The Application Of Multi-grid Method In The Structural Dynamics Response Finite Element Analysis

In the field of engineering and science technology, people can educemathematics models for many mechanics and the physics problem, which are thebasic equations(ordinary differential equations or partial differential equations)should be followed and the relevant solutions qualifications. But only fewequations which are easy and have quite inerratic geometry figures can be solvedaccurately using analytic method. For most problems, because of the nonlinearproperty of equations, or the complicated geometry figure of the solution region,they can be solved only by numerical method.Developed numerical method of partial differential equations can becompartmentalized into two species. One is finite difference method;the other isfinite element method. Both the two kinds of methods are dividing the solutionregion first, then distributing the differential equations, at last educing a set oflinear algebra equations or nonlinear ones, and then solving the algebra equationsof the meshes selection by direct method or iterative method. Usually we do notknow the function property or the function action of the solution region, so we cannot confirm a kind of appropriate meshes selection in advance. If it is divided toodensely, it will result in big algebra equations, and excessive compute time.Multi-grid method can remedy the defect of these two kinds of method, and cancombine with the distribute process and the solution process well.Multi-grid method bases on this thought: the distribute process and thesolution process is unattached in the traditional finite element method, whichleads to division model must be carried through solutions every time, in terms ofstructural dynamics problems or nonlinear problems, it means prodigiouscomputer time;and it can not take full advantage of the result of the original finiteelement model. So it will greatly reduce the computer time, if we can combine thedistribute process and the solution process, especially if we can take fulladvantage of the result of the original finite element model.In the real life, many engineering structure is usually influenced bydynamical load following with time, such as nuclear power stations﹑large damsand high architectures enduring earthquake load﹑offshore structures enduringimpact of ocean wave. For the sake of the assurance the structural in gear and onthe safe side, it is important to carry through dynamics analysis in engineeringdesign. Dynamics analysis problem is important in the numerical method, at thesame time it uses more computer time than structural static do, so many scholarshave studied in this field, and there are many developed methods to analyzing orsolving the dynamic response of a vibration systems now. However, thesemethods all have a definite application scope and localization.At present, there is a certain evolution in the study of the multi-grid methodabout structural static problems, the dynamic problems about the naturalfrequency of a structure vibration system have been studied by domestic scholars,but there is little study on the aspect of dynamic response problem.The main content of this thesis are:1. Expounding the keystone and application rules of the multi-grid method,and summing up the domestic and foreign applications and productions of themulti-grid method.2. Putting forward a multi-grid method which is fit for analyzing dynamicresponse problem in the finite element method cooperating with the senior. Themethod takes full advantage of the result of the original meshes and takes bilinearinterpolation technique approximation to obtain new mesh's displacement vectorin the changed meshes. And then it accomplishes the solution of the dynamicresponse problem by multi-grid iteration procedure. In the former smooth processwe select Conjugate Gradient method in order to improve the rate of convergence.In the coarse grid correction we deduct a formula to solve the errors of thedisplacement vectors approximately. In the after smooth process we select Jacobiiterations in order to improve the slippery of solution. The method takes fulladvantage of multi-grid's characteristic that it can combine the grid discretizationwith the numerical solution and establish a fast reanalysis procedure afterremeshing.3. Compiling the procedure which combines the multi-grid method with thestructural of dynamics response finite element analysis applying language ofFortran90, and validating the procedure through solving the problems of elasticbeam, and comparing the result with ANSYS which is more popular software inanalyzing the finite element problems, then educing the result of the multi-gridmethod to analyze the structural of dynamic response problems is more accuratethan ANSYS. We also solve the lamella dynamic response problem, and comparethe iterative time with traditional Newmark keeping the premise of the sameaccuracy, then educe that the compute efficiency is high if adopting multi-gridmethod. At last, we compare the multi-grid procedure's CPU runtime with that ofthe ANSYS, then we educe that adopting multi-grid method can greatly reduce theCPU runtime. The procedure mostly accomplishes hereinafter affair: computingcell rigid matrix and assemble collectivity rigid matrix;computing cell massmatrix and assembling collectivity mass matrix;implementing the basicarithmetic to solve the dynamic response by Newmark;implementing thearithmetic of multi-grid method, including restriction subprogram andprolongation subprogram. This procedure core is the arithmetic of multi-gridmethod. In the process of procedure compilation, we took full advantage of themain characteristic of the Fortran90 that use of the finite element accounting, andused programmer's structured get down from the crest and modularizationideology, which improved the generalization and naturalization.