Research Of Structural Dynamic Analysis And Shape Optimization Based On RKPM

Meshless method is an emerging numerical computational method in engineering. In recent years, meshless method has attracted wide concern from the engineering fields as a result that it needs only nodes and no elements when it generates the shape function. It does not need initial mesh generation and re-meshes, and it can avoid the difficulties arising from the mesh distortion of finite element during the shape optimization. In this way, the computational accuracy is ensured and the computational complexity is also decreased. Reproducing Kernel Particle Method (RKPM) is a meshless method that generates the shape function using reproducing kernel approximation, and it is widely applied in structural dynamics.In this paper, the structural dynamic analysis and sensitivity analysis were studied by using RKPM, and then the structural dynamic shape optimization was performed on the basis of the above analysis. The main research contents are summarized as follows:Firstly, according to the free vibration theory of the elastic structure (for example, beam,plate,shell and so on), the Lagrange multiplier method was used in implementation essential boundary conditions, and the visibility criterion was used to dispose the discontinuity of field function. The MATLAB programs of modal analysis based on RKPM were written and the modal analysis of the perforated thin plates with different boundaries were performed. The dimensions of the stiffness matrices and mass matrices were reduced through the orthogonal transformation, and then the computational efficiency was improved greatly.Secondly, according to the basic theory of meshless method and structural dynamics, the penalty method was employed into imposing the essential boundary conditions, and the dynamic equations based on RKPM for beam and plate were deduced. The correlative MATLAB programs were written, and then the dynamic response analysis of damped beam and plate under different boundaries and complicated types of load were performed. The effects of the penalty factor and time integration step on the computational accuracy and efficiency were discussed and emphasized, and then the appropriate range of the penalty factor was gained in dynamic problem.Thirdly, the dynamic sensitivity analysis equations based on RKPM were proposed by using Direct Differentiation Method (DDM) and discrete derivatives on the basis of structural dynamic analysis, and the differences between RKPM and FEM were indicated. The MATLAB programs of dynamic sensitivity analysis were written and the dynamic sensitivity analysis of the beam and arc were performed, and then the numerical results obtained were compared with those obtained using Semi-Analytical Method (SAM) and Global Finite Differences Method (GFDM). Finally, by integrating the optimization algorithm with parameterized descriptive method of boundary shape, the structural dynamic shape optimization of two examples in engineering were performed, and a new numerical method was presented for structural dynamic optimization design.