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... Acoustic topology optimization (ATO) problems have been solved, using the BESO method, in the works of Vicente et al. [11] and Picelli et al. [12], but their main focus was on the structural part of the problem and, in many situations, even neglected the effects of the acoustic domain. Kook [13] and Dilgen et al. [14] also used the bidirectional optimization method, together with a mixed u/p formulation, in order to solve classical acoustic-structure problems. ...

... Eqn. (12) states the topology optimization problem as the minimization of the average square acoustic pressure amplitude at Ω o [7-9, 13] subject to restrictions stated in Eqns. (13), (14) and (15). ...

... If the case AR > AR max happens, only some elements with the lowest α i will be turned to air in order to respect the AR = AR max restriction. This fact also implies that some elements with the highest α i will be turned to rigid, fulfilling V it+1 [12]. ...

In the past few years, acoustic-mechanical devices have become widely used, which increased the demand for noise control solutions. One of the approaches to solve such
problems consists in designing noise barriers. However, finding the best topology for these barriers can be a complex task. In this work it is proposed a methodology to design periodic noise barriers, composed of rigid materials, using the bi-directional evolutionary structural optimization (BESO) method. The acoustic problem is modeled using the Helmholtz equation and solved by the finite element procedure, while a material interpolation scheme is used for switching acoustic and rigid elements. The optimization problem is defined as the minimization of the average square pressure amplitude in a specific region of the acoustic domain, while the volume of the barrier is reduced. The sensitivity analysis was carried out by the gradient of the objective function with respect to the design variable. Two cases are presented in order to show the capabilities of the proposed approach. In the first one, periodic conditions are imposed in the entire system, while in the second non-periodic conditions are considered. The results showed that, although the barrier volume was reduced by 35% in both cases, the objective function decreased at least 68.80%.

... The schematic piston design problem from Fig. 5 is considered for optimization. To the best of the authors' knowledge, this problem was first explored by Bourdin and Chambolle (2003) and later used by several other works for a range of different methods (Sigmund and Clausen 2007;Lee and Martins 2012;Picelli et al. 2015b;Emmendoerfer et al. 2018). This schematic problem is explored here to verify the topology solution offered by our proposed method. ...

... The topology solution from Fig. 6 agrees well with the existing solutions, e.g., by Lee and Martins (2012) and Wang et al. (2016) using the SIMP method and Picelli et al. (2015b) using the BESO method. They have the central holes converging to the bottom center support, convex shapes of the top structural members (resembling seashells), and the curved shapes of the structures touching the lateral R. Picelli et al. ...

... The pressurized chamber design problem was solved in the pioneering works by Hammer and Olhoff (2000) and Chen and Kikuchi (2001). Besides these two, only Zhang et al. (2008) and Picelli et al. (2015b) solved this problem using the SIMP and the BESO methods, respectively. Therefore, this example is explored less in the literature and to the best of the authors' knowledge, it has not been solved with the LSTO method. ...

A few level-set topology optimization (LSTO) methods have been proposed to address complex fluid-structure interaction. Most of them did not explore benchmark fluid pressure loading problems and some of their solutions are inconsistent with those obtained via density-based and binary topology optimization methods. This paper presents a LSTO strategy for design-dependent pressure. It employs a fluid field governed by Laplace’s equation to compute hydrostatic fluid pressure fields that are loading linear elastic structures. Compliance minimization of these structures is carried out considering the design-dependency of the pressure load with moving boundaries. The Ersatz material approach with fixed grid is applied together with work equivalent load integration. Shape sensitivities are used. Numerical results show smooth convergence and good agreement with the solutions obtained by other topology optimization methods.

... The geometric and material properties of the suspension often result in a strong structure-acoustic interaction with the surrounding air, which makes it essential to consider the acoustic field in such an optimization. Only a few studies in the literature concern topology optimization where the structure-acoustic interface is modified [3][4][5][6][7][8][9][10][11], which have arisen mainly within the past decade. The challenge in applying density-based topology optimization in structure-acoustic interaction problems lies in the fact that the governing equations on the two considered media (solid material and air) are different and have different primary variables (pressure for air and displacement for solid, when using the standard Eulerian pressure formulation [12]), which poses the question of how to formulate the interpolation between the two. ...

