# Static

Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.

With this approach, the time step is often small due to stability condition. For the static solution of structural mechanical problems as the steady state is a part of the transient response for a temporal-step load, the use of explicit scheme is usually possible if the computation time remains reasonable. However, in static or slow dynamic computations as duration of the study is large, many cycles are necessary to carry out the simulation.

To resolve static problems, an alternative to explicit method is the implicit time-integration scheme. In this method, a system of nonlinear equations is obtained and then resolved by Newton-Raphson method. It can be shown that the implicit scheme is always stable. That results in a large time step with the explicit method. However, as a global stiffness matrix should be assembled and inverted, the method is relatively high cost per loading step.

The primary difference between the explicit and implicit methods is that an explicit algorithm obtains the next value from known previous values. An implicit method assumes a solution to a problem and solves the equations simultaneously. As the global equilibrium equation is generally nonlinear, an iterative numerical resolution is generally used.

The implicit method might fail when:
• The material law is highly nonlinear. Complicated material behavior is easier to accommodate using an explicit method.
• The number of elements is too large.
• Explicit method does not require large matrix inversion, the I/O is less important and the memory required is also less.
• Matrices must be re-evaluated at each time step and for most of the iterations.
In such cases the CPU time of an explicit solution becomes competitive:
• The problem includes several contacts. Contact algorithms are very efficient in explicit programs.
• The static analysis is a pre-loading case before a fully dynamic behavior phase. In this case, the coupling of two phases is very common.
• Explicit approaches furnish an alternative to the previous cases.

As of Radioss V5 both implicit and explicit methods are available to study the static behavior of systems. The choice a method depends on the nature of the problem and the engineer's feeling. The explicit approach is especially attractive for problems with highly nonlinear geometric and material behavior as all quantities may be treated as vectors, resulting in low storage requirements. The number of cycles to achieve convergence may be quite large, but global efficiency is generally observed. The implicit method is introduced to study efficiently static applications such as spring back in sheet metal forming or gravity loading or other initial state computations before/after dynamic simulations.