Solving Modelica Models

Integration Methods

By default OpenModelica transforms a Modelica model into an ODE representation to perform a simulation by using numerical integration methods. This section contains additional information about the different integration methods in OpenModelica. They can be selected by the method parameter of the simulate command or the -s simflag.

The different methods are also called solver and can be distinguished by their characteristic:

  • explicit vs. implicit

  • order

  • step size control

  • multi step

A good introduction on this topic may be found in [CK06] and a more mathematical approach can be found in [HNorsettW93].

DASSL

DASSL is the default solver in OpenModelica, because of a severals reasons. It is an implicit, higher order, multi-step solver with a step-size control and with these properties it is quite stable for a wide range of models. Furthermore it has a mature source code, which was originally developed in the eighties an initial description may be found in [Pet82].

This solver is based on backward differentiation formula (BDF), which is a family of implicit methods for numerical integration. The used implementation is called DASPK2.0 (see 1) and it is translated automatically to C by f2c (see 2).

The following simulation flags can be used to adjust the behavior of the solver for specific simulation problems: jacobian, noRootFinding, noRestart, initialStepSize, maxStepSize, maxIntegrationOrder, noEquidistantTimeGrid.

IDA

The IDA solver is part of a software family called sundials: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [HBG+05]. The implementation is based on DASPK with an extended linear solver interface, which includes an interface to the high performance sparse linear solver KLU [DN10].

The simulation flags of DASSL are also valid for the IDA solver and furthermore it has the following IDA specific flags: idaLS, idaMaxNonLinIters, idaMaxConvFails, idaNonLinConvCoef, idaMaxErrorTestFails.

CVODE

The CVODE solver is part of sundials: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [HBG+05]. CVODE solves initial value problems for ordinary differential equation (ODE) systems with variable-order, variable-step multistep methods.

In OpenModelica, CVODE uses a combination of Backward Differentiation Formulas (varying order 1 to 5) as linear multi-step method and a modified Newton iteration with fixed Jacobian as non-linear solver per default. This setting is advised for stiff problems which are very common for Modelica models. For non-stiff problems an combination of an Adams-Moulton formula (varying order 1 to 12) as linear multi-step method together with a fixed-point iteration as non-linear solver method can be choosen.

Both non-linear solver methods are internal functions of CVODE and use its internal direct dense linear solver CVDense. For the Jacobian of the ODE CVODE will use its internal dense difference quotient approximation.

CVODE has the following solver specific flags: cvodeNonlinearSolverIteration, cvodeLinearMultistepMethod.

Basic Explicit Solvers

The basic explicit solvers are performing with a fixed step-size and differ only in the integration order. The step-size is based on the numberOfIntervals, the startTime and stopTime parameters in the simulate command: \mbox{stepSize} \approx \cfrac{\mbox{stopTime} - \mbox{startTime}}{\mbox{numberOfIntervals}}

  • euler - order 1

  • heun - order 2

  • rungekutta - order 4

Basic Implicit Solvers

The basic implicit solvers are all based on the non-linear solver KINSOL from the SUNDIALS suite. The underlining linear solver can be modified with the simflag -impRKLS. The step-size is determined as for the basic explicit solvers.

  • impeuler - order 1

  • trapezoid - order 2

  • imprungekutta - Based on Radau IIA and Lobatto IIIA defined by its Butcher tableau where the order can be adjusted by -impRKorder.

Experimental Solvers

The following solvers are marked as experimental, mostly because they are till now not tested very well.

  • cvode - experimental implementation of SUNDIALS CVODE solver - BDF or Adams-Moulton method - step size control, order 1-12

  • rungekuttaSsc - Runge-Kutta based on Novikov (2016) - explicit, step-size control, order 4-5

  • irksco - Own developed Runge-Kutta solver - implicit, step-size control, order 1-2

  • symSolver - Symbolic inline solver (requires --symSolver) - fixed step-size, order 1

  • symSolverSsc - Symbolic implicit inline Euler with step-size control (requires --symSolver) - step-size control, order 1-2

  • qss - A QSS solver

DAE Mode Simulation

Beside the default ODE simulation, OpenModelica is able to simulate models in DAE mode. The DAE mode is enabled by the flag --daeMode. In general the whole equation system of a model is passed to the DAE integrator, including all algebraic loops. This reduces the amount of work that needs to be done in the post optimization phase of the OpenModelica backend. Thus models with large algebraic loops might compile faster in DAE mode.