... The series of papers recently published by Vicente et al. [9,10] describes a topology optimization methodology based on the Bi-directional Evolutionary Structural Optimization (BESO) method applied to fluid-structure interaction problems. BESO is a discrete optimization technique, i.e. intermediate elements are not allowed during the optimization; therefore, the FSI is well-defined at all stages, which allows for calculating the system response using the segregated displacement/pressure formulation. ...

... The formulation of the system matrices is described in the subsections below for each of the two considered methods, yielding the forms in Eqs. (5) and (9). The objective function can be formulated as ...

Topology optimization is a powerful, versatile tool that has been applied successfully in many engineering fields. In vibro-acoustic design problems, the fact that the interface between the solid and the acoustic domains varies during the optimization poses an extra challenge for the sensitivity calculation. In this paper, two topology optimization methods for structure-acoustic interaction problems that have been proposed in the literature are compared in the context of hearing aid suspension design, a system where the structural-acoustic coupling is strong due to the suspension shape and material properties. The first method, referred to as "Mixed-MMA", uses a mixed formulation of the fluid-structure interaction problem where both the structural and the acoustic domains are governed by the same equations and present displacement and pressure primary variables. This allows for converting solid elements into fluid ones, and vice-versa, by varying their material properties, which can be done in a smooth way by allowing intermediate elements during the optimization. The second method, referred to as "Segregated-BESO", uses the more conventional formulation of the problem where the solid and the acoustic domains are described by different equations and primary variables; therefore, the fluid-structure interface must be well-defined at all stages, and an optimization strategy that uses discrete variables is used. A drawback of the second method is that the sensitivities cannot be calculated accurately, since an interpolation scheme between solid and acoustic elements is not available; however, attractive features are the ease of implementation in commercial FE softwares and the compactness of the segregated formulation. The performance of the two methods is evaluated on a 2D suspension design problem for different degrees of the structure-acoustic coupling strength, which shows that the Segregated-BESO method is challenged due to the sensitivity errors when the coupling is strong.

... Although so far, it has only been used to solve the acoustic-mechanical interaction problems that also need to track the interface boundaries explicitly during the optimization process. The segregated finite element model combined with the discrete topology optimization methods, such as the BESO method [31,32], the TOBS method [33], to solve the interface coupling problems between different physical fields, where the coupling interface during the optimization process is naturally defined and heuristically updated by discrete 0/1 design variables. Nevertheless, those discrete topology optimization methods have also shown some difficulties for the strong coupling problems [34,35], e.g., acoustic-mechanical interaction problems, since the accurate calculation of sensitivities (gradients) needs the material interpolation scheme which can effectively simulate the continuous transition from structural elements to acoustic elements, or vice versa. ...

... Nevertheless, those discrete topology optimization methods have also shown some difficulties for the strong coupling problems [34,35], e.g., acoustic-mechanical interaction problems, since the accurate calculation of sensitivities (gradients) needs the material interpolation scheme which can effectively simulate the continuous transition from structural elements to acoustic elements, or vice versa. In the discrete topology optimization methods [31,33], the material interpolation scheme for structural elements only was employed to switch elements between solid and void. To capture the coupling effect between solid and acoustic media, structural void elements were gradually converted to acoustic elements in a layer-by-layer fashion. ...

The poroelastic material is often used as a sandwich core due to its lightweight and excellent sound insulation and absorption, and the research revealed that poroelastic materials with macro- or meso-holes could further improve its sound insulation performance. This study proposes a topology optimization approach to design poroelastic cores of sandwich structures for sound insulation. The mixed displacement/pressure (u/p) formulation based on Biot's theory and the ersatz material model are used to overcome the potential difficulties in topology optimization. The floating projection topology optimization method (FPTO) for maximizing sound transmission loss (TL) of sandwich structures filling with poroelastic materials is established based on the floating projection constraint, which simulates 0/1 constraints of the design variables. Some 2D and 3D numerical examples are given to demonstrate the capability and effectiveness of the proposed topology optimization algorithm. The results show that the distribution of the poroelastic material often concentrates at some locations. The traditional sandwich structures often have periodic cores, considering the ease of manufacture and the resistance of uncertain external loads. Therefore, topology optimization of periodic structures for sound insulation is also investigated in this paper. The numerical examples indicate that optimized periodic cores of sandwiches make the poroelastic material to be distributed evenly within the design domain but slightly scarify the sound insulation performance.

