Improved Lanczos Method forthe Eigenvalue Analysis of Structures2002 2002 4 13Byoung-Wan Kim1), Woon-Hak Kim2) and In-Won Lee3)1) Graduate Student, Dept. of Civil and Environmental Eng., KAIST2) Professor, Dept. of Civil Engineering, Hankyong National Univ.3) Professor, Dept. of Civil and Environmental Eng., KAIST
Introduction Matrix-powered Lanczos method Numerical examples ConclusionsContents
Introduction BackgroundDynamic analysis of structures- Direct integration method- Mode superposition method Eigenvalue analysis
Eigenvalue analysis- Subspace iteration method- Determinant search method- Lanczos method
The Lanczos method is very efficient.
Literature reviewThe Lanczos method was first proposed in 1950.
Erricson and Ruhe (1980):Lanczos algorithm with shifting
Smith et al. (1993):Implicitly restarted Lanczos algorithm
Gambolati and Putti (1994):Conjugate gradient scheme in Lanczos method
In the fields of quantum physics, Grosso et al. (1993)modified Lanczos recursion to improve convergence.
ObjectiveApplication of Lanczos method using the power techniqueto the eigenproblem of structures in structural dynamics
Matrix-powered Lanczos method
Eigenproblem of structure Matrix-powered Lanczos method
Modified Gram-Schmidt process of Krylov sequence
Modified Lanczos recursion
Reduced tridiagonal standard eigenproblem
Summary of algorithm and operation countn = order of M and K, m = half-bandwidth of M and Kq = the number of calculated Lanczos vectors or order of Tsj = the number of iterations of jth step in QR iteration
OperationCalculationNumber of operationsFactorizationIteration i = 1 qSubstitutionMultiplicationMultiplicationReorthogonalizationMultiplicationDivisionRepeatReduced eigensolution
Numerical examples Simple spring-mass system (Chen 1993) Plan framed structure (Bathe and Wilson 1972) Three-dimensional frame structure (Bathe and Wilson 1972) Three-dimensional building frame (Kim and Lee 1999) Structures Physical error norm (Bathe 1996)
Simple spring-mass system (DOFs: 100) System matrices
Failure in convergence due tonumerical instability of high matrix power Number of operations
No. of eigenpairs = 1 = 2 = 3 = 4 2 4 6 810 38663 78922120458157649214729 29823 58529 85712117587154418 26954 47567 7304010305513812223653441226939199550
Plane framed structure (DOFs: 330) Geometry and propertiesA = 0.2787 m2I = 8.63110-3 m4E = 2.068107 Pa = 5.154102 kg/m3
Number of operationsFailure in convergence due tonumerical instability of high matrix power
No. of eigenpairs = 1 = 2 = 3 = 4 612182430 10908273 20855865 27029145 31581179102944376 742905013578945186762092251653365994807 707245211688377165085072016479754112986 66335361123762516047093
Three-dimensional frame structure (DOFs: 468) Geometry and propertiesE = 2.068107 Pa = 5.154102 kg/m3: A = 0.2787 m2, I = 8.63110-3 m4: A = 0.3716 m2, I = 10.78910-3 m4: A = 0.1858 m2, I = 6.47310-3 m4: A = 0.2787 m2, I = 8.63110-3 m4Column in front buildingColumn in rear buildingAll beams into x-directionAll beams into y-direction
Number of operations
No. of eigenpairs = 1 = 2 = 3 = 41020304050 71602154 181780512 307269560 6841622221024104917 50687925124269611215884077453454527656188310 48705515116680070192064376378770940553972908 46214349108715163182518601356596304504420108
Geometry and properties Three-dimensional building frame (DOFs: 1008)A = 0.01 m2I = 8.310-6 m4E = 2.11011 Pa = 7850 kg/m3
Number of operationsFailure in convergence due tonumerical instability of high matrix power
No. of eigenpairs = 1 = 2 = 3 = 4 20 40 60 80100 3950790201196316954304557829533987467933536190824 278717178 801878160199310812825091254743625240574
ConclusionsMatrix-powered Lanczos method has not only the better convergence but also the less operation count than the conventional Lanczos method.
The suitable power of the dynamic matrix that gives numerically stable solution in the matrix-powered Lanczos method is the second power.
