For both the explicit and the implicit time integration procedures, equilibrium is defined in terms of the external applied forces, , the internal element forces, , and the nodal accelerations:

ABAQUS/Standard uses automatic incrementation based on the full Newton iterative solution method. Newton's method seeks to satisfy dynamic equilibrium at the end of the increment at time and compute displacements at the same time. The time increment, , is relatively large compared to that used in the explicit method because the implicit scheme is unconditionally stable. For a nonlinear problem each increment typically requires several iterations to obtain a solution within the prescribed tolerances. Each Newton iteration solves for a correction, , to the incremental displacements, . Each iteration requires the solution of a set of simultaneous equations,

However, if the model contains highly discontinuous processes, such as contact and frictional sliding, quadratic convergence may be lost and a large number of iterations may be required. Cutbacks in the time increment size may become necessary to satisfy equilibrium. In extreme cases the resulting time increment size in the implicit analysis may be on the same order as a typical stable time increment for an explicit analysis, while still carrying the high solution cost of implicit iteration. In some cases convergence may not be possible using the implicit method.

Each iteration in an implicit analysis requires solving a large system of linear equations, a procedure that requires considerable computation, disk space, and memory. For large problems these equation solver requirements are dominant over the requirements of the element and material calculations, which are similar for an analysis in ABAQUS/Explicit. As the problem size increases, the equation solver requirements grow rapidly so that, in practice, the maximum size of an implicit analysis that can be solved on a given machine often is dictated by the amount of disk space and memory available on the machine rather than by the required computation time.

For many analyses it is clear whether ABAQUS/Standard or ABAQUS/Explicit should be used. For example, earlier in this chapter we saw that ABAQUS/Explicit is the clear choice for a wave propagation analysis; on the other hand, ABAQUS/Standard is more efficient for solving smooth nonlinear problems. There are, however, certain static or nearly static problems that can be simulated well with either program. Typically, these are problems that usually would be solved with ABAQUS/Standard but may have difficulty converging because of contact or material complexities, resulting in a large number of iterations. Such analyses are expensive in ABAQUS/Standard because each iteration is expensive and requires a large set of linear equations to be solved.

Whereas ABAQUS/Standard must iterate to determine the solution to a nonlinear problem, ABAQUS/Explicit determines the solution without iterating by explicitly advancing the kinematic state from the previous increment. Even though a static analysis requires a large number of time increments using the explicit method, the analysis can be more efficient in ABAQUS/Explicit if the same analysis in ABAQUS/Standard would require many expensive iterations.

Another advantage of ABAQUS/Explicit is that it requires much less disk space and memory than ABAQUS/Standard for the same simulation. For some problems in which the computational cost of the two programs may be comparable, the substantial disk space and memory savings of ABAQUS/Explicit make it attractive.

Using the explicit method, the computational cost is proportional to the number of elements and roughly inversely proportional to the smallest element dimension. Mesh refinement, therefore, increases the computational cost by increasing the number of elements and reducing the smallest element dimension. As an example, consider a three-dimensional model with uniform, square elements. If the mesh is refined by a factor of two in all three directions, the computational cost increases by a factor of 2 × 2 × 2 as a result of the increase in number of elements and by a factor of 2 as a result of the decrease in the smallest element dimension. The total computational cost of the analysis increases by a factor of 2⁴, or 16, by refining the mesh. Disk space and memory requirements are proportional to the number of elements with no dependence on element dimensions; thus, these requirements increase by a factor of 8.

Whereas predicting the cost increase with mesh refinement for the explicit method is rather straightforward, cost is more difficult to predict when using the implicit method. The difficulty arises from the problem-dependent relationship between element connectivity and solution cost, a relationship that does not exist in the explicit method. Using the implicit method, experience shows that for many problems the computational cost is roughly proportional to the square of the number of degrees of freedom. Consider the same example of a three-dimensional model with uniform, square elements. Refining the mesh by a factor of two in all three directions increases the number of degrees of freedom by approximately 2³, causing the computational cost to increase by a factor of roughly (2³)², or 64. The disk space and memory requirements increase in the same manner, although the actual increase is difficult to predict.

The explicit method shows great cost savings over the implicit method as the model size increases, as long as the mesh is relatively uniform. Figure 3–15 illustrates the comparison of cost versus model size using the explicit and implicit methods. For this problem the number of degrees of freedom scales with the number of elements.