Level Set-based Shape Optimization Research Articles

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.

Shape optimization of the time-harmonic response of vibroacoustic devices using cut elements

This article presents a method for generalized shape optimization of time-harmonic vibroacoustic problems. The modeling approach utilizes an immersed boundary cut element method in conjunction with a level set representation of the geometry. The cut element method utilizes a fixed background mesh, a dimensionless contrast parameter and an integration scheme to realize complex geometries and obtain accurate physical solutions to the governing problem. The design parameterization is obtained using a nodal level set description, directly linked to the mathematical design variables, and the gradients of the objective and the constraints are obtained with the discrete adjoint approach. The framework is applied to the optimization of three 2D examples. A study on the effect of initial guess for the proposed optimization procedure is presented on a benchmark example of the design of an acoustic partitioner. Further optimization examples include design of a wave splitter to realize prescribed frequency dependent directivity for emitted acoustic waves and a suspension structure design to improve the performance of a simplified 2D model of a hearing instrument. The results demonstrate that, even though the final topology is strongly dictated by the initial design, modifying the shape allows for a significant improvement of the system behavior. • A generalized shape optimization methodology for 2D time-harmonic vibroacoustic problems using cut elements. • A simple, yet accurate, crisp interface cut element method without the need for stabilization. • A numerically stable level set based shape optimization procedure based on density methods. • Three numerical examples that demonstrate the strengths and weaknesses of the proposed method.

A wavelength selective emitter design method using hyperbolic tangent level set-based shape optimization

A wavelength selective emitter is one of the key elements of the TPV generation system. Recently, a wavelength selective emitter with periodic microstructure on the surface has attracted attention. In this study, we propose a nonparametric design method of the wavelength selective emitter, which utilizes a shape optimization technique. The proposed method provides a design example of a 1D grating wavelength selective emitter that shows near unity emissivity at target wavelength. The obtained optimal configuration utilizes two physical mechanisms for the emissivity improvement: the excitation of the surface plasmon polariton and the cavity resonance. Furthermore, by considering multiple wavelengths, the proposed method can also provide an emitter configuration which has higher wavelength selectivity performance.

Level set based shape optimization using trimmed hexahedral meshes

This paper presents a new approach for level set based shape optimization using trimmed hexahedral meshes that geometrically conform to shape and topology changes during the optimization process. The marching cubes algorithm is employed to generate trimmed hexahedral meshes at each optimization iteration by splitting a regular background hexahedral mesh with the zero-isosurface of a level set function. A simple method is proposed to handle nonconforming polyhedral elements at the interface due to the face ambiguity arising in the marching cubes algorithm. The motion of the zero-isosurface of a level set function is obtained from the solution of the Hamilton–Jacobi transport equation using a normal velocity field defined on the regular background hexahedral mesh. Numerical results show that the present approach provides a natural and effective tool to solve shape optimization problems relying on explicitly discretized meshes of the shapes throughout the optimization process.

Structural optimization under internal porosity constraints using topological derivatives

Porosity is a well-known phenomenon occurring during various manufacturing processes (casting, welding, additive manufacturing) of solid structures, which undermines their reliability and mechanical performance. The main purpose of this article is to introduce a new constraint functional of the domain which controls the negative impact of porosity on elastic structures in the framework of shape and topology optimization. The main ingredient of our modeling is the notion of topological derivative, which is used in a slightly unusual way: instead of being an indicator of where to nucleate holes in the course of the optimization process, it is a component of a new constraint functional which assesses the influence of pores on the mechanical performance of structures. The shape derivative of this constraint is calculated and incorporated into a level set based shape optimization algorithm. Our approach is illustrated by several two- and three-dimensional numerical experiments of topology optimization problems constrained by a control on the porosity effect.

Level set-based shape optimization for temperature minimization problems in nanostructures

Level set-based shape optimization for temperature minimization problems in nanostructures

Self-consistent gradient flow for shape optimization

We present a model for image segmentation and describe a gradient-descent method for level-set based shape optimization. It is commonly known that gradient-descent methods converge slowly due to zig–zag movement. This can also be observed for our problem, especially when sharp edges are present in the image. We interpret this in our specific context to gain a better understanding of the involved difficulties. One way to overcome slow convergence is the use of second-order methods. For our situation, they require derivatives of the potentially noisy image data and are thus undesirable. Hence, we propose a new method that can be interpreted as a self-consistent gradient flow and does not need any derivatives of the image data. It works very well in practice and leads to a far more efficient optimization algorithm. A related idea can also be used to describe the mean-curvature flow of a mean-convex surface. For this, we formulate a mean-curvature Eikonal equation, which allows a numerical propagation of the mean-curvature flow of a surface without explicit time stepping.