Thank you, Mr. Chairman.Good afternoon, Ladies and Gentlemen.My name is Byoung-Wan Kim and the title of my presentation is Matrix power Lanczos method and its application to the eigensolution of structures.This research was conducted by Research Assistant Professor Hyun-Jo Jung, Professor In-Won Lee of the Korea Advanced Institute of Science and Technology, and myself.The outline of my presentation is as follows:Introduction, matrix power Lanczos method, numerical examples, and conclusions.The dynamic analysis is divided into two solution methods; direct integration method and mode superposition method.Eigenvalue analysis is an essential step when the mode superposition method is used.Subspace iteration method, determinant search method and Lanczos method are widely used in eigenvalue analysis.Among them, Lanczos method is very efficient. Lanczos method was first proposed in 1950 by Lanczos.To improve the method many researchers have studied various procedures as follows:In 1980, Erricson accelerated Lanczos algorithm with shifting technique.In 1993, Smith proposed restarted Lanczos algorithm.In 1994, Gambolati employed conjugate gradient scheme in Lanczos algorithm to improve convergence.Recently, my advisor Lee developed Lanczos-based algorithm for nonclassical damping system, 1999.In the fields of quantum physics, the following Lanczos recursion is used to obtain the eigenstate of quantum systems.In 1993, Grosso modified the Lanczos recursion to improve convergence with second-powered operator like this.wherea and b are scalar coefficientsf is basis functionsH is a given operatorEt is trial energy which corresponds to shiftn is Lanczos step number
That power technique is not applied to structural dynamics yet.The objective of this study is to apply the powered Lanczos method to the eigensolution in structural dynamics.Matrix power Lanczos method is proposed as the name in this study.Now, Ill present the algorithm for matrix power Lanczos method.The eigenproblem of structure can be expressed as.whereM and K are symmetric mass and stiffness matrix, respectively.lambdai and phii are ith eigenpair.n is order of M and K.Lanczos schemed this Gram-schmidt process about Krylov sequence to calculate Lanczos vectors.I apply the power technique to the dynamic matrix in Krylov sequence like this.wheredelta is a positive integer, upsilon is scalar coefficient, x0 is a trial vector, x is Lanczos vector, kmyuinversem is dynamic matrix.kmyu is k minus myum, myu is shift.Modified process contains Lanczos vectors with deltai iterated Krylov sequence, so it has better convergence than conventional process.From the modified Gram-Schmidt process, following modified Lanczos recursion can be derived.xibar,alphai,the next Lanczos vector xi+1 and betai are calculated as follows:Then, we can solve this reduced tridiagonal eigenproblem.whereX is a set of Lanczos vectors and T is a tridiagonal matrix like this.QR iteration combined with inverse iteration can be used to solve the reduced eigenproblem.This table shows summary of algorithm and operation count for matrix power Lanczos method.To verify the effectiveness of the matrix power Lanczos method, a simple spring-mass system, a plan frame structure, a three-dimensional frame structure and a three-dimensional building frame are analyzed.Physical error norm like this is used to check convergence.
The first example is a simple spring-mass system with ten hundred degrees of freedom.System matrices are shown.This table and this figure show the number of operations for calculating eigenpairs of simple spring-mass system.As you can see, matrix power Lanczos method has better convergence than conventional Lanczos method.In some cases, high matrix power causes failure in convergence due to numerical instability.Asterisk means such convergence failure.Convergence failure occurs in fourth power in this example.The second example is a plane framed structure with three hundred and thirty degrees of freedom.Geometry and properties are shown in this figure.This table and this figure show the number of operations for calculating eigenpairs of plane framed structure.As you can see, matrix power Lanczos method has better convergence than conventional Lanczos method.Convergence failure occurs in fourth power in this example.The third example is a three-dimensional frame structure with four hundred and sixty eight degrees of freedom.Geometry and properties are shown in this figure.This table and this figure show the number of operations for calculating eigenpairs of three-dimensional frame structure.As you can see, matrix power Lanczos method has better convergence than conventional Lanczos method.Convergence failure doesn't occur in this example.The last example is a three-dimensional building frame with one thousand eight eight degrees of freedom.Geometry and properties are shown in this figure.This table and this figure show the number of operations for calculating eigenpairs of three-dimensional building frame.Matrix power Lanczos method has better convergence than conventional Lanczos method.Convergence failures occur in third and fourth power in this example.Finally, I present the conclusions of the study.First, The convergence of matrix power Lanczos method is better than that of the conventional Lanczos method.Second, The optimal power of dynamic matrix that reduces the number of operations and gives numerically stable solution in matrix power Lanczos method is the second power.