Level Set-based Topology Optimization Method Research Articles

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.

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.

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.

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.

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.

Topology optimization for acoustic structures considering viscous and thermal boundary layers using a sequential linearized Navier–Stokes model

This study proposes a level set-based topology optimization method for designing acoustic structures with viscous and thermal boundary layers in perspective. Acoustic waves propagating in a narrow channel are damped by viscous and thermal boundary layers. To estimate these viscothermal effects, we first introduce a sequential linearized Navier–Stokes model based on three weakly coupled Helmholtz equations for viscous, thermal, and acoustic pressure fields. Then, the optimization problem is formulated, where a sound-absorbing structure comprising air and an isothermal rigid medium is targeted, and its sound absorption coefficient is set as an objective function. The adjoint variable method and the concept of the topological derivative are used to approximately obtain design sensitivity. A level set-based topology optimization method is used to solve the optimization problem. Two-dimensional numerical examples are provided to support the validity of the proposed method. Moreover, the mechanisms that lead to the high absorption coefficient of the optimized design are discussed.

Topology Optimization of High Resolution Lattice Structures Using a PDE-filter

Lattice-like structures can be observed in many natural systems and industrial applications. Motivated by the need of such porous structure in the design engineering, the present work uses a single scale approach and proposes a level set-based topology optimization method for the lattice designing. The key idea is to introduce a maximum length-scale constraint, realized by a PDE-filter, into the reaction-diffusion equation based LSM. The proposed TO technique can be applied in the unstructured and/or adaptive mesh-based framework, and it can be coupled with the distributed computing technique to achieve high resolution designs. A numerical test case is presented to demonstrate the proposed methodology.

有限変形を考慮したコンプライアントモーフィングフラップのトポロジー最適設計

In order to consider a finite deformation for the compliant morphing flap, this study resolves a numerical instability, which is caused by an artificial weak material under excessively large deformations. The remeshing method which adopts more precise structural model without using such weak materials is integrated into a level set-based topology optimization method. Then, the optimal design problem is formulated for the compliant morphing flap which deforms to the target shape under the applied load. Through numerical examples, validity of the proposed method is demonstrated.

Topology optimization design and experimental validation of an acoustic metasurface for reflected wavefront modulation

Based on a level set-based topology optimization method, this study designs an acoustic metasurface capable of tailoring reflection wave arbitrarily. The present metasurface is composed of a periodic array of supercells and the supercell is constructed by eight inhomogeneous units possessing a full 2π reflection phase coverage with an interval of π/4. In the optimization, each unit in the supercell is designed through the topology optimization method to achieve a specific reflection phase, i.e., reaching a target reflection coefficient corresponding to the desired reflection phase. During the optimization process, the boundary of the unit is updated via solving the level set function, while the normal velocity is derived from the sensitivity analysis. The optimal unit configurations have relatively smooth and distinct boundaries and sample structures. The reflection fields of the designed optimal metasurface under normal and oblique plane wave incidences are investigated numerically and experimentally, and a generally good agreement between them is achieved. Results reveal that a desired reflection wave manipulation is realized by the designed metasurface based on diffraction theory, demonstrating the effectiveness and efficiency of the present topology optimization design method. Furthermore, applications of these optimal units to acoustic flat focusing and acoustic self-bending beam generation are successfully demonstrated by numerical simulations. The presented design approach is quite promising for metasurface design and may be generally used in a variety of metamaterials with a prescribed set of properties.

Level set-based topology optimization for compliance and stress minimization of shell structures using trimmed quadrilateral shell meshes

In this paper, a novel level set-based topology optimization method for shell structures is presented to minimize both compliance and stress under a volume constraint. During the optimization process, trimmed quadrilateral shell meshes are generated by cutting a background quadrilateral shell mesh with the zero-isolines of a level set function. Polygonal shell elements with assumed strains are used for the trimmed quadrilateral shell elements created along the boundaries of shell structures. A new sensitivity field for the evolution of a level set function is proposed to minimize both compliance and stress of shell structures. A p -norm function is employed to aggregate local stresses into a single global stress measure in the sensitivity analysis. Several numerical examples are carried out to show the validity and effectiveness of the proposed method.

Level set-based topology optimization for graded acoustic metasurfaces using two-scale homogenization

This paper proposes a level set-based topology optimization method for acoustic metasurfaces consisting of multiple types of unit cells. As a type of acoustic metasurface, we focus on a two-layered metasurface, where each layer contains various types of unit cell designs. To handle this metasurface, we first propose a two-scale homogenization method based on previous studies targeting metasurfaces composed of a single type of unit cell structure. As a result of homogenization, each layer of the array of unit cells is replaced with an interface with a nonlocal transmission condition. Next, an optimization problem is formulated. We set an objective function to achieve negative refraction, and macroscopic responses obtained by the homogenization method are used to define the objective function. The material distributions in each unit cell of the metasurface are set as the design variables. Based on these settings, we perform a sensitivity analysis based on the concept of the topological derivative. A level set-based topology optimization method is introduced to solve the optimization problem, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. An optimal design of a two-layered acoustic metasurface that induces negative refraction is proposed, and a discussion of its mechanism is provided.

