Level Set-based Topology Optimization Research Articles

Shape optimization with level set-based method using a reaction diffusion equation for 2D sound barrier

A level set-based method using a reaction diffusion equation is applied for optimization problems of two dimensional (2D) sound barriers. The level set method is employed to implicitly represent the sound barrier structure, which distinguishes the material and void domains by the value of the level set function. The boundary element method is employed to solve acoustic problems governed by Helmholtz equation. Topological derivatives are computed by the boundary integral equation combined with the adjoint variable method. The distribution of level set function is iteratively updated based on the reaction diffusion equation to find the optimal structure. For the existent floating scatterers in the optimization process and the sharp and narrow parts on the surface of the sound barrier, we propose a filtering algorithm to remove floating scatterers and develop a method to achieve a smooth surface of the sound barrier. The shape optimization of sound barriers is achieved using these techniques, integrating the level set-based topology optimization method. Numerical tests are provided to demonstrate the validity and effectiveness of the proposed methods.

A level set based topology optimization for elastodynamic problems using BEM

The paper presents a topology optimization methodology for 2D elastodynamic problems using the boundary element method (BEM). The topological derivative is derived based on the variation method and the adjoint variable method. The level set method is employed for the representation of the material domain and voids within a specified design domain. Thus, the boundaries can easily be generated, following the zero isocontour of the level set function. Numerical implementation is carried out to demonstrate the effectiveness of the proposed topology optimization methodology in wave isolation and waveguide problems.

Acoustic metasurface development for transmitted wavefront manipulation using a level set-based topology optimization approach

A customized transmission-type metasurface design is developed for high-efficiency wavefront modulation by employing a level set-based topology optimization approach. The metasurface consists of periodically arranged supercells made of discrete unit cells aiming to form a precisely linear phase gradient of the transmitted wave ranging from 0 to 2π with high transmittances. The configuration of each unit is optimized to generate a desired phase and amplitude response of the transmitted wave at the interface. The transmitted wavefront steering realized by the phase-gradient metasurface is numerically and experimentally verified according to the diffraction theory, demonstrating that the level set-based topology optimization approach is a viable and effective technique for developing a transmitted phase-gradient metasurface. Afterwards, the designed optimal unit cells are elaborately arranged to form desired phase shift distributions for realizing fascinating acoustic functionalities, including acoustic energy focusing and acoustic Airy-like beam generation, further demonstrating the ability of the optimal units to achieve unprecedented transmitted wavefront control. This study presents a promising approach for customized transmission-type metasurface design based on the level set-based topology optimization, which is expected to be significantly valuable for acoustic device development.

Level-set based topology optimization for bi-linear type elasto-plastic problems

Level-set based topology optimization for bi-linear type elasto-plastic problems

Level set-based topology optimization for programmable acoustic structures

Controlling acoustic waves is essential in noise-reduction applications. Acoustic structures, which can efficiently manipulate acoustic waves based on their structural design, have been used to achieve the desired wave propagation behaviors. To meet various demands on sound propagation, it is crucial to design acoustic structures with multiple functionalities. In this research, we propose a level set-based topology optimization method for acoustic structures with desired acoustic performances that depend on their deformation. Firstly, we introduce a mathematical model that represents acoustic wave propagation based on the Helmholtz equation and the coordinate transformation. Next, we formulate the optimization problem based on the framework of a level set-based topology optimization method. A two-dimensional numerical example is provided to demonstrate the validity of the proposed method. As an example, we optimize a sound reflective structure with the desired acoustic responses that are dependent on the deformation of the optimized structure.

A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al. (Int J Solids Struct 45(14–15):4130–4146, 2008) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our Hilbertian projection method chooses a normal velocity for the level set function as a linear combination of (1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity and (2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.

