YAPISAL MÜHENDİSLİKTE KULLANILAN TOPOLOJİ ENİYİLEMESİ YÖNTEMLERİ ÜZERİNE GENEL BİR BAKIŞ
Kırklareli university journal of engineering and science
Any mechanical performance measure of a structure is strongly related with its topology. Size and shape optimization cannot give the best structural performance, since these methods cannot change the structure's topology. Hence, topology optimization should be employed to obtain the best performance. In this paper, a review of topology optimization is provided. At first, the general topology optimization problem is defined. Then, modern topology optimization methods are presented and discussed.
... nted and discussed. YAPISAL MÜHENDİSLİKTE KULLANILAN TOPOLOJİ ENİYİLEMESİ YÖNTEMLERİ ÜZERİNE GENEL BİR BAKIŞ Özet Bir yapının gösterdiği mekanik performans, o yapının topolojisi ile çok yakından alakalıdır. Boyut ve şekil eniyilemeleri sonucunda, yapının topolojisinde bir değişiklik olmadığı için, en iyi performans elde edilemez. Netice itibariyle, en iyi performansın elde edilebilmesi için topoloji eniyilemesinden faydalanılması gerekmektedir. Bu çalışmada, topoloji eniyilemesi yöntemleri hakkında bir derleme sunulmuştur. İlk olarak, genel topoloji eniyilemesi problemi tanıtılmış, ardından modern topoloji eniyilemesi yöntemleri tartışılmıştır. An Overview On Topology Optimization Methods Employed In Structural Engineering 160 INTRODUCTION Structural optimization is a subspace of the general optimization field which concerns with obtaining optimized structures to achieve certain needs while satisfying some constraints. Three types of structural optimization problems are encountered in the field: size, shape and topology optimization. In size optimization, the geometry of the structure is known, but certain sizes (dimensions) are the design variables to be determined. In shape optimization, the topology is known, but the boundary curves/surfaces of the structure either parameterized or discretized design variables are to be found. Whereas in topology optimization, usually the whole design domain is discretized, then the boundary curves/surfaces as well as number of holes and the location of the holes inside the design domain can be altered. Topology optimization [1, 2] is basically the determination of optimum material distribution in a design space which minimizes (or maximizes) an objective function while satisfying some constraints. The objective function can be compliance minimization (i.e., stiffness maximization) for static problems and, fundamental frequency or frequency gap maximization for dynamic problems  . A topology optimization problem is solved in a certain design space in which applied loads and boundary conditions are specified. In this design space, a volume fraction value, which indicates the ratio of solid volume to total design space volume, is also defined. Volume fraction determines how much of the design space will include solid material. In order to conduct topology optimization, a finite element model is needed. For two dimensional structures, a finite element model can be constructed by employing large number of square elements, which can be considered as pixels. Then, the topology optimization problem can be summarized as determining which pixels will include material and which pixels will be void, for the optimized design. Topology optimization is the most comprehensive method to be employed in structural design, since it involves simultaneous size and shape optimization, as well. Because of this promise, it is widely studied in the structural engineering field  . In this study, a brief review on topology optimization fundamentals and modern topology optimization methods utilized in optimum structural design is provided. An Overview On Topology Optimization Methods Employed In Structural Engineering 161 2. TOPOLOGY OPTIMIZATION 2.1. Description of Topology Mathematically, topology is concerned with the deformable objects, and all distortions are considered as transformations or reversibly unique mappings . The topological transformations or topological mappings are the transformations which do not change the neighborhood relations, in other words, topological transformations cause topologically equivalent domains, therefore, topological property of a domain is invariant for all topological mappings . Topological domains consist of all subsets of R 3 . The degree of connection of domains determines its topology class. Domains belonging to the same topology class are considered as topologically equivalent  (see Figure 1a ). Naming of topology classes can be done as follows: n-fold connected domains require (n-1) cuts from one boundary to another, to convert them into a simply connected domain  (see Figure 1b ). According to the above definitions, size and shape optimizations cause topological mappings that result in topologically equivalent structures. However, topology optimization transforms the structure from one topology class to another. Topology Optimization Problem Although exact analytical solutions of topology optimization problems reveal basic characteristics of optimal designs, they are only capable of solving problems having simple load and support conditions . Therefore, for more realistic problems, it is necessary to use a discretized design domain. Most of the studies in the literature use finite element formulation with a fixed mesh as the discretized model . Here, the general form of topology optimization problem is provided for this model as: subject to: , ≥ 0 for = 1, 2, ... . . , = 0 or 1 for = 1, 2, ... . . , where x is the design variable vector, u is the state field, H is the objective function, G j is the jth constraint, P is the total number of constraints, x e is the eth structural member that constitutes the design variable vector x, N is the total number of design variables (structural members). One of Derleme An Overview On Topology Optimization Methods Employed In Structural Engineering 168 3.2. Density Methods Density methods [25-27] are introduced after the homogenization methods. In these methods, for an individual finite element (i.e., microstructure) there exists only one design variable which is the element density (x e ). A microstructure's material property contribution to the whole structure depends on this density design variable which can take any value between 0 and 1. With this formulation, various well-known optimization algorithms (see ) can be utilized to solve the topology optimization problem. Moreover, density methods involve penalization parameters, which penalize intermediate densities to obtain manufacturable solid-void designs with clear material boundaries instead of the perforated ones acquired with homogenization methods. Various density methods can be found in the literature [9, 19]. Here, the most popular of them, -Solid Isotropic Microstructure with Penalization‖ (SIMP) method [25, 26] is addressed. The SIMP method [25, 26] is a