... Allaire and Jouve [1] utilized a level-set method to design twoand three-dimensional structures having maximal fundamental frequencies. Picelli et al. [19] applied an evolutionary topology optimization method to maximize the first natural frequency in free-vibration problems considering acoustic-structure interaction. Zuo et al. [28] employed a reanalysis method for eigenvalue problem to reduce computation cost during a design optimization by a genetic algorithm for minimizing the weight of structures with frequency constraints. ...

... where i ≡ √ −1, and ω denotes an angular frequency. From the governing equations of Eq. (19), using the principle of virtual work and substituting Eq. (20), we obtain the following generalized eigenvalue problem. ...

This paper presents a configuration design optimization method for three-dimensional curved beam built-up structures having maximized fundamental eigenfrequency. We develop the method of computation of design velocity field and optimal design of beam structures constrained on a curved surface, where both designs of the embedded beams and the curved surface are simultaneously varied during the optimal design process. A shear-deformable beam model is used in the response analyses of structural vibrations within an isogeometric framework using the NURBS basis functions. An analytical design sensitivity expression of repeated eigenvalues is derived. The developed method is demonstrated through several illustrative examples.

... Allaire and Jouve [1] utilized a level-set method to design twoand three-dimensional structures having maximal fundamental frequencies. Picelli et al. [19] applied an evolutionary topology optimization method to maximize the first natural frequency in free-vibration problems considering acoustic-structure interaction. Zuo et al. [28] employed a reanalysis method for eigenvalue problem to reduce computation cost during a design optimization by a genetic algorithm for minimizing the weight of structures with frequency constraints. ...

... where i ≡ √ −1, and ω denotes an angular frequency. From the governing equations of Eq. (19), using the principle of virtual work and substituting Eq. (20), we obtain the following generalized eigenvalue problem. ...

This paper presents an isogeometric configuration design optimization of curved beam structures for maximizing fundamental eigenfrequency. A shear-deformable beam model is used in the response analyses of structural vibrations within an isogeometric framework using the NURBS basis functions. An analytical design sensitivity of repeated eigenvalues is used. A special attention is paid to the computation of design velocity field and optimal design of beam structures constrained on a curved surface, where both designs of the embedded beams and the curved surface are simultaneously varied during the optimal design process. The developed design optimization method is demonstrated through several illustrative examples.

... Structural optimization procedures have also been used in various multi-physical problems such as the frequency response of fluid-structure systems [7], the automotive muffler acoustic topology optimization [8] and the frequency maximization problems for acoustic-structure interactions [9]. More closely related to the present research, numerous studies have been written for the topology optimization of piezoelectric materials and its applications as follows: Sigmund et. ...

... It can be seen in Fig. 6a that when = 3, where is the mechanical stiffness' penalization factor, the objective function minimizes more. This is consistent with literature, especially for the compliance minimization of two-dimensional structures [4,6] [9]. Furthermore, after an exhaustive analysis of different penalization factor combinations, not limited to those shown in Fig. 6a, it was found that = 1 and = 2 yield the best results for the optimization. ...

Due to developments in additive manufacturing, the production of piezoelectric materials with complex geometries is becoming viable and enabling the manufacturing of thicker harvesters. Therefore, in this study a piezoelectric harvesting device is modelled as a bimorph cantilever beam with a series connection and an intermediate metallic substrate using the plane strain hypothesis. On the other hand, the thickness of the harvester's piezoelectric material is structurally optimized using a discrete topology optimization method. Moreover, different optimization parameters are varied to investigate the algorithm's convergence. The results of the optimization are presented and analyzed to examine the influence of the harvester's geometry and its different substrate materials on the harvester's energy conversion efficiency.

... These techniques consider Heaviside functions [16,17] or morphology-based operators [18,19] to project filtered densities into 0/1 solution space while aiming length scale control. 30 There are some gradient-based methods created for obtaining 0/1 optimal solutions. One class consists of boundary-description based methods, e.g. the level-set method (LSM), in which an implicit function is used to describe the structure with clearly defined boundaries. ...

