Vector-Valued Rational Approximate Method For Structural Dynamic Reanalysis

With the development of science and technology, many large-complicated structures need to be designed. In structural design or optimization, the procedures are generally iterative and require repeated analysis as the structures are progressively modified. Each resign involves extensive calculations. This difficulty motivates extensive studies on reanalysis techniques. The object of reanalysis is to evaluate the structural response for successive modifications in the design without solving the set of the modified implicit equations so that the computational cost is significantly reduced. In order to avoid a fresh analysis after each modification, many reanalysis techniques have been devised. In this thesis, a vector-valued rational approximate method in structural dynamic reanalysis problems is developed. The thesis is composed of two aspects. In the first part, we program the perturbation method for the matrix eigenvalue problems by using MATLAB language. It is well known that if a power series of converges slowly or has a finite radius of convergence, it is not suitable to represent the function in a large range of . Therefore, one need to develop efficient summation methods to achieve, or improve the convergence of power series. In the second part of the thesis, we present a vector-valued rational approximate method. This method is based on combining the power-series expansion with a rational approximation. This method provides high quality approximations of the structural behavior for large changes in the design variables. The vector-valued rational approximate method is as follows: Compute the perturbative vibration modes and eigenvalues of modified structure by using vibration modes and eigenvalues of original structure: , , 2,Compute matrix perturbation results: (1) (2)3,Compute vibration modes of modified structure by applying vector-valued rational approximate method: (3)where is the partial sum of truncated power series by using the first coefficients (4)Here, are scalars and satisfy a linear system of equations of the form (5) where are defined as the dinner product (6) The equations in (5) involving are typical equation in least-squares problems of the following form where denotes the norm. Numerical examples have shown that high quality approximations of the structural behavior for large changes in the design variables can be achieved. In addition, the overall computational effort is considerably reduced. Therefore it is an efficient method to solve structural dynamic reanalysis problems.