Automated Multi-Level Substructuring CHAPTER 4 : AMLS METHOD. Condensation. Exact condensation
|
|
- Warren Boone
- 6 years ago
- Views:
Transcription
1 Automated Multi-Level Substructuring CHAPTER 4 : AMLS METHOD Heinrich Voss voss@tu-harburg.de Hamburg University of Technology AMLS was introduced by Bennighof (1998) and was applied to huge problems of frequency response analysis. The large finite element model is recursively divided into very many substructures on several levels based on the sparsity structure of the system matrices. Assuming that the interior degrees of freedom of substructures depend quasistaticay on the interface degrees of freedom, and modeling the deviation from quasistatic dependence in terms of a sma number of selected substructure eigenmodes the size of the finite element model is reduced substantiay yet yielding satisfactory accuracy over a wide frequency range of interest. Recent studies in vibro-acoustic analysis of passenger car bodies where very large FE models with more than six miion degrees of freedom appear and several hundreds of eigenfrequencies and eigenmodes are needed have shown that AMLS is considerably faster than Lanczos type approaches. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Condensation Exact condensation Partition degrees of freedom into variables x i to be kept (for substructurings: interface DoF) and variables x l to be droped (local DoF). After reordering problem (1) obtains the foowing form Given (a finite element model of a structure, e.g.) Kx = λmx (1) where K R n n and M R n n are symmetric and M is positive definite. Aim: Reduce the number of unknowns by some sort of elimination. ( ) ( ) ( ) ( ) K K li xl M M = λ li xl K ii x i M ii x i K il Solving the first equation for x l yields M il x l = (K λm ) 1 (K li λm li )x i (2) and substituting in the second equation one gets the exactly condensed eigenproblem T (λ)x i = K ii x i + λm ii x i + (K il λm il )(K λm ) 1 (K li λm li )x i Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
2 Static condensation Substructuring Linearizing the exactly condensed problem at ω = 0 yields the staticay condensed eigenproblem (introduced independently by Irons (1965) and Guyan (1965)) K ii x i = λ M ii x i (3) where K ii = K ii K il K 1 K li M ii = M ii K il K 1 M li M li K 1 K li + K il K 1 M K 1 K li For vibrating structures this means that the local degrees of freedom are assumed to depend quasistaticay on the interface degrees of freedom, and the inertia forces of the substructures are neglected. Consider the vibrations of a structure which is partitioned into r substructures connecting to each other through the variables on the interfaces only. Then ordering the unknowns appropriately the stiffness matrix obtains the foowing block form K 1 O... O K li1 O K 2... O K li2 K = O O... K ssr K smr K il1 K il2... K msr K ii and M has the same block form. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Substructuring ct. For the staticay condensed problem we obtain Example FEM model of a container ship: DoF, bandwidth: 1072 K ii = K ii r j=1 K msj K 1 ssj K smj M ii = M ii r M mmj, j=1 where M mmj = K msj K 1 ssj M smj + M msj K 1 ssj K smj K msj K 1 ssj M ssj K 1 ssj K smj. The submatrices corresponding to the individual substructures can be determined independently from smaer subproblems and in parael Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
3 Example ct. 10 substructures; condensation to 1960 interface DoF Example ct. Container ship: relative errors of static condensation # eigenvalue nodal cond e e e e e e e e e e e e e e e e e e e e e e e e-01 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 A projection approach We transform the matrix K to block diagonal form using block Gaussian elimination, i.e. we apply the congruence transformation with ( I K 1 P = K ) li 0 I to the pencil (K, M) obtaining the equivalent pencil (( (P T KP, P T K 0 MP) = 0 Kii Here K and M stay unchanged, and ) ( )) M Mli,. (4) M il Mii K ii = K ii K il K 1 K li is the Schur complement of K M li = M li M K 1 K li = M il T M ii = M ii M il K 1 K li K il K 1 M li + K il K 1 M K 1 K li. static condensation revisited Neglecting in (4) a rows and columns corresponding to local degrees of freedom, ( i.e. projecting problem (1) to the subspace spanned by columns of K 1 K ) li one obtains the method of static condensation I K ii y = λ M ii y To model the deviation from quasistatic behavior thereby improving the approximation properties of static condensation we consider the eigenvalue problem K Φ = M ΦΩ, Φ T M Φ = I, (5) where Ω is a diagonal matrix containing the eigenvalues. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