Once a model is compiled in DAE mode the simulation can be only performed with SUNDIALS/IDA integrator and with enabled -daeMode simulation flag. Both are enabled automatically by default, when a simulation run is started.

Initialization

To simulate an ODE representation of an Modelica model with one of the methods shown in Integration Methods a valid initial state is needed. Equations from an initial equation or initial algorithm block define a desired initial system.

Choosing start values

Only non-linear iteration variables in non-linear strong components require start values. All other start values will have no influence on convergence of the initial system.

Use -d=initialization to show additional information from the initialization process. In OMEdit Tools->Options->Simulation->OMCFlags, in OMNotebook call setCommandLineOptions("-d=initialization")

_images/piston.png

Figure 30 piston.mo

model piston
  Modelica.Mechanics.MultiBody.Parts.Fixed fixed1 annotation(
    Placement(visible = true, transformation(origin = {-80, 70}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  Modelica.Mechanics.MultiBody.Parts.Body body1(m = 1)  annotation(
    Placement(visible = true, transformation(origin = {30, 70}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  Modelica.Mechanics.MultiBody.Parts.FixedTranslation fixedTranslation1(r = {0.3, 0, 0})  annotation(
    Placement(visible = true, transformation(origin = {-10, 70}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  Modelica.Mechanics.MultiBody.Parts.FixedTranslation fixedTranslation2(r = {0.8, 0, 0})  annotation(
    Placement(visible = true, transformation(origin = {10, 20}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
  Modelica.Mechanics.MultiBody.Parts.Fixed fixed2(animation = false, r = {1.1, 0, 0})  annotation(
    Placement(visible = true, transformation(origin = {70, -60}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
  Modelica.Mechanics.MultiBody.Parts.Body body2(m = 1)  annotation(
    Placement(visible = true, transformation(origin = {30, -30}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  inner Modelica.Mechanics.MultiBody.World world annotation(
    Placement(visible = true, transformation(origin = {-70, -50}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  Modelica.Mechanics.MultiBody.Joints.Prismatic prismatic(animation = true)  annotation(
    Placement(visible = true, transformation(origin = {30, -60}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  Modelica.Mechanics.MultiBody.Joints.RevolutePlanarLoopConstraint revolutePlanar annotation(
    Placement(visible = true, transformation(origin = {-50, 70}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
  Modelica.Mechanics.MultiBody.Joints.Revolute revolute1(a(fixed = false),phi(fixed = false), w(fixed = false))  annotation(
    Placement(visible = true, transformation(origin = {10, 48}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
  Modelica.Mechanics.MultiBody.Joints.Revolute revolute2 annotation(
    Placement(visible = true, transformation(origin = {10, -10}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
equation
  connect(prismatic.frame_b, fixed2.frame_b) annotation(
    Line(points = {{40, -60}, {60, -60}, {60, -60}, {60, -60}}, color = {95, 95, 95}));
  connect(fixed1.frame_b, revolutePlanar.frame_a) annotation(
    Line(points = {{-70, 70}, {-60, 70}, {-60, 70}, {-60, 70}}));
  connect(revolutePlanar.frame_b, fixedTranslation1.frame_a) annotation(
    Line(points = {{-40, 70}, {-20, 70}, {-20, 70}, {-20, 70}}, color = {95, 95, 95}));
  connect(fixedTranslation1.frame_b, revolute1.frame_a) annotation(
    Line(points = {{0, 70}, {10, 70}, {10, 58}, {10, 58}}, color = {95, 95, 95}));
  connect(revolute1.frame_b, fixedTranslation2.frame_a) annotation(
    Line(points = {{10, 38}, {10, 38}, {10, 30}, {10, 30}}, color = {95, 95, 95}));
  connect(revolute2.frame_b, prismatic.frame_a) annotation(
    Line(points = {{10, -20}, {10, -20}, {10, -60}, {20, -60}, {20, -60}}));
  connect(revolute2.frame_b, body2.frame_a) annotation(
    Line(points = {{10, -20}, {10, -20}, {10, -30}, {20, -30}, {20, -30}}, color = {95, 95, 95}));
  connect(revolute2.frame_a, fixedTranslation2.frame_b) annotation(
    Line(points = {{10, 0}, {10, 0}, {10, 10}, {10, 10}}, color = {95, 95, 95}));
  connect(fixedTranslation1.frame_b, body1.frame_a) annotation(
    Line(points = {{0, 70}, {18, 70}, {18, 70}, {20, 70}}));
end piston;
>>> loadModel(Modelica);
>>> setCommandLineOptions("-d=initialization");
>>> buildModel(piston);
"[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body2.r_CM[3] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body2.r_CM[2] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body2.r_CM[1] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body1.r_CM[3] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body1.r_CM[2] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body1.r_CM[1] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
Warning: Assuming fixed start value for the following 2 variables:
         $STATESET1.x:VARIABLE(start = /*Real*/($STATESET1.A[1]) * $START.revolute1.w + /*Real*/($STATESET1.A[2]) * $START.revolute2.w fixed = true )  type: Real
         $STATESET2.x:VARIABLE(start = /*Real*/($STATESET2.A[1]) * $START.revolute1.phi + /*Real*/($STATESET2.A[2]) * $START.revolute2.phi fixed = true )  type: Real
"

