Product: ABAQUS/Standard
This example demonstrates the use of the iterative linear equation solver in ABAQUS/Standard for a simple static perturbation analysis of a heterogeneous cube. It also demonstrates the effect that material properties have on the convergence of the iterative solver. It should be noted that this problem is much too small for the iterative solver to outperform the direct equation solver and is used only for demonstration purposes.
The problem consists of a cube made of two elastic materials and fixed on one side with a uniform load applied to the opposite side. A structured mesh of C3D8 elements, with 20 elements in each of the three dimensions, is used, leading to approximately 28000 total degrees of freedom. The cube is divided in half along a plane parallel to the fixed and loaded faces. The material region adjacent to the fixed side has a Young's modulus of 
, and the other half has a Young's modulus of 
, such that 
. The ratio of the two materials is varied, and the number of iterations for the iterative solver to converge to a true residual in the neighborhood of 10–6 is studied. Recall that the global residual is defined as 
; where K is the global stiffness matrix, f is the applied load vector, and u is the approximate displacement solution computed by the iterative solver (see “Iterative linear equation solver,” Section 6.1.5 of the ABAQUS Analysis User's Manual, for more details). Since this cube is relatively small and the default number of domains for this problem would be too small to be meaningful, we arbitrarily choose to have the iterative solver decompose the model into 10 domains using the *SOLVER CONTROLS option.
The results are summarized in Table 1.1.13–1. Since this is a relatively small model, the number of linear solver iterations necessary for convergence is much lower than for real models; typically, more than 100 iterations will be needed to converge. However, the trend that is observed is representative of what may be observed for larger problems. As the ratio of the Young's moduli becomes large, the number of iterations for convergence increases dramatically, which is directly related to the performance of the iterative solver. Compared to the direct solver, which takes the same amount of time to solve the system of equations regardless of the material properties, the iterative solver time is directly impacted (for this model the time spent in the iterative solver for 
= 103 is a factor of two greater than that for = 1, the homogeneous case). The number of iterations may vary slightly depending on the platform on which the job is run, although the same general trend should be observed.
= 1.
= 101.
= 102.
= 103.
= 104.
