## Group Oberwolfach

Group Description |
Oberwolfach model reduction benchmark collection ================================================================================ ==> bone010/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Trabecular Bone Micro-Finite Element Models (bone* problems) -------------------------------------------------------------------------------- B. van Rietbergen, E. B. Rudnyi, J. G. Korvink 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. Fig. 1 sketches the micro finite element analysis [1]. Micro computed tomography (CT) is employed to make 3D high-resolution images (~50 microns) of a bone. Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone voxels to equally sized 8-node brick elements. This results in finite element (FE) models with a very large number of elements. Such models can be used, for example to study differences in bone tissue loading between healthy and osteoporotic human bones during quasi static loading [2]. Fig. 1: see http://www.imtek.de/simulation/benchmark/ Fig. 1. Micro finite element analysis. There is increasing evidence, however, that bone responds in particular to dynamic loads [3]. It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic. In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading. This would be possible with the micro-FE analysis, but then such an analysis need to be a dynamic one. The present benchmark presents 6 bone models varying in dimension from about 200 thousand to 12 millions equations with the goal to research on scalability of model reduction software. Each model represent a second order system in the form M d2x/dt2 + K x = B y = Cx where the matrices M and K are symmetric and positive definite. The goal of model reduction is to speed up harmonic response analysis in the frequency range 1-100 Hz. The matrix properties are given in Table 1 below. Table 1. Bone micro-finite element models. BS01 BS10 B010 B025 B050 B120 number of elements 20098 192539 278259 606253 1378782 3387547 number of nodes 42508 305066 329001 719987 1644848 3989996 number of DoFs 127224 914898 986703 2159661 4934244 11969688 nnz in half M 1182804 9702186 12437739 27150810 61866069 151251738 nnz in half K 3421188 28191660 36326514 79292769 180663963 441785526 Fig. 2 and 3 : see http://www.imtek.de/simulation/benchmark/ Fig. 2. Bone models for BS01 and BS10. It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four. The connectivity in the last four models is about four times higher. This can be seen by comparing models BS10 and B010. Although models look similar by number of nonzeros in the system matrices, the model B010 is much harder to solve: the number of nonzero elements in the factor for model B010 is about four times more than for BS10. The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements. As a result, a file describing the node indices for each element is enough to assemble the global matrix. Each node has three degrees of freedom (UX, UY, UZ) and it contributes three consecutive entries to the state vector. The node numbering is natural from the first to the last. The assembly procedure as a pseudo-code is presented below (indices start from one). It is assumed that the last 300 degrees of freedom are fixed as zero Dirichlet boundary conditions. For simplicity, the pseudo-code does not take into account that the matrix is symmetric. The data file for each model contains the number of elements, nel, and the number of nodes, nnod, in the first line and then nel lines with eight numbers for node indices in each line. The input matrix contains a single column with B(1) = 1. The output matrix takes first three components of the state vector, that is, three displacements UX, UY and UZ for the first node. The archive assemble.tar.gz contains the element mass and stiffness matrices as well as the sample code in C++ to assemble the dynamic system. The code can write the dynamic system in the Matrix Market format or can be used as a hook to transform the global matrices to an appropriate format. The gzipped data files for element assembly as described above can be downloaded from Table 1. Model reduction for models BS010 and BS10 was performed in [4]. The benchmarking of the parallel MUMPS direct solver [5] for the stiffness matrices is described in [6]. References [1] B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models. J. Biomechanics, v. 28, N 1, p. 69-81, 1995. [2] B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur, J. Bone Mineral Research, v. 18, N 10, p. 1781-1787, 2003. [3] L. E. Lanyon, C. T. Rubin, Static versus dynamic loads as an influence on bone remodelling, Journal of Biomechanics, v. 17, p. 897-906, 1984. [4] 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. [5] P. R. Amestoy and A. Guermouche and J.-Y. L'Excellent, S. Pralet, Hybrid scheduling for the parallel solution of linear systems. Parallel Computing, v. 32, N 2, pp. 136-156 (2006). http://graal.ens-lyon.fr/MUMPS/ [6] E. B. Rudnyi, B. van Rietbergen, J. G. Korvink. Model Reduction for High Dimensional Micro-FE Models. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 14 - 16 June 2006. ==> bone010/bone010.C.names <== Node1_UX Node1_UY Node1_UZ ==> boneS01/boneS01.C.names <== Node1_UX Node1_UY Node1_UZ ==> boneS10/boneS10.C.names <== Node1_UX Node1_UY Node1_UZ ================================================================================ ==> chipcool0/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Convective Thermal Flow Problems (chipcool*, flowmeter* problems) -------------------------------------------------------------------------------- Christian Moosmann, moosmann@imtek.uni-freiburg.de Andreas Greiner, greiner@imtek.uni-freiburg.de Many thermal problems require simulation of heat exchange between a solid body and a fluid flow. The most elaborate approach to this problem is computational fluid dynamics (CFD). However, CFD is computationally expensive. A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model. However, caution has to be taken to select the film coefficient. An intermediate level is to include a flow region with a given velocity profile, that adds convective transport to the model. Compared to convection boundary conditions this approach has the advantage that the film coefficient has not to be specified and that information about the heat profile in the flow can be obtained. A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation, because the solution accuracy when employing upwind finite element schemes depends on the element size. While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one. So this problem type can be used as a benchmark for problems containing un-symmetric matrices. Fig. 1a. Fig. 1b. see http://www.imtek.de/simulation/benchmark Fig. 1. Convective heat flow examples: 2D anemometer model (left), 3D cooling structure (right) Two different designs are tested: a 2D model of an anemometer-like structure mainly consisting of a tube and a small heat source (Fig 1 left) [1]. The solid model has been generated and meshed in ANSYS. Triangular PLANE55 elements have been used for meshing and discretizing by the finite element method, resulting in 19 282 elements and 9710 nodes. The second design is a 3D model of a chip cooled by forced convection (Fig 1 right) [2]. In this case the tetrahedral element type SOLID70 was used, resulting in 107 989 elements and 20542 nodes. Since the implementation of the convective term in ANSYS does not allow for definition of the fluid speed on a per element, but on a per region basis, the flow