Simulation engine for index-1 DAEs and other mathematical utilities for Modia and Modia3D
Author ModiaSim
21 Stars
Updated Last
1 Year Ago
Started In
August 2018


ModiaMath is outdated and should not be used. The package is currently redesigned. For more details, see ModiaSim.

Travis AppVoyer Stable Latest The MIT License

ModiaMath provides a simulation engine and other mathematical utilities for packages Modia and Modia3D that are used to model physical systems such as electrical circuits, robots, or vehicles. The recommended way is to use ModiaMath via Modia or Modia3D. However, ModiaMath is self-contained and can be also used without Modia/Modia3D.

The central part of ModiaMath is a simulation engine to solve implicit index one Differential Algebraic Equations (DAEs) with and without time and state events. The theory is partially described in (Otter/Elmqvist, 2017). In particular it is shown, that a large class of DAEs can be transformed automatically to this form (including multibody systems with kinematic loops). As integrator currently IDA of the Sundials integrator suite is used (via the Julia Sundials interface package). It is planned to adapt ModiaMath to Julia package DifferentialEquations and use IDA and other appropriate integrators via this package in the future.

Additionally, ModiaMath provides functions to perform plotting in a convenient way, to generate and use rotation matrices and quaternions for kinematic transformations in 3D, and to provide an infrastructure for DAE variables as needed by Modia3D.


The package is registered in METADATA.jl and can be installed in the following way (Julia >= 1.0 is required):

julia> ]add ModiaMath

ModiaMath uses PyPlot for plotting. If PyPlot is not available in your current Julia environment an information message is printed and all ModiaMath.plot(..) calls are ignored. In order that plot windows are displayed, you need to add PyPlot to your current environment via ]add PyPlot. Often this automatic installation fails and it is recommended to follow instead the instructions Installing PyPlot in a robust way.


  • STABLEdocumentation of the last released version.
  • LATESTin-development version of the documentation.


To define a model

(note, it is simpler and less error prone to define a model with Modia or Modia3D):

  using ModiaMath
  using StaticArrays

  @component Pendulum(;L=1.0, m=1.0, d=0.1, g=9.81) begin
     phi = RealScalar(start=pi/2, unit="rad"    , fixed=true,               numericType=ModiaMath.XD_EXP)
     w   = RealScalar(start=0.0 , unit="rad/s"  , fixed=true, integral=phi, numericType=ModiaMath.XD_EXP)
     a   = RealScalar(            unit="rad/s^2",             integral=w  , numericType=ModiaMath.DER_XD_EXP)
     r   = RealSVector{2}(        unit="m"      ,                           numericType=ModiaMath.WC)

  function ModiaMath.computeVariables!(p::Pendulum, sim::ModiaMath.SimulationState)
     L = p.L; m = p.m; d = p.d; g = p.g; phi = p.phi.value; w = p.w.value

     p.a.value = (-m*g*L*sin(phi) - d*w) / (m*L^2)

     if ModiaMath.isStoreResult(sim)
        p.r.value = @SVector [L*sin(phi), -L*cos(phi)]

  simulationModel = ModiaMath.SimulationModel(Pendulum(L=0.8, m=0.5, d=0.2), stopTime=5.0);

To simulate a model and plot results:

  result = ModiaMath.simulate!(simulationModel; log=true);
  ModiaMath.plot(result, [(:phi, :w) :a])


To run examples and tests

  # run examples
  import ModiaMath
  include("$(ModiaMath.path)/examples/Simulate_Pendulum.jl")         # ODE as index-0 DAE
  include("$(ModiaMath.path)/examples/Simulate_FreeBodyRotation.jl") # index-1 DAE
  include("$(ModiaMath.path)/examples/withoutMacros_withoutVariables/Simulate_PendulumDAE.jl") # index-3 DAE
  include("$(ModiaMath.path)/examples/withoutMacros_withoutVariables/Simulate_IdealClutch.jl") # Dirac impulses

  # run all tests


The ModiaMath version number is 0.5.2, for details see the release notes, and functionality and robustness is planned to be improved for the 1.0 version, see Plans for ModiaMath version 1.0.

Issues and Contributions

Contributions are welcome, as are feature requests and suggestions. Please open an issue in this case and also if you encounter problems.

Main Developer

Martin Otter, DLR - Institute of System Dynamics and Control

License: MIT (expat)