Comparison of implicit and explicit time integration procedures

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

M⁢u¨=P-I,
where M is the mass matrix. Both procedures solve for nodal accelerations and use the same element calculations to determine the internal element forces. The biggest difference between the two procedures lies in the manner in which the nodal accelerations are computed. In the implicit procedure a set of linear equations is solved by a direct solution method. The computational cost of solving this set of equations is high when compared to the relatively low cost of the nodal calculations with the explicit method.

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 t+Δ⁢t and to compute displacements at the same time. The time increment, Δ⁢t, 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, cj, to the incremental displacements, Δ⁢uj. Each iteration requires the solution of a set of simultaneous equations,

K^j⁢cj=Pj-Ij-Mj⁢u¨j,

which is an expensive procedure for large models. The effective stiffness matrix, K^j, is a linear combination of the tangent stiffness matrix and the mass matrix for the iteration. The iterations continue until several quantities—force residual, displacement correction, etc.—are within the prescribed tolerances. For a smooth nonlinear response Newton's method has a quadratic rate of convergence, as illustrated below:

Iteration Relative Error
1 1
2 10−2
3 10−4
. .
. .
. .

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.