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 2012 1 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 2 / 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 2012 3 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 4 / 45
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 2012 5 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 6 / 45 Substructuring ct. For the staticay condensed problem we obtain Example FEM model of a container ship: 35262 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. 40 20 0 10 0 10 0 50 100 150 200 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 7 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 8 / 45
Example ct. 10 substructures; condensation to 1960 interface DoF Example ct. Container ship: relative errors of static condensation # eigenvalue nodal cond. 1 1.2555112888e-01 5.02e-05 2 1.4842667377e-01 2.36e-05 3 1.8859647898e-01 6.32e-05 4 8.2710672903e-01 1.06e-04 5 1.4571047916e+00 3.98e-04 6 1.8843144791e+00 6.16e-04 7 2.4004294125e+01 5.47e-03 8 5.2973437588e+01 2.11e-02 9 5.6869743387e+01 2.49e-02 10 1.7501327597e+02 8.41e-02 11 2.0806150033e+02 1.08e-01 12 2.8210662009e+02 1.25e-01 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 9 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 10 / 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 2012 11 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 12 / 45
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 1 0 0 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 2012 13 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 14 / 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 2012 15 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 16 / 45
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 50 2183). 329 eigenvalues of the substructure problems were less than our threshold, and the dimension of the resulting projected problem was 2289. 10 2 CMS: cut off frequency 20000 10 3 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 10 4 10 5 10 6 10 7 10 8 0 10 20 30 40 50 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 2012 17 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 18 / 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 1 0 0 0 I ˆM12 0 ˆM14 0 Γ 1 0 0 0 0 Ω 2 0, ˆM 21 I ˆM23 0 0 ˆM32 I ˆM34 (13) 0 0 0 Γ 2 ˆM 41 0 ˆM43 I Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 19 / 45... 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 2012 20 / 45
Relative errors CMS and AMLS(1) Multi-Level Substructuring: Level 0 10 1 CMS and AMLS(1): cut off frequency 20000 10 2 10 3 relative error 10 4 10 5 10 6 10 7 10 8 0 10 20 30 40 50 number of eigenvalue Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 21 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 22 / 45 Multi-Level Substructuring: Level 1 Multi-Level Substructuring: Level 2 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 23 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 24 / 45
Multi-Level Substructuring: Level 3 Multi-Level Substructuring: Level 4 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 25 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 26 / 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 1 2 3 4 K r 5 6 K m 7 K s Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 27 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 28 / 45
AMLS - Algorithm ct. Congruence transformation with U = I 1 Ks K sr Ks 1 O I O O O I yields K s 0 0 0 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 2012 29 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 30 / 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 2012 31 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 32 / 45
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 2012 33 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 34 / 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. 10 0 10 1 10 2 10 3 10 4 10 5 Example FEM model of 2D problem in vibrational analysis with linear Lagrangean elements. n = 68 862 degrees of freedom. AMLS Method 10 eigvals 50 eigvals 200 eigvals Arnoldi 10.7 37.8 221.1 secs Jacobi-Davidson 42.2 148.3 901.9 secs ω c t red t solve n c max.err. 10 max.err. 50 max.err. 200 10 λ 50 205.7 1.7 418 0.14% 3.63% 25.2% 40 λ 50 209.1 7.5 1407 0.014% 0.37% 2.67% 50 λ 50 209.2 10.3 1720 0.0097% 0.24% 1.75% 65 λ 50 211.3 14.7 2166 0.0068% 0.15% 1.05% 10 6 10 7 0 5 10 15 20 25 30 35 40 45 50 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 35 / 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 2012 36 / 45
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 = 517161 DoFs is used. The AMLS method is applied with cut-off frequency ω c = 4 10 9. 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 2012 37 / 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 2012 38 / 45 Connected beams ct. The foowing table contains substructuring information for AMLS level n sub Avg.EVP size Avg. # CNM (Σ) 1 1 681 39.00 (39) 2 2 537 11.50 (23) 3 4 376 7.00 (28) 4 9 548 9.67 (87) 5 27 425 4.70 (127) 6 55 309 2.80 (154) 7 110 268 1.04 (114) 8 220 160 0.19 (41) 9 440 87 0 (0) 10 882 126 0 (0) 11 1060 112 0 (0) 12 953 153 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 2012 39 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 40 / 45
Connected beams ct. Memory aocation profile of AMLS method Connected beams ct. 700 600 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] 500 400 300 200 100 0 0 50 100 150 200 250 300 350 400 450 500 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 2012 41 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 42 / 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 = 1 951 170 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 = 5 431 seconds (approximately 1.5 hours). Bisections are used for the substructuring which results in n sub = 13 694 substructures over n level = 14 levels. Another discretization has been computed with quadratic Lagrange elements and has n = 1 270 947 degrees of freedom. Here, the interface problems are larger than for the linear Lagrange element system. For instance the highest level has 2 202 degrees of freedom and the average size of eigenvalue problems on the fourth level is 1 019. The reduction over 13 levels and n sub = 8 223 substructures takes t red = 4 161 seconds. Significantly raising the cut-off frequency to ω c = 10 11 results in a n c = 10 240 dimensional system. Notice that the computational time increases by less than 3%. Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 43 / 45 Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 44 / 45
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 517 161 4 10 8 218 3 763 12 499 sec Linear 517 161 4 10 9 613 3 763 12 502 sec Quadratic 1 270 947 4 10 9 653 8 223 13 4 161 sec Quadratic 1 270 947 1 10 11 10 240 8 223 13 4 232 sec Linear 1 951 170 4 10 9 648 13 694 14 5 431 sec Linear 2 297 175 4 10 9 651 15 283 14 7 928 sec Heinrich Voss (Hamburg University of Technology) AMLS Eigenvalue problems 2012 45 / 45