The concept of structural optimization has been more and more widely accepted in many engineering fields during the past several decades, because the optimization can result in a much more reasonable and economical structure design with even less material consumption.; A significant limitation of the conventional level set method in topology optimization is that it can not create new holes in the design domain. Therefore, the topological derivative approach is proposed to overcome this problem. In this part of the thesis, we investigated the use of the topological derivative in combination with the level set method for topology optimization of solid structures. The topological derivative can indicate the appropriate location to create new holes so that the strong dependency of the optimal topology on the initial design can be alleviated. We also develop an approach to evolve the level set function by replacing the gradient item with a Delta function in the standard Hamilton-Jacobi equation. We find that this handling can create new holes in the solid domain, grow a structure from an empty domain, and improve the convergence rate of the optimization process. The success of our approach is demonstrated by several numerical examples.; In the second part of this thesis, we implement another variational level set method, the piecewise constant level set (PCLS) method. This method was first proposed by Lie-Lysaker-Tai in the interface problem field for such tasks as image segmentation and denoising problems. In this approach, by defining a piecewise density function over the whole design domain, the sensitivity of the objective function in respect to the design variable, the level set surface, can be explicitly obtained. Thus, the piecewise density function can be viewed as a bridge establishing the relationship between the implicit level set function and the performance function defined on the design domain. This proposed method retains the advantages of the implicit level set representation, such as the capability of the interface to develop sharp corners, break apart and merge together in a flexible manner. Because the PCLS method is implemented by an implicit iteration differential scheme rather than solving the Hamilton-Jacobi equation, it is not only free of the CFL condition and the reinitialization scheme, but it is also easy to implement. These favorable properties lead to a great timesaving advantage over the conventional level set method. Two other meaningful advantages are the natural nucleation property with which the proposed PCLS method need not incorporate any artificial nucleation scheme and the dependence of the initial design is greatly alleviated.; In the third part of this thesis, we apply a parametric scheme by combining the conventional level set method with radial basis functions (RBFs). This method is introduced because the conventional level set function has no analytical form then the entire design domain must be made discrete in an artificial manner using a rectilinear grid for level set processing - often through a distance transform. The classical level set method for structural topology optimization requires a careful choice of an upwind scheme, extension velocity and a reinitialization algorithm. With the versatile tool, RBF, the original problem can be converted to a parametric optimization problem. Therefore, the costly Hamilton-Jacobi PDE solving procedure can be easily replaced by a standard gradient method or another mature conventional optimization method in the parameter space such as MMA, OC, mathematic programming and so on.; Following those methods some numerical implementation issues are discussed, and numerical examples of 2D structural topology optimization problems of minimum compliance design are given and combined with a comparative study where the efficiency, convergence and accuracy of the present methods are highlighted. Finally, conclusions are given.; Keywords: structural optimiza
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