The Investigation Of Multigrid Method For Structural Dynamic Response Analysis

It is a problom which the designer considerin the dynamic response analysis of positive linear system during the last three decades. For general dynamic system, we have owned a lot of methods in its time. The application of these numerical technique in engineering practice has enable satisfactory solutions to obtained for many engineering problems.A lot of large-scale practical engineering problems would not be solved today without these numerical technique and modern computer technology. It is inevitable for these numerical technique that they have the discretization error. Sometimes the error is so great that the results are unreliable. The above methods show the defectives both in sensitivity of time step and in lower accuracy simultaneously.The adaptive finite element methods have been developed to control the discretization error yielded by finite element methods. The finite element mesh have change frequently as the structural dynamic problem is solved by adaptive finite method. It is also imcomplete to analysis adaptive process using traditional finite method,such as subspace iteration method, because it include two process , one is discrete process and the other is numerical study process, which are independent each other without interal instruction, and it yeild much waste. In the discrete process, however, predicition of a suitable solution and auxiliary solution to every position is not received. The result that the local smoothing solution is not efficiently used is that too fine meshes induce that the order which obtained in algebraic equation group is unnecessaryly higher, but the accuracy is still lower. This deficiency is more obvious in dynamic problem.The multigrid method prosthesis the deficiency of traditional finite element method. It allow a effecient procedure which discrete process is coupled with numerical study process. The main benifit is that accomplish iteration solution of residual equation in the coarse mesh layer until the solution is converge, and recovery the result of error correction to the fine mesh layer by interplolated approach. Because the iteration procedure is only effected on the high-frequency element of error vector, so that the multigrid method take simple iteration method as smoothing dose. It can not eliminate all frequency error but smooth (or decrease) the high-frequency element of error vector at current mesh layer, and transfer the low-frequency element of error vector to the later coarse mesh to continue smoothing. The multigrid method therefore require that the process is implemented hierarchically under a set of different size meshes, the iteration of every layer coarse mesh would provide a more accurate result of error correction for the next layer fine mesh. The performance of the restriction operator is to transfer the error residuals from fine mesh to coarse mesh in order to smooth iteration error, and the performance of the interpolation operator is to recover the result of error correction from coarse mesh to fine mesh in order to add them to approximate solution of the equation.In this dissertation we will investigate and discuss on four aspects: multigrid method for structural dynamic response analysis, adaptive finite element procedure of multigrid method and application of structural dynamic response solved by multigrid method in the identification of modes parameter and the damage detection of structure. The context concerns mainly with some aspects as follows:1. The effective multigrid method for solving structural dynamic response in finite element analysis. The method utilized sufficiently the results of the initial mesh and took bilinear interpolation technique to obtain new displacement vectors in the changed mesh. It accomplished the solution of the structural dynamic response problems by multigrid iteration procedure. Conjugate gradient method was applied in the smoothing process of multigrid iteration procedure in order to improve the rate of convergence. An equation was adopted for estimating the error of approximate displacement vector in the innovation process of the coarse mesh. The post-smoothing process of multigrid iteration procedure was solved by Jacobi relaxation iterations in order to improve the evenness of the solution. The discretization of mesh was combined with the numerical solution, and a fast reanalysis procedure was established after mesh refinement. Numerical examples were presented to show the efficiency of the method for both computation and application as compared with the traditional finite element method. It can be used as a particularly effective tool for solving structural dynamic response problems by adaptive finite element method.2. Adaptive method of space meshes and time stepping about multigrid finite element method is also presented. A method is presented for obtaining accurate response using finite element analysis of multigrid method based on superconvergent patch recovery. The improvement of accuracy would depend on the error estimate of the discretization about both space anf time. The adaptive multigrid method is presented where both space and time discretization errors are reduced and iteratively converges to a solution of desired accuracy though adaptive refinement. The efficiency of computation is ensured simultaneously.3. The artificial identification of modes parameter based on the structural dynamic response solved by multigrid method. The power spectrum density function of response at all points is obtained by discretization Fourier translation under the condition of simulating ambient vibration. The response at all points is provided by multigrid method. The natural frequencies, mode shape and damping ratios is identified by the combining about the power spectrum density function and the results of practical measurements about the structure. And the dynamic behaviour of the structure is received.4. The damage detection of structure is solved by using modes parameter which was received in artificial identification and experiment analysis. The modes parameter which was received in emulation identification and experiment analysis is compared according to Modal Assurance Criterion (MAC) and Coordinate Modal Assurance Criterion (COMAC) after the identification of modes has been finished. So the certain damaged location is ascertained. It provide a favorable theoretic basic for the evaluation of structure state.The works in this dissertation are supported by National Natural Science Foundation of China (19872029).