Level Set Topology Optimization Research Articles

Design of metamaterial-based heat manipulators using isogeometric level-set topology optimization

We exploit level-set topology optimization to find the optimal material distribution for metamaterial-based heat manipulators. The level-set function, geometry, and solution field are parameterized using the Non-Uniform Rational B-Spline (NURBS) basis functions to take advantage of easy control of smoothness and continuity. In addition, NURBS approximations can produce conic geometries exactly and provide higher efficiency for higher-order elements. The values of the level-set function at the control points (called expansion coefficients) are utilized as design variables. For optimization, we use an advanced mathematical programming technique, Sequential Quadratic Programming. Taking into account a large number of design variables and the small number of constraints associated with our optimization problem, the adjoint method is utilized to calculate the required sensitivities with respect to the design variables. The efficiency and robustness of the proposed method are demonstrated by solving three numerical examples. We have also shown that the current method can handle different geometries and types of objective functions. In addition, regularization techniques such as Tikhonov regularization and volume regularization have been explored to reduce unnecessary complexity and increase the manufacturability of optimized topologies.

An efficient MATLAB code for stress-constrained level set topology optimization method

An efficient MATLAB code for stress-constrained level set topology optimization method

A novel discrete adjoint-based level set topology optimization method in B-spline space

A novel discrete adjoint-based level set topology optimization method in B-spline space

Level-set topology optimization for Ductile and Brittle fracture resistance using the phase-field method

This work presents a rigorous mathematical formulation for topology optimization of a macro structure undergoing ductile failure. The prediction of ductile solid materials which exhibit dominant plastic deformation is an intriguingly challenging task and plays an extremely important role in various engineering applications. Here, we rely on the phase-field approach to fracture which is a widely adopted framework for modeling and computing the fracture failure phenomena in solids. The first objective is to optimize the topology of the structure in order to minimize its mass, while accounting for structural damage. To do so, the topological phase transition function (between solid and void phases) is introduced, thus resulting in an extension of all the governing equations. Our second objective is to additionally enhance the fracture resistance of the structure. Accordingly, two different formulations are proposed. One requires only the residual force vector of the deformation field as a constraint, while in the second formulation, the residual force vector of the deformation and phase-field fracture simultaneously have been imposed. An incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. Thereafter, to obtain optimal topology to enhance the structural resistance due to fracture, the level-set-based formulation is formulated. The level-set-based topology optimization is employed to seek an optimal layout with smooth and clear boundaries. Sensitivities are derived using the analytical gradient-based adjoint method to update the level-set surface for both formulations. Here, the evolution of the level-set surface is realized by the reaction–diffusion equation to maximize the strain energy of the structure while a certain volume of design domain is prescribed. Several three-dimensional numerical examples are presented to substantiate our algorithmic developments.

Multiphysics Design Optimization via Generative Adversarial Networks

Abstract This work presents a method for generating concept designs for coupled multiphysics problems by employing generative adversarial networks (GANs). Since the optimal designs of multiphysics problems often contain a combination of features that can be found in the single-physics solutions, we investigate the feasibility of learning the optimal design from the single-physics solutions, to produce concept designs for problems that are governed by a combination of these single physics. We employ GANs to produce optimal topologies similar to the results of level set topology optimization (LSTO) by finding a mapping between the sensitivity fields of specific boundary conditions, and the optimal topologies. To find this mapping, we perform image-to-image translation GAN training with a combination of structural, heat conduction, and a relatively smaller number of coupled structural and heat conduction data. We observe that the predicted topologies using GAN for coupled multiphysics problems are very similar to those generated by level set topology optimization, which can then be used as the concept designs for further detailed design. We show that using a combination of multiple single-physics data in the training improves the prediction of GAN for multiphysics problems. We provide several examples to demonstrate this.

Multi-material topology optimization of coated structures using level set method

The coated structures are widely applied in engineering, whose coating material covers the surface to protect substrate structure or realize certain functionalities. Currently, the coated structures are generally filled with uniform material. Moreover, the multi-material infill structures can further improve their performances. Therefore, this paper proposes a level set topology optimization method to design the coated structures with multiple infill materials. The coating material with uniform thickness is required to attach the outside of the substrate structure, which can be seen as a manufacturing constraint. In particular, we demonstrate how to construct the multi-material infill structures and formulate the multi-material interpolation model by combining multiple level set functions. Besides, the shape derivatives for the multi-material optimization model of the coated structures are derived for the first time. Finally, numerical examples confirm that the coated structures with multiple infill materials can be effectively obtained by using the proposed method.

Level set-based topology optimization for the design of labyrinthine acoustic metamaterials

We present a topology optimization method for the design of labyrinthine metamaterials with a large path length for acoustic wave propagation. An objective function maximizes the pressure gradient in an air-filled region at a low frequency, and a level set-based optimization method enables the optimal design of the labyrinthine structure with high computational efficiency owing to the concept of topological derivative. First, we explain the design settings of labyrinthine metamaterials. Next, the objective functions to obtain labyrinthine structures are introduced, and the corresponding topological derivatives are obtained. The optimization problem is formulated in the framework of the level set topology optimization method, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. Based on the frequency responses of the effective parameters and dispersion analysis, the optimized designs are characterized by a negative refractive index over a wide frequency interval, which can be controlled by a parameter introduced in the objective function.

