The standard integrator

One very common problem that arises in simulation of engineering systems is that of numerical stiffness numerical stiffness . A stiffness problem stiff problem is one in which there is a time constant which is extremely small compared to the simulation range i.e. Final time - Start time. The famous Gear’s method was specifically developed to solve such problems. Most other integration algorithms are impossibly slow when applied to such problems.

Another common problem is discontinuities. These are points at which there is a switch from a set of one or more governing equation(s) to another completely different set. One extreme example occurs in the modeling of a mass where there are physical limitations on movement. These are often referred to as endstops. Simcenter Amesim Run uses two techniques to model endstops:

1. they are modeled as elastic with a spring force and a damping force, or,

2. they are modeled as inelastic so that the mass collides with the endstop coming instantaneously to rest.

The second alternative is often described as an ideal endstop. With an ideal endstop the mass reaches an extreme position and it is modeled as coming instantaneously to rest – a true discontinuity. The mass is then fixed in the extreme position until a force of the correct sign and of sufficient magnitude to overcome static friction moves it away from the extreme position. A more detailed description of problems arising from discontinuities is given in the Submodel Editor manual to which the interested reader should refer.

Gear’s method is very intolerant of discontinuities unless special code is inserted to deal with them. Unfortunately the equations defining the model of many engineering systems are stiff and contain discontinuities. If discontinuity-handling code is not provided or is provided but not used, Gear’s method may fail on these problems.

Simple Runge-Kutta algorithms are relatively tolerant of discontinuities and can perform well on some problems but they are very unsuitable for stiff problems. However, many simulations are performed on stiff problems rich in discontinuities using these methods. Often a solution can be obtained, but the run times can be hours or days. The answer here is to use a Gear integrator with good discontinuity handling. The saving in computing times can be spectacular.

The standard Simcenter Amesim Run integrator does not give the user a choice of integration algorithm. Instead the characteristics of the equations governing the model are used to select automatically the most appropriate algorithm. If the model contains any implicit variables the differential algebraic equation integration algorithm DASSL DASSL is used otherwise the ordinary differential equation integration algorithm LSODA LSODA is used.

LSODA

LSODA uses a collection of non-stiff integration (Adams-Moulton multistep) methods and the same collection of stiff integration (backward differentiation formulae multistep) methods as are employed in Gear’s method. LSODA monitors the characteristics of the governing equations and switches between the non-stiff and stiff integrators as is necessary. By this means LSODA is generally a very efficient solver irrespective of the characteristics of the model ordinary differential equations.

DASSL

DASSL is probably the best differential algebraic equation integration algorithm currently available and is certainly the only one that is widely used. It uses a collection of integrators of the same type as those employed in Gear’s method. Differential algebraic equations often behave like ordinary differential equations with a time constant that tends to zero so this is hardly surprising. Non-stiff integrators do not work well with differential algebraic equations.

Implicit systems are usually a set of constraint equations and/or implicit equations. Solving this kind of system requires the use of specific Newton-base iterative methods. These methods need to solve a linear system to evaluate the amount by which an iterative variable will be changed to converge towards the solution at a given time.

To solve this kind of system the user can choose two types of method:

• The LU algorithm which is based on the well-known Gaussian elimination method, which will solve the given system in a direct way.

• Krylov

A Krylov method, which is based on minimizing residuals on Krylov subspaces, and then iterating over subspaces towards the real solution of the initial system.

The default choice for the implicit integrator is the direct method, which is the most commonly-used method for solving this kind of system. An iterative matrix is computed and factored, then used in as many steps as possible.

Nevertheless, it is possible that integration of nontrivial Differential Algebraic Equations (DAE) will require the use of a preconditioner (the iteration matrix becomes more and more badly-conditioned as the step-size reduced). Then a preconditioner matrix (an approximation of the initial iteration matrix) is computed and used in as many steps as possible. This last feature is only available with the iterative method, and is appropriate for large system integration.

These two methods differ only in the way the linear systems arising in integration are solved, as initialization and time stepping algorithms remain the same.

Note

If preconditioning will generally make the integration more robust, it will also prove to be less efficient and more time-consuming, requiring many more operations than direct evaluations.

If some DAE integrations require the use of the iterative method, you can enable this feature by putting the SIMP00 submodel on the sketch and specify the method in Parameter mode.

Figure 243: SIMP00 Parameters

DASSL and LSODA as implemented in Simcenter Amesim Run are substantially different from the original algorithms.