Reordering for the Iterative Solution of Hybrid Elastic Structures

The Finite Element Method (FEM) is widely used in civil engineering to analyze the behavior of complex structures where the coupling of different elements, such as beams, trusses, and shells, is often required. FEM leads to symmetric positive definite (SPD) matrices that, depending on the type of discretization, may prove quite unsuitable for the solution by the Preconditioned Conjugate Gradient (PCG) method. Preconditioning techniques based on the incomplete Cholesky decomposition may fail when far nodes are connected. In this case the native ordering of the nodal unknowns yields a large matrix bandwidth, and a poor quality preconditioner which is also very expensive to be calculated. Numerical experiments are planned and presented where the effect of reordering on the PCG performance for the solution of a realistic bridge structure is explored and discussed. The preconditioner used is a variant of the incomplete Cholesky factorization with variable fill-in. It is shown that some reordering specifically designed and implemented for direct elimination methods can be very helpful as they lead to both a faster PCG convergence and a cheaper preconditioner computation. A main disadvantage is the need for an appropriate degree of fill-in which turns out to be problem dependent and must be found empirically. On the other hand, direct solvers perform usually better than PCG and moreover do not require the selection of an user-specified parameter like the most appropriate degree of fill-in