern property of the final matrix, so it is not adequate for this problem, as can be verified from execution time in table 5. The second is the most conventional iterative method applied in solution of sparse unsymmetric linear systems. The computational cost of the method is the matrix-vector product carried out along each iteration, and the sparsity pattern generated in the final matrix is not a relevant factor for the convergence in the method. But the strong dependence of the GMRES, SAAD & SCHULTZ (1986), is due to the condition number of the matrix, and the application of BEM in elastostatic problems, in general, does not generate a well-conditioned matrix property, principally, in the zone and BEM-FEM coupling methods. In the Harwell sparse matrix package MA28, DUFF (1977), based on Gauss elimination, the data structures of the matrix is a determinant factor for a rapid execution of code, due essentially to the fill-in effect. However, the sparsity pattern generator software was not used for reducing the number of operations. Despite this, the execution time obtained by MA28, DUFF (1977), showed a good efficiency in solving problems based on the mixed BEM-FEM formulation. Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007
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SUMÁRIO Análise experimental de p
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2 Alexandre Araújo Bertini & Libâ
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Inércia equivalente das estruturas
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ISSN 1809-5860 OTIMIZAÇÃO DE COMP
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