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Structural shape and topology optimization has become an effective design tool for obtaining more efficient structures. The topology optimization, as a conceptual design tool, has been regarded as a powerful structural optimization method because of its ability in producing structures with the highest performance. On the other hand, structural topology optimization has been identified as one of the most challenging tasks in structural design since it is not easy to handle the topological changes in structures. Since the birth of the finite element (FE) based topology optimization at the end of 1980s, various techniques and approaches have been developed during the past decade.
One main approach to structural design for variable topologies is the method of homogenization introduced in [i] (see also [ [ii] , [iii] ]), in which a material model with micro-scale voids is introduced and the topology optimization problem is solved by seeking the optimal porosity of such a porous medium using one of the optimality criteria. By transforming the difficult topology design problem into a relatively easier "sizing" problem, the homogenization technique is capable of producing internal holes without prior knowledge of their existence. However, the homogenization method may not yield the intended results for some objectives in the mathematical modeling of structural design. It often produces designs with infinitesimal pores in the materials that make the structure not manufacturable. Furthermore, numerical instabilities may introduce "non-physical" artifacts in the results and make the designs sensitive to variations in the loading.
A number of variations of the homogenization method have been investigated to deal with these issues by penalization of intermediate densities, especially the "solid isotropic material with penalization" (SIMP) approach for its conceptual simplicity [ [iv] ,5 [v] ]. Material properties are assumed constant within each element used to discretize the design domain and the design variables are the element densities. The material properties are modeled to be proportional to the relative material density raised to some power. The power law-based approach has been widely applied to topology optimization problems with multiple constraints, multiple physics, and multiple materials. However, numerical instability and unsmooth structure boundary remain to be the major difficulties for realistic requirements.
A simple method for shape and layout optimization, called "evolutionary structural optimization" (ESO), has been proposed by Xie and co-workers [ [vi] , [vii] ], which is based on the concept of gradually removing material to achieve an optimal design. The method was developed for various problems of structural optimization including stress considerations, frequency optimization, and stiffness constraints. The ESO method uses a fixed model with standard finite elements to represent the initial design domain while the so-called optimum design is found as a subset of the initial set of finite elements. A key process of this method is to use an appropriate criterion to assess the contribution of each element to the specified behavior (response) of the structure and subsequently to remove some elements with the least contribution (usually known as hard kill). This approach is essentially based on an evolutionary strategy focusing on local consequences but not on the global optimum. It is typically computationally expensive.
A similar approach called "reverse adaptivity" was proposed by Reynolds et al. [ [viii] ], in which a fixed percentage of relatively under-stressed material is removed to find approximately fully stressed structures. Essentially, both ESO and reverse adaptivity are homotopy methods based on material hard kills. In reverse adaptivity, finite element meshes near the boundary during the design procedure are refined to reduce computational cost or increase resolution.
Another related approach is called "bubble method" which is proposed by Eschenauer and co-workers [ [ix] , [x] ]. In this method, so-called characteristic functions of the stresses, strains and displacements are employed to determine the placements or insertion of holes in known shape at optimal positions in the structure, thus modifying the structural topology in a prescribed manner. In such case, the design for a given topology is settled before its further changes.
All the methods mentioned above focus on material and take material properties as design variables. Differently, the "level set method" is based on boundary variations. In this method, the design variable is actually the exterior and interior boundaries of the structure. Bound- aries are represented by the level set function and are propagated by the level set equation. Since the level set method can handle the merging and separating of interfaces naturally and flexibly, it offers a tool for simultaneous shape and topology optimization. A short introduction will be given in the next section.
Level Set Methods for Structural Optimization
The level set method is introduced into the structural optimization field first by Sethian and Wiegmann [ [xi] ]. In their method, the boundaries are allowed to move according to the stresses on the boundaries.
A level set method is employed for tracking the motion of the structural boundaries under a speed function and handling the presence of potential topological changes. An explicit jump immersed interface method is used for computing the solution of the elliptic problem (the Lame equations) in complex geometries with a regular mesh. Osher and Santosa [ [xii] ] propose a level set method for frequencies optimization problems. They use functional gradients to calculate the velocity of the level set and deal with optimization problems with geometrical constraints.