4 Craig Bampton form Changing the basis for the local degrees of freedom to a modal one, i.e. applying the further congruence transformation diag{φ, I} to problem (4) one gets (( ) ( )) Ω 0 I Φ T Mli,. (6) M il Φ Mii 0 Kii In structural dynamics (6) is caed Craig Bampton form of the eigenvalue problem (1) corresponding to the partitioning (2). In terms of linear algebra it results from block Gaussian elimination to reduce K to block diagonal form, and diagonalization of the block K using a spectral basis. Component Mode Synthesis (CMS) Selecting some eigenmodes of problem (5), and dropping the rows and columns in (6) corresponding to the other modes one arrives at the component mode synthesis method (CMS) introduced by Hurty (1965) and Craig & Bampton (1968). If the diagonal matrix Ω 1 contains in its diagonal the eigenvalues to drop and Φ 1 the corresponding eigenvectors, and if Ω 2 and Φ 2 contain the eigenvalues and eigenvectors to keep, respectively, then the eigenproblem (6) can be rewritten as Ω x 1 I 0 Mli1 x 1 0 Ω 2 0 x 2 = λ 0 I Mli2 x 2 (7) 0 0 Kii with x 3 M il1 Mil2 Mii M smj = Φ T j (M li M K 1 K li) = M T msj, j = 1, 2, x 3 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 CMS ct. and the CMS approximations to the eigenpairs of (1) are obtained from the reduced eigenvalue problem ( Ω2 0 0 Kii ) ( ) I Mli2 y = λ y (8) M il2 Mii Usuay the eigenvectors according to eigenvalues which do not exceed a cut off threshold are kept. In vibration analysis of a structure this choice is motivated by the fact that the high frequencies of a substructure do not influence the wanted low frequencies of the entire substructure very much. Notice however that in a recent paper Bai and Liao (2006) suggested a different choice based on a moment matching analysis. Container ship We consider the structural deformation caused by a harmonic excitation at a frequency of 4 Hz which is a typical forcing frequency stemming from the engine and the propeer. Since the deformation is sma the assumptions of the linear theory apply, and the structural response can be determined by the mode superposition method taking into account eigenfrequencies in the range between 0 and 7.5 Hz (which corresponds to the 50 smaest eigenvalues for the ship under consideration). To apply the CMS method we partitioned the FEM model into 10 substructures as shown before. This substructuring by hand yielded a much smaer number of interface degrees of freedom than automatic graph partitioners which try to construct a partition where the substructures have nearly equal size. For instance, our model ends up with 1960 degrees of freedom on the interfaces, whereas Chaco ends up with a substructuring into 10 substructures with 4985 interface degrees of freedom. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
5 Container ship ct. We solved the eigenproblem by the CMS method using a cut-off bound of 20,000 (about 10 times the largest wanted eigenvalue λ ). 329 eigenvalues of the substructure problems were less than our threshold, and the dimension of the resulting projected problem was CMS: cut off frequency Reducing interface DoF The number of interface degrees of freedom may sti be very large, and therefore the dimension of the reduced problem (8) may be very high. It can be reduced further by modal reduction of the interface degrees of freedom in the foowing way: Considering the eigenvalue problem K ii Ψ = M ii ΨΓ, Ψ T Kii Ψ = Γ, Ψ T Mii Ψ = I, (10) relative error number of eigenvalue and applying the congruence transformation to the pencil in (6) with P = diag{i, Ψ}, we obtain the equivalent pencil with (( ) ( )) Ω O I ˆMli, O Γ I ˆM T li (11) ˆM li = Φ T (M li M K 1 K li)ψ = ˆM il. T (12) Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Reducing interface DoF ct. Selecting eigenmodes of (5) and of (10) and neglecting rows and columns in (11) which correspond to the other modes one gets a reduced problem which is the one level version of the automated multilevel substructuring method, introduced by Bennighof (1992). Similarly as for the CMS method we partition the matrices Γ and Ψ into ( ) Γ1 0 Γ = and Ψ = (Ψ 0 Γ 1, Ψ 2 ) 2 and rearranging the rows and columns beginning with the modes corresponding to Φ 1 and Ψ 1 to be dropped foowed by the ones corresponding to Φ 2 and Ψ 2 problem (11) obtains the form Ω I ˆM12 0 ˆM14 0 Γ Ω 2 0, ˆM 21 I ˆM ˆM32 I ˆM34 (13) Γ 2 ˆM 41 0 ˆM43 I Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / where Reducing interface DoF ct. ˆM 12 = Φ T 1 (M li M K 1 K li)ψ 1 = ˆM T 21 ˆM 14 = Φ T 1 (M li M K 1 K li)ψ 2 = ˆM T 41 ˆM 32 = Φ T 2 (M li M K 1 K li)ψ 1 = ˆM T 23 ˆM 34 = Φ T 2 (M li M K 1 K li)ψ 2 = ˆM T 43. Then the single level approximations of AMLS to eigenpairs are obtained from ( ) ( ) Ω2 0 I ˆM34 y = λ y. (14) 0 Γ 2 I For the container ship we reduced the interface degrees of freedom as we with the same cut-off bound 20,000. This reduced the dimension of the projected eigenproblem further from 2289 to 436. ˆM 43 The next picture shows the relative errors of CMS and the single level version of AMLS. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
6 Relative errors CMS and AMLS(1) Multi-Level Substructuring: Level CMS and AMLS(1): cut off frequency relative error number of eigenvalue Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Multi-Level Substructuring: Level 1 Multi-Level Substructuring: Level 2 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
7 Multi-Level Substructuring: Level 3 Multi-Level Substructuring: Level 4 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Multi-Level Substructuring: Level 5 AMLS - Algorithm (Kx = λmx) Reorder System (using Graph Partitioner): K s K sm K s1 sr Ksm T K m K mr with K s = K... Ksr T Kmr T K r K sn K r 5 6 K m 7 K s Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