profile has to be approximated by piece-wise step functions. The approximation used for this benchmarks is shown in figure 1. The Dirichlet boundary conditions are applied to the original system. In both models the reference temperature is set to 300 K, Dirichlet boundary conditions as well as initial conditions are set to 0 with respect to the reference. The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the unsymmetric case (with forced convection). Table 1 shows the output nodes specified for the two benchmarks, table 2 links the filenames according to the different cases. Matrices are in the Matrix Market format. The matrix name is used as an extension of the matrix file. *.C.names contains a list of ouput names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4ansys. Table 1: Output nodes for the two models Model Number Code Comment Flow Meter 1 out1 outlet position 2 out2 outlet position 3 SenL left sensor position 4 Heater within the heater 5 SenR right sensor position cooling Structure 1 out1 outlet position 2 out2 outlet position 3 out3 outlet position 4 out4 outlet position 5 Heater within the heater Table 2: Provided files Model fluid speed (m/s) name Flow Meter 0 flow_meter_model_v0.tgz renamed flowmeter0 0.5 flow_meter_model_v0.5.tgz renamed flowmeter5 cooling Structure 0 chip_cooling_model_v0.tgz renamed chipcool0 0.1 chip_cooling_model_v0.1.tgz renamed chipcool1 Further information on the models can be found in [3] where model reduction by means of the Arnoldi algorithm is also presented. Bibliography 1 H. Ernst : High-Resolution Thermal Measurements in Fluids, PhD thesis, University of Freiburg, Germany(2001). 2 C. A. Harper : Electronic packaging and interconnection handbook, New York McGraw- Hill, USA (1997) 3 C. Moosmann, E. B. Rudnyi, A. Greiner, J. G. Korvink: Model Order Reduction for Linear Convective Thermal Flow, Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems, THERMINIC2004, 29 Sept - 1 Oct , 2004, Sophia Antipolis, France. Preprint is at http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ ==> chipcool0/chip_cooling_model_v0.C.names <== out1 out2 out3 out4 Heater ==> chipcool1/chip_cooling_model_v0.1.C.names <== out1 out2 out3 out4 Heater ==> flowmeter0/flow_meter_model_v0.C.names <== out1 out2 SenL Heater SenR ==> flowmeter5/flow_meter_model_v0.5.C.names <== out1 out2 SenL Heater SenR ================================================================================ ==> filter2D/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Tunable Optical Filter (filter2D, filter3D problems) -------------------------------------------------------------------------------- Dennis Hohlfeld, hohlfeld@imtek.uni-freiburg.de Tamara Bechtold, bechtold@imtek.uni-freiburg.de Hans Zappe, zappe@imtek.uni-freiburg.de The DFG project AFON (funded under grant ZA 276/2-1) aimed at the development of an optical filter, which is tunable by thermal means. The thin-film filter is configured as a membrane in order to improve thermal isolation. Fabrication is based on silicon technology. Wavelength tuning is achieved through thermal modulation of resonator optical thickness, using metal resistor deposited onto the membrane. The devices features low power consumption, high tuning speed and excellent optical performance [1]. Fig. 1. see http://www.imtek.de/simulation/benchmark Fig. 1. Tunable optical filter. The benchmark contains a simplified thermal model of a filter device. It helps designers to consider important thermal issues, such as what electrical power should be applied in order to reach the critical temperature at the membrane or homogeneous temperature distribution over the membrane. The original model is the heat transfer partial differential equation. There are two different benchmarks, 2D model and 3D model (see Table 1). Due to modeling differences, their simulation results cannot be compared with each other directly. Table 1: Tunable optical filter benchmarks Code comment dimension nnz(A) nnz(E) filter2D 2D, linear elements, PLANE55 1668 6209 1668 filter3D 3D, linear elements, SOLID90 108373 1406808 1406791 The device solid models have been made, meshed and discretized in ANSYS 6.1 by the finite element method. All material properties are considered as temperature independent. Temperature is assumed to be in Celsius with the initial state of 0 C. The Dirichlet boundary conditions of T = 0 C have been applied at the bottom of the chip. The output nodes for the models are described in Table 2 and schematically displayed in Fig 2. Output 1 is located at the very center of the membrane. By simulating itÕs temperature one can prove what input power is needed to reach the critical membrane temperature for each wavelength. Furthermore, the output 2 to 5 must be very close to output 1 (homogenous temperature distribution) in order to provide the same optical properties across the complete diameter of the laser beam. Table 2: Outputs for the optical filter model Number Code Comment 1 Memb1 Membrane center 2 Memb2 Membrane node with radius 25E-6, theta 90 3 Memb3 Membrane node with radius 50E-6 theta 90 4 Memb4 Membrane node with radius 25E-6, theta 135 5 Memb5 Membrane node with radius 50E-6 theta 135 Fig. 2. see http://www.imtek.de/simulation/benchmark Fig. 2. Schematical position of the chosen output nodes. The benchmark contains a constant load vector. The input function equal to 1 corresponds to the constant input power of of 1 mW for 2D model and 10 mW for 3D model. The linear ordinary differential equations of the first order are written as: E dT/dt = A T + B u y = C T where E and A are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), B is the load vector, C is the output matrix, and T is the vector of unknown temperatures. The output of the transient simulation for node 1 over the rise time of the device (0.25 s) for 3D model can be find in Filter3DTransResults. The results can be used to compare the solution of a reduced model with the original one. The time integration has been performed in ANSYS with accuracy of about 0.1 %. The results are given as matrices where the first row is made of times, the second of the temperatures. Download matrices in the Matrix Market format: filter2D.tar.gz, 106502 bytes; filter3D.tar.gz, 37417415 bytes. The matrix name is used as an extension of the matrix file. File *.C.names contains a list of ouput names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4ansys. The discussion of electro-thermal modeling related to the benchmark can be found in [2]. Bibliography 1. D. Hohlfeld, H. Zappe, All-dielectric tunable optical filter based on the thermo-optic effect, Journal of Optics A: Pure and Applied Optics, 6(6), 504- 511 (2003). 