In a series of papers [ [xiii] - [xiv] ], the theories and algorithms of level set based structural optimization method are developed gradually and the technique is implemented into more general problems. As stated in [ [xv] ], a boundary-based method with the capability of handling topology changes has the most promising potential. It is a more direct approach than material-based methods. For example, in general it allows more explicit representation of any features to be incorporated in the design.
There are some variations of the level set based optimization method, which usually focus on the solution method for the level set equation. In [ [xvi] , [xvii] ], radial basis functions (RBFs) are used to discretize the level set function. By means of the method of lines, the authors separate the dependence of the level set function on time and space and transform the partial differential equation (PDE) into a system of ordinary differential equations (ODEs).
RBFs are also employed in [ [xviii] - [xix] ] but in a different manner. Instead of solving the level set equation, this kind of methods parameterize the level set method. The level set equation, which has been discretized using RBF, is substituted into the shape derivative formulation. By virtue of the chain rule, design sensitivities with respect to parameters are derived, and the level set function can be updated by varying parameters according to sensitivities. This method differs from other level set methods in that it needs boundary velocity only and requires boundary integration.
A piecewise constant level set (PCLS) method is implemented to solve the structural optimization problems in [ [xx] , [xxi] , [xxii] ]. In this approach, a piecewise density function is defined over the design domain. This function is regarded as the link between the level set function and the objective function. The PCLS method retains advantages of the conventional level set method and it is free of the Courant-Friedrichs-Lewy (CFL) condition and reinitialization. More importantly, this method allows new holes to nucleate so it is useful in two-dimensional topology optimization.
Finite Element Based Level Set Methods
The level set equation is a hyperbolic PDE. If the standard Galerkin finite elements are used to solve it, numerical instabilities may arise. There are usually two categories of methods to overcome this difficulty. The first category is to use some stabilized finite element methods which are suitable for hyperbolic or advection dominated equations. The second category changes the level set equation to what can be solved by using the standard Galerkin finite element method. Most of the finite element based level set methods fall in the first category, which are introduced below.
Barth and Sethian are the first to discretize the level set equation on unstructured triangular meshes using finite element techniques [ [xxiii] ]. They use the stabilized Petrov-Galerkin method to approximate the Hamilton-Jacobi equation. To remove small oscillations sometimes presenting near slope discontinuities, a discontinuity capturing operator [ [xxiv] ] is employed. This method is subsequently applied in [ [xxv] ] to treat the growth of cracks. Petrov-Galerkin method is also used, with different formulations, to solve the level set problem in some special applications. For example, the incompressible multiphase flow is simulated in [ [xxvi] , [xxvii] ] and the geodesic contours problem in image processing is handled in [ [xxviii] ]. In it the Eikonal equation is combined with the level set equation, and both equations are solved simultaneously. No reinitialization is needed in this case. To improve the efficiency, the authors use a banded algorithm which restrict computation to the vicinity of the zero set of the level set function. It is worthwhile to note that there are some different names of the Petrov-Galerkin method for advection dominated equations in the literature, such as the streamline upwind/Petrov-Galerkin (SUPG) [ [xxix] ], and the streamline diffusion method.
The Galerkin least squares (GLS) finite element methods [ [xxx] ], which coincide with SUPG, have also been implemented to solve level set equations [ [xxxi] ]. In , both the level set equation and the velocity extension equation are discretized with GLS and the shock capturing operator [  ] is added to prevent numerical oscillations at sharp corners in the interface. Formulations for reinitialization are not proposed in this paper. In [ [xxxii] ], the same method is used to solve the level set equation for modeling thermal oxidation of Silicon. The reinitialization is not discussed either.
The discontinuous Galerkin (DG) method is originally developed to provide an approximation exhibiting a better behavior in the presence of discontinuous solutions. This method has also been implemented for level set problems in [ [xxxiii] ]. The least squares finite element method (LSFEM) [ [xxxiv] ] is another stable FEM for hyperbolic problems. It is applied to solve the level set equation on curvilinear coordinates. This FE-based level set technique is subsequently used to optimize the shell structure.