8 AMLS - Algorithm ct. Congruence transformation with U = I 1 Ks K sr Ks 1 O I O O O I yields K s ˆKm ˆKmr, 0 T ˆK mr ˆKr K mr M s ˆMsm ˆMsr ˆM sm T ˆM m ˆMmr ˆM sr T ˆM mr T ˆM r Solving of substructure EVPs AMLS - Algorithm ct. K s Φ s = M s Φ s Ω s, Φ T s M s Φ s = I and projecting on a subset of Φ s (usuay corresponding to eigenvalues not exceeding a cut-off frequency) yields Ω s 0 0 I s ˆMsm ˆMsr 0 ˆKm ˆKmr, ˆM sm T ˆM m ˆMmr T 0 ˆK mr ˆKr ˆM r ˆM T sr ˆM T mr Notice that K s is block-diagonal, and determining Ks 1 K sr means that a large number of linear system of sma dimension have to solved. Moreover, the congruence transformation consists of block matrix multiplications for blocks of sma dimension. This first step of AMLS was introduced already by Hurty (1965) and by Craig and Bampton (1968), and it is caed Component Mode Synthesis (CMS). Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 AMLS - Algorithm ct. Once substructures on the lowest level have been transformed and reduced by modal projection they are assembled to parent substructures on the next level. AMLS - Algorithm ct. Interface and local degrees of freedom are identified, and the substructure models are transformed similarly as on the lowest level. Ω 1 O O O O Ω 2 O O O O K ii Kir O O K ir H K rr z 1 z 2 z 3 z 4 I O M1i M1r = λ O I M2i M2r M 1i H M 1r H M 2i H M 2r H M ii M ir H Mir M rr z 1 z 2 z 3 z 4, Block-elimination of K jr yields Ω 1 O O O O Ω w 1 I O M1i ˆM1r 2 O O O O K w 2 ii O w 3 = λ O I M2i ˆM2r M H O O O ˆK 1i M 2i H M ii ˆMir rr w 4 ˆM H 1r ˆM H 2r ˆM H ir ˆM rr w 1 w 2 w 3 w 4, Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
9 AMLS - Algorithm ct. To perform the modal reduction of the interior degrees of freedom of the current substructure one would have to solve the eigenvalue problem Ω 1 O O O Ω 2 O w 1 I O M1i w 1 w 2 = ω O I M2i w 2. O O K ii w 3 w 3 However, since the number of interior degrees of freedom of substructures grows too large in the course of the algorithm, we reduce the dimension only taking advantage of the eigenvalue problem corresponding to the right lower diagonal block, i.e. K ii Φ i = M ii Φ i Ω i, Φ H i M ii Φ i = I. M H 1i M H 2i M ii AMLS - Algorithm ct. Treating coarser levels one after the other in the same way one gets a projected eigenvalue problem of significantly lower dimension K c x = λm c x with K c spd and diagonal and M c spd in generalized arrowhead structure. Massmatrix of AMLS Applying the congruence transformation with T = diag{i, I, Φ i, I} and dropping a rows and columns in the third block if the corresponding eigenvalue exceeds the cut-off frequency we further reduce the dimension of the eigenproblem. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Container ship We substructured the FE model of the container ship by Metis with 4 levels of substructuring. Neglecting eigenvalues exceeding 20,000 and 40,000 on a levels AMLS produced a projected eigenvalue problem of dimension 451 and 911, respectively Example FEM model of 2D problem in vibrational analysis with linear Lagrangean elements. n = degrees of freedom. AMLS Method 10 eigvals 50 eigvals 200 eigvals Arnoldi secs Jacobi-Davidson secs ω c t red t solve n c max.err. 10 max.err. 50 max.err λ % 3.63% 25.2% 40 λ % 0.37% 2.67% 50 λ % 0.24% 1.75% 65 λ % 0.15% 1.05% Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Even this quite sma sized eigenvalue problems demonstrates that AMLS becomes competitive if a large number of eigenvalues is wanted the accuracy of which need not be too high. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
10 Connected beams We report on the performance of AMLS for a FE model of a structure of connected beams Connected beams ct. For the foowing analysis a discretization with linear Lagrangean elements with n = DoFs is used. The AMLS method is applied with cut-off frequency ω c = Due to the linear elements the matrices are relatively sparse resulting in sma interfaces over a levels. Consequently, the eigenvalue problems are sma as we, which can be seen in the average size of the eigenvalue problems on each level. The computer used is a 32-bit workstation with a 3.0 GHz Pentium and 1.5 GByte memory. AMLS is implemented (by Kolja Elssel) in C using METIS for computing graph partitions and LAPACK. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 The distribution of component normal modes over the levels is typical for large scale problems. The average number of component normal modes (CNM) for the interface and substructure eigenvalues problems that are below the cut-off frequency decreases on lower levels. Quite commonly no CNMs are used for the lowest level substructures. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Connected beams ct. The foowing table contains substructuring information for AMLS level n sub Avg.EVP size Avg. # CNM (Σ) (39) (23) (28) (87) (127) (154) (114) (41) (0) (0) (0) (0) Connected beams ct. The limiting factors for the applicability of the algorithm are the computational time and the memory requirements. For the computations discussed external storage was used to store contemporary data and data needed for subsequent calculations such as the computation of Ritz vectors. The foowing figure shows the memory consumption and the temporary storage needed by the algorithm. The large peak at the beginning of the calculation and in the middle are due the graph partitioner which computes partitions for the graph corresponding to the system matrices. For larger systems this becomes a limiting factor. For systems which have a denser structure the size of the interface problems become larger and cause problems with memory consumption and the solution of the interface eigenvalue problems. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