Note how OpenModelica will inform the user about relevant and irrelevant start values for this model and for which variables a fixed default start value is assumed. The model has four joints but only one degree of freedom, so one of the joints revolutePlanar or prismatic must be initialized.

So, initializing phi and w of revolutePlanar will give a sensible start system.

model pistonInitialize
  extends piston(revolute1.phi.fixed = true, revolute1.phi.start = -1.221730476396031, revolute1.w.fixed = true, revolute1.w.start = 5);
equation
end pistonInitialize;
>>> setCommandLineOptions("-d=initialization");
>>> simulate(pistonInitialize, stopTime=2.0);
"[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body2.r_CM[3] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body2.r_CM[2] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body2.r_CM[1] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body1.r_CM[3] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body1.r_CM[2] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
[/var/lib/jenkins/ws/OpenModelica_master_2/build/lib/omlibrary/Modelica 4.0.0+maint.om/Mechanics/MultiBody/Parts/Body.mo:14:3-15:65:writable] Warning: Parameter body1.r_CM[1] has no value, and is fixed during initialization (fixed=true), using available start value (start=0.0) as default value.
"
_images/piston.svg

Figure 31 Vertical movement of mass body2.

Homotopy Method

For complex start conditions OpenModelica can have trouble finding a solution for the initialization problem with the default Newton method.

Modelica offers the homotopy operator 3 to formulate actual and simplified expression for equations, with homotopy parameter \lambda going from 0 to 1:

actual \cdot \lambda + simplified \cdot (1-\lambda).

OpenModelica has different solvers available for non-linear systems. Initializing with homotopy on the first try is default if a homotopy operator is used. It can be switched off with noHomotopyOnFirstTry. For a general overview see [SCO+11], for details on the implementation in OpenModelica see [OB13].

The homotopy methods distinguish between local and global methods meaning, if \lambda affects the entire initialization system or only local strong connected components. In addition the homotopy methods can use equidistant \lambda or and adaptive \lambda in [0,1].

Default order of methods tried to solve initialization system

If there is no homotopy in the model
  • Solve without homotopy method.

If there is homotopy in the model or solving without homotopy failed
  • Try global homotopy approach with equidistant \lambda.

The default homotopy method will do three global equidistant steps from 0 to 1 to solve the initialization system.

Several compiler and simulation flags influence initialization with homotopy: --homotopyApproach, -homAdaptBend, -homBacktraceStrategy, -homHEps, -homMaxLambdaSteps, -homMaxNewtonSteps, -homMaxTries, -homNegStartDir, -homotopyOnFirstTry, -homTauDecFac, -homTauDecFacPredictor, -homTauIncFac, -homTauIncThreshold, -homTauMax, -homTauMin, -homTauStart, -ils.

Algebraic Solvers

If the ODE system contains equations that need to be solved together, so called algebraic loops, OpenModelica can use a variety of different linear and non-linear methods to solve the equation system during simulation.

For the C runtime the linear solver can be set with -ls and the non-linear solver with -nls. There are dense and sparse solver available.

