The Moore-Penrose Methods And Applications For Static Reanalysis Of Structural Topology Modifications

The design of an engineering structure is an iterative process of modificationsand analyses. In this process, an engineer always encounters the fast reanalysis ofstructural modifications. In the process of large-scale engineering structures'design, it costs a lot of computational efforts due to huge number of iterations. Forthe complexity of engineering designs, there are various forms in structuralvariations. According to the pattern modified the structural modifications fall intothree types: the parameter modification, the shape modification and the topologymodification. The reanalysis method is to evaluate not only the response of system, such asthe displacement, stress, etc., but also the change of natural frequencies or theirfunction. Thus, it is necessary to seek a faster computational method for reanalysis.Reanalysis method can be broadly divided into approximate and exact methods.The approximate reanalysis methods, intended to reduce the computational cost,are usually suitable for moderate changes in numerous design variables, whereasthe exact methods can achieve its efficiency when modifications take place onlywithin a relatively small proportion of the structure.The traditional structural modification emphasize particularly on the certainparameter modification of structure, that we modify some parameters, such as areaof cross section, mass, stiffness, etc., based on the fixed topology and the fixedgeometry shape of structure. In recent decades, the methods of parametermodification of structure have been developed. There are many domestics andoverseas scholars have made a great deal of research and developed some effectivemethods. However, in many practical engineering problem, the change of structuretake place not only in the parameters but also in the geometry shape or topology ofstructure, such as the addition or deletion of pole in the frame. Only using theparameters cannot describe the change clearly. Under the circumstance of fixedstructural topology, the modification of the geometry shape of structure is calledshape modification, such as the modification of nodes' coordinate;the modificationof structural topology, such as the addition or deletion of the articles and nodes, iscalled as topological modification. At present, the application of topologicalmodification drop behind the progress of theory, but as the development ofcomputer technology, it will be applied more and more frequently.The structural topological modification is a new subject. Although we canobtain the good geometry scale and shape of structure by modifying structuralparameters and shapes, it is more important to achieve a good topology ofstructure.In this paper, the Moore-Penrose inverse structural topological variationtheory and method are presented. This method can be used in the static reanalysisof the topological modifications and parameter modifications structures. There aresome details as follows:1. In chapter 2, some fundamental matrix theories are introduced, includingthe formulae for computing the Moore-Penrose inverse of a partitioned matrix.These formulae will be used in later chapters.2. A topological modification method for structural variations of the symmetrystructure of shaft is presented using the Moore-Penrose inverse theory and a newfactorization of a two dimensional constant strain triangular orbicular element. Setsof explicit formulations of structural variations are obtained, which are suitable forthe analysis of a structure after the structural topology and the parameters aremodified locally.3. We will extend the Moore-Penrose inverse topological variation theory tothe space structures and give a set of explicit topological variation formulations forsolving the static responses. This method is especially suitable for local staticmodifications of the three dimensions elastomeric structures.4. The part calculation formula of the Moore -Penrose inverse areprogrammed using Mat lab. Thus , the Moore-Penrose inverse methods of thestructural topological modification can be implemented. A numerical example isgiven to illustrate the application of the approach presented in this study.