Implicit vs. Explicit Solutions

Finite Element Analysis (FEA) involving simulating short-time large deformation dynamics, quasi-static problems with large deformations and multiple nonlinearities, or complex contact/impact problems requires the use of either implicit or explicit solution techniques. Examples of these types of simulations are crashworthiness analysis, drop testing, deep drawing, rolling, extruding, pipe whip, bird strike, fan containment and many more.

The ANSYS program with the addition of the LS-DYNA program includes the ability to address both implicit as well as explicit solutions. Significantly, at Revision 5.4 the ANSYS program added the ability to perform implicit-explicit or explicit-implicit sequential solution simulations. A major step forward in the state-of-the-art in predicting results of many complex processes in a highly efficient way. An example of an explicit-implicit analysis would be sheetmetal forming and springback that follows. An example of an implicit-explicit analysis would be drop testing of a preloaded or packaged consumer product.

Let’s look at the differences between the two techniques in a condensed summary form. Both processes involve a numerical time integration scheme to solve for the unknown displacement solution, which is the basis for calculating resulting strains and stresses. Implicit integration schemes (ANSYS uses a Newmark Forward Differencing method) assume a constant average acceleration over each time step, between tn and tn+1. The value tn is time at the beginning of each time step and the value tn+1 is time at the end of each time step. The governing equation is evaluated and the resulting accelerations and velocities at tn+1 are calculated. Then the unknown displacements at tn+1 are determined. Explicit integration schemes (LS-DYNA uses a Central Differance method) assume a linear change in displacement over each time step. The governing equation is evaluated and the resulting accelerations and velocities at tn are calculated. Then the unknown displacements at tn+1 are determined.

There is one major difference between the two techniques in the equations that are used to solve for displacements at tn+1. The implicit solution method requires matrix inversion of the structural stiffness matrix, the explicit solution does not. However, unlike the implicit solution scheme, which is unconditionally stable for large time steps, the explicit scheme is stable only if the time step size is smaller than the critical time step size for the structure being simulated. The undamped critical time step size is 2/wn (where wn is the largest natural circular frequency), which is usually a very small value. This very small time step size requirement for stability thereby makes explicit solutions useful only for very short transients. But, even though the number of time steps in an explicit solution may be orders of magnitude greater than that of an implicit solution, it is significantly more efficient than an implicit solution since no matrix inversion is required. Neither an implicit nor explicit solution is the clear winner in all cases.

ANSYS users can have the best of both worlds in an integrated program that allows them to transfer geometry and results to perform implicit-explicit or explicit-implicit sequential analysis. Combined with the power of the ANSYS GUI (Graphical User Interface) to postprocess their sequential analysis results, they have the tools to solve some of the FEA world's most demanding simulations. Give us a call if you’re interested in learning more about ANSYS/LS-DYNA.
 

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