Conventional Topology Optimization Methods Research Articles

Metamodeling approach for concurrent optimization considering structural topology and its circumstances of mechanical component

Conventional topology optimization method only optimizes a single machine component structure while its circumstances are fixed. Concurrent optimization of topology and circumstances widens the effective range of topology optimization in mechanical design. We have developed a method for the concurrent optimization, which includes problem decomposition and metamodeling. However, the method requires further verification, especially, for cases where a relationship between circumstance variables and results of topology optimization becomes complex. In this research, we examine whether the method can be successfully applied to complex case if a metamodeling approach is switched. Case studies of three kinds of metamodeling techniques are performed. In the case studies, we test following metamodeling approaches: polynomial approximation, radial basis function(RBF) interpolation, polynomial approximation with spatial reduction. It shows that the proposed method can be applied even for complex cases by appropriate selection of metamodeling.

Revised Level Set-Based Method for Topology Optimization and Its Applications in Bridge Construction

To cure imperfections such as low accuracy and the lack of ability to nucleate hole in the conventional level set-based topology optimization method, a novel method using a trapezoidal method with discrete design variables is proposed. The proposed method can simultaneously accomplish topology and shape optimization. The finite element method is employed to obtain element properties and provide data for calculating design and topological sensitivities. With the aim of performing the finite element method on a non-conforming mesh, a relation between the level set function and the element densities field has to be clearly defined. The element densities field is obtained by averaging the Heaviside function values. The Lagrange multiplier method is exploited to fulfill the volume constraint. Based on topological and design sensitivity and the trapezoidal method, the Hamilton-Jacobi partial differential equation is updated recursively to find the optimal layout. In order to stabilize the iterations and improve the efficiency of the algorithm, re-initiation of the level set function is necessary. Then, the detailed process of a cantilever design is illustrated. To demonstrate the applications of the proposed method in bridge construction, two numerical examples of a pylon bridge design are introduced. It is shown that the results match practical designs very well, and the proposed method is a helpful tool in bridge design.

Stiffness design of machine tool structures by a biologically inspired topology optimization method

This paper introduces a novel approach for designing the stiffener layout inside large machine tools by applying the self-optimal growth principle of plant ramifications in nature. Firstly, numerical studies are carried out in order to confirm the potential of leaf venation as concept generators for creating the optimal load-bearing topology for stiffened machine tool structures. Then, a mathematical model explaining the optimality of plant morphogenesis is presented. Based on this, an evolutionary algorithm is developed, which uses three growth strategies to determine the candidate stiffeners to grow or atrophy with respect to the loads applied. The proposed growth-based method could generate a distinct stiffener layout, which is different to those produced by the conventional topology optimization methods, and thus offers unique possibilities of improving the design efficiency and commonality for machine tool development. The suggested approach is finally applied to the re-design of an actual grinding machine column, on which the numerical analyses and experimental tests conducted exemplify the performance enhancement, and therefore is a good choice for the stiffener layout design of machine tool structures.

2210 慣性力を考慮した構造衝突部材のトポロジー最適化(建築・構造)

Topology optimization on density method on a structural member subject to large inertial force and plastic deformation is discussed. The sensitivities are calculated from the energy difference between two analyses, the current densities and the densities with a slight change for the target element. Maximum energy absorption and load-displacement curve optimization on 1000x100 mm^2 two-dimensional beam subjected to 20 m/s enforced velocity are performed with/without the consideration of inertial force. As a result, the convergence of the objective function are significantly improved and the final topology of the member is different from the one derived from conventional topology optimization method without inertial force.

Topology optimization of continuum structures with bi-modulus materials

Since the elasticity of bi-modulus materials is stress dependent, it is difficult to apply most conventional topology optimization methods to such bi-modulus structures owing to great computational expense. Therefore, this study employs the material-replacement method to improve the computational efficiency for topology optimization of bi-modulus structures. In this method, first, the bi-modulus material is replaced by two isotropic materials which have the same tensile or compressive modulus. Secondly, the isotropic materials for finite elements are determined by the local stress/strain states. The local elemental stiffness can be modified according to the current modulus and stress state of the element. Thirdly, the relative densities of elements, acting as the design variables, are updated using the optimality criterion method. Finally, the distributions of elemental densities and moduli are obtained for further applications. Several typical numerical examples are used to demonstrate the effectiveness of the proposed method.

Multidiscipline Topology Optimization of Stiffened Plate/Shell Structures Inspired by Growth Mechanisms of Leaf Veins in Nature

Biological structures with preeminent performance in nature endow inexhaustible inspiration for creative design in engineering. In this paper, based on the observation of the natural morphogenesis of leaf veins, we put forward a simple and practical multidiscipline topology optimization method to produce the stiffener layout for plate/shell structures. This method simulates the emergence of complex branching patterns copying the self-optimization of leaf veins which always try to grow into a configuration with global optimal performances. Unlike the conventional topology optimization methods characterized by “subtraction mode,” the proposed method is based on the “addition mode,” giving great potential for designers to achieve more clear stiffener layout patterns rather than vague material distributions and, consequently, saving computational resources as well as enhancing availability of design outputs. Numerical studies of both static and dynamic problems considered in this paper clearly show the suitability of the proposed method for the optimal design of stiffened plate/shell structures.