11 Connected beams ct. Memory aocation profile of AMLS method Connected beams ct Memory Harddisk The computational time for this problem can be roughly divided into three parts. With 60% the largest part of the computational time is spent on matrix multiplications resulting from the variable transformations in step 2 of the AMLS algorithm. Memory Aocation [MByte] Time [seconds] The second largest part is with 20% due to the eigenvalue solver, foowed by the solution of linear systems of equations with 15%. The remaining five percent consist of matrix partitioning (about 3%), matrix substructuring and algorithmic overhead. Note, the matrix multiplications originating from the eigensolver and the linear system solver are included into their respective percentages and are not included in the percentage of the matrix multiplications. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Connected beams ct. Connected beams ct. To compare the scalability other discretization of the same model have been computed. One of the largest systems which has been reduced with the AMLS method has been discretized with linear Lagrange elements and has n = degrees of freedom (about 78 miion non-zeros in the stiffness matrix and 26 miion in the mass matrix). The computational time for this discretization is t red = seconds (approximately 1.5 hours). Bisections are used for the substructuring which results in n sub = substructures over n level = 14 levels. Another discretization has been computed with quadratic Lagrange elements and has n = degrees of freedom. Here, the interface problems are larger than for the linear Lagrange element system. For instance the highest level has degrees of freedom and the average size of eigenvalue problems on the fourth level is The reduction over 13 levels and n sub = substructures takes t red = seconds. Significantly raising the cut-off frequency to ω c = results in a n c = dimensional system. Notice that the computational time increases by less than 3%. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
12 Connected beams ct. Results of AMLS applied to large linear eigenvalue problems Elements n ω c n c n sub n level t red Linear sec Linear sec Quadratic sec Quadratic sec Linear sec Linear sec Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems / 45
Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem
Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem Introduction Equations of dynamic equilibrium eigenvalue problem K x = ω M x The eigensolutions of this problem are written in
More informationTechnical University Hamburg { Harburg, Section of Mathematics, to reduce the number of degrees of freedom to manageable size.
Interior and modal masters in condensation methods for eigenvalue problems Heinrich Voss Technical University Hamburg { Harburg, Section of Mathematics, D { 21071 Hamburg, Germany EMail: voss @ tu-harburg.d400.de
More informationPreconditioning Subspace Iteration for Large Eigenvalue Problems with Automated Multi-Level Sub-structuring
Preconditioning Subspace Iteration for Large Eigenvalue Problems with Automated Multi-Level Sub-structuring Heinrich Voss 1, and Jiacong Yin 2 and Pu Chen 2 1 Institute of Mathematics, Hamburg University
More informationT(λ)x = 0 (1.1) k λ j A j x = 0 (1.3)
PROJECTION METHODS FOR NONLINEAR SPARSE EIGENVALUE PROBLEMS HEINRICH VOSS Key words. nonlinear eigenvalue problem, iterative projection method, Jacobi Davidson method, Arnoldi method, rational Krylov method,
More informationhave invested in supercomputer systems, which have cost up to tens of millions of dollars each. Over the past year or so, however, the future of vecto
MEETING THE NVH COMPUTATIONAL CHALLENGE: AUTOMATED MULTI-LEVEL SUBSTRUCTURING J. K. Bennighof, M. F. Kaplan, y M. B. Muller, y and M. Kim y Department of Aerospace Engineering & Engineering Mechanics The
More informationSolving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices
Solving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices Peter Gerds and Lars Grasedyck Bericht Nr. 30 März 2014 Key words: automated multi-level substructuring,
More informationMultilevel Methods for Eigenspace Computations in Structural Dynamics
Multilevel Methods for Eigenspace Computations in Structural Dynamics Ulrich Hetmaniuk & Rich Lehoucq Sandia National Laboratories, Computational Math and Algorithms, Albuquerque, NM Joint work with Peter
More informationEigenvalue Problems CHAPTER 1 : PRELIMINARIES
Eigenvalue Problems CHAPTER 1 : PRELIMINARIES Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Preliminaries Eigenvalue problems 2012 1 / 14
More informationReduction in number of dofs
Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole
More informationKeeping σ fixed for several steps, iterating on µ and neglecting the remainder in the Lagrange interpolation one obtains. θ = λ j λ j 1 λ j σ, (2.
