Determination of Natural Frequencies and Mode Shapes of Structures Using Subspace Iteration Method with Accelerated Starting Vectors Byoung-Wan Kim1; Sang-Won Cho2; Chun-Ho Kim3; and In-Won Lee4 Abstract: Natural frequencies and mode shapes of structures are determined from eigenvalue analysis. This paper proposes a subspace iteration method with an accelerated Lanczos starting subspace for the efficient eigenvalue analysis of structures. The proposed method uses accelerated Lanczos vectors as starting vectors in order to reduce the number of subspace iterations. Accelerated Lanczos starting vectors are generated by employing the repeated forward reduction and back substitution. The proposed method has less computing time than the subspace iteration method with a conventional Lanczos starting subspace when the number of required eigenpairs is relatively small. The efficiency of the proposed method is verified through numerical examples. DOI: 10.1061/共ASCE兲0733-9445共2005兲131:7共1146兲 CE Database subject headings: Natural frequency; Eigenvalues; Vector analysis; Structural analysis.
Introduction Natural frequencies and mode shapes of structures are determined from eigenvalue analysis. Various methods are available for eigenvalue analysis. Among them, the subspace iteration method is widely used for structural problems. The subspace iteration method was proposed by Bathe and Wilson 共1972兲. Various improved versions of the method have been employed. Akl et al. 共1979兲 employed an over-relaxation technique. Bathe and Ramaswamy 共1980兲 presented the subspace iteration method with Lanczos starting subspace. They also used shifting techniques. Wilson and Itoh 共1983兲 proposed an efficient strategy for largescaled problems. All of the improved methods are numerically efficient. Among them, the subspace iteration method with Lanczos starting subspace 共Bathe and Ramaswamy 1980兲 is also efficient. The method uses Lanczos vectors as starting iteration vectors. In 1 Senior Researcher, Ocean Development System Research Division, Korea Ocean Research and Development Institute/Korea Research Institute of Ships and Ocean Engineering, Yuseong P.O. Box 23, Daejeon 305-600, Republic of Korea. E-mail:
[email protected] 2 Postdoctoral Researcher, Dept. of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Daejeon 305-701, Republic of Korea. E-mail:
[email protected] 3 Professor, Dept. of Civil Engineering, Joongbu Univ., 101 Daehaklo, Chubu-myeon, Geumsan-gun, Chungnam 312–702, Republic of Korea. E-mail:
[email protected] 4 Professor, Dept. of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Daejeon 305-701, Republic of Korea. E-mail:
[email protected] Note. Associate Editor: Barry Thomas Rosson. Discussion open until December 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this technical note was submitted for review and possible publication on February 2, 2004; approved on November 29, 2004. This technical note is part of the Journal of Structural Engineering, Vol. 131, No. 7, July 1, 2005. ©ASCE, ISSN 0733-9445/2005/7-1146–1149/$25.00.
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quantum problems, Grosso et al. 共1993兲 modified the Lanczos algorithm to more efficiently obtain the eigenstate of quantum systems. The modified algorithm has been adopted by Cordelli 共1994兲 and Bevilacqua et al. 共1996兲. The modified algorithm is obtained by squaring the conventional operator to accelerate convergence. The similar squaring technique was proposed in the simultaneous inverse iteration process in the modified subspace iteration method 共Lam and Bertolini 1994; Qian and Dhatt 1995; Wang and Zhou 1999兲. This paper applies the technique to the subspace iteration method with Lanczos starting subspace. By employing the squared dynamic matrix in the generation of Lanczos starting vectors, the convergence is improved. Detailed procedures of the proposed scheme are described in the next section.
