Level Set Method For Optimization Research Articles

2-D shape optimization of auxiliary anode for ICCP

In this paper, to determine the influence of the shape of the auxiliary anode on the protection potential distribution in an impressed current cathodic protection (ICCP) system, an electrochemical and a level-set optimization method were combined to optimize the shape of the auxiliary anode. To accomplish this, the square of the deviation between the potential of the structure and the protection potential was used as the objective function, and the level set theory was employed to develop the anode shape. Following a set number of iterations, the optimized anode shape was obtained. Using the optimized anode, a current was injected into the anode surface, and the potential distribution was calculated. The experimental results showed that the potential distribution of the protected surface was uniform, and it could fully protect the protected structure’s surface, which helped to avoid overprotection or underprotection.

Parametric plate lattices: Modeling and optimization of plate lattices with superior mechanical properties

Recent advancements in additive manufacturing have made it possible to fabricate complex structures at the microscale, leading to increased interest in lattice structures for achieving high-performance materials. Among lattice structures, plate lattices have shown superior mechanical properties. However, conventional plate lattices are typically designed based on cubic crystals and their mixing structures, which limits the range of available morphologies and properties. In this study, we propose a parametric plate lattice (PPL), which is a unified parametric model integrated with shape optimization to generate plate lattices with desired mechanical properties. The parametric model allows for the generation of plate lattices with controlled topology continuity and quadrilateral mesh directly. When combined with a level-set optimization method, the parametric model thoroughly explores the property space of plate lattices and can achieve the target mechanical properties. We validate the performance of PPL through a comprehensive evaluation using both simulations and physical experiments. The results demonstrate that PPL can generate a large set of novel plate lattices with mechanical performance surpassing that of traditional lattices, both in isotropic and orthotropic conditions. These novel PPLs exhibit a larger property space and higher stiffness, which may approach or even exceed the theoretical stiffness upper bound.

A Parametric Level Set Method for Topology Optimization Based on Deep Neural Network

Abstract This paper proposes a new parametric level set method for topology optimization based on deep neural network (DNN). In this method, the fully connected DNN is incorporated into the conventional level set methods to construct an effective approach for structural topology optimization. The implicit function of level set is described by fully connected DNNs. A DNN-based level set optimization method is proposed, where the Hamilton–Jacobi partial differential equations (PDEs) are transformed into parametrized ordinary differential equations (ODEs). The zero-level set of implicit function is updated through updating the weights and biases of networks. The parametrized reinitialization is applied periodically to prevent the implicit function from being too steep or too flat in the vicinity of its zero-level set. The proposed method is implemented in the framework of minimum compliance, which is a well-known benchmark for topology optimization. In practice, designers desire to have multiple design options, where they can choose a better conceptual design base on their design experience. One of the major advantages of the DNN-based level set method is capable to generate diverse and competitive designs with different network architectures. Several numerical examples are presented to verify the effectiveness of the proposed DNN-based level set method.

Topology optimization of elastic structures with the aim of maximizing the buckling load factor using the level set method

This paper presents a study on Topology optimization of continuum structures with the aim of maximizing the lowest buckling load factor. In the structural shape and topology optimization problems, stability and buckling issues are not usually considered therefore in some cases, long and thin members are obtained in an optimized configuration that can lead to the instability of the structure. In this article level set method incorporating a fictitious interface, energy is applied to find the optimal configuration. One of the main problems in traditional continuum structural optimization methods, which are based on the removal or alterations in element density, is the creation of pseudo buckling modes in the optimization process. These pseudo buckling modes should be identified and removed. The level set optimization method, which is grounded on moving the structure's boundary, has a high ability to control topological complexities. Therefore, the idea of using the level set method has led to the elimination of pseudo buckling modes and the resolving of this problem. Derivation of the required speed term in the level set method is complicated and this derivation term for the buckling load factor is the innovation of this research. Numerical examples are illustrated to prove the effectiveness of this method.

