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Project
Nonlinear and Parametric Model Order Reduction for Second Order Dynamical Systems by the Dominant Pole Algorithm
Noise and vibration performance is a key parameter for assessing the quality of automotive and
aerospace products.
In order to gain competitive advantage, manufacturers are continually striving to reduce noise and
vibrations levels.
The numerical analysis of the acousticbehaviour results in huge mathematical models, in particular,
for higher frequency analyses.
This leads to high requirements regarding computational and storage resources.
Furthermore, the cost increases dramatically when the model has a large number of design variables
that have to be taken into account for the development of the optimal design.
This motivates the importance of reducing the size of the models in order to reduce the simulation cost.
One choice for building such reduced models is algebraic
Model Order Reduction (MOR) for linear dynamical systems. The aim of MOR is to reduce the system matrix
in such a way that the reduced system has similar input/output behaviour as the original system.
The goal of this thesis is to use the Dominant Pole algorithm (DPA) for computing a truncated modal
representation of a large scale parametric linear second order dynamical system and also large
scale dynamical systems whose matrix has non-linear frequency dependency.
First, we adapted the DPA for reducing systems that have an infinite number of poles.
Deflation is an important ingredient for this type of methods in order to prevent eigenvalues to be computed more than once.
Because of the nonlinearity frequency dependency, classical deflation approaches are
not applicable. Therefore we propose an alternative technique that essentially
removes computed poles from the systems input and output vectors. This method appears to be reliable
for computing a large number of dominant poles of the system.
Next, we apply the DPA to parametric second order
dynamical system,whose system matrix depends on parameters.
We will iteratively compute the parametric dominant poles. We consider two
approaches. In the first approach, we compute the parameter dependent poles
one by one, i.e., all parameters are taken into account together. We will use
interpolation in the parameter space to achieve this. In the second approach,
the dominant eigenpairs are computed for a selection of interpolation points
in the parameter space, independently from each other. As the eigenvectors
are continuous functions of the parameters, we use the already computed
eigenvectors from previous parameter values for computing starting values of
the DPA.
aerospace products.
In order to gain competitive advantage, manufacturers are continually striving to reduce noise and
vibrations levels.
The numerical analysis of the acousticbehaviour results in huge mathematical models, in particular,
for higher frequency analyses.
This leads to high requirements regarding computational and storage resources.
Furthermore, the cost increases dramatically when the model has a large number of design variables
that have to be taken into account for the development of the optimal design.
This motivates the importance of reducing the size of the models in order to reduce the simulation cost.
One choice for building such reduced models is algebraic
Model Order Reduction (MOR) for linear dynamical systems. The aim of MOR is to reduce the system matrix
in such a way that the reduced system has similar input/output behaviour as the original system.
The goal of this thesis is to use the Dominant Pole algorithm (DPA) for computing a truncated modal
representation of a large scale parametric linear second order dynamical system and also large
scale dynamical systems whose matrix has non-linear frequency dependency.
First, we adapted the DPA for reducing systems that have an infinite number of poles.
Deflation is an important ingredient for this type of methods in order to prevent eigenvalues to be computed more than once.
Because of the nonlinearity frequency dependency, classical deflation approaches are
not applicable. Therefore we propose an alternative technique that essentially
removes computed poles from the systems input and output vectors. This method appears to be reliable
for computing a large number of dominant poles of the system.
Next, we apply the DPA to parametric second order
dynamical system,whose system matrix depends on parameters.
We will iteratively compute the parametric dominant poles. We consider two
approaches. In the first approach, we compute the parameter dependent poles
one by one, i.e., all parameters are taken into account together. We will use
interpolation in the parameter space to achieve this. In the second approach,
the dominant eigenpairs are computed for a selection of interpolation points
in the parameter space, independently from each other. As the eigenvectors
are continuous functions of the parameters, we use the already computed
eigenvectors from previous parameter values for computing starting values of
the DPA.
Date:24 Nov 2008 → 13 Sep 2013
Keywords:Model Reduction Techniques
Disciplines:Analysis, Applied mathematics in specific fields, General mathematics, History and foundations, Other mathematical sciences and statistics
Project type:PhD project