Research On Fast Calculating The Defective System Modification And Key Technique Of Optimization By Adaptive CA Method

Vibration analysis is a very important theory during the design of large and complexstructures, which can be applied to many fields, such as CAE analysis, vibration measurement,modal identification and so on. It is the main research content of solid mechanics.When the objective of vibration analysis is a large and complex structure, thecomputational complexity caused by modifying the design variables continually, will be huge.One can solve the problem from two aspects: One is to enhance the performance of computers,and the other is to search for effective methods of structural reanalysis. Therefore, it isnecessary to explore available algorithms for dynamic reanalysis, which is helpful to improvethe computational efficiency and shorten the design cycle.Based on the theories of vibration analysis, the characteristics and reanalysis methods ofvibration systems should be researched respectively according to different kinds of systems,such as non-defective system and defective system. For the non-defective system, there existsa cluster of linearly independent eigenvectors, which means the Schmidt orthogonalizationcan be used. Hence, each two eigenvectors satisfy the orthogonality and the theory of modalexpansion can be applied well, which leads to easy and intuitive process of analysis andcalculation. As a result, most of existing references assume the system is non-defective.However, some actual engineering problems, such as non-proportionally dampingsystem or flutter analysis of aero-elasticity, sometimes appear defective. Furthermore, largeflexible space structures, such as the space shuttles, usually have low natural frequencies closely distributing in a narrow frequency domain. The eigenvectors of these structures withclose frequencies are extremely instable. When the modification of structure is small, there isgenerally a mutual transformation between systems with close frequencies and defectivesystems. At this moment, for there is no complete eigenvectors in the defective systems, theconventional methods and theories for non-defective systems (such as the theory of modalexpansion) will be invalid. Therefore, effective methods of reanalyzing defectives systemsand systems with close frequencies are focused on.This paper is supported by the National Natural Science Foundation of China titled "Fastoptimization of cross-sectional parameters for simplified car body multi-elements framestructure based on reanalysis theory"(No.50975121), and the Specialized Research Fund forthe Doctoral Program of Higher Education titled "The research of adaptive reanalysisalgorithm of the defective vibration system"(No.20090061110022). The methods for fastsolving the eigenproblems of modified defective systems and the systems with closefrequencies are investigated. The mainly contents can be concluded as follows.(1) Classification of vibration systems and methods for solving eigenproblems ofmodified non-defective systems.First, the classification of vibration systems is discussed. The definations and theories ofnon-defective systems and defective systems are given. Specially, the characteristic ofsystems with close frequencies which are non-defective, is summarized. Finally, thesensitivity method and matrix perturbation method, applied for solving eigenproblems ofmodified non-defective systems, are introduced respectively.(2) Fast method of reanalyzing defective systems.A recursion technique for fast solving the modified eigenproblem is constructed bycalculating the sensitivity of eigenvectors. According to the theories of generalizedeigenvectors and adjoint matrices, some new independent vectors are added in order toconstruct the linearly independent basement of eigenspace. So the motion of defective systembased on this basement can be described correctly. Methods of fast calculating the modifiedeigenvectors for both general defective systems and defective systems with N repeated eigenvalues, are given respectively. By the methods, the combined coefficients of generalizedeigenvectors can be determined by recursion. Meanwhile, the state space of defective systemcan be decoupled well. Thereby, based on eigensolutions of original system and the combinedcoefficients, the eigenvectors of modified defective system can be calculated rapidly withoutsolving large-scale equations. This technique improves the computation effectiveness, as wellas keeping high precision and stability.(3) Adaptive technique for determining the number of basis vectors in CA method.The technique for selecting numbers of basis vectors is researched for improving thecomputation effectiveness of CA method. It has been proved that CA method and PCGmethod provide theoretically identical results in the Krylov subspace, which means thenumber of basis vectors in CA method equals to the iterations of PCG method. Therefore,pre-estimating the number of basis vectors can be transformed into calculating the iterationsof PCG method. Based on the present references, the approximate method of calculating theK-condition number of CA method is presented after deducing the relationship betweenK-condition number and spectral norm. At this moment, the number of basis vectors can beautomatically pre-estimated by the modification of structure. This paper suggests that thenumber of basis vectors should be less than10%of the degrees of freedom, which means theoperations of CA method will be no more than those of Gauss method. This technique isbenefit to improve the efficiency, avoid the sickness of reduced equations and keep thecomputation stability of CA method.(4) Fast reanalysis for systems with close frequencies based on the adaptive CAmethod.The definition and behaviors of systems with close frequencies are introduced. Based onthe instability of eigenvectors, the difficulties of reanalysis and methods of identification forsystems with close frequencies are discussed. When the modification is very small, one canimprove the stability and efficiency of computation by not only considering structures withclose frequencies as the ones with repeated frequencies approximately, but also applying theCA method. Specially, when the system with repeated frequencies is defective, the eigenvectors are linearly dependent and the computation condition of CA method will not besatisfied. So the theories of generalized eigenvectors are applied to expand a complete statespace, as well as a relaxation factor is taken to keep the reversibility of state matrix in order tosatisfy the computation condition of CA method. Meanwhile, the adaptive CA techniqueproposed in this paper, which can estimate the number of basis vectors according to themodifications in advance, is used for solving the perturbed eigenproblem rapidly andimproving the efficiency highly. The numerical examples demonstrate the correctness andvalidity for both the theoretical derivation and the adaptive CA technique in this paper.