A parallel parameterized level set topology optimization framework for large-scale structures with unstructured meshes

In addition to the requirements of full-scale optimization, the adaptability to structures with arbitrary geometries and complex boundary conditions is also important to topology optimization in practical engineering applications. A parallel parameterized level set topology optimization framework for large-scale structures with unstructured meshes is proposed in this work, in which the full-scale optimization is realized with distributed memory parallel computing technology while the arbitrary geometries and complex boundary conditions are conveniently handled with the usage of unstructured meshes. To realize the combination of distributed memory parallel computing technology and parameterized level set topology optimization using unstructured meshes, several means are taken: (1) the shape functions in finite element analysis are employed to parameterize the level set function; (2) the data structure called directed acyclic graph is adopted to represent the unstructured mesh; (3) the passive domain and boundary conditions are imposed directly on the geometry entities of the structures; (4) a multiple averaging filter is introduced to reduce the tiny structural members in the optimized results for the requirement of manufacturability. Several computing tests are presented in this paper, which verify the stability, efficiency, scalability, and the potential to discover new structure styles of the framework.

Level set topology optimization of elasto-plastic materials with local stress constraints

Level set topology optimization of elasto-plastic materials with local stress constraints

A parameter space approach for isogeometrical level set topology optimization

AbstractIn this article, a new isogeometrical level set topology optimization is introduced. In previous studies, the level set function is approximated by b‐splines basis functions and the control net is updated during the optimization process. Since control points are not basically on the level set surface, a discrepancy between zero level of function that represents boundaries of the structure, and zero level of its control netis appeared. In this article, the idea is solving level set equation (LSE) over the parameter space of b‐splines basis functions defined in the IGA model. Afterwards, the updated level set function is mapped into the physical space. The level set function is approximated over a grid defined in parameter space which is constantly a unit square. By doing so, control net of the IGA model and the grid for solving LSE are separated. Two well‐known radial basis functions (RBF) and reaction–diffusion (RD) based methods are employed to solve the LSE. The proposed method is applied to minimize the mean compliance when certain amount of material is used and also for weight minimization subject to local stress constraints. Several numerical examples are presented to demonstrate performance and accuracy of the proposed method.

Minimum feature size control in level set topology optimization via density fields

A level set topology optimization approach that uses an auxiliary density field to nucleate holes during the optimization process and achieves minimum feature size control in optimized designs is explored. The level set field determines the solid-void interface and the density field describes the distribution of a fictitious porous material using the solid isotropic material with penalization. These fields are governed by two sets of independent optimization variables which are initially coupled using a penalty for hole nucleation. The strength of the density field penalization and projection is gradually increased during the optimization process to promote a 0–1 density distribution. In addition, a second penalty regulates the evolution of the density field in the void phase. The treatment of the density field combined with the second penalty mitigate the appearance of small design features. The minimum feature size of optimized designs is controlled by the radius of the linear filter applied to the density optimization variables. The structural response is predicted by the extended finite element method, the sensitivities by the adjoint method, and the optimization variables are updated by a gradient-based optimization algorithm. Numerical examples investigate the robustness of this approach with respect to algorithmic parameters and mesh refinement. The results show the applicability of the combined density level set topology optimization approach for both optimal hole nucleation and for minimum feature size control in 2D and 3D. This comes, however, at the cost of a more complex problem formulation and additional computational cost due to an increased number of optimization variables.

Topology optimization for maximizing linear buckling load based on level set method

Stability and buckling have attracted extensive attention in the design of structural elements, especially in the design of thin-walled structures since they may naturally have poor stability and be prone to buckling failure. This paper proposes a level-set based topology optimization (TO) method that can maximize the lowest linear buckling load under a mean compliance constraint. First, we conduct the linearized buckling analysis and formulate the optimum design problem. Second, we derive the design sensitivity and revisit the reaction-diffusion equation-based level-set topology optimization. Finally, we solve several two-dimensional benchmark problems and the design results are presented to validate the proposed method.

Level set topology optimization of synchronous reluctance machines using a body-fitted mesh representation

Level set topology optimization of synchronous reluctance machines using a body-fitted mesh representation

Level set topology optimization with nodally integrated reproducing kernel particle method

A level set topology optimization (LSTO) using the stabilized nodally integrated reproducing kernel particle method (RKPM) to solve the governing equations is introduced in this paper. This methodology allows for an exact geometry description of a structure at each iteration without remeshing and without any interpolation scheme. Moreover, useful characteristics of the RKPM such as the easily controlled order of continuity and the ability to freely place particles in a design domain wherever needed are illustrated through stress based and design-dependent surface loading examples. The numerical results illustrate the effectiveness and robustness of the methodology with good optimization convergence behavior and ability to handle large topological changes. Furthermore, it is shown that different particle distributions can be used to increase efficiency without additional complexity.