Topology optimization of natural convection using porous metal foam based on the adjoint lattice Boltzmann method and level set method

In this work, the level-set based topology optimization (TO) method for optimizing the distribution of porous metal foam in the fully-coupled natural convection system is developed. The respective adjoint lattice Boltzmann (LB) model and the sensitivity expression are derived. The forward LB model is validated by natural convection in cavity partially filled with porous medium and the TO method is validated by the pipe bend optimization problem. The validated TO method is utilized to design the two-dimensional passive heat sink made up of metal foam. It is found that the optimized design can approach the heat transfer enhancement of the conventional design with 100% filling while using only 40% material. For the base case, the average Nusselt number can be increased by 3.96∼6.74 times with the optimized designs mounted on the heating element when the Grashof number ranges from 15.7 to 2.8 × 104. Effects of increasing the Darcy number, decreasing the thermal conductivity ratio and decreasing the maximum foam volume fraction on the optimized designs and the heat transfer performance are discussed in detail. The two-dimensional TO structures are further stretched to three-dimension for further verification, and a 13.1%∼41.4% lower temperature than the conventional design is achieved by the optimized designs, indicating the superiority of the TO method.

Level set-based shape optimization of deformable structures for achieving desired sound transmission and reflective responses

In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based on coordinate transformation and the Eulerian approach. Next, we formulate the shape optimization problem and perform sensitivity analysis based on the shape derivative concept. We then construct an optimization algorithm based on the framework of a level set-based shape and topology optimization method. Finally, we provide two-dimensional optimization examples that demonstrate the effectiveness of our proposed method in providing optimized designs with desired functionality, while considering the structural deformation.

A fully coupled level set-based topology optimization of flexible components in multibody systems

A fully coupled level set-based topology optimization of flexible components in multibody systems is considered. Thereby, using the floating frame of reference approach, the flexible components are efficiently modeled and incorporated in multibody systems. An adjoint sensitivity analysis is utilized to obtain the gradient of the objective function with respect to a set of density-like design variables assigned to elements included in the underlying finite element model. The utilized adjoint sensitivity analysis provides a gradient, which is within numerical limits exact. In this process, the parametrization of material properties of finite elements has a significant influence on the calculated gradient, in particular for poorly filled elements. These influences are studied in detail. As an application example, the compliance minimization problem of a flexible piston rod in a transient slider-crank mechanism is considered. For this model, the influence of different parametrization methods on the obtained gradient is discussed, and a gradient strategy is proposed to overcome numerical issues included in different parametrization laws. Using this gradient strategy within a level set-based algorithm, a topology optimization of the flexible piston rod is performed. The corresponding results are then compared with optimization results provided by the method of moving asymptotes (MMA). Moreover, the computational effort of the sensitivity analysis is high and scales with the number of design variables. In this work, a gradient approximation is introduced using radial basis functions (RBFs). This helps to develop an appropriate gradient for a level set-based topology optimization of the flexible components in multibody systems, where the RBF-based design space reduction decreases the computational effort of the utilized sensitivity analysis. Finally, the efficiency gain obtained by the introduced design space reduction is demonstrated by optimization examples.

Hole and dot sensitivity analysis and level set-based topology optimization of superconducting systems operating under critical current density

Hole and dot sensitivity analysis and level set-based topology optimization of superconducting systems operating under critical current density

A level set-based interface-enriched topology optimization for the design of phononic crystals with smooth boundaries

Phononic crystals can be designed to show band gaps—ranges of frequencies whose propagation is strongly attenuated in the material. In essence, their working principle is based on destructive interference of waves reflecting from the periodic arrangement of material interfaces (i.e., Bragg scattering). Consequently, capturing accurately the behavior at material interfaces requires appropriate numerical modeling and computational design techniques. However, the commonly used density-based representation in popular topology optimization methods results in a diffuse staircased boundary. The heavily refined finite element meshes required to compensate for this boundary description results in exceedingly large and expensive optimization problems. In this paper, we demonstrate the adverse effect of the density-based boundary description. Furthermore, we propose a level set-based topology optimization procedure with an enriched finite element method that shows improved performance when compared to the density-based approach.