2. T. Bechtold, Model Order Reduction of Electro-Thermal MEMS, PhD thesis, University of Freiburg, Germany (In preparation). ==> filter2D/filter2D.C.names <== Memb1 Memb2 Memb3 Memb4 Memb5 ==> filter3D/filter3D.C.names <== Memb1 Memb2 Memb3 Memb4 Memb5 ================================================================================ ==> gas_sensor/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Microhotplate Gas sensor (gas_sensor problem) -------------------------------------------------------------------------------- Jürgen Hildenbrand, hildenbr@imtek.uni-freiburg.de Tamara Bechtold, bechtold@imtek.uni-freiburg.de Jürgen Wöllenstein, woellen@ipm.fhg.de The goal of European project Glassgas (IST-99-19003) was to develop a novel metal oxide low power microhotplate gas sensor [1]. In order to assure a robust design and good thermal isolation of the membrane from the surrounding wafer, the silicon microhotplate is supported by glass pillars emanating from a glass cap above the silicon wafer, as shown in Fig 1. In present design, four different sensitive layers can be deposited on the membrane. The thermal management of a microhotplate gas sensor is of crucial importance. Fig. 1. see http://www.imtek.de/simulation/benchmark/ Fig. 1. Micromashined metal oxide gas sensor array; Bottom view (left), top view (right). The benchmark contains a thermal model of a single gas sensor device with three main components: a silicon rim, a silicon hotplate and glass structure [2]. It allows us to simulate important thermal issues, such as the homogeneous temperature distribution over gas sensitive regions or thermal decoupling between the hotplate and the silicon rim. The original model is the heat transfer partial differential equation. The device solid model has been made and then meshed and discretized in ANSYS 6.1 by means of the finite element method (SOLID70 elements were used). It contains 68000 elements and 73955 nodes. Material properties were considered as temperature independent. Temperature is assumed to be in degree Celsius with the initial state of 0 C. The Dirichlet boundary conditions of T = 0 C is applied at the top and bottom of the chip (at 7038 nodes). The output nodes are described in Table 1. In Fig 2 the red marked nodes are positioned on the silicon rim. Their temperature should be close to the initial temperature in the case of good thermal decoupling between the membrane and the silicon rim. The black marked nodes are placed on the sensitive layers above the heater and are numbered from left to right row by row, as schematically shown in Fig 2. They allow us to prove whether the temperature distribution over the gas sensitive layers is homogeneous (maximum difference of 10C is allowed by design). Table 1: Outputs for the gas sensor model Number Code Comment 1 aHeater within a heater, to be used for nonlinear input 2-7 SiRim1 to SiRim7 silicon rim 8-28 Memb1 to Memb21 gas sensitive layer Fig. 2. see http://www.imtek.de/simulation/benchmark/ Fig. 2. Masks disposition (left) and the schematical position of the chosen output nodes (right). The benchmark contains a constant load vector. The input function equal to 1 corresponds to the constant input power of 340 mW. One can insert a weak input nonlinearity related to the dependence of heater's resistivity on temperature given as: R(T)=R{0}(1 + alpha T) where alpha =1.469e-3 K^-1. To this end, one has to multiply the load vector by a function: {U^2 274.94 (1 + alpha T)}/{0.34 (274.94 (1 + alpha T)+148.13)^2} where U is a desired constant voltage. The temperature in equation above should be replaced by the temperature at the output 1. The linear ordinary differential equations of the first order are written as: E dT/dt = A T + B u y = C T where E and A are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), B is the load vector, C is the output matrix, and T is the vector of unknown temperatures. The dimension of the system is 66917, the number of nonzero elements in matrix E is 66917, in matrix A is 885141. The outputs of the transient simulation at output 18 (Memb11) over the rise time of the device of 5 s for the original linear (with constant input power of 340 mW) and nonlinear (with constant voltage of 14 V) model are placed in files LinearResults and NonlinearResults respectively. The results can be used to compare the solution of a reduced model with the original one. The time integration has been performed in ANSYS with accuracy of about 0.1 %. The results are given as matrices where the first row is made of times, the second of the temperatures. Download matrices in the Matrix Market format: GasSensor.tar.gz, 8407057 bytes. The matrix name is used as an extension of the matrix file. File *.C.names contains a list of ouput names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4ansys. The discussion of electro-thermal modeling related to the benchmark including the nonlinear input function can be found in [3]. Bibliography 1. J. Wöllenstein H. Böttner, J. A. Pláza, Carlos Carné, Y. Min, H. L. Tuller : A novel single chip thin film metal oxide array, Sensors and Actuators B: Chemical 93 (1-3) 350-355 (2003). 2. J. Hildenbrand : Simulation and Characterisation of a Gas sensor and Preparation for Model Order Reduction, Diploma Thesis, University of Freiburg, Germany (2003). 3. T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink : Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor, Proceedings of 5th International conference on thermal and mechanical simulation and experiments in microelectronics and microsystems, EUROSIME2004, May 10-12, 2004, Brussels, Belgium, p. 263-267. Preprint is at http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ ==> gas_sensor/GasSensor.C.names <== aHeater SiRim1 SiRim2 SiRim3 SiRim4 SiRim5 SiRim6 Memb1 Memb2 Memb3 Memb4 Memb5 Memb6 Memb7 Memb8 Memb9 Memb10 Memb11 Memb12 Memb13 Memb14 Memb15 Memb16 Memb17 Memb18 Memb19 Memb20 Memb21 ================================================================================ ==> gyro/README.txt <== ================================================================================ Oberwolfach model reduction benchmark collection ------------------------------------------------ Contributions by Evgenii Rudnyi and Dag Billger. See also: http://www.imtek.de/simulation/benchmark http://www.imtek.de/simulation/benchmark/wb/38847 http://www.imtek.de/simulation/benchmark/wb/35889 -------------------------------------------------------------------------------- IMEGO gyroscope -------------------------------------------------------------------------------- The Butterfly Gyro Dag Billger, The Imego Institute, Arvid Hedvalls Backe 4, SE-411 33, Goteborg, Sweden, Tel. +46 317 501 853, Fax. +46 317 501 801, Email: dag dot billger at imego dotcom Figure 1: see http://www.imtek.de/simulation/benchmark/wb/35889/fig1.jpg Fig. 1. The Butterfly and micro-SIC mounted together. The Butterfly gyro is developed at the Imego Institute in an ongoing project with Saab Bofors Dynamics AB. The Butterfly is a vibrating micro-mechanical gyro that has sufficient theoretical performance characteristics to make it a promising candidate for use in inertial navigation applications. The goal of the current project is to develop a micro unit for inertial navigation that can be commercialized in the high-end segment of the rate sensor market. This project has reached the final stage of a three-year phase where the development and research efforts have ranged from model based signal processing, via electronics packaging to design and prototype manufacturing of the sensor element. The project has also included the manufacturing of an ASIC, named micro-SIC, that has been especially designed for the sensor (see Fig. 1). The gyro chip consists of a three-layer silicon wafer stack, in which the middle layer contains the sensor element. The sensor consists of two wing pairs that are connected to a common frame by a set of beam elements (see Fig. 2 and 3); this is the reason the gyro is called the Butterfly. Since the structure is manufactured using an anisotropic wet-etch process, the connecting beams are slanted. This makes it possible to keep all electrodes, both for capacitive excitation and detection, confined to one layer beneath the two wing pairs. The excitation electrodes are the smaller dashed areas shown in Fig. 2. The detection electrodes correspond to the four larger ones. Figure 2: see http://www.imtek.de/simulation/benchmark/wb/35889/fig2.jpg Fig. 2. Schematic layout of the Butterfly design. By applying DC-biased AC-voltages to the four pairs of small electrodes, the wings are forced to vibrate in anti-phase in the wafer plane. This is the excitation mode. As the structure rotates about the axis of sensitivity (see Fig. 2), each of the masses will be affected by a Coriolis acceleration. This acceleration can be represented as an inertial force that is applied at right angles with the external angular velocity and the direction of motion of the mass. The Coriolis force induces an anti-phase motion of the wings out of the wafer plane. This is the detection mode. The external angular velocity can be related to the amplitude of the detection mode, which is measured via the large electrodes. When planning for and making decisions on future improvements of the Butterfly, it is of importance to improve the efficiency of the gyro simulations. Repeated analyses of the sensor structure have to be conducted with respect to a number of important issues. Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases and the effect of force-feedback. The use of model order reduction indeed decreases runtimes for repeated simulations. Moreover, the reduction technique enables a transformation of the FE representation of the gyro into a state space equivalent formulation. This will prove helpful in testing the model based Kalman signal processing algorithms that are being designed for the Butterfly gyro. The structural model of the gyroscope has been done in ANSYS using quadratic tetrahedral elements (SOLID187, see Fig. 3). The model shown is a simplified one with a coarse mesh as it is designed to test the model reduction approaches. It includes the pure structural mechanics problem only. The load vector is composed from time-varying nodal forces applied at the centres of the excitation electrodes (see Fig. 2). The amplitude and frequency of each force is equal to 0.055 micro-Newtons and 2384 Hz, respectively. The Dirichlet boundary conditions have been applied to all degree of freedom of the nodes belonging to the top and bottom surfaces of the frame. The output nodes are listed in Table 2 and correspond to the centres of the detection electrodes (see Fig. 3). Figure 3: see http://www.imtek.de/simulation/benchmark/wb/35889/fig3.jpg Fig. 3. Finite element mesh of the gyro with a background photograph of the gyro wafer pre-bonding. The structural model M d<sup>2</sup>x/dt<sup>2</sup> + E dx/dt + K x = B y = C x contains the mass M and stifness matrices K. The damping matrix E can be modeled as E = alpha M + beta K, where the typical values of alpha and beta are 0 and 1e-6 respectively. The nature of the damping matrix is in reality more complex (squeeze film damping, thermo elastic damping, etc.) but this simple approach has been chosen with respect to the model reduction test. B is the load vector, C is the output matrix. The dynamic model has been converted to (http://math.nist.gov/MatrixMarket/) Matrix Market format by means of (http://www.imtek.de/simulation/mor4ansys/) mor4ansys. The statistics for the matrices is shown in Table 1. Table 1. System matrices for the gyroscope. matrix m n nnz Is Symmetric? M 17361 17361 178896 yes K 17361 17361 519260 yes B 17361 1 8 no C 12 17361 12 no Table 2. Outputs for the Butterfly Gyro Model # Code Comment 1-3 det1m_Ux, det1m_Uy, det1m_Uz Displacements of detection electrode 1, (bottom left large electrode of Fig. 2) 4-6 det1p_Ux, det1p_Uy, det1p_Uz Displacements of detection electrode 2, (bottom right large electrode of Fig. 2) 7-9 det2m_Ux, det2m_Uy, det2m_Uz Displacements of detection electrode 3, (top left large electrode of Fig. 2) 10-12 det2p_Ux, det2p_Uy, det2p_Uz Displacements of detection electrode 4, (top right large electrode of Fig. 2) (Note: the original matrices, in Matrix Market format, are at http://www.imtek.de/simulation/benchmark). The model reduction of the gyroscope model by means of (http://www.imtek.de/simulation/mor4ansys/) mor4ansys is described in Ref [1]. [1] Jan Lienemann, Dag Billger, Evgenii B. Rudnyi, Andreas Greiner, and Jan G. Korvink, MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices, the Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA Preprint is at http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ ==> gyro/gyro.C.names <== det1pUX det1pUY det1pUZ det1mUX det1mUY det1mUZ det2pUX det2pUY det2pUZ det2mUX det2mUY det2mUZ ================================================================================ ==> inlet/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Active Control of a Supersonic Engine Inlet (Inlet problem) -------------------------------------------------------------------------------- Karen Willcox, Guillaume Lassaux, and David Gratton http://web.mit.edu/kwillcox/www/home.htm ================================================================================ ==> LF10/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Linear 1D Beam Models (LF* problems) -------------------------------------------------------------------------------- Jan Lienemann <lieneman (at) imtek (dot) de>, Andreas Greiner <greiner (at) imtek (dot) de>, Jan G. Korvink <korvink (at) imtek (dot) de> Introduction Moving structures are an essential part for many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures. Several actuation principles can be employed on microscopic length scales, the most frequent certainly the electromagnetic forces. While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and fabrication expense. While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge. This challenge is raised by the fact that many of these devices show a nonlinear behavior. Especially for electrostatic structures, a further difficulty is the large reach of the electrostatic forces, leading to a strong spatial coupling of charges. Accurate modelling of such a system typically leads to high order models. The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximate it by a lower order model. A application of electrostatic moving structures are e.g. RF switches or filters. Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space. Model description Setup of beam This model describes a slender beam with four degrees of freedom per node: x Axial displacement thetax Axial rotation y Flexural displacement thetaz Flexural rotation Degree of Freedom x Degree of Freedom theta x Degrees of Freedom y and theta z The beam is supported either on the left side or on both sides. For the left side (fixed) support, the force is applied on the rightmost node in y direction, whereas for the support on both sides (simply supported), a node in the middle is loaded. The damping matrix is calculated by a linear combination of the mass matrix M and the stiffness matrix K. Benchmark examples Based on the finite element discretization presented in [1], an interactive matrix generator has been created using Wolfram Research's webMathematica. It is available online at http://www.imtek.uni-freiburg.de/simulation/mstkmpkt/models/LinearBeam.html, "Linear beam model". However, models produced by this generator are in the DSIF format, which allows for nonlinear terms. For the purpose of the benchmark collection, we have precomputed four systems and converted them to the Matrix market format which is easier to import in standard computer algebra packages. All examples are made for a steel beam with the following properties: Property Value Beam length (l) 0.1 m Material density (rho) 