Classification approach for reliability-based topology optimization using probabilistic neural networks

This research explores the usage of classification approaches in order to facilitate the accurate estimation of probabilistic constraints in optimization problems under uncertainty. The efficiency of the proposed framework is achieved with the combination of a conventional topology optimization method and a classification approach- namely, probabilistic neural networks (PNN). Specifically, the implemented framework using PNN is useful in the case of highly nonlinear or disjoint failure domain problems. The effectiveness of the proposed framework is demonstrated with three examples. The first example deals with the estimation of the limit state function in the case of disjoint failure domains. The second example shows the efficacy of the proposed method in the design of stiffest structure through the topology optimization process with the consideration of random field inputs and disjoint failure phenomenon, such as buckling. The third example demonstrates the applicability of the proposed method in a practical engineering problem.

Topology Optimization Incorporating Level Set Boundary Expressions Using a Particle Method

Topology optimization for structures has been applied to nonlinear structural problems, however conventional topology optimization methods for structures with geometrical nonlinearity encounter difficulties during nonlinear analysis using the FEM (Finite Element Method), due to the use of a mesh. In this study, we propose a new level set-based topology optimization method considering geometrical nonlinearity using a mesh-free/particle technique, for optimizing elastic structures that undergo large deformation. In the proposed method, the MPS (Moving Particle Semiimplicit) method, a particle method, is used for the response analysis, since it does not rely on a mesh for geometrically nonlinear analysis. In this paper, first, a topology optimization problem is formulated based on the level set method and a method for regularizing the optimization problem by the Tikhonov regularization method is explained. The reactiondiffusion equation that updates the level set function is then derived and an optimization algorithm, which uses the FEM to solve the equilibrium equations and the reaction-diffusion equation when updating the level set function, is constructed. Next, the particle interaction model and the treatment of geometrical nonlinearity in the MPS method are shown, and the implementation of combining the level set-based topology optimization and the MPS methods is explained. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed method of topology optimization for geometrically nonlinear problems.

110 レベルセット法を用いた車体構造の最適化

In this paper we propose a new method to design a reinforcement for a given thin shell structure such as body structure of a vehicle by utilizing a structural optimization method based on the level set method. In this method the allowed design space is defined around the given structure and the optimal reinforcement is created utilizing the level set based optimization method. The proposed method can give a designer more useful information of topology and shape of the reinforcement than the conventional topology or shape structural optimization methods. We demonstrate the efficiency of the method by showing a result where the optimal reinforcement is found for a simple given structure made by thin shell.

Topology Synthesis of Extrusion-Based Nonlinear Transient Designs

Many practical structural designs require that the structure is easily manufactured. Design concepts synthesized using conventional topology optimization methods are typically not easily manufactured, in that multiple finishing processes are required to construct the component. A manufacturing technique that requires only minimal effort is extrusion. Extrusion is a manufacturing process used to create objects of a fixed cross-sectional profile. The result of using this process is lower costs for the manufacture of the final product. In this paper, a hybrid cellular automaton algorithm is developed to synthesize constant cross section structures that are subjected to nonlinear transient loading. The novelty of the proposed method is the ability to generate constant cross section topologies for plastic-dynamic problems since the issue of complex gradients can be avoided. This methodology is applied to extrusions with a curved sweep along the direction of extrusion as well. Three-dimensional examples are presented to demonstrate the efficiency of the proposed methodology in synthesizing these structures. Both static and dynamic loading cases are studied.

Node-Wise Topological Shape Optimum Design for Structural Reinforced Modeling of Michell-Type Concrete Deep Beams

This study presents an associated structural design to continuous material topology optimization and a particular case of shape optimization using node-wise densities as design parameters. The generation of optimal shapes and topologies represented in this study is based on a three-dimensional density function bilinearly interpolated by element shape functions and nodal densities. The material interface between void and solid regions is described by a specific 0.5 cut-off level of continuous and smooth iso-lines of the nodal density function on a fixed mesh. This approach allows us to perform a simultaneous node-wise topology and shape optimization, which can be easily implemented by existing gradient-based optimization codes. Contrary to those of conventional material topology optimization methods, these optimal solutions are similar to ideal optimal solutions from analytical optimization techniques. Numerical examples for structural reinforced modeling of Michell-type concrete deep beams are used to demonstrate the efficiency and superiority of the resolutions of the present method.

Minimum Averaged Compliance Density Based Topology Optimization of Structures

Usually, an optimal topology is obtained by optimizing the material distribution within a prescribed domain; for example, a rectangular domain with a specified length and width for a plane problem. However, the dimensions (i.e. aspect ratio) of a rectangular design domain have significant influence on the resultant optimal topology. In this paper, a minimum Averaged Compliance Density (ACD) based method for topology optimization of structures is proposed. Unlike the conventional topology optimization method, the ACD is taken as the objective function, and the topology and domain dimensions of the structure are optimized simultaneously. As an example, the topology of a cantilever beam with large aspect ratio will be optimized, which is often difficult for traditional topology optimization algorithms. Through optimizing the topology and the dimensions of the design domain, a base structure is obtained, which is repeated to yield the whole, assembled beam. The influence of the relative values of shear force and moment is analyzed numerically. Results show that as the value of the bending moment increases relative to the shear force, the optimal topology changes from a truss‐like structure to a vertically stiffened box‐like structure.