M-VCUT level set method for optimizing cellular structures

This paper presents a multiple variable cutting (M-VCUT) level set method for the topology optimization of cellular structures. This method first divides the reference design domain into cells, and then in each cell it uses multiple basic level set functions combined with their cutting functions to describe the microstructure. The basic level set functions are used to describe microstructure prototypes that are kept fixed during the course of optimization. Each basic level set function is cut by its own cutting function to achieve shape and topology variation of microstructure. After the cutting operations, multiple virtual microstructures are obtained in each cell, and they are further combined together through a union operation to generate an actual microstructure. Based on the multiple virtual microstructures, multiple combination options are available, thus resulting in an enlarged design space and performance improvement as compared to the VCUT level set method developed in our previous study. Moreover, as a generalization of the VCUT level set method, connections between microstructures in neighboring cells are naturally guaranteed without any extra constraints.

A level set topology optimization method using a biharmonic equation based on plate theory

This study aims to propose a new level set optimization method capable of adjusting the complexity of resulting structure without loss of its optimality. The key idea is the deformational behavior of plates. We consider the level set function as the deflection of a fictitious plate on an elastic foundation subjected to the accumulative topological derivative load. The governing equation of this plate contains linear and biharmonic operators. The linear operator, arising from elastic foundation, directly connects the topological derivative to the level set function and plays the main role in creating new holes over the working domain based on the value of the topological derivative. While the biharmonic operator, arising from physical behavior of plate, performs as a filter and adjusts complexity of optimized structure. In the present paper, the topological derivative at a certain point is obtained by measuring the influence of removing element on an objective function. To impose a constraint, the classical controllers are added to the procedure. The results of several numerical examples confirm the validity and efficiency of the proposed optimization method.

Fuzzy time dependent structural reliability analysis using alpha level set optimization method based on genetic algorithm

Fuzzy time dependent structural reliability analysis using alpha level set optimization method based on genetic algorithm

An Efficient Image Co-segmentation Algorithm based on Active Contour and Image Saliency

Image co-segmentation is the problem of extracting common objects from multiple images and it is a very challenging task. In this paper we try to address the co-segmentation problem by embedding image saliency into active contour model. Active contour is a very famous and effective image segmentation method but performs poor results if applied directly to co-segmentation. Therefore, we can introduce additional information to improve the segmentation results, such as saliency which can show the region of interest. In order to optimize the model, we propose an efficient level-set optimization method based on super-pixels, hierarchical computation and convergence judgment. We evaluated the proposed method on iCoseg and MSRC datasets. Compared with other methods, our method yielded better results and demonstrated the significance of using image saliency in active contour.

A level set-based topology optimization incorporating arbitrary Lagrangian Eulerian method for wavelength filter using extraordinary optical transmission

This paper deals with structural optimization for designing periodic structures in a hole array wavelength filter. The hole array wavelength filter that consists of metallic thin film and dielectric enables to transmit narrow bandwidth light. It is known that transmission spectrum can be changed not only by the periodicity of hole array but also by the shape of holes. For optimizing the hole shape, the level set method is used in this study. In the ordinary level set method, the boundaries are implicitly expressed by the zero level set of a scalar function, called the level set function, within a fixed mesh. Therefore, the material interpolation becomes numerically awkward within the elements across the implicit zero level set because those elements inevitably take on intermediate material properties even if the boundary of zero level set are mathematically clear. As the result, the optimization is likely to yield wrong solution. Here, a new level set optimization method incorporating Arbitrary Lagrangian Eulerian method is proposed to eliminate intermediate values on the interfaces perfectly. As a result, the proposed method can successfully perform the structural optimization of hole shape without intermediate values.

Level set method for optimization of contact problems

This paper deals with the numerical solution of topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is described by an elliptic variational inequality of the second order governing a displacement field. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The shape of this boundary and its evolution is described using the level set approach. Level set methods are numerically efficient and robust procedures for the tracking of interfaces. They allow domain boundary shape changes in the course of iteration. The evolution of the domain boundary and the corresponding level set function is governed by the Hamilton–Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative of an appropriately defined functional with respect to the free boundary. In this paper the necessary optimality condition is formulated. The level set method, based on the classical shape gradient, is coupled with the bubble or topological derivative method, which is precisely designed for introducing new holes in the optimization process. The holes are supposed to be filled by weak phase mimicking voids. Since both methods capture a shape on a fixed Eulerian mesh and rely on a notion of gradient computed through an adjoint analysis, the coupling of these two method yields an efficient algorithm. Moreover the finite element method is used as the discretization method. Numerical examples are provided and discussed.

Radial basis functions and level set method for structural topology optimization

AbstractLevel set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.