8000 kg/m3 Cross-sectional area (A) 7.854e-7 m2 Moment of inertia (I) 4.909e-14 m4 Polar moment of inertia (J) 9.817e-14 Modulus of elasticity (E) 2e11 Pa Poisson ratio (nu) 0.29 Contribution of M to damping 1e2 Contribution of K to damping 1e-2 Support Simple, both sides The following examples are available (all files are compressed .zip archives, Units: SI): File Degrees of freedom Number of nodes Number of equations File size/Compressed size LF10.zip flexural (y and thetaz) 10 18 LF10000.zip flexural (y and thetaz) 10000 19998 LFAT5.zip flexural (y and thetaz), axial, torsional 5 14 LFAT5000.zip flexural (y and thetaz), axial, torsional 50000 19994 The zip files contain matrices M, E, K, B and E for the following system of equations: M x" + E x' + K x = B u y = C x where B is a n×1 matrix and C is a 1×n matrix with the only nonzero entry at the y DOF of the middle node. Details of the implementation are available in a separate report (PDF, 88 kB). A typical input to this system is a step response; periodic on/off switching is also possible. The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics. [1] William Weaver, Jr., Stephen P. Timoshenko, and Donovan H. Young, Vibration problems in engineering, 5th ed., Wiley (1990) Last modified: Wed Sep 8 16:53:01 CEST 2004 ================================================================================ ==> piston/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Axi symmetric infinite element model for circular piston (piston problem) -------------------------------------------------------------------------------- Karl Meerbergen Free Field Technologies, 16, place de l'Université, 1348 Louvain-la-Neuve, Belgium This example is a model of the form \begin{displaymath} M \ddot{x} + E \dot{x} + K x = f \end{displaymath} with $M$, $C$, and $K$ non-symmetric matrices and $M$ singular. This is thus a differential algebraic equation. It is shown that it has index one [1]. The input of the system is $f$, the output is the state vector $x$. The motivation for using model reduction for this type of problems is the reduction of the computation time of a simulation. This is an example from an acoustic radiation problem discussed in [3]. Consider a circular piston subtending a polar angle $0 < \theta < \theta_p$ on a submerged massless and rigid sphere of radius $\delta$. The piston vibrates harmonically with a uniform radial acceleration. The surrounding acoustic domain is unbounded and is characterized by its density $\rho$ and sound speed $c$. We denote by $p$ and $a_r$ the prescribed pressure and normal acceleration respectively. In order to have a steady state solution $\tilde p(r,\theta,t)$ verifying \begin{displaymath} \tilde p(r,\theta,t) = \mathcal R \mbox{e} \left( p(r,\theta) e^{i\omega t} \right), \end{displaymath} the transient boundary condition is chosen as: \begin{displaymath} a_r = \left. \frac{-1}{\rho} \frac{\partial p(r,\theta)}{\pa... ...e \theta_p, \\ 0, & \theta > \theta_p.\\ \end{array}\right. \end{displaymath} The axisymmetric discrete finite-infinite element model relies on a mesh of linear quadrangle finite elements for the inner domain (region between spherical surfaces $r=\delta$ and $r = 1.5 \delta$). The numbers of divisions along radial and circumferential directions are $5$ and $80$, respectively. The outer domain relies on conjugated infinite elements of order $5$. For this example we used $\delta=1 \mathrm{(m)}$, $\rho = 1.225 \mathrm{(kg/m^3)}$, $c=340 \mathrm{(m/s)}$, $a_0 = 0.001 \mathrm{(m/s^2)}$ and $\omega = 1000 \mathrm{(rad/s)}$. The matrices $K$, $C$, $M$ and the right-hand side $f$ are computed by MSC.Actran [2]. The dimension of the second-order system is $N = 2025$. ================================================================================ ==> rail_79841/README.txt <== ================================================================================ -------------------------------------------------------------------------------- A Semi-discretized Heat Transfer Problem for Optimal Cooling of Steel Profiles (rail* problems) -------------------------------------------------------------------------------- Summary Several generalized state-space models arising from a semi-discretization of a controlled heat transfer process for optimal cooling of steel profiles are presented. The models oder differs due to different refinements applied to the computational mesh. The model equations We consider the problem of optimal cooling of steel profiles. This problem arises in a rolling mill when different steps in the production process require different temperatures of the raw material. To achieve a high production rate, economical interests suggest to reduce the temperature as fast as possible to the required level before entering the next production phase. At the same time, the cooling process, which is realized by spraying cooling fluids on the surface, has to be controlled so that material properties, such as durability or porosity, achieve given quality standards. Large gradients in the temperature distributions of the steel profile may lead to unwanted deformations, brittleness, loss of rigidity, and other undesirable material properties. It is therefore the engineers goal to have a preferably even temperature distribution. The scientific challenge here is to give the engineers a tool to pre-calculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose. We can only briefly introduce the model here for details we refer to [Saa03] or [BS04]. We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model. Exploiting the symmetry of the workpiece, the computational domain Ω⊂R2 is chosen as the half of a cross section of the rail profile. The heat distribution is modeled by the instationary linear heat equation on Ω (1) cρ ∂t x(t,ξ) - λΔx(t,ξ) = 0 in R>0 × Ω, x(0,ξ) = x0(ξ) in Ω, λ ∂νx(t,ξ) = gi on R>0× Γi, ∂Ω = ∪i Γi where x is the temperature distribution (∈H1([0,∞],X) with X:=H1(Ω) the state space), c the specific heat capacity, λ the heat conductivity and ρ the density of the rail profile. We split the boundary into several parts Γi on which we have different boundary functions gi, allowing us to vary controls on different parts of the surface. By ν we denote the outer normal on the boundary. Figure 1: Initial mesh, partitioning of the boundary, and a picture of a cooling plant. Grobgitter Kühlbett see http://www.imtek.de/simulation/benchmark/ We want to establish the control by a feedback law, i.e. we define the boundary functions gi to be functions of the state x and the control ui, where (ui)i=:u=Fy for a linear operator F which is chosen such that the cost functional J(x_0,u):=∫0∞ (Qy,y)Y +(Ru,u)U dt with y=Cx (2) is minimized. Here, Q and R are linear selfadjoint operators on the output space Y and the control space U with Q≥0, R>0 and C∈ L(X,Y). The variational formulation of (1) with gi(t,ξ) = qi(ui- x(ξ,t)) leads to: (∂tx,v)=-∫Ω α∇x ∇v dξ + ∑k { qk uk ∫Γk (cρ)-1 v dσ -∫Γk qk(cρ)-1 xv dσ} (3) for all v∈C0∞(Ω). Here the uk are the exterior (cooling fluid) temperatures used as the controls, qk are constant heat transfer coefficients (i.e. parameters for the spraying intensity of the cooling nozzles)[2] and α:=λ/(cρ). Note that q0=0 gives the Neumann isolation boundary condition on the artificial inner boundary on the symmetry axis. In view of (3) we can apply a standrrd Galerkin approach for discretizing the heat transfer model in space, resulting a first-order ordinary differential equation. This idescribed in the following section. The discretized mathematical