The Mechanical APDL solver uses an implicit solution scheme based on Newton-Raphson iterations to solve the nonlinear mechanical equilibrium equation in static and full transient analyses. Sometimes the solution does not converge within the convergence criterion and no solution is achieved. The semi-implicit method (SEMIIMPLICIT command) can alleviate this problem.
The following semi-implicit analysis topics are available:
The following assumptions and restrictions apply to the semi-implicit method:
This method is available only for static (ANTYPE,STATIC) and full transient (ANTYPE,TRANS) analysis types.
Since the semi-implicit method uses the central difference method to solve the transient dynamics for part of the solution, each node must have a mass associated with it. Therefore, each node must belong to at least one element that has material density defined.
The method is available only for structural degrees of freedom. It does not support any element with non-structural degrees of freedom.
Some element formulations are not supported. For a complete list, see Semi-Implicit Method Limitations in the Advanced Analysis Guide.
Nonlinear mesh adaptivity (NLADAPTIVE) and manual rezoning are not supported if the mesh changes occur before or during the transition to the semi-implicit solve phase, but these capabilities are supported if the mesh changes occur after the semi-implicit solution has transitioned back to the implicit solve phase.
When an implicit solution fails to converge, the semi-implicit method converts the implicit problem (static or transient dynamic) into a transient dynamic problem, and the semi-discrete equation of motion (in the most general form) is given as:
 | (15–265) |
where:
| [M] = structural mass matrix |
| [C] = damping matrix (only present if damping is defined in the initial run) |
 = nodal acceleration |
 = nodal velocity |
 = internal load vector |
 = applied load vector |
The semi-implicit method then solves the mechanical equilibrium using central difference time integration and, hence, does not require Newton-Raphson iterations (and ensuing convergence checks).
More specifically, the acceleration for substep n is solved
as:
 | (15–266) |
And the velocity at the midstep 
and the displacements 
for the next substep n+1 are updated
as:
 | (15–267) |
 | (15–268) |
 | (15–269) |
After solving the equations in the semi-implicit scheme for some amount of time (which is user-specified) and overcoming the hurdle that was causing difficulty in convergence, the method transitions back to the implicit time integration scheme for the remainder of the solution. In the semi-implicit method, the problem seamlessly transitions between pure implicit and semi-implicit time integration of the equilibrium equation. The semi-implicit method is a hybrid method with some features of traditional implicit finite element analysis, including the solution of a coupled system of equations to obtain nodal acceleration by matrix inversions, the ability to bisect when a solution is unstable, and the ability to support Lagrange multiplier based constraints and mixed u-P type solid elements. The method also has some features of traditional explicit finite element analysis, namely the use of central difference time integration—thus, the name semi-implicit method.
Central difference time integration is conditionally stable. The stability of the central difference scheme is determined by the Courant condition. For stability, the time increment Δt should be less than the time associated with the maximum frequency ωmax of the structure, which in turn can be approximated by the smallest time it takes for sound to traverse through an element; that is:
 | (15–270) |
where:
 = element characteristic length |
| c = speed of sound |
The speed of sound c is in turn given by the material properties, density (ρ) and elastic modulus (E). Thus, the stable time increment depends upon the element size and material density and moduli. For example, the stable time increment for a 1D element (such as LINK180) with linear elastic moduli is given by:
 | (15–271) |
where:
| me = element mass |
The stable time increment increases with increasing mass and increasing mesh size.
With increased mass, the inertia force increases. If the base problem was a quasi-static problem (ANTYPE,STATIC), this is especially undesirable as it is assumed that the inertia affects are very small for the physical problem. However, to be able to solve quasi-static problems using central difference time integration, the stable time increment needs to be reasonably large. This problem is addressed by a selective mass scaling technique (Olovsson et al. [437]). The selective mass scaling (SMS) increases the mass such that the stable time increment increases while minimizing the effect of scaled mass on the inertia force, thus keeping the problem closer to quasi-static analysis as intended.
Bulk viscosity is applied by default during the semi-implicit solve phase for all elements that contribute to the mass matrix, except gasket elements. The force due to bulk viscosity consists of three parts. A linear term (Landshoff [438]) is introduced to eliminate the unphysical oscillations in the element:
 | (15–272) |
where:
 = trace of strain rate tensor |
 = linear damping ratio (defaults to 1.5) |
A quadratic term is used to prevent shock zone collapse under an extremely high velocity gradient (Von Neumman and Richtmyer [439]):
 | (15–273) |
where:
 = quadratic damping ratio (defaults to 0.06) |
For shell elements, bulk viscosity force proportional to the rotational DOF is also introduced:
 | (15–274) |
where:
| h = shell thickness |
 = trace of curvature rate tensor |
 = rotational damping ratio (defaults to 0.06) |
An upper limit of bulk viscosity is set such that the force due to bulk viscosity always stays within a percentage of the element internal force.