Vibration reduction Design of Permanent Magnet Motor Using Level Set Based Shape Optimization Method

This paper presents a new optimization method to design a rotor structure for a permanent magnet motor that can reduce vibration. The optimization problem is formulated with a multi-objective function to minimize the fluctuation of the radial magnetic force and the torque ripple. To obtain the optimal rotor shape consisting of a permanent magnet and ferromagnetic material, a multiple level set model is employed to express the structural boundaries and magnetic properties of each material. The updating process of the level set function based on the adjoint sensitivity and the time evolutional equation makes it possible to obtain a novel rotor configuration of the motor. To verify the usefulness of the proposed method, a rotor design example of the surface-mounted permanent magnet motor for electric power steering system is performed.

Driving force profile design in comb drive electrostatic actuators using a level set-based shape optimization method

Electrostatic actuators, actuators actuated by electrostatic forces, are now widely used as sensors and switches, especially in Micro-Electro-Mechanical Systems (MEMS). Among different kinds of electrostatic actuators, the comb drive type is one of the most popular because it has a relatively large range of displacement. In design problems for electrostatic actuators, the driving force profile is of primary engineering importance. In this paper, we develop a structural optimization method for comb drive electrostatic actuators that achieves prescribed driving force profiles, based on a level set-based shape optimization method that provides optimal configurations with clear boundaries, solutions that are valid in an engineering sense. Accurate calculation of the electrostatic forces that occur on the structural boundaries during optimization is important for developing actuators that operate with prescribed driving forces. In the conventional level set-based shape optimization methods, inaccuracies in the calculation of these electrostatic forces occur because the structural boundaries are seldom aligned with the finite element method (FEM) nodes. To precisely calculate the electrostatic forces, we developed a mesh adaptation scheme by which the finite element nodes are brought into alignment with the structural boundaries at every iteration of the optimization procedure. In the following, we explain the details of the proposed level set-based shape optimization method, in which a multi-objective optimization problem is formulated to achieve a prescribed driving force profile. The sensitivity is derived using the adjoint variable method. Four numerical examples are provided, to examine the suitability of the proposed optimization method.

Level Set Based Shape Optimization of Linear Structures using Topological Derivatives

Abstract Using a level set method and topological derivatives, a topological shape optimization method that is independent of an initial design is developed for linearly elastic structures. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in the level set function, which facilitates to handle complicated topological shape changes. The “Hamilton-Jacobi(H-J)” equation and computationally robust numerical technique of “up-wind scheme” lead the initial implicit boundary to an optimal one according to the normal velocity field while minimizing the objective function of compliance and satisfying the constraint of allowable volume. Based on the asymptotic regularization concept, the topological derivative is considered as the limit of shape derivative as the radius of hole approaches to zero. The required velocity field to update the H-J equation is determined from the descent direction of Lagrangian derived from optimality conditions. It turns out that the initial holes are not required to get the optimal result since the developed method can create holes whenever and wherever necessary using indicators obtained from the topological derivatives. It is demonstrated that the proper choice of control parameters for nucleation is crucial for efficient optimization process.

A level set-based shape optimization method for periodic sound barriers composed of elastic scatterers

This paper presents a level set-based topology optimization method for noise barriers formed from an assembly of scatterers. The scattering obstacles are modeled by elastic bodies arranged periodically along the wall. Due to the periodicity, the problem can be reduced to that in a unit cell. The interaction between the elastic scatterers and the acoustic field is described in the context of the level set analysis. The semi-infinite acoustic wave regions located on the both sides of the barrier are represented by impedance matrices. The objective function is defined by the energy transmission passing the barrier. The design sensitivity is evaluated analytically by the aid of adjoint equations. The dependency of the optimal profile on the stiffness of scatterers and on the target frequency band is examined. The feasibility of the developed optimization method is proved through numerical examples.

2101 レベルセット法による形状表現に基づく等断面形状制約を考慮したトポロジー最適設計法(OS7-1 設計と最適化I～形状・形態設計～,OS7 設計と最適化)

Level set-based shape optimization methods are proposed as new type of structural optimization method and we proposed a new topology optimization method based on the level set method and concept of the phase field method. This paper proposes a level set-based topology optimization method with an uniform cross-section surface constraint based on our proposed level set-based topology optimization method. First, basic details of the level set-based topology optimization method are briefly discussed. Next, based on the regularization method, the topology optimization method considering a uniform cross-section surface constraint is formulated. Finally, three-dimensional example is provided to confirm the validity and utility of the proposed topology optimization method.