Level set-based BEM topology optimization method for maximizing total potential energy of thermal problems

A topology optimization for maximizing total potential energy by combining the level set method and boundary element method is proposed. The objective function is considered a function of temperature and thermal flux defined on boundaries with Neumann boundary condition. An adjoint field is constructed for the topology optimization problem of total potential energy maximization with previously derived topology sensitivity, which is verified by finite difference approximation. The effectiveness of the present method is validated by comparing it with the result in the literature. The present method proves to initial configuration independent by placing circular and rectangular holes in initial configurations, respectively. An example of topology optimization on replacing the boundary conditions on certain boundaries is studied, with the optimization result a different material distribution. It will help with making versatile configurations by simply changing boundary conditions on existing boundaries. The major novel aspect of this paper, in summarization, is a level set-based topology optimization method is developed for maximizing the total potential energy of thermal problems, as well as the enlightenment of obtaining versatile results by changing boundary conditions.

Topology optimization of acoustic metasurfaces by using a two-scale homogenization method

In this paper, we propose a level set-based topology optimization method for the unit-cell design of acoustic metasurfaces by using a two-scale homogenization method. Based on previous works, we first propose a homogenization method for acoustic metasurfaces that can be combined with topology optimization. In this method, a nonlocal transmission condition depending on the unit cell of the metasurface appears in a macroscale problem. Next, we formulate an optimization problem within the framework of a level set-based topology optimization method, wherein an objective functional is expressed as the macroscopic responses obtained through the homogenization, and material distributions in the unit cell are set as design variables. A sensitivity analysis is conducted based on the concept of the topological derivative. To confirm the validity of the proposed method, two-dimensional numerical examples are provided. First, we provide a numerical example that supports the validity of the homogenization method, and we then perform optimization calculations based on the waveguide settings of the acoustic metasurfaces. In addition, we discuss the mechanism of the obtained optimized structures.

Structural design for modular integrated construction with parameterized level set-based topology optimization method

Modular integrated construction (MIC) can realize modular production in factories and has the advantages of increasing material utilization, speeding up construction, and improving construction environment, which make it more and more widely accepted in engineering practice. But a critical issue for MIC is how to design standard structural modular to maximize its construction productivity. In this paper, a structural topology optimization method based on a parameterized level set is adopted to realize the modular design for stiffness maximum problem of structures. By introducing the symmetry and pattern repetition (SPR) constraints in the optimization model and combining the finite element-based numerical analysis method, the modular design of a structure can be obtained through topology optimization. As a result, the structure can be assembled with repeated modular of the same topology to realize the MIC. Several two-dimensional (2D) and three-dimensional (3D) numerical examples are studied to illustrate the effectiveness of the proposed method in building designs following the MIC concept.

Topology-optimized lattice structures with simultaneously high stiffness and light weight fabricated by selective laser melting: Design, manufacturing and characterization

Abstract In this study, the level set-based topology optimization method was adopted to design novel lattice structures. The Ti-6Al-4V topology-optimized lattice structures with 30 % volume of solid materials were successfully fabricated by selective laser melting (SLM). The study on mechanical properties and energy absorption capability of the topology-optimized structures was also conducted through the quasi-static compressive testing. Results showed that as the unit cell number of lattice structures increases, the structural failure mechanism changes from the layer-by-layer fracture to the 45° inclined fracture. Topology-optimized lattice structures have a better relative elastic modulus of 0.037 than that of the majority of reported lattice structures, and the as-designed lattice structures possess relatively high energy absorption efficiency of 67.9 % at 0.15 strain. The findings illustrated that the proposed topology optimization method combined with SLM processing is an effective way to obtain novel lattice structures with high load-bearing and energy absorption capacities.

A parameterized level set method combined with polygonal finite elements in topology optimization

Level set-based topology optimization methods have received increased attention in the last decade due to the ease with which they can handle topology changes by splitting and merging boundaries. However, most implementations are limited to box-like design domains and cannot easily handle irregular domains that are common in practical engineering applications. In this paper, a modified level set model, which is parameterized and updated with radial basis functions (RBFs), is introduced to solve topology optimization problems in complex design domains. Because RBFs are not necessarily located in a structured manner, the parametrized level set method can be combined effortlessly with unstructured polygonal finite element meshes to easily handle complex design domains and also achieve the high accuracy characteristic of polygonal discretizations. Furthermore, the nucleation capability of the proposed approach along with an approximate reinitialization is shown to obtain more flexible topology changes of the optimal design, and smooth boundaries can be obtained without introducing smoothing scheme. Several numerical examples illustrate the effectiveness of the proposed model.

Parametric Topology Optimization Toward Rational Design and Efficient Prefabrication for Additive Manufacturing

The significant advance in the boosted fabrication speed and printing resolution of additive manufacturing (AM) technology has considerably increased the capability of achieving product designs with high geometric complexity and provided tremendous potential for mass customization. However, it is also because of geometric complexity and large quantity of mass-customized products that the prefabrication (layer slicing, path planning, and support generation) is becoming the bottleneck of the AM process due to the ever-increasing computational cost. In this paper, the authors devise an integrated computational framework by synthesizing the parametric level set-based topology optimization method with the stereolithography (SLA)-based AM process for intelligent design and manufacturing of multiscale structures. The topology of the design is optimized with a distance-regularized parametric level set method considering the prefabrication computation. With the proposed framework, the structural topology optimization not only can create single material structure designs but also can generate multiscale, multimaterial structures, offering the flexibility and robustness of the structural design that the conventional methods could not provide. The output of the framework is a set of mask images that can be directly used in the AM process. The proposed approach seamlessly integrates the rational design and manufacturing to reduce the numerical complexity of the computationally expensive prefabrication process. More specifically, the prefabrication-friendly design and optimization procedure are devised to drastically eliminate the redundant computations in the traditional framework. Two test examples, including a free-form 3D cantilever beam and a multiscale meta-structure, are utilized to demonstrate the performance of the proposed approach. Both the simulation and experimental results verified that the new rational design could significantly reduce the prefabrication computation cost without affecting the original design intent or sacrificing the original functionality.

Level set-based topology optimization with overhang constraint: Towards support-free additive manufacturing

This paper presents a level set-based topology optimization method considering the overhang constraint in additive manufacturing (AM) processes. Though the combination of the topology optimization and AM shows a promising potential and high design flexibility, there are still certain limitations. The overhang constraint is one of the major issues that need to be considered in the design stage. It requires the inclination angles of structural downward-facing surfaces to be larger than a given lower bound, so as to prevent the structure from warping or collapsing during the AM process. We propose a new form of overhang constraint in the level set framework, which is expressed as a single domain integral instead of point-wise constraints. This domain integral form facilitates the detection of overhang constraint violation. The shape derivative of the overhang constraint is derived by using the signed distance property of the level set function. The proposed method is capable of dealing with constraints with different minimum overhang angles. Theoretically, it allows the optimization to proceed from an arbitrary structural layout, without the need to satisfy the overhang constraint in the initial design. Several numerical examples are given to show the validity and effectiveness of the proposed method. It is seen in these examples that the overhang constraint is satisfied mainly by adjusting the local shape of structural members violating the overhang constraint during the optimization process. Thus, the overhang angle constrained optimization can generate similar load paths as in conventional optimal designs in most cases, without significantly worsening the structural stiffness.

Parameterized level-set based topology optimization method considering symmetry and pattern repetition constraints

Symmetry and repetitive patterns are very common in both natural and artificial structures. Design with symmetry and pattern repetition (SPR) can be very useful in practical engineering. In this paper, we discuss a new optimal structural design method considering SPR constraints. The basic idea consists in combining a parameterized level-set method together with radial basis function (RBF) interpolation strategy, which allows for decoupling RBF knots from the finite element (FE) analysis mesh. The merits of the decoupled manner are that RBF knots can be mapped independently on regular or irregular design domain, and SPR constraints can be imposed by simply building a bijective relationship between RBF knots regardless of analysis mesh. Furthermore, due to parameterization of the level-set function, there are fewer numerical manipulations that need to be implemented than the conventional level-set method during the optimization process. Several numerical examples are presented to show feasibility and effectiveness of the proposed method.

Topology optimization for hyperbolic acoustic metamaterials using a high-frequency homogenization method

In this paper, we propose a level set-based topology optimization method for the design of hyperbolic acoustic metamaterials using a high-frequency homogenization method. To estimate the properties of acoustic metamaterials, we first introduce the high-frequency homogenization method proposed in previous work, through which the macroscopic wave behavior of periodic structures can be obtained even if the wavelength of impinging acoustic waves is comparable to the size of the unit cell. We propose a new level set-based topology optimization method based on the high-frequency homogenization method that can provide unit cell designs that exhibit hyperbolic dispersion. We compute the topological derivatives using an equation that expresses the relationship between the topological and shape derivatives. Numerical examples are provided to confirm the utility of the newly proposed optimization method. The obtained optimized unit cell designs, configurations of air and aluminum, exhibit hyperbolic dispersion by exploiting resonances induced in the unit cell.