Linear solvers
  • default : Lapack with totalpivot as fallback [ABB+99]

  • lapack : Non-Sparse LU factorization using [ABB+99]

  • lis : Iterative linear solver [Nis10]

  • klu : Sparse LU factorization [Nat05]

  • umfpack : Sparse unsymmetric multifrontal LU factorization [Dav04]

  • totalpivot : Total pivoting LU factorization for underdetermined systems

Non-linear solvers
  • hybrid : Modified Powell hybrid method from MINPACK [DJS96]

  • kinsol : Combination of Newton-Krylov, Picard and fixed-point solver [T+98]

  • newton : Newton-Raphson method [CK06]

  • mixed : Homotopy with hybrid as fallback [Kel78] [BBOR15]

  • homotopy : Damped Newton solver with fixed-point solver and Newton homotopy solver as fallbacks

In addition, there are further optional settings for the algebraic solvers available. A few of them are listed in the following:

General: -nlsLS

Newton: -newton -newtonFTol -newtonMaxStepFactor -newtonXTol

Sparse solver: -nlssMinSize -nlssMaxDensity

Enable logging: -lv=LOG_LS -lv=LOG_LS_V -lv=LOG_NLS -lv=LOG_NLS_V

References

ABB+99(1,2)

Edward Anderson, Zhaojun Bai, Christian Bischof, L Susan Blackford, James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, Sven Hammarling, Alan McKenney, and others. LAPACK Users' guide. SIAM, 1999.

BBOR15

Bernhard Bachmann, W Braun, L Ochel, and V Ruge. Symbolical and numerical approaches for solving nonlinear systems. In Annual OpenModelica Workshop, volume 2015. 2015.

CK06(1,2)

Francois E. Cellier and Ernesto Kofman. Continuous System Simulation. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006. ISBN 0387261028.

DN10

T. A. Davis and E. Palamadai Natarajan. Algorithm 907: klu, a direct sparse solver for circuit simulation problems. ACM Trans. Math. Softw., 37(3):36:1–36:17, September 2010. URL: http://doi.acm.org/10.1145/1824801.1824814, doi:10.1145/1824801.1824814.

Dav04

Timothy A Davis. Algorithm 832: umfpack v4. 3—an unsymmetric-pattern multifrontal method. ACM Transactions on Mathematical Software (TOMS), 30(2):196–199, 2004.

DJS96

John E Dennis Jr and Robert B Schnabel. Numerical methods for unconstrained optimization and nonlinear equations. SIAM, 1996.

HNorsettW93

E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag Berlin Heidelberg, 2nd rev. ed. 1993. corr. 3rd printing 2008 edition, 1993. ISBN 978-3-540-56670-0. doi:10.1007/978-3-540-78862-1.

HBG+05(1,2)

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward. SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Transactions on Mathematical Software (TOMS), 31(3):363–396, 2005.

Kel78

Herbert B Keller. Global homotopies and newton methods. In Recent advances in numerical analysis, pages 73–94. Elsevier, 1978.

Nat05

Ekanathan Palamadai Natarajan. KLU–A high performance sparse linear solver for circuit simulation problems. PhD thesis, University of Florida, 2005.

Nis10

Akira Nishida. Experience in developing an open source scalable software infrastructure in japan. In International Conference on Computational Science and Its Applications, 448–462. Springer, 2010.

OB13

Lennart A Ochel and Bernhard Bachmann. Initialization of equation-based hybrid models within openmodelica. In Proceedings of the 5th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools; April 19; University of Nottingham; Nottingham; UK, number 084, 97–103. Linköping University Electronic Press, 2013.

Pet82

L.R. Petzold. Description of dassl: a differential/algebraic system solver. 1982.

SCO+11

Michael Sielemann, Francesco Casella, Martin Otter, Christop Clauß, Jonas Eborn, Sven Erik Matsson, and Hans Olsson. Robust initialization of differential-algebraic equations using homotopy. In Proceedings of the 8th International Modelica Conference; March 20th-22nd; Technical Univeristy; Dresden; Germany, number 063, 75–85. Linköping University Electronic Press, 2011.

T+98

Allan G Taylor and others. User documentation for kinsol, a nonlinear solver for sequential and parallel computers. Technical Report, Lawrence Livermore National Lab., CA (United States), 1998.

Footnotes

1

DASPK Webpage

2

Cdaskr source

3

Modelica Association, Modelica® - A Unified Object-Oriented Language for Systems Modeling Language Specification - Version 3.4, 2017 - Section 3.7.2.4