Subspace Iteration Method with Lanczos Starting Subspace Natural frequencies and mode shapes of structures are obtained by solving the eigenproblem, K j = jM j. M and K are symmetric mass and stiffness matrices of order n, respectively. j is the jth eigenvalue which is the squared value of the jth natural frequency, and j is the jth eigenvector which describes the jth mode shape of structures. The steps of the subspace iteration method to solve the eigenproblem are summarized as follows. The first step in the subspace iteration method is to establish a starting subspace. Starting subspace takes the form of a matrix whose columns are starting iteration vectors. The subspace iteration method with Lanczos starting subspace uses Lanczos vectors as starting iteration vectors, and the Lanczos vectors are generated from the Lanczos algorithm. The Lancozs algorithm was first proposed by Lanczos 共1950兲 and the applicability was studied by some researchers 共Paige 1972; Parlett and Scott 1979; Ericsson and Ruhe 1980; Nour-Omid et al. 1983; Simon 1984; Matthies 1985; Chen and Taylor 1988; Rajakumar 1993; Smith et al. 1993兲. The Lanczos algorithm combines the inverse iteration and Gram– Schmidt orthogonalization. The algorithm can be summarized as follows. The first Lanczos vector is established by normalizing an
Table 1. Number of Iterations 1,008 DOFs p
Fig. 1. Three-dimensional building frames
arbitrary vector. Then, the following inverse iteration is performed ¯xi = K−1Mxi
共1兲
where xi = ith Lanczos vector; and K−1M = dynamic matrix. ¯xi is not obtained by direct calculation of the dynamic matrix. It is obtained by forward reduction and back substitution. Orthogonalization is performed by the recursive formula, ˜xi = ¯xi − ␣ixi − i−1xi−1, where ␣i = ¯xTi Mxi. The next Lanczos vector is obtained ˜ Ti Mx ˜ i兲0.5. In addition to by normalization, xi+1 = ˜xi / i, with i = 共x the basic steps, the reorthogonalization process is required to retain the orthogonality of Lanczos vectors. In this paper, reorthogonalization algorithm presented by Bathe and Ramaswamy 共1980兲 is used. If the number of required eigenpairs 共eigenvalue and the corresponding eigenvector兲 is p, the number of required Lanczos vectors is generally 2p and the starting subspace will be ⌽1 = 关x1x2 ¯ x2p兴. Since Lanczos vectors are good approximate eigenvectors, only a few iterations are generally required. The next step is to perform subspace iteration which includes simultaneous inverse iteration, system reduction, eigenvalue analysis for reduced system and computation of updated subspace. The ¯ simultaneous inverse iteration is performed by calculating ⌽ k+1 −1 = K M⌽k. Then, the reduced system is obtained by calculating
Conventional
5,040 DOFs Proposed
5 13 13 10 12 8 15 10 2 20 5 1 25 9 1 30 14 1 35 7 1 40 3 1 45 21 1 50 14 1 55 16 1 60 15 1 65 10 1 70 4 1 75 1 1 80 1 1 85 1 1 90 1 1 95 1 1 100 1 1 Note: DOF= degrees of freedom; and
p
Conventional
Proposed
10 8 4 20 16 5 30 12 1 40 5 1 50 23 1 60 13 1 70 7 1 80 12 1 90 13 1 100 13 1 110 7 1 120 15 1 130 3 1 140 1 1 150 1 1 160 1 1 170 1 1 180 1 1 190 1 1 200 1 1 p = number of required eigenpairs.
¯ T K⌽ ¯ ¯T ¯ Kk+1 = ⌽ k+1 and Mk+1 = ⌽k+1M⌽k+1. The eigensolution for k+1 the reduced system is obtained by solving Kk+1Qk+1 = Mk+1Qk+1⌳k+1. The generalized Jacobi method can be effectively used for the reduced system. Using the eigensolution of reduced system, the updated subspace is obtained by ⌽k+1 ¯ Q . As k increases, diagonal entries of ⌳ and columns =⌽ k+1 k+1 k+1 of ⌽k+1 converge to exact eigenpairs. In this paper, the error measure, j = 储K j − jM j储 / 储K j储 共Bathe and Ramaswamy 1980兲, is used for checking convergence and the tolerance is 10−6 which yields a sufficient accuracy.
Proposed Method In this paper, a modified Lanczos algorithm is proposed to generate accelerated Lanczos vectors to improve convergence of the subspace iteration method with Lanczos starting subspace. The modified Lanczos algorithm employs squared dynamic matrix to generate accelerated Lanczos vectors while the conventional algorithm uses nonsquared dynamic matrix in Eq. 共1兲. The modified form of Eq. 共1兲 is
Fig. 2. Natural frequencies of three-dimensional building frames JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JULY 2005 / 1147
Fig. 3. Comparison of computing time
¯xi = 共K−1M兲2xi
共2兲
¯xi is not calculated by the direct square of the dynamic matrix. It is obtained by repeated forward reduction and back substitution. The steps of orthogonalization, normalization, and reorthogonalization are identical to those of conventional method. Since ¯xi in Eq. 共2兲 is calculated from the repeated inverse iteration, it can separate approximate eigenvalues more rapidly than ¯xi in Eq. 共1兲. Therefore, modified starting subspace with accelerated Lanczos vectors are closer to exact eigenvector space than conventional starting subspace, resulting in a reduction in the number of iterations. Of course, the squared dynamic matrix requires an additional cost for forward reduction and back substitution. However, the degree of cost reduction due to less iteration is greater than that of the cost increase due to additional forward reduction and back substitution. The simultaneous inverse iteration, system reduction, eigenvalue analysis for the reduced system, and updating subspace are the same as the conventional method
Numerical Examples Two numerical examples are considered to verify the efficiency of the proposed method. The number of iterations and computing time are examined. Fig. 1 shows example structures that are three-dimensional building frames. Young’s modulus is 2.1 ⫻ 1011 Pa and mass density is 7,850 kg/ m3. The area and moment of inertia are 0.01 m2 and 8.3⫻ 10−6 m4, respectively. The first structure has 400 beam elements and 180 nodal points resulting in 1,008 degrees of freedom 共DOFs兲. The second structure is composed of 2,170 beam elements and 864 nodal points and its DOF is 5,040. Natural frequencies of example structures are plotted in Fig. 2. Table 1 and Fig. 3 summarize the number of iterations and computing time, respectively, for obtaining required eigenpairs. As shown in Table 1, the subspace iteration method, with proposed Lanczos starting subspace, has a smaller number of iterations than the subspace iteration method with conventional Lanczos starting subspace. Fig. 3 shows that the proposed method is much more efficient when the number of required eigenpairs is small. When the number of required eigenpairs is large, conventional Lanczos starting subspace has already good approximations of exact eigenvector space because the size of the starting subspace is sufficiently large. In that case, the proposed starting subspace has no gains. However, in practical dynamic analysis or seismic design, the number of required eigenpairs is generally small because a few lower modes are dominant. Therefore, it could be concluded that the proposed method is practically useful. In Fig. 3, H 关=兺 pj=1sT共Tj s兲共M j兲 / 共sTs兲兴 is a measure for the con1148 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JULY 2005
tribution of eigenvectors to external load 共Wilson et al. 1982兲. s is a vector describing the spatial distribution of external load. If a horizontal earthquake is considered in this example, H of the first structure is 99.991% when the number of eigenpairs is 75 共Ⰶ1008兲. That implies that only a few lower modes are dominant in structural problems.