RATIONAL KRYLOV FOR NONLINEAR EIGENPROBLEMS, AN ITERATIVE PROJECTION METHOD ELIAS JARLEBRING AND HEINRICH VOSS Key words. nonlinear eigenvalue problem, rational Krylov, Arnoldi, projection method AMS subject
More informationA METHOD FOR PROFILING THE DISTRIBUTION OF EIGENVALUES USING THE AS METHOD. Kenta Senzaki, Hiroto Tadano, Tetsuya Sakurai and Zhaojun Bai
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 3A, pp. 839-853, June 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ A METHOD FOR PROFILING THE DISTRIBUTION OF EIGENVALUES USING THE
More informationAn Arnoldi Method for Nonlinear Symmetric Eigenvalue Problems
An Arnoldi Method for Nonlinear Symmetric Eigenvalue Problems H. Voss 1 Introduction In this paper we consider the nonlinear eigenvalue problem T (λ)x = 0 (1) where T (λ) R n n is a family of symmetric
More informationA Jacobi Davidson Method for Nonlinear Eigenproblems
A Jacobi Davidson Method for Nonlinear Eigenproblems Heinrich Voss Section of Mathematics, Hamburg University of Technology, D 21071 Hamburg voss @ tu-harburg.de http://www.tu-harburg.de/mat/hp/voss Abstract.
More informationAn Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems
An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems WEIGUO GAO Fudan University XIAOYE S. LI and CHAO YANG Lawrence Berkeley National Laboratory and ZHAOJUN BAI University
More informationComputation of Smallest Eigenvalues using Spectral Schur Complements
Computation of Smallest Eigenvalues using Spectral Schur Complements Constantine Bekas Yousef Saad January 23, 2004 Abstract The Automated Multilevel Substructing method (AMLS ) was recently presented
More informationIterative projection methods for sparse nonlinear eigenvalue problems
Iterative projection methods for sparse nonlinear eigenvalue problems Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Iterative projection
More informationCOMPUTATIONAL ASSESSMENT OF REDUCTION METHODS IN FE-BASED FREQUENCY-RESPONSE ANALYSIS
11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver
More informationTHE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES. Lou Menglin* SUMMARY
264 THE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES Lou Mengl* SUMMARY In this paper, the static substructure method based on the Ritz vector direct superposition method is suggested
More informationEffect of Mass Matrix Formulation Schemes on Dynamics of Structures
Effect of Mass Matrix Formulation Schemes on Dynamics of Structures Swapan Kumar Nandi Tata Consultancy Services GEDC, 185 LR, Chennai 600086, India Sudeep Bosu Tata Consultancy Services GEDC, 185 LR,
More informationVibration Transmission in Complex Vehicle Structures
Vibration Transmission in Complex Vehicle Structures Christophe Pierre Professor Matthew P. Castanier Assistant Research Scientist Yung-Chang Tan Dongying Jiang Graduate Student Research Assistants Vibrations
More informationIncomplete Cholesky preconditioners that exploit the low-rank property
anapov@ulb.ac.be ; http://homepages.ulb.ac.be/ anapov/ 1 / 35 Incomplete Cholesky preconditioners that exploit the low-rank property (theory and practice) Artem Napov Service de Métrologie Nucléaire, Université
More information2C9 Design for seismic and climate changes. Jiří Máca
2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-dof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More informationEigenvalue Problems in Surface Acoustic Wave Filter Simulations
Eigenvalue Problems in Surface Acoustic Wave Filter Simulations S. Zaglmayr, J. Schöberl, U. Langer FWF Start Project Y-192 3D hp-finite Elements: Fast Solvers and Adaptivity JKU + RICAM, Linz in collaboration
More informationThis appendix gives you a working knowledge of the theory used to implement flexible bodies in ADAMS. The topics covered include
Appendix D Theoretical Background This appendix gives you a working knowledge of the theory used to implement flexible bodies in ADAMS. The topics covered include modal superposition component mode synthesis,
More informationDomain decomposition on different levels of the Jacobi-Davidson method
hapter 5 Domain decomposition on different levels of the Jacobi-Davidson method Abstract Most computational work of Jacobi-Davidson [46], an iterative method suitable for computing solutions of large dimensional
More informationEfficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures
Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures Yung-Chang Tan Graduate Student Research Assistant Matthew P. Castanier Assistant Research Scientist
More informationOutline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices
Outline in MDOF Systems Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano May 8, 014 Additional Today we will study the properties of structural matrices, that is the operators that
More informationITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 11 : JACOBI DAVIDSON METHOD
ITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 11 : JACOBI DAVIDSON METHOD Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation
More informationA Jacobi Davidson-type projection method for nonlinear eigenvalue problems
A Jacobi Davidson-type projection method for nonlinear eigenvalue problems Timo Betce and Heinrich Voss Technical University of Hamburg-Harburg, Department of Mathematics, Schwarzenbergstrasse 95, D-21073
More informationChapter 4 Analysis of a cantilever
Chapter 4 Analysis of a cantilever Before a complex structure is studied performing a seismic analysis, the behaviour of simpler ones should be fully understood. To achieve this knowledge we will start
More informationBEYOND AMLS: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING
EYOND AMLS: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING VASSILIS KALANTZIS, YUANZHE XI, AND YOUSEF SAAD Abstract. This paper proposes a rational filtering domain decomposition technique for the solution
More informationStructural Matrices in MDOF Systems
in MDOF Systems http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 9, 2016 Outline Additional Static Condensation
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit VII Sparse Matrix
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit VII Sparse Matrix Computations Part 1: Direct Methods Dianne P. O Leary c 2008
More informationDavidson Method CHAPTER 3 : JACOBI DAVIDSON METHOD
Davidson Method CHAPTER 3 : JACOBI DAVIDSON METHOD Heinrich Voss voss@tu-harburg.de Hamburg University of Technology The Davidson method is a popular technique to compute a few of the smallest (or largest)
More informationITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 3 : SEMI-ITERATIVE METHODS
ITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 3 : SEMI-ITERATIVE METHODS Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation
More informationNumerical Methods in Matrix Computations
Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices
More informationBEYOND AUTOMATED MULTILEVEL SUBSTRUCTURING: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING
SIAM J SCI COMPUT Vol 0, No 0, pp 000 000 c XXXX Society for Industrial and Applied Mathematics EYOND AUTOMATED MULTILEVEL SUSTRUCTURING: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING VASSILIS KALANTZIS,
More informationStructural Dynamics Lecture 7. Outline of Lecture 7. Multi-Degree-of-Freedom Systems (cont.) System Reduction. Vibration due to Movable Supports.