... The method has been effectively applied to a wide range of problems like nonlinear structures [28], natural frequency maximization [29,30], material opti- 50 mization and multiscale problems [31, 32,33,34,35,36], multiphysics problems [37,38], etc. Another discrete topology optimization method was proposed by Svanberg and Werme [39] where the authors effectively proposed a sequential integer linear programming approach, where one starts with a coarse mesh to solve an optimization problem and uses the final solution of this problem as the 55 initial solution for optimization on a refined mesh and so on. ...

This work proposes an improved method for gradient-based topology optimization in a discrete setting of design variables. The method combines the features of BESO developed by Huang and Xie [1] and the discrete topology optimization method of Svanberg and Werme [2] to improve the effectiveness of binary variable optimization. Herein the objective and constraint functions are sequentially linearized using Taylor's first order approximation, similarly as carried out in [2]. Integer Linear Programming (ILP) is used to compute globally optimal solutions for these linear optimization problems, allowing the method to accommodate any type of constraints explicitly, without the need for any Lagrange multipliers or thresholds for sensitivities (like the modern BESO [1]), or heuristics (like the early ESO/BESO methods [3]). In the linearized problems, the constraint targets are relaxed so as to allow only small changes in topology during an update and to ensure the existence of feasible solutions for the ILP. This process of relaxing the constraints and updating the design variables by using ILP is repeated until convergence. The proposed method does not require any gradual refinement of mesh, unlike in [2] and the sensitivities every iteration are smoothened by using the mesh-independent BESO filter. Few examples of compliance minimization are shown to demonstrate that mathematical programming yields similar results as that of BESO for volume-constrained problems. Some examples of volume minimization subject to a compliance constraint are presented to demonstrate the effectiveness of the method in dealing with a non-volume constraint. Volume minimization with a compliance constraint in the case of design-dependent fluid pressure loading is also presented using the proposed method. An example is presented to show the effectiveness of the method in dealing with displacement constraints. The results signify that the method can be used for topology optimization problems involving non-volume constraints without the use of heuristics, Lagrange multipliers and hierarchical mesh refinement.

... These different topology optimization methods are applied in different areas and purposes. Academic research includes topological optimization of general composite laminate shell structures [25], minimizing the frequency responses [26] of multiscale systems composed of macro and micro phases [27,28], multi-material topology optimization methods considering isotropic and anisotropic materials and their combination [29], optimization of trussstructures [30,31,32], transient design [33], topology optimization considering movable non-design domain [34], and topology optimization considering fluid-structure iteration [35,36,37]. The advances presented in academic research are then applied to problems found in the industrial area, such as the methods in aerospace structures [38,39]. ...

Structural-engineered wood products are used in construction as a sustainable alternative to concrete and steel. The most commonly used engineered wood products are Glued-Laminated Timber (GLULAM) and Cross-Laminated Timber (CLT), which in most productions are made of layers of wood glued under pressure. In the GLULAM, the wood fiber directions are parallel stacked in layers, while in CLT, the layers have a typical rotation of 90◦. This work develops a topology optimization method with an objective function in displacement and volume constraint applied to the core of engineered wood products such as GLULAM and CLT structures, seeking new product designs with reduced material consumption. The optimization considers the wood’s orthotropic nature and the layers’ stacking. Two numerical examples are performed, the first evaluating the core of a GLULAM structure with and without periodicity constraints and the second on the core of a CLT structure with periodicity constraints. The results show that the proposed procedure can be effectively applied to the core of the engineered wood products. In addition, applying periodic constraint result in optimized topologies that help the manufacturability of the new designs. Furthermore, the proposed procedure highlights the structural differences in layer importance.

... The BESO method has become a widely used design technique in both academic research (e.g., thermal conduction [15], biomechanics [16,17], acoustics [18], microstructural materials [19,20], and nano-photonic designs [21]) and industrial applications (e.g., architecture [22], automotive [23], aircraft [24] and railway vehicles [25]). ...