density method in which elastic properties of a microstructure is proportional to an exponential multiple of an element density (x e ) function. Exponential multiplier of an element density function (penalization number, p) causes the continuous and differentiable variation of the microstructure property of an individual finite element, therefore the topology optimization problem is converted into a continuous problem (like Equation 2). This allows one to use the gradient based mathematical programming methods. Moreover, in the presence of a volume constraint, the use of penalization number (p) values larger than 1 (i.e., p > 1) penalizes microstructures with intermediate densities . When p > 1, the microstructures with intermediate densities contribute to structure's elasticity less than the structure's mass. In other words, they do not provide enough stiffness for the structure compared to their mass. So it becomes uneconomical for the structure to hold them. Consequently microstructures (finite elements) with intermediate densities are penalized and they are forced to vanish through the optimization process. As a result, a void-solid (0-1) type of a final design is obtained. On the other hand, the common numerical instabilities in topology optimization problems, such as local minima problem, checkerboard pattern formation and more importantly, mesh dependency issues  , can also occur in the SIMP method with its current formulation, as well. In Section 2.2, the need to transform the original problem in Equation (1) to Equation (2) is mentioned. One way to successfully achieve this is the homogenization approach (see Section An Overview On Topology Optimization Methods Employed In Structural Engineering 169 3.1). On the other hand, the relaxation attained by the SIMP method with its current form is not satisfactory enough to make the problem well-posed . The nonexistence of solutions forces the optimization to converge to different results for different finite element mesh discretizations. This phenomenon is known as the mesh dependency of solutions in the literature . To close the design space in an appropriate sense, i.e., to make the optimization problem well-posed, problem regularization is needed . Thus, lack of convergence related with mesh dependency can be discarded. There exist many problem regularization techniques in the literature [30-33]. Discrete Methods Usage of continuous density design variables in the topology optimization problems results in porous or grey regions in the optimized structure if proper penalization or problem regularization is not applied. Discrete methods concern with Equation (1) instead of Equation (2) in an attempt to obtain directly 0-1 (i.e., void-solid) designs. For this discrete problem, a natural choice is to use non-gradient optimization approaches such as Genetic Algorithms, Simulated Annealing, Particle Swarms, Ant Colonies, etc. However, these methods are inefficient for topology optimization problems in which large number of design variables are involved  . They can only solve problems with coarse finite element meshes  . Moreover, these algorithms also need some type of problem regularization or filtering to obtain physically reliable designs. There exists another type of approach so-called evolutionary methods  . Inspired from the natural evolutionary process, evolutionary structural optimization (ESO) method  is suggested. In the ESO method, the optimum is sought by removing some of the inefficient material from the design domain in every iteration step. The determination of inefficient material depends on the type of objective function and constraints. However, as the material can only be removed (not added) during optimization, sometimes non-optimized designs are obtained . To cope with this problem, -bi-directional evolutionary structural optimization‖ (BESO) method  is suggested. In the BESO method, material is being added in the vicinity of overstressed elements while in the inefficient areas, material is being removed. Despite its conceptual simplicity, ESO/BESO methods are subjected to criticism    . These include, intuitive nature of the procedure, obtaining non optimal designs at the end of the process and difficulties encountered with multiple constraints  . As a consequence of these, a new Derleme An Overview On Topology Optimization Methods Employed In Structural Engineering 170 BESO algorithm is suggested recently [41, 42]. In the new BESO algorithm, design variable sensitivities are utilized with penalization in order to sort the elements with respect to their efficiency. One further novelty in the method is, none of the elements are completely removed from the design space, instead, inefficient elements get a predetermined minimum value. By this, it is claimed that, void elements can become solid in any iteration step of the algorithm. It is argued in the literature that, the new BESO algorithm can be categorized as a discrete update version of the standard SIMP scheme . 3.4. Boundary Variation Methods Boundary variation methods are the recent development in the topology optimization field. They emerged from the need to obtain crisp and smooth edged structures as a result of optimization procedure. These methods have their roots in shape optimization techniques, however, they are fundamentally different, because they allow formation and disappearance of void regions inside the design domain in addition to allowing structural boundary movements . There exist numerous boundary variation methods in the literature, but here, only the most popular level set methods are considered. For other alternative approaches, refer to [9, 19]. In the level set approach [43-45], the level set function determines where the boundaries, void and solid sections take place. Material boundaries exist where the level set function takes a certain constant value (usually it is 0). Where the level set function values are below this constant there exist void regions, whereas solid regions take place where the level set function values are greater than that constant. Changes in structure's topology are achieved by level set function updating, and mostly it is performed by solving the augmented Hamilton-Jacobi equation with diffusive and reactive terms . In this manner, the original Hamilton-Jacobi equation is utilized for shape updating, reactive term is used for nucleation of new holes and diffusive term is employed to smooth the level set field . Level set methods are mainly composed of level set function parameterization, mechanical model and optimization parts . Firstly, design variables are parameterized via level set function. Then, geometry described by the level set function is mapped onto the mechanical structural model. Finally, design update is performed . Depending on the level set method type used, in each of these three parts, regularization techniques can be employed to regularize ill-posedness of Derleme