Topology optimization method with nonlinear diffusion

This paper is concerned with topology optimization based on a level set method using (doubly) nonlinear diffusion equations. Topology optimization using the level set method is called level set-based topology optimization, which is possible to determine optimal configurations that minimize objective functionals by updating level set functions. In this paper, as an update equation for level set functions, (doubly) nonlinear diffusion equations with reaction terms are derived, and then the singularity and degeneracy of the diffusion coefficient are applied to obtain fast convergence of configurations and damping oscillation on boundary structures. In particular, the reaction terms in the proposed method do not depend on the topological derivatives, and therefore, sensitivity analysis to determine a descent direction for objective functionals is relaxed. Furthermore, a numerical algorithm for the proposed method is constructed and applied to typical minimization problems to show numerical validity. This paper is a justification and generalization of the method using reaction–diffusion equations developed by one of the authors in Yamada et al. (2010).

Two-phase topology optimization for metamaterials with negative Poisson’s ratio

Although recent developments in 3D printing technology have made it possible to fabricate metamaterials with characteristic mechanical properties, it is not easy to fabricate complex shapes containing cavities. In this study, a composite structure comprising two types of materials without a cavity was optimized. Moreover, a mechanical metamaterial with a negative Poisson’s ratio that can be fabricated using an additive manufacturing method was developed. First, a homogenization method that characterizes the properties of composite structures was briefly described. Then, an optimization problem to realize a negative Poisson’s ratio was formulated, and a level set-based topology optimization method was proposed to solve the abovementioned problem. Next, three-dimensional numerical examples are presented to confirm the effectiveness of the proposed method, and the deformation behaviors of the optimized designs are numerically examined. Furthermore, a sample containing the optimized design with negative Poisson’s ratio for the tensile test was fabricated using a 3D printer. We conducted some experiments to evaluate its mechanical performance.

Enhancing level set-based topology optimization with anisotropic graded meshes

We propose a new algorithm for the design of topologically optimized lightweight structures, under a minimum compliance requirement. The new process enhances a standard level set formulation in terms of computational efficiency, thanks to the employment of a computational mesh customized to the problem at hand. We pursue a twofold goal, i.e., to deliver a final layout characterized by a smooth contour and reliable mechanical properties. The smoothness of the optimized structure is ensured by the employment of an anisotropic adapted mesh, which sharply captures the material/void interface. A robust mechanical performance is guaranteed by a uniform tessellation of the internal part of the optimized configuration. A thorough numerical investigation corroborates the effectiveness of the proposed algorithm as a reliable and computationally affordable design tool, both in two- and three-dimensional contexts.

XFEM level set-based topology optimization for turbulent conjugate heat transfer problems

Solving conjugate heat transfer design problems is relevant for various engineering applications requiring efficient thermal management. Heat exchange between fluid and solid can be enhanced by optimizing the system layout and the shape of the flow channels. As heat is transferred at fluid/solid interfaces, it is crucial to accurately resolve the geometry and the physics responses across these interfaces. To address this challenge, this work investigates for the first time the use of an eXtended Finite Element Method (XFEM) approach to predict the physical responses of conjugate heat transfer problems considering turbulent flow. This analysis approach is integrated into a level set-based optimization framework. The design domain is immersed into a background mesh and the geometry of fluid/solid interfaces is defined implicitly by one or multiple level set functions. The level set functions are discretized by higher-order B-splines. The flow is predicted by the Reynolds Averaged Navier–Stokes equations. Turbulence is described by the Spalart–Allmaras model and the thermal energy transport by an advection–diffusion model. Finite element approximations are augmented by a generalized Heaviside enrichment strategy with the state fields being approximated by linear basis functions. Boundary and interface conditions are enforced weakly with Nitsche’s method, and the face-oriented ghost stabilization is used to mitigate numerical instabilities associated with the emergence of small integration subdomains. The proposed XFEM approach for turbulent conjugate heat transfer is validated against benchmark problems. Optimization problems are solved by gradient-based algorithms and the required sensitivity analysis is performed by the adjoint method. The proposed framework is illustrated with the design of turbulent heat exchangers in two dimensions. The optimization results show that, by tuning the shape of the fluid/solid interface to generate turbulence within the heat exchanger, the transfer of thermal energy can be increased.