Outline of Multi-Degree-of-Freedom Systems (cont.) System Reduction. Truncated Modal Expansion with Quasi-Static Correction. Guyan Reduction. Vibration due to Movable Supports. Earthquake Excitations.
More informationParallelization of Multilevel Preconditioners Constructed from Inverse-Based ILUs on Shared-Memory Multiprocessors
Parallelization of Multilevel Preconditioners Constructed from Inverse-Based ILUs on Shared-Memory Multiprocessors J.I. Aliaga 1 M. Bollhöfer 2 A.F. Martín 1 E.S. Quintana-Ortí 1 1 Deparment of Computer
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT -09 Computational and Sensitivity Aspects of Eigenvalue-Based Methods for the Large-Scale Trust-Region Subproblem Marielba Rojas, Bjørn H. Fotland, and Trond Steihaug
More informationMatrix Iteration. Giacomo Boffi.
http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 12, 2016 Outline Second -Ritz Method Dynamic analysis of MDOF
More informationDISPENSA FEM in MSC. Nastran
DISPENSA FEM in MSC. Nastran preprocessing: mesh generation material definitions definition of loads and boundary conditions solving: solving the (linear) set of equations components postprocessing: visualisation
More informationStructured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries
Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries Fakultät für Mathematik TU Chemnitz, Germany Peter Benner benner@mathematik.tu-chemnitz.de joint work with Heike Faßbender
More informationINPUT-OUTPUT BASED MODEL REDUCTION FOR INTERCONNECTED SYSTEMS
11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver
More informationA substructuring FE model for reduction of structural acoustic problems with dissipative interfaces
A substructuring FE model for reduction of structural acoustic problems with dissipative interfaces PhD started October 2008 Romain RUMPLER Conservatoire National des Arts et Métiers - Cnam Future ESR
More informationMatrix Assembly in FEA
Matrix Assembly in FEA 1 In Chapter 2, we spoke about how the global matrix equations are assembled in the finite element method. We now want to revisit that discussion and add some details. For example,
More informationPFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers
PFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers James Kestyn, Vasileios Kalantzis, Eric Polizzi, Yousef Saad Electrical and Computer Engineering Department,
More informationAPVC2009. Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations
Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations Kiyong Park, Gunhee Jang* and Chanhee Seo Department of Mechanical Engineering,
More informationPartitioned Formulation with Localized Lagrange Multipliers And its Applications **
Partitioned Formulation with Localized Lagrange Multipliers And its Applications ** K.C. Park Center for Aerospace Structures (CAS), University of Colorado at Boulder ** Carlos Felippa, Gert Rebel, Yong
More informationAdaptive Coarse Space Selection in BDDC and FETI-DP Iterative Substructuring Methods: Towards Fast and Robust Solvers
Adaptive Coarse Space Selection in BDDC and FETI-DP Iterative Substructuring Methods: Towards Fast and Robust Solvers Jan Mandel University of Colorado at Denver Bedřich Sousedík Czech Technical University
More informationA comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control
A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control B. Besselink a, A. Lutowska b, U. Tabak c, N. van de Wouw a, H. Nijmeijer a, M.E. Hochstenbach
More informationITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 1 : INTRODUCTION
ITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 1 : INTRODUCTION Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation TUHH
More informationApplication of Lanczos and Schur vectors in structural dynamics
Shock and Vibration 15 (2008) 459 466 459 IOS Press Application of Lanczos and Schur vectors in structural dynamics M. Radeş Universitatea Politehnica Bucureşti, Splaiul Independenţei 313, Bucureşti, Romania
More informationParallel Eigensolver Performance on High Performance Computers
Parallel Eigensolver Performance on High Performance Computers Andrew Sunderland Advanced Research Computing Group STFC Daresbury Laboratory CUG 2008 Helsinki 1 Summary (Briefly) Introduce parallel diagonalization
More informationOn correction equations and domain decomposition for computing invariant subspaces
On correction equations and domain decomposition for computing invariant subspaces Bernard Philippe Yousef Saad February 1, 26 Abstract By considering the eigenvalue problem as a system of nonlinear equations,
More information822. Non-iterative mode shape expansion for threedimensional structures based on coordinate decomposition
822. Non-iterative mode shape expansion for threedimensional structures based on coordinate decomposition Fushun Liu, Zhengshou Chen 2, Wei Li 3 Department of Ocean Engineering, Ocean University of China,
More informationstiffness to the system stiffness matrix. The nondimensional parameter i is introduced to allow the modeling of damage in the ith substructure. A subs
A BAYESIAN PROBABILISTIC DAMAGE DETECTION USING LOAD-DEPENDENT RIT VECTORS HOON SOHN Λ and KINCHO H. LAW y Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 9435-42,U.S.A.
