## A new approach for numerical simulation of fluid power circuits using Rosenbrock methods

##### Esque, S. (2008)

Esque, S.

Tampere University of Technology

2008

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**Julkaisun pysyvä osoite on**

http://urn.fi/URN:NBN:fi:tty-200812291140

##### Tiivistelmä

The mathematical formulation of the dynamics observed in fluid power systems in-volves the numerical solution of differential equations. Because of the intrinsic characteris-tics and physics of fluid power circuits, the numerical integrators employed to solve such system of equations must retain certain properties in order to guarantee the accuracy, stability and efficiency of the numerical solution. In this thesis, different classes of numerical integration methods used for stiff systems have been analyzed and tested in order to quantitatively and qualitatively assess their performance against the numerical stiffness, high non-linearities and discontinuities typically shown in the differential equations arisen in fluid power circuits.

Numerical integration methods of the Rosenbrock class although rarely employed in the simulation of fluid power circuits have shown excellent numerical stability proper-ties and also above-the-average efficiency (solution accuracy to number of integration steps ratio) when compared to other popular single and multiple-step integration formulas. At the same time, the formulation of Rosenbrock methods involves a reduced number of linear algebra operations, which makes them computationally inexpensive. The main drawback of employing a Rosenbrock formula is the fact that an accurate Jacobian evaluation of the ODE system needs to be provided at each integration step in order to maintain the accuracy and stability of the formula. In order to solve this disadvantage, a method is presented in this thesis for the systematic modelling of fluid power components and systems as ODEs, following an object-oriented and modular approach. By following this methodology, the analytical form of the Jacobian matrix can be automatically generated and fed to the integration formula for any given fluid power system. This has the advantage that the Jacobian evaluation is done with a fraction of computational cost and also more accurately than a Jacobian obtained with numerical techniques.

The tests conducted in this thesis have confirmed that Rosenbrock formulas are good candidates for being used in real-time simulations (fixed integration step size) and in offline simulations (variable integration step size) of fluid power circuits. Their easy implementation, good stability, high efficiency and low computational costs make them, in most of the cases tested, superior to other popular codes.

Numerical integration methods of the Rosenbrock class although rarely employed in the simulation of fluid power circuits have shown excellent numerical stability proper-ties and also above-the-average efficiency (solution accuracy to number of integration steps ratio) when compared to other popular single and multiple-step integration formulas. At the same time, the formulation of Rosenbrock methods involves a reduced number of linear algebra operations, which makes them computationally inexpensive. The main drawback of employing a Rosenbrock formula is the fact that an accurate Jacobian evaluation of the ODE system needs to be provided at each integration step in order to maintain the accuracy and stability of the formula. In order to solve this disadvantage, a method is presented in this thesis for the systematic modelling of fluid power components and systems as ODEs, following an object-oriented and modular approach. By following this methodology, the analytical form of the Jacobian matrix can be automatically generated and fed to the integration formula for any given fluid power system. This has the advantage that the Jacobian evaluation is done with a fraction of computational cost and also more accurately than a Jacobian obtained with numerical techniques.

The tests conducted in this thesis have confirmed that Rosenbrock formulas are good candidates for being used in real-time simulations (fixed integration step size) and in offline simulations (variable integration step size) of fluid power circuits. Their easy implementation, good stability, high efficiency and low computational costs make them, in most of the cases tested, superior to other popular codes.

##### Kokoelmat

- Väitöskirjat [3921]