Multiscale topology optimization of electromagnetic metamaterials using a high-contrast homogenization method

This study proposes a multiscale topology optimization method for electromagnetic metamaterials using a level set-based topology optimization method that incorporates a high-contrast homogenization method. The high-contrast homogenization method can express wave propagation behavior in metamaterials for various frequencies. It can also capture unusual properties caused by local resonances, which cannot be estimated by conventional homogenization approaches. We formulated multiscale topology optimization problems where objective functions are defined by the macroscopic wave propagation behavior, and microstructures forming a metamaterial are set as design variables. Sensitivity analysis was conducted based on the concepts of shape and topological derivatives. As numerical examples, we offer optimized designs of metamaterials composed of multiple unit cell structures working as a demultiplexer based on negative permeability. The mechanism of the obtained metamaterials is discussed based on homogenized coefficients.

Level set-based topology optimization for thermal-fluid system based on the radial basis functions

This work proposes a novel topology optimization methodology derived from a parametric level set model to the weakly-coupled heat flow system. In proposed approach, traditional level set function is explicitly through compactly supported radial basis functions and adjoint variable method (AVM) is utilized to derive the sensitivity analysis. The optimal design goal is treated as maximizing heat exchange and minimizing total power dissipation, simultaneously. To achieve comparable results within fewer iterations and more stable convergence process than solving Ordinary Differential Equations, the method of moving asymptotes is employed to drive design variables update. Earth Mover’s Distance which can weigh similarity between two structures, has been chosen as the index instead of visual observation to estimate the dependence of the developed method and density-based method on the initial guesses. Three dimensional results and distribution characteristics of each physical fields are presented.

Parametrized level set method for a coupled thermal–fluid problem using radial basis functions

This paper develops a parametric level set-based topology optimization method for the micro-channel heat sink design. In this study, the optimal objective functional is formulated as a bi-objective functional consisting of heat transfer maximization and pressure drop minimization based on the two- and three-dimensional steady-state Navier–Stokes and energy balance equations. The normal velocity of the interface propagation, which is derived from the steepest descent method and adjoint variable method, is used to drive the evolution of the level set function. Influence of the governing equations on the optimized heat sink structure and the distribution characteristics of adjoint variables are analysed through numerical experiments, which proved that governing equations can improve the heat dissipation capacity of the optimal structure. Based on Earth Mover’s Distance, the developed method is proved to have less dependence on initial structures compared with the density-based topology optimization method. Finally, through the experimental tests and numerical simulations, the effectiveness of the developed model is confirmed.

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 level set-based topology optimization approach for thermally radiating structures

The need for efficient thermally radiating structures is apparent in many aerospace system designs including satellites, launch vehicles, and hypersonic aircraft. This paper presents a novel level set-based topology optimization approach for designing thermally efficient radiating structures. In this paper, we derive a shape sensitivity of the thermal heat power radiated objective function using the adjoint method. This sensitivity is a necessary ingredient for our gradient-based algorithm. We apply an augmented Lagrangian method to solve an example 2D problem where the goal is to maximize heat power rejected subject to a material volume constraint. The radiating surface is kept fixed during the optimization to maintain a design-independent boundary condition, while the conducting region is optimized. Several solutions are illustrated with varying initial conditions. We also present a case study indicating that maximizing the thermal compliance functional is not sufficient for solving this class of problems.