Isogeometric level set topology optimization for elastoplastic plane stress problems

Isogeometric level set topology optimization for elastoplastic plane stress problems

Level set topology optimization of load carrying battery packs

In this study, the optimum design of load carrying battery packs that can also exchange heat with a coolant is presented. The level set topology optimization method is used as the design tool. The flow is modeled using the Darcy potential flow, and the heat transfer is modeled using the convection-diffusion equation. The displacement field is computed using a thermo-elasticity model. A multi-objective optimization is used that minimizes a linear combination of structural compliance, average temperature of the battery cells, and the pressure drop across the coolant. As numerical examples, we present a design study of a load carrying battery pack module that is placed inside an aircraft wing. The efficiency of the optimized designs are demonstrated by showcasing their multifunctional nature—they reduce the temperature in the battery cells by exchanging the heat generated with a coolant all the while sustaining a given load.

Topology Optimization Considering Porosity Defects in Metal Additive Manufacturing

In this paper, considering the porosity defects of Additive Manufacturing (AM), a level set topology optimization method for AM with porosity constraints is proposed. The concept of topological sensitivity is used to formulate a global porosity constraint function in the proposed method, and a level set topology optimization model considering porosity defects is obtained. To improve the robustness of the algorithm, the topology optimization model is solved in two phases. At first, the classical level set method without the porosity constraint is used to initially optimize the structure. During this process, the hole nucleation method combining bi-directional evolutionary structural optimization (BESO) and the topological sensitivity is used. Secondly, the topology optimization considering the effects of porosity is implemented on the preliminary optimization results. After performing the two-step optimization, a robust structure that alleviates the harmful impact of porosity defects is obtained. Finally, the robustness and effectiveness of the proposed method are validated by several two-dimensional numerical examples.

Level-set topology optimization for effective control of transient conductive heat response using eigenvalue

Designing thermally efficient devices for an instantaneous heating condition is a challenging task, since a structure that simultaneously satisfies a lower peak temperature as well as a slower thermal response time must be achieved. In order to design a structure for both of the requirements, a transient analysis-based structural topology optimization method can be usually employed. However, the overall thermal response time of a heat conduction system can be modelled simply via the fundamental thermal eigenvalue, which is a physical quantity. This suggests that a thermal eigenvalue analysis can be adopted instead of transient analysis for an effective design of the structure controlling its thermal response time. In this study, we aim at developing a level-set topology optimization method to effectively optimize the overall response time of heat conduction systems using the thermal eigenvalue. The proposed level-set thermal eigenvalue topology optimization updates the structural boundary based on the common fixed grid approach, and the adjoint variable method is applied to compute shape sensitivity. Numerical examples clearly show the controllability of overall heat response time through the thermal eigenvalue, and especially, it is demonstrated that the proposed method is useful in designing an effective cooling structure for an instantaneous heating.

Transient level-set topology optimization of a planar acoustic lens working with short-duration pulse.

With the advance of additive manufacturing, many researchers are increasingly interested in planar acoustic lenses that are not only easier to fabricate than typical convex/concave lenses, but also have excellent imaging performance. However, the planar acoustic lenses reported so far cannot work for a short-duration pulse used in conventional imaging systems due to their inherent dispersive characteristics. This study addresses the challenge by devising a transient topology optimization formulation to design a planar acoustic lens that works effectively for a short-duration pulse. A planar lens consists of two materials where optimal combination and distribution are obtained with a crisp interface via the level-set method. Design is based on the transient acoustic responses, which are calculated from a time-dependent acoustic model solved by the Newmark method. The proposed method uses the area-fraction approach to compute the acoustic properties of a cut element by the interface. A localizing time-window function is introduced so that acoustic energy can be focused within the desired time range as much as possible. We obtain optimum design solutions designed with the proposed method and verify its effectiveness through the numerical investigations.

A Hole Nucleation Method Combining BESO and Topological Sensitivity for Level Set Topology Optimization.

As is known to all, the incapacity to nucleate holes automatically in the design domain is one of the main issues of the classical level set topology optimization method. To solve the issue of hole nucleation, this paper employs the bi-directional evolutionary structural optimization (BESO) method based on the material removal scheme and the frequently used topological sensitivity and proposes the combining BESO and topological sensitivity (CBT) method for level set topology optimization. This method can replace the existing hole nucleation method for level set topology optimization. First, the topological sensitivity is combined with BESO, and the BESO method based on topological sensitivity is proposed. Second, the method is integrated into level set topology optimization to solve the issue of hole nucleation. Two sensitivity thresholds are defined depending on the evolutionary volume ratio and boundary topological sensitivity, respectively, and the smaller one is used as the sensitivity threshold for hole nucleation. The material is removed from the design domain to nucleate holes based on this threshold. Three classical two-dimensional numerical examples are used to validate the proposed hole nucleation method.