model For the discretization we use the ALBERTA-1.2 fem-toolbox (see [SS00] for details). We applied linear Lagrange elements and used a projection method for the curved boundaries. The initial mesh (see Fig. 1. on the left) was produced by MATLABs pdetool which implements a Delaunay triangulation algorithm. The finer discretizations were produce by global mesh refinement using a bisection refinement method. The discrete LQR problem is then: minimize (2) with respect to (4) E d/dt x(t) = A x(t) + B u(t), with t>0, x(0) = x0, y(t) = C x(t). This benchmark includes four different mesh resolutions. The best fem-approximation error that one can expect (under suitable smoothness assumptions on the solution) is of order O(h2) where h is the maximum edge size in the corresponding mesh. This order should be matched in a model reduction approach. The following table lists some relevant quantities for the provided models. matrix dimension num. nonzero in A num. nonzero in E max. mesh width 1357 8985 8997 5.5280e-2 5177 35185 35241 2.7640e-2 20209 139233 139473 1.3820e-2 79841 553921 554913 6.9100e-3 Note that A is negative definite while E is positive definite, so that the resulting linear time-invariant system is stable. The data sets are named rail_(problem dimension)_C60.(matrix name). Here C60 is a specific output matrix which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in Fig 1.) and keep temperature gradients small. The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g. temperature difference between nodes 83 and 34. Again refer to Figure 1. for the nodes used. the definitions of other output matrices that we tested can be found in [Saa03]. The problem resides at temperatures of approximately 1000 degrees centigrade down to about 500-700 degrees depending on calculation time. The state values are scaled to 1000 being equivalent to 1.000. This results in a scaling of the time line with factor 100, meaning that calculated times have to be divided by 100 to get the real time in seconds. Acknowledgments This benchmark example serves as a model problem for the project A15: Efficient numerical solution of optimal control problems for instationary convection-diffusion-reaction-equations of the Sonderforschungsbereich SFB393 Parallel Numerical Simulation for Physics and Continuum Mechanics, supported by the Deutsche Forschungsgemeinschaft. It was motivated by the model described in TU01. A very similar problem is used as model problem in the LYAPACK software package Pen00b. Bibliography BS04 P. Benner and J. Saak. Efficient numerical solution of the LQR-problem for the heat equation. Proc. Appl. Math. Mech., 2004. submitted. Pen00 T. Penzl. LYAPACK Users Guide. Technical Report SFB393/00-33, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 09107 Chemnitz, FRG, 2000. Available from http://www.tu-chemnitz.de/sfb393/sfb00pr.html . Saa03 J. Saak. Effiziente numerische Lösung eines Optimalsteuerungsproblems für die Abkühlung von Stahlprofilen. Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universität Bremen, D-28334 Bremen, September 2003. Available from http://www-user.tu-chemnitz.de/∼saak/Data/index.html . SS00 A. Schmidt and K. Siebert. ALBERT: An adaptive hierarchical finite element toolbox. Preprint 06/2000 / Institut für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, edition: albert-1.0 edition, 2000. Available from http://www.mathematik.uni-freiburg.de/IAM/ALBERT/doc.html. TU01 F. Tröltzsch and A. Unger. Fast solution of optimal control problems in the selective cooling of steel. Z. Angew. Math. Mech., 81:447-456, 2001. ================================================================================ ==> spiral/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Spiral Inductor PEEC Model (spiral models) -------------------------------------------------------------------------------- Figure: Spiral inductor with part of overhanging copper plane; see http://www.imtek.de/simulation/benchmark/ Excerpt from report "PEEC model of a spiral inductor generated by Fasthenry" at http://www.imtek.de/simulation/benchmark/ : This inductor which first appeared in [KWW00] is intended as an integrated RF passive inductor. To make it also a proximity sensor, a 0.1 um plane of copper is added 45 um above the spiral. The spiral is also copper with turns 40 um wide, 15um thick, with a separation of 40um. The spiral is suspended 55um over the substrate by posts at the corners and centers of the turns in order to reduce the capacitance to the substrate. (Note that neither the substrate nor the capacitance is modeled in this example.) The overall extent of the suspended turns is 1.58mm x 1.58mm. ... The frequency dependence of the resistance shows two effects, first a rise due to currents induced in the copper plane and then a much sharper rise due to the skin effect. Capturing the rise due to the skin effect while also maintaining the low frequency response is a challenge for many model reduction algorithms. Authors: Jing-Rebecca Li, INRIA-Rocquencourt, Projet Ondes, Domaine de Voluceau - Rocquencourt - B.P. 105, 78153 Le Chesnay Cedex, France jingrebecca.li at inria.fr Mattan Kamon, Coventor, Inc. 625 Mt. Auburn St, Cambridge, Ma 02138, USA matt at coventor.com. "E" dx/dt = "A"x+"B"u, y = "B"^Tx. The complex impedence is Z(w) = Resis(w)+i*w*Induc(w) = (G(i*w))^(-1)=(B^T(-A+i*w*E)^(-1)B)^(-1) Download short matlab files from www.imtek.de to 1. plot Resis(w) and Induc(w) 2. perform a PRIMA reduction of order 50 3. produce symmetrized standard state-space system: dx/dt = "A_symm"x+"B_symm"u, y = "B_symm"^Tx, where A_symm is symmetric. Last modified: Wed Jul 6 17:38:31 CEST 2005 [KWW00] Mattan Kamon, Frank Wang, and Jacob White. Generating nearly optimally compact models from krylov-subspace based reduced order models. IEEE Transactions On Circuits and Systems-II:Analog and Digital Signal Processing, 47(4):239-248, April 2000. ================================================================================ ==> t2dah/README.txt <== ================================================================================ Oberwolfach model reduction benchmark collection ------------------------------------------------ Contributions by Evgenii Rudnyi and Dag Billger. See also: http://www.imtek.de/simulation/benchmark http://www.imtek.de/simulation/benchmark/wb/38847 http://www.imtek.de/simulation/benchmark/wb/35889 -------------------------------------------------------------------------------- Micropyros Thruster -------------------------------------------------------------------------------- E. B. Rudnyi, rudnyi AT imtek dot uni-freiburg dot de The goal of the European project Micropyros (http://www.laas.fr/Micropyros/) was to develop a microthruster array shown in fig. 1. It is based on the co-integration of solid fuel with a silicon micromachined system [1]. In addition to space applications, the device can be also used for gas generation or as a highly-energetic actuator. When the production of a bit-impulse is required, the fuel is ignited by heating a resistor at the top of a particular microthruster. Design requirements and modeling alternatives are described in Ref [2]. The discussion of electro-thermal modeling related to the benchmark can be found in Ref [3]. Figure 1: see http://www.imtek.de/simulation/benchmark/wb/38847/fig1.jpg Fig. 1. Firing a micro thruster in an 4x4 array. Illustration courtesy of C. Rossi, LAAS-CNRS. The benchmark contains a simplified thermal model of a single microthruster to help with a design problem to reach the ignition