More informationArnoldi Methods in SLEPc
Scalable Library for Eigenvalue Problem Computations SLEPc Technical Report STR-4 Available at http://slepc.upv.es Arnoldi Methods in SLEPc V. Hernández J. E. Román A. Tomás V. Vidal Last update: October,
More informationA robust multilevel approximate inverse preconditioner for symmetric positive definite matrices
DICEA DEPARTMENT OF CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING PhD SCHOOL CIVIL AND ENVIRONMENTAL ENGINEERING SCIENCES XXX CYCLE A robust multilevel approximate inverse preconditioner for symmetric
More informationCourse Notes: Week 1
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues
More informationPresentation of XLIFE++
Presentation of XLIFE++ Eigenvalues Solver & OpenMP Manh-Ha NGUYEN Unité de Mathématiques Appliquées, ENSTA - Paristech 25 Juin 2014 Ha. NGUYEN Presentation of XLIFE++ 25 Juin 2014 1/19 EigenSolver 1 EigenSolver
More informationModal Analysis: What it is and is not Gerrit Visser
Modal Analysis: What it is and is not Gerrit Visser What is a Modal Analysis? What answers do we get out of it? How is it useful? What does it not tell us? In this article, we ll discuss where a modal
More informationA priori verification of local FE model based force identification.
A priori verification of local FE model based force identification. M. Corus, E. Balmès École Centrale Paris,MSSMat Grande voie des Vignes, 92295 Châtenay Malabry, France e-mail: corus@mssmat.ecp.fr balmes@ecp.fr
More information11 th International Computational Accelerator Physics Conference (ICAP) 2012
Eigenmode Computation For Ferrite-Loaded Cavity Resonators Klaus Klopfer*, Wolfgang Ackermann, Thomas Weiland Institut für Theorie Elektromagnetischer Felder, TU Darmstadt 11 th International Computational
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More informationThe quadratic eigenvalue problem (QEP) is to find scalars λ and nonzero vectors u satisfying
I.2 Quadratic Eigenvalue Problems 1 Introduction The quadratic eigenvalue problem QEP is to find scalars λ and nonzero vectors u satisfying where Qλx = 0, 1.1 Qλ = λ 2 M + λd + K, M, D and K are given
More informationProgram System for Machine Dynamics. Abstract. Version 5.0 November 2017
Program System for Machine Dynamics Abstract Version 5.0 November 2017 Ingenieur-Büro Klement Lerchenweg 2 D 65428 Rüsselsheim Phone +49/6142/55951 hd.klement@t-online.de What is MADYN? The program system
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 19: Computing the SVD; Sparse Linear Systems Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationproblem Au = u by constructing an orthonormal basis V k = [v 1 ; : : : ; v k ], at each k th iteration step, and then nding an approximation for the e
A Parallel Solver for Extreme Eigenpairs 1 Leonardo Borges and Suely Oliveira 2 Computer Science Department, Texas A&M University, College Station, TX 77843-3112, USA. Abstract. In this paper a parallel
More informationContour Integral Method for the Simulation of Accelerator Cavities
Contour Integral Method for the Simulation of Accelerator Cavities V. Pham-Xuan, W. Ackermann and H. De Gersem Institut für Theorie Elektromagnetischer Felder DESY meeting (14.11.2017) November 14, 2017
More informationBlock Iterative Eigensolvers for Sequences of Dense Correlated Eigenvalue Problems
Mitglied der Helmholtz-Gemeinschaft Block Iterative Eigensolvers for Sequences of Dense Correlated Eigenvalue Problems Birkbeck University, London, June the 29th 2012 Edoardo Di Napoli Motivation and Goals
More informationSolving Regularized Total Least Squares Problems
Solving Regularized Total Least Squares Problems Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation Joint work with Jörg Lampe TUHH Heinrich Voss Total
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationMultiple Degree of Freedom Systems. The Millennium bridge required many degrees of freedom to model and design with.
Multiple Degree of Freedom Systems The Millennium bridge required many degrees of freedom to model and design with. The first step in analyzing multiple degrees of freedom (DOF) is to look at DOF DOF:
More informationEIGIFP: A MATLAB Program for Solving Large Symmetric Generalized Eigenvalue Problems
EIGIFP: A MATLAB Program for Solving Large Symmetric Generalized Eigenvalue Problems JAMES H. MONEY and QIANG YE UNIVERSITY OF KENTUCKY eigifp is a MATLAB program for computing a few extreme eigenvalues
More informationVariational Principles for Nonlinear Eigenvalue Problems
Variational Principles for Nonlinear Eigenvalue Problems Heinrich Voss voss@tuhh.de Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Variational Principles for Nonlinear EVPs
More informationA Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation
A Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation Tao Zhao 1, Feng-Nan Hwang 2 and Xiao-Chuan Cai 3 Abstract In this paper, we develop an overlapping domain decomposition
More informationCommunities, Spectral Clustering, and Random Walks
Communities, Spectral Clustering, and Random Walks David Bindel Department of Computer Science Cornell University 3 Jul 202 Spectral clustering recipe Ingredients:. A subspace basis with useful information
More informationENERGY FLOW MODELS FROM FINITE ELEMENTS: AN APPLICATION TO THREE COUPLED PLATES
FIFTH INTERNATIONAL CONGRESS ON SOUND DECEMBER 15-18, 1997 ADELAIDE, SOUTH AUSTRALIA AND VIBRATION ENERGY FLOW MODELS FROM FINITE ELEMENTS: AN APPLICATION TO THREE COUPLED PLATES P.J. Shorter and B.R.
