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*E. B. Rudnyi.*

**Thermal and Mechanical System Simulation**.

ANSYS Conference & 27. CADFEM Users Meeting,
18 - 20 November 2009, Congress Center Leipzig.

Slides

Mechanical and civil engineers for long time have been using the Guyan and mode superposition methods in order to find a low-dimensional subspace. Our results presented in papers below show that implicit moment matching has advantages in many scenarios.

*E. B. Rudnyi, J. Lienemann, A. Greiner, and J. G. Korvink,*

mor4ansys: Generating Compact Models Directly from ANSYS Models.

Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show,
Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA, vol. 2, p.
279-282.

Final paper at NSTI.

Model reduction of linear large-scale dynamic systems is already quite an established area. In a number of papers, the advantages of model reduction have been demonstrated. In the present paper, we describe a software tool to perform moment-matchingmodel reduction via the Arnoldi algorithm directly to ANSYS finite element models. We discuss the application of the tool to a structural mechanical problem with a second order linear differential equation (ODE). Its successful application to the first order case of electro-thermal modeling is demonstrated elsewhere.

*J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, and J. G.
Korvink,*

MEMS Compact Modeling Meets
Model Order Reduction: Examples of the Application of Arnoldi Methods to
Microsystem Devices.

Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show,
Nanotech 2004, March 7-11, 2004, Bosten, Massachusetts, USA, vol. 2, p.
303-306.

Final paper at NSTI.

Modeling and simulation of the behavior of a system consisting of many single devices is an essential requirement for the reduction of design cycles in the development of microsystem applications. Analytic solutions for the describing partial differential equations of each component are only available for simple geometries. For complex geometries, either approximations or numerical methods can be used. However, the numerical treatment of the PDEs of thousands of interconnected single devices with each exhibiting a complex behavior is almost impossible without reduction of the order of unknowns to a lower-dimensional system. We present a fully automatic method to generate a compact model of second-order linear systems based on the Arnoldi process, and provide an example of successfull model order reduction to a gyroscope.

*J. Lienemann, A. Greiner, E. B. Rudnyi, J. G. Korvink, L. Ferrario, M.
Zen*,

Automatic Order Reduction
for Finite Element Models.

Books of abstract of the 9th National conference on Sensor and Microsystems,
AISEM 2004, Ferrara, Italy, February 8-11, p. 94.

In the process of physical modelling of microsystems operating on various energy domains, the engineer is used to apply Finite Element techniques for the discrete representation of the functionality of the device under investigation in a simulation environment. There are many commercial products that help the engineer in performing this task. The common feature of all these simulation tools is that the discrete representation consists of a system of ordinary differential equations. The dimension of this system is directly connected to the number of degrees of freedom for the respective problem. For a spatial displacement field, e.g., the degrees of freedom are three times the number of discretization nodes. The higher the requirements for precision of the simulation results, the more discretization nodes are usually introduced. Nevertheless, the results the engineer will use are in most cases of low dimensional order. In other words, the characteristic features of the required functionality of the device under developement are well represented in low dimensional subspace of the entire solution space of a very fine Finite Element model. Moreover, the requirement for system behaviour simulation makes it impossible to couple large-scale models for this task. Therefore there is a demanding need for reducing the dimension of the mathematical representation of sub-systems. An approach to this task will be presented in this work, together with the modeling of an RF-microswitch as an example.

*J. S. Han, E. B. Rudnyi, J. G. Korvink.*

**Efficient optimization of transient dynamic problems in MEMS devices
using model order reduction.**

Journal of Micromechanics and Microengineering 2005, v. 15, N 4, p. 822-832.

Paper at IOP.

One of the main obstacles to including transient dynamic effects into the performance functions of a structural optimization for microelectromechanical systems (MEMS) is the high computational cost of each time-dependent response simulation. This paper focuses on the application of model order reduction techniques to optimal design so as to reduce the transient analysis time for the optimization process. To do this, our open-source software mor4ansys performs model order reductions via the block Arnoldi algorithm directly to ANSYS finite element models. We adopt a micro accelerometer as an example to demonstrate the advantages of this approach. The harmonic and transient results of a reduced-order model of the accelerometer yield very good agreement with that from the original high-dimensional ANSYS model. The use of reduced-order models within the optimization iterations produces almost the same results as those without model order reduction, and speeds up the total computation by at least an order of magnitude.

*J. S. Han, E. B. Rudnyi, J. G. Korvink.*

Efficient Vibrational Simulation
of a Knuckle Using Model Order Reduction.

Proceedings of the Kyungbuk branch of the KSME 2005 Spring Annual Meeting,
pp. 18-22, April 16, 2005 (in Korean).

Currently most practical vibrational and structural problems in automotive suspensions require the use of the finite element method to obtain their structural responses. When the finite element model has a very large number of degrees of freedom, the harmonic and dynamic analyses are computationally too expensive to repeat within a feasible design process time. To alleviate the computational difficulty, this paper presents a moment-matching based model order reduction (MOR) which reduces the number of degrees of freedom of the original finite element model and speeds up the necessary simulations with the reduced-size models. The moment-matching model reduction via the Arnoldi process is performed directly to ANSYS finite element models by software mor4ansys. Among automotive suspension components, a knuckle is taken as an example to demonstrate the advantages of this approach for vibrational simulation.

*E. B. Rudnyi, B. van Rietbergen, J. G. Korvink.*

Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction.

12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", University of Ulm, 20-21 July 2005. Proceedings of the 12th FEM Workshop, p. 61-68. ISBN: 3-9806183-8-2.

Three-dimensional serial reconstruction techniques allow us to develop very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture. However, such models are of very high dimension and, at present, simulation is limited to a linear elastic analysis only. In the present paper, we suggest to use model reduction in order to enable harmonic simulation for micro-FE models. We take two bone models of dimensions 130 000 and 900 000 and report results for implicit moment matching based via the Arnoldi process. We demonstrate that for the fist model a low-dimensional subspace of dimension 10 allows us to accurately describe frequency response up to 190 Hz. For the second model, a low-dimensional subspace of dimension 25 is enough to accurately describe frequency response up to 30 Hz. We show that the time to perform model reduction and then to simulate the low-dimensional model is orders of magnitude less than that needed for harmonic simulation of the original model.

*J. S. Han.*

**Efficient Vibration Simulation Using Model Order Reduction.**

Transactions of the KSME, A, Vol. 30, No. 3, pp. 310-317, 2006

*J. S. Han.*

**Eigenvalue and Frequency Response Analyses of a Hard Disk Drive Actuator Using Reduced Finite Element Models.**

Transactions of the KSME, A, Vol. 31, No. 5, pp. 541-549, 2007.

Evgenii B. Rudnyi

Designed by

Masha Rudnaya