temperature within the fuel and at the same time not to reach the critical temperature at neighboring microthrusters, that is, at the border of the computational domain. At the same time, the resistor temperature during the heating pulse should not become too high as this leads to the destruction of the membrane. The benchmark suite has been made with the Micropyros software developed by IMTEK (http://www.imtek.uni-freiburg.de/simulation/pyros/). There are four different test cases described in Table 1 with the goal to cover different cases of different computational complexity. Note that the results from different models cannot be compared directly with each other as the output nodes are located in slightly different geometrical positions and there is some difference in modeling for the 3D and 2D-axisymmetric cases. Table 1. Microthruster benchmarks. Code comment dimension nnz(A) nnz(E) T2DAL 2D-axisymmetric, linear elements 4257 20861 4257 T2DAH 2D-axisymmetric, quadratic elements 11445 93781 93781 T3DL 3D, linear elements 20360 265113 20360 T3DH 3D, quadratic elements 79171 2215638 2215638 The device solid model has been made and meshed in ANSYS. The material properties assumed to be constant. The system matrices have been converted to the (http://math.nist.gov/MatrixMarket/) Matrix Market format by means of mor4ansys. Temperature is assumed to be in Celsius with the initial state of 0 C. The output nodes are described in Table 2. Nodes 2 to 5 show the fuel temperature distribution and nodes 6 and 7 characterize temperature in the wafer, nodes 5 and 7 being the most faraway from the resistor. Table 2. Outputs for the microthruster models. # Code Comment 1 aHeater within the heater 2 FuelTop fuel just below the heater 3 FT-100 fuel 0.1 mm below the heater 4 FT-200 fuel 0.2 mm below the heater 5 FuelBot fuel bottom 6 WafTop1 wafer top (touching fuel) 7 WafTop2 wafer top (end of computational domain) The benchmark files contain a constant load vector, corresponding to the constant power input of 150 mW. In order to insert a weak nonlinearity related to the dependence of the resistivity on temperature, one has to multiply the load vector by a function 1 + 0.0009 T + 3E-07 T^2 assuming the constant current. It is necessary to replace the temperature in the equation above by the temperature at the node 1. The linear ordinary differential equations of the first order are written as E dT/dt = A T + B y = C T where E and A are the system matrices (both are symmetric), B is the load vector, C is the output matrix, and T is the vector of unknown temperatures. (Note: the original matrices, in Matrix Market format, are at http://www.imtek.de/simulation/benchmark). The ANSYS results for the original models as well as the reduced models obtained by mor4ansys can be found at the (http:///www.imtek/de/simulation/pyros/) Micropyros software page: choose EleThermo for T2DAL and T2DAH or EleThermo3D for T3DL and T3DH. The model reduction of the microthruster model by means of (http://www.imtek.ed/simulation/mor4ansys/) mor4ansys is described in Ref [4]. [1] C. Rossi, B. Larangot, T. Camps, M. Dumonteil, D. Lagrange, P. Q. Pham, D. Briand, N. F. de Rooij, M. Puig-Vidal, P. Miribel, E. Montane, E. Lopez, J. Samitier, E. B. Rudnyi, T. Bechtold, J. G. Korvink, Review of solid propellant microthrusters on silicon, Journal Propulsion and Power (2004), to be published. [2] E. B. Rudnyi, T. Bechtold, J. G. Korvink, C. Rossi, Solid Propellant Microthruster: Theory of Operation and Modelling Strategy, Nanotech 2002 - At the Edge of Revolution, September 9-12, 2002, Houston, USA, AIAA Paper 2002-5755. [3] G. Korvink, E. B. Rudnyi, Keynote: Computer-aided engineering of electro-thermal MST devices: moving from device to system simulation, EUROSIME'03, 4th international conference on thermal and mechanical simulation and experiments in micro-electronics and micro-systems Aix-en-Provence, France, March 30 - April 2, 2003. [4] T. Bechtold, E. B. Rudnyi, J. G. Korvink and C. Rossi, Efficient Modelling and Simulation of 3D Electro-Thermal Model for a Pyrotechnical Microthruster. International Workshop on Micro and Nanotechnology for Power Generation and Energy Conversion Applications PowerMEMS 2003, Makuhari,Japan, 4-5 December 2003. Preprints are at http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ ==> t2dah/T2DAH.C.names <== aHeater FuelTop FT-100 FT-200 FuelBot WafTop1 WafTop2 ==> t2dal/T2DAL.C.names <== aHeater FuelTop FT-100 FT-200 FuelBot WafTop1 WafTop2 ==> t3dh/T3DH.C.names <== aHeater FuelTop FT-100 FT-200 FuelBot WafTop1 WafTop2 ==> t3dl/T3DL.C.names <== aHeater FuelTop FT-100 FT-200 FuelBot WafTop1 WafTop2 ================================================================================ ==> t2dal_bci/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Boundary Condition Independent Thermal Model (t2dal_bci_* problems) -------------------------------------------------------------------------------- E. B. Rudnyi, rudnyi@imtek.uni-freiburg.de J. G. Korvink, korvink@imtek.uni-freiburg.de One of important requirements for a compact thermal model is that it should be boundary condition independent. This means that a chip producer does not know conditions under which the chip will be used and hence the chip compact thermal model must allow an engineer to research on how the change in the environment influences the chip temperature. The chip benchmarks representing boundary condition independent requirements are described in [1]. Mathematically, the problem is that the thermal problem is modeled by the heat transfer partial differential equation when the heat exchange through device interfaces is modeled by convection boundary conditions. The latter contains the film coefficient, h_i, to describe the heat exchange for the i-th interface. After the discretization of both equations one obtains a system of ordinary differential equations as follows E dT/dt = (A + Sum_i h_i A_i)T + B where E and A are the device system matrices, A_i is the diagonal matrix due to the discretization of convection boudnary condition for the i-th interface, T is the vector with unknown temperatures. In terms of the equation above, the engineering requirements read as follows. A chip producer specifies the system matrices but the film coefficient, h_i, is controlled later on by another engineer. As such, any reduced model to be useful should preserve h_i in the symbolic form. This problem can be mathematically expressed as parametric model reduction [2,3,4]. Unfortunately, the benchmark from [1] is not available in the computer readable format. For research purposes, we have modified a microthruster benchmark [5] (see Fig 1). In the context of the present work, the model is as a generic example of a device with a single heat source when the generated heat dissipates through the device to the surroundings. The exchange between surrounding and the device is modeled by convection boundary conditions with different film coefficients at the top, h_top, bottom, h_bottom, and the side, h_side. From this vewpoint, it is quite similar to a chip model used as a benchmark in [1]. The goal of parametric model reduction in this case is to preserve h_top, h_bottom, and h_side in the reduced model in the symbolic form. Fig. 1. see http://www.imtek.de/simulation/benchmark Figure 1. A 2D-axisymmetrical model of the micro-thruster unit (not scaled). The axis of the symmetry on the left side. A heater is shown by a red spot. We have used a 2D-axisymmetric microthruster model (T2DAL in [5]). The model has been