Conclusions The subspace iteration method with an accelerated Lanczos starting subspace is proposed for the efficient eigenvalue analysis of structures. From a numerical analysis, the characteristics of the proposed method can be summarized as follows. The subspace iteration method with the proposed Lanczos starting subspace has a smaller number of iterations than the subspace iteration method with a conventional Lanczos starting subspace because the squared dynamic matrix in the proposed algorithm can accelerate convergence. The number of required eigenpairs is generally small in practical analysis, because a few lower modes are dominant. The proposed method has less computing time than the conventional method when the number of required eigenpairs is small. Therefore, the proposed method is practically efficient. Akl, F. A., Dilger, W. H., and Irons, B. M. 共1979兲. “Over-relaxation and subspace iteration.” Int. J. Numer. Methods Eng., 14, 629–630. Bathe, K. J., and Wilson, E. L. 共1972兲. “Large eigenvalue problems in dynamic analysis.” J. Eng. Mech., 98共6兲, 1471–1485. Bathe, K. J., and Ramaswamy, S. 共1980兲. “An accelerated subspace iteration method.” Comput. Methods Appl. Mech. Eng., 23, 313–331. Bevilacqua, G., Martinelli, L., and Parravicini, G. P. 共1996兲. “Jahn-Teller effect in ZnS : Fe2+ revisited with a modified Lanczos-type algorithm.” Phys. Rev. B, 54, 7626–7629. Chen, H. C., and Taylor, R. L. 共1988兲. “Solution of eigenproblems for damped structural systems by the Lanczos algorithm.” Comput. Struct., 30, 151–161. Cordelli, A. 共1994兲. “Application of accelerated-convergence technique to modified Lanczos calculations.” Nuovo Cimento D, 16, 45–53. Ericsson, T., and Ruhe, A. 共1980兲. “The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems.” Math. Comput., 35, 1251–1268. Grosso, G., Martinelli, L., and Parravicini, G. P. 共1993兲. “A new method for determining excited states of quantum systems.” Nuovo Cimento D, 15, 269–277. Lam, Y. C., and Bertolini, A. F. 共1994兲. “Acceleration of the subspace iteration method by selective repeated inverse iteration.” Finite Elem. Anal. Design, 18, 309–317. Lanczos, C. 共1950兲. “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.” J. Res. Natl. Bur. Stand., 45, 255–282. Matthies, H. G. 共1985兲. “A subspace Lanczos method for the generalized
symmetric eigenproblem.” Comput. Struct. 21, 319–325. Nour-Omid, B., Parlett, B. N., and Taylor, R. L. 共1983兲. “Lanczos versus subspace iteration for solution of eigenproblems.” Int. J. Numer. Methods Eng., 19, 859–871. Paige, C. C. 共1972兲. “Computational variants of the Lanczos method for the eigenproblem.” J. Inst. Math. Appl., 10, 373–381. Parlett, B. N., and Scott, D. S. 共1979兲. “The Lanczos algorithm with selective orthogonalization.” Math. Comput., 33, 217–238. Qian, Y. Y., and Dhatt, G. 共1995兲. “An accelerated subspace method for generalized eigenproblems.” Comput. Struct., 54, 1127–1134. Rajakumar, C. 共1993兲. “Lanczos algorithm for the quadratic eigenvalue problem in engineering applications.” Comput. Methods Appl. Mech. Eng., 105, 1–22.
Simon, H. D. 共1984兲. “The Lanczos algorithm with partial reorthogonalization.” Math. Comput., 42, 115–142. Smith, H. A., Sorensen, D. C., and Singh, R. K. 共1993兲. “A Lanczosbased technique for exact vibration analysis of skeletal structures.” Int. J. Numer. Methods Eng., 36, 1987–2000. Wang, X., and Zhou, J. 共1999兲. “An accelerated subspace iteration method for generalized eigenproblems.” Comput. Struct., 71, 293– 301. Wilson, E. L., and Itoh, T. 共1983兲. “An eigensolution strategy for large systems.” Comput. Struct., 16, 259–265. Wilson, E. L., Yuan, M. W., and Dickens, J. M. 共1982兲. “Dynamic analysis by direct superposition of Ritz vectors.” Earthquake Eng. Struct. Dyn., 10, 813–821.
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