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationEML4507 Finite Element Analysis and Design EXAM 1
2-17-15 Name (underline last name): EML4507 Finite Element Analysis and Design EXAM 1 In this exam you may not use any materials except a pencil or a pen, an 8.5x11 formula sheet, and a calculator. Whenever
More informationActive Integral Vibration Control of Elastic Bodies
Applied and Computational Mechanics 2 (2008) 379 388 Active Integral Vibration Control of Elastic Bodies M. Smrž a,m.valášek a, a Faculty of Mechanical Engineering, CTU in Prague, Karlovo nam. 13, 121
More informationMatrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland
Matrix Algorithms Volume II: Eigensystems G. W. Stewart University of Maryland College Park, Maryland H1HJ1L Society for Industrial and Applied Mathematics Philadelphia CONTENTS Algorithms Preface xv xvii
More informationThe Finite Element Method
Page he Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. o. Dr. Michael Havbro bo Faber Dr. Nebojsa Mojsilovic Swiss Federal Institute of EH Zurich, Switzerland Contents of
More informationFast transient structural FE analysis imposing prescribed displacement condition. by using a model order reduction method via Krylov subspace
Journal of Applied Mechanics Vol.13 Vol.13, (August pp.159-167 21) (August 21) JSCE JSCE Fast transient structural FE analysis imposing prescribed displacement condition by using a model order reduction
More informationPerturbation of periodic equilibrium
Perturbation of periodic equilibrium by Arnaud Lazarus A spectral method to solve linear periodically time-varying systems 1 A few history Late 19 th century Emile Léonard Mathieu: Wave equation for an
More informationPreface to the Second Edition. Preface to the First Edition
n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................
More informationParallel scalability of a FETI DP mortar method for problems with discontinuous coefficients
Parallel scalability of a FETI DP mortar method for problems with discontinuous coefficients Nina Dokeva and Wlodek Proskurowski University of Southern California, Department of Mathematics Los Angeles,
More informationSparse solver 64 bit and out-of-core addition
Sparse solver 64 bit and out-of-core addition Prepared By: Richard Link Brian Yuen Martec Limited 1888 Brunswick Street, Suite 400 Halifax, Nova Scotia B3J 3J8 PWGSC Contract Number: W7707-145679 Contract
More informationAssembling Reduced-Order Substructural Models
. 21 Assembling Reduced-Order Substructural Models 21 1 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS 21 2 21.1 NTRODUCTON The previous chapter has devoted to the reduced-order modeling o a
More informationLARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems
LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization General Tools for Solving Large Eigen-Problems
More informationAA 242B / ME 242B: Mechanical Vibrations (Spring 2016)
AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally
More informationIMPROVEMENT OF A STRUCTURAL MODIFICATION METHOD
MPROVEMENT OF A STRUCTURAL MODFCATON METHOD USNG DATA EXPANSON AND MODEL REDUCTON TECHNQUES Mathieu Corus,Etienne Balmès EDF DR&D, 1 Avenue du Général De Gaule, 92141 Clamart Cedex, France ECP, MSSMat,
More informationABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in s
ABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in system identification, finite element model updating,
More informationLARGE SPARSE EIGENVALUE PROBLEMS
LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization 14-1 General Tools for Solving Large Eigen-Problems
More informationDomain decomposition for the Jacobi-Davidson method: practical strategies
Chapter 4 Domain decomposition for the Jacobi-Davidson method: practical strategies Abstract The Jacobi-Davidson method is an iterative method for the computation of solutions of large eigenvalue problems.
More informationPERFORMANCE ENHANCEMENT OF PARALLEL MULTIFRONTAL SOLVER ON BLOCK LANCZOS METHOD 1. INTRODUCTION
J. KSIAM Vol., No., -0, 009 PERFORMANCE ENHANCEMENT OF PARALLEL MULTIFRONTAL SOLVER ON BLOCK LANCZOS METHOD Wanil BYUN AND Seung Jo KIM, SCHOOL OF MECHANICAL AND AEROSPACE ENG, SEOUL NATIONAL UNIV, SOUTH
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationMultispace and Multilevel BDDC. Jan Mandel University of Colorado at Denver and Health Sciences Center
Multispace and Multilevel BDDC Jan Mandel University of Colorado at Denver and Health Sciences Center Based on joint work with Bedřich Sousedík, UCDHSC and Czech Technical University, and Clark R. Dohrmann,
More information