made in ANSYS and system matrices have been extracted by means of mor4ansys [6]. The benchmark contains a constant load vector. The input function equal to one corresponds to the constant input power of 15 mW. The linear ordinary differential equations of the first order are written as: E T=(A - h_top A_top - h_bottom A_bottom - h_side A_side) T + B u y=Cx where E and A are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), B is the load vector, C is the output matrix, A_top, A_bottom, and A_side are the diagonal matrices from the discretization of the convection boundary conditions and T is the vector of unknown temperatures. The numerical values of film coefficients, h_top, h_bottom, and h_side can be from 1 to 1e9. Typical important sets film coefficients can be found in [1]. The allowable approximation error is 5 % [1]. The benchmark has been used in [7,8] where the problem is also described in more detail. Download matrices in the Matrix Market format: T2DAL_BCI.tar.gz, 223938 bytes. The matrix name is used as an extension of the matrix file. File T2DAL_BCI.C.names contains a list of ouput names written consecutively. Bibliography 1. C. J. M. Lasance, Two benchmarks to facilitate the study of compact thermal modeling phenomena, IEEE Transactions on Components and Packaging Technologies, 24, 559-565 (2001). 2. D. S. Weile, E. Michielssen, E. Grimme, K. Gallivan, A method for generating rational interpolant reduced order models of two-parameter linear systems, Applied Mathematics Letters, 12, 93-102 (1999). 3. P. K. Gunupudi, R. Khazaka, M. S. Nakhla, T. Smy, and D. Celo, Passive parameterized time-domain macromodels for high-speed transmission-line networks, IEEE Transactions on Microwave Theory and Techniques, 51, 2347-2354 (2003). 4. L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J. White, A Multiparameter Moment-Matching Model-Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models, IIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 23, 678-693 (2004). 5. E. B. Rudnyi, Micropyros Thruster, http://www.imtek.uni-freiburg.de/simulation/benchmark/. 6. E. B. Rudnyi and J. G. Korvink, Model Order Reduction of MEMS for Efficient Computer Aided Design and System Simulation, MTNS2004, Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 5-9, 2004. http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ 7. L. Feng, E. B. Rudnyi, J. G. Korvink, Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model, THERMINIC 2004, 10th International Workshop on Thermal Investigations of ICs and Systems, 29 September - 1 October 2004, Sophia Antipolis, Cªte d'Azur, France. Preprint is at http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ 8. L. Feng, E. B. Rudnyi, J. G. Korvink, Preserving the film coefficient as a parameter in the compact thermal model for fast electro-thermal simulation, 2004, to be submitted. ==> t2dal_bci/T2DAL_BCI.C.names <== aHeater FuelTop FT-100 FT-200 FuelBot WafTop1 WafTop2 ================================================================================ ==> windscreen/README.txt <== ================================================================================ -------------------------------------------------------------------------------- Structural model of a car windscreen (windscreen problem) -------------------------------------------------------------------------------- Karl Meerbergen Free Field Technologies, Louvain-la-Neuve, Belgium. Karl.Meerbergen@fft.be This is an example for a model in the frequency domain of the form \begin{eqnarray*} K_d x - \omega^2 M x & = & f \\ y & = & f^* x \end{eqnarray*} where $f$ represents a unit point load in one unknown of the state vector. $M$ is a symmetric positive-definite matrix and $K_d = (1+i\gamma) K$ where $K$ is symmetric positive semi-definite. The test problem is a structural model of a car windscreen. This is a 3D problem discretized with 7564 nodes and 5400 linear hexahedral elements (3 layers of $60 \times 30$ elements). The mesh is shown in Figure 1. The material is glass with the following properties: the Young modulus is $7\,10^{10} \mathrm{N}/\mathrm{m}^2$, the density is $2490 \mathrm{kg}/\mathrm{m}^3$, and the Poisson ratio is $0.23$. The natural damping is $10\%$, i.e. $\gamma=0.1$. The structural boundaries are free (free-free boundary conditions). The windscreen is subjected to a point force applied on a corner. The goal of the model reduction is the fast evaluation of y. Modelreduction is used as a fast linear solver for a sequence of parametrized linear systems. The discretized problem has dimension $n=22,692$. The goal is to estimate $x(\omega)$ for $\omega\in[0.5,200]$. In order to generate the plots the frequency range was discretized as $\{\omega_1,\ldots,\omega_m\} = \{0.5j,j=1,\ldots,m\}$ with $m=400$. Figure 1: Mesh of the car windscreen and frequency response function see http://www.imtek.de/simulation/benchmark/ The archive contains files windscreen.K, windscreen.M and windscreen.B representing Kd, M and f accordingly. Karl Meerbergen 2004-11-17 |
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Displaying collection matrices

**1 - 20**of**38**in totalId | Name | Group | Rows | Cols | Nonzeros | Kind | Date | Download File |
---|---|---|---|---|---|---|---|---|

1202 | gyro_k | Oberwolfach | 17,361 | 17,361 | 1,021,159 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1203 | gyro_m | Oberwolfach | 17,361 | 17,361 | 340,431 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1204 | t2dah_a | Oberwolfach | 11,445 | 11,445 | 176,117 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1205 | t2dah_e | Oberwolfach | 11,445 | 11,445 | 176,117 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1206 | t2dal_a | Oberwolfach | 4,257 | 4,257 | 37,465 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1207 | t2dal_e | Oberwolfach | 4,257 | 4,257 | 4,257 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1208 | t3dh_a | Oberwolfach | 79,171 | 79,171 | 4,352,105 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1209 | t3dh_e | Oberwolfach | 79,171 | 79,171 | 4,352,105 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1210 | t3dl_a | Oberwolfach | 20,360 | 20,360 | 509,866 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1211 | t3dl_e | Oberwolfach | 20,360 | 20,360 | 20,360 | Duplicate Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1428 | chipcool0 | Oberwolfach | 20,082 | 20,082 | 281,150 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1429 | chipcool1 | Oberwolfach | 20,082 | 20,082 | 281,150 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1430 | filter2D | Oberwolfach | 1,668 | 1,668 | 10,750 | Model Reduction Problem | 2006 | MATLAB Rutherford Boeing Matrix Market |

1431 | filter3D | Oberwolfach | 106,437 | 106,437 | 2,707,179 | Model Reduction Problem | 2006 | MATLAB Rutherford Boeing Matrix Market |

1432 | flowmeter0 | Oberwolfach | 9,669 | 9,669 | 67,391 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1433 | flowmeter5 | Oberwolfach | 9,669 | 9,669 | 67,391 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1434 | gas_sensor | Oberwolfach | 66,917 | 66,917 | 1,703,365 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1435 | gyro | Oberwolfach | 17,361 | 17,361 | 1,021,159 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |

1436 | inlet | Oberwolfach | 11,730 | 11,730 | 328,323 | Model Reduction Problem | 2006 | MATLAB Rutherford Boeing Matrix Market |

1437 | LF10000 | Oberwolfach | 19,998 | 19,998 | 99,982 | Model Reduction Problem | 2004 | MATLAB Rutherford Boeing Matrix Market |