In another kind of stabilized FEMs, temporal discretization precedes the spatial one. The unknown variable is often expanded by Taylor series in time and then the time derivatives are replaced by using the advection equation. This procedure introduces into the equation some additional terms that add the stabilizing diffusion in the streamline di- rection. The characteristic-Galerkin method and the Taylor-Galerkin method belong to this kind. Although these two methods are developed differently, they are very similar to each other.
Next, we introduce the second category in which the hyperbolic level set equation is modified first. The new equation can be solved with the standard Galerkin finite element method. The first method in this category is to add a diffusion term to the level set equation. Then the hyperbolic equation becomes a advection-diffusion equation, which can be solved with the standard Galerkin FEM because the stability is guaranteed by the artificial dissipation. This idea is realized in [ [xxxv] ] and the authors couple the level set method with structural topology optimization via the FEMLAB package. It is pointed in [ [xxxvi] ] that, however, adding artificial diffusion term is not a good choice since this terms usually causes too much dissipations in the crosswind direction.
The second method in the second category is to assume that the level set function is always a signed distance function. Consequently, the advection term disappear and the level set equation becomes a ordinary differential equation. This is method is called the assumedgradient method and the same idea is used in [ [xxxvii] ] too. In this method, one obtains a very simple level set equation at the expense of the task to maintain a signed distance function strictly. Ordinarily, the level set function used in structural optimization is not sensitive to its slope. In most cases, we reinitialize the level set function after several steps. Moreover, it is not necessary to obtain a strict signed distance function. However, in the assumed-gradient method, the level set function needs to be fully reinitialized before each step. In [ [xxxviii] ], the reinitialization requires locating the closest-point projection of each node onto the interface.
Contributions and Organization of this Dissertation
The advantages of the level set method for structural optimization have been extensively discussed in literature. In most of the applications, the level set method is implemented with the finite difference method (FDM). This method works well on a structured grid, but difficulties happen if the problem involves complex geometries and boundaries, where spatial discretization with the structured grid is impossible. However, the finite element method (FEM) handles these problems flexibly. This is one of our motivations for implementing the level set method with the FEM.
The second motivation is related to the procedure of the structural optimization. There are generally two stages in a level set based structural optimization procedure: the stress analysis stage and the boundary evolution stage involving level set methods. The first one is typically carried out with FEMs as often in industrial applications. Therefore, our aim is to unify the techniques of both stages within a uniform framework. In this dissertation, a finite element based level set method is introduced for structural topology optimization. The streamline diffusion finite element method (SDFEM) is used to solve the level set equation and the reinitialization equation, and this SDFEM-based level set technique is combined with the structural optimization in the first time. The reason that we employ SDFEM for level set methods may be summarized as follows:
This method is relatively simple compared with other stabilized FEMs.
In this method, the coefficient matrix of the discretized level set equation is symmetric and positive definite. Moreover, we point out that this matrix is similar to the mass matrix in structural dynamics. Therefore, the mass lumping technique is borrowed. Numerical results show that using the lumped coefficient matrix improves efficiency significantly.
We have also discussed the accuracy of the proposed method and compared it with the finite difference method (FDM) commonly used in conventional level set methods. The presented method possesses the same order of accuracy as ENOl, the first-order accurate upwinding FDM. Although there are some higher-order schemes in FDM, we show that, in this study, the accuracy of the presented method is enough for structural optimization problems.
While the reinitialization equation is solved, numerical errors or added diffusion term will cause the boundary to move. We use the Lagrangian multiplier method or the penalty method to fix the boundary. It turns out that this is a natural procedure within the FEM framework.
Since the performance of the level set method depends highly on the velocity field, the velocity extension aiming at structural optimization problems is discussed. Some related issues, such as the influence of stresses singularities and stresses smoothing, are also discussed.
This dissertation is organized as follows. In Chapter 2, the background knowledge of the structural optimization and the level set method are introduced. Some algorithms for level set based structural optimization are discussed. In what follows we present the finite element based level set method in Chapter 3. Formulations are derived in details and parameters are defined explicitly. Some test cases demonstrate the performance of the proposed method. The velocity extension is discussed in Chapter 4, where two methods are introduced and compared. Chapter 5 exhibits numerical examples including problems in regular and irregular domain, with structured and free mesh. Results illustrate the feasibility of the presented method. Conclusions and future work are discussed in the last chapter.