The bi-directional evolutionary structural optimisation (BESO) has attracted much interest in recent decades. However, the high computational cost of the topology optimisation method hinders its applications in large-scale industrial designs. In this study, a parallel BESO method is developed to solve high-resolution topology optimisation problems. An open-source computing platform, FEniCS, is used to parallelise the finite element analysis (FEA) and optimisation steps. Significant improvements in efficiency have been made to the FEA and the filtering process. An iterative solver, a reanalysis approach and a hard-kill option in
BESO have been developed to reduce the computational cost of the FEA. An isotropic filter scheme is used to eliminate the time-consuming elemental adjacency search process. The efficiency and effectiveness of the developed method are demonstrated by a series of numerical examples in both 2D and 3D. It is shown that the parallel BESO can efficiently solve problems with more than 100 million tetrahedron elements on a 14-core CPU server. This work holds great potential for high-resolution design problems in engineering and architecture.
Keywords: Topology optimisation, Bi-directional evolutionary structural optimisation, FEniCS, Parallel computing, High-resolution

... Even though the BESO can add or remove elements, it is common to start with a design full of Material 1 to avoid a topology predefinition and ensure that each of the elements has the same probability of being part of the final topology. The four elements filled with Material 2 at the center of the base cell are introduced to avoid the uniform sensitivity distribution and allow the algorithm to initialize the design process (Huang et al. 2011Picelli et al. 2015). Two different cases are presented for the NTE and the PTE problems to compare the final topology of metamaterials with different values of negative thermal expansion and mesh dependency. ...

The use of computational evolutionary strategies in the design of metamaterials with desired thermal expansion coefficients is uncommon due to the discrete nature of the design variables. This work presents a Bi-directional Evolutionary Structural Optimization (BESO) based methodology for designing orthotropic metamaterials with a specific thermal expansion coefficient using an objective function considering only the thermal expansion coefficients, with no constraints on geometry or stiffness. Topologies of the metamaterials, composed of two material phases and a void, are obtained using a material interpolation between neighboring material phases and three easy-to-implement numerical strategies to stabilize the evolutionary process. Two are on the sensitivity calculation and one is on the addition ratio’s value. The strategies applied to the sensitivity numbers are proposed to avoid the positive and negative values of the elemental sensitivity numbers and the element change between no neighboring materials. Additionally, the addition ratio’s value reduction strategy assures the convergence of the thermal expansion properties to the desired value. The homogenization method is used to obtain the equivalent thermal expansion properties of the designed materials. Some numerical examples are presented to show the potential and effectiveness of the proposed methodology.

... Apart from the density framework to describe the topology, eigenfrequency topology optimization methods have been developed within the framework of level set method (Xia et al., 2011) and bidirectional evolutionary structural optimization approach (Zuo et al., 2010). Meanwhile, eigenfrequency topology optimization is also extended and further developed to solve multiphysics and multi-scale problems (Picelli et al., 2015;Zhang et al., 2020). It should be noted that compared to static compliance topology optimization, most of existing frequency topology optimization methods are concentrated on handling numerical issues such as (1) mode switching and (2) high computational complexity. ...

Purpose
The purpose of this paper is to develop a topology optimization algorithm considering natural frequencies.
Design/methodology/approach
To incorporate natural frequency as design criteria of shell-infill structures, two types of design models are formulated: (1) type I model: frequency objective with mass constraint; (2) type II model: mass objective with frequency constraint. The interpolation functions are constructed by the two-step density filtering approach to describe the fundamental topology of shell-infill structure. Sensitivities of natural frequencies and mass with respect to the original element densities are derived, which will be used for both type I model and type II model. The method of moving asymptotes is used to solve both models in combination with derived sensitivities.
Findings
Mode switching is one of the challenges faced in eigenfrequency optimization problems, which can be overcome by the modal-assurance-criterion-based mode-tracking strategy. Furthermore, a shifting-frequency-constraint strategy is recommended for type II model to deal with the unsatisfactory topology obtained under direct frequency constraint. Numerical examples are systematically investigated to demonstrate the effectiveness of the proposed method.
Originality/value
In this paper, a topology optimization method considering natural frequencies is proposed by the author, which is useful for the design of shell-infill structures to avoid the occurrence of resonance in dynamic conditions.

... The topology optimization community has been working on this topic for three decades, see e.g. [82,29,196,420,302,140,244,90,157,406,305,229,359,412,216]. There are various optimization objectives in this topic, for example: maximizing the specific eigenfrequency [82,242]; maximizing the band gap of adjacent eigenfrequencies [164]; obtaining eigenfrequencies close to desired frequencies [241]; imposed resonant peaks constraints to the resonant structure [77]. ...

The objective of this thesis is to develop density based-topology optimization methods for several challenging dynamic structural problems. First, we propose a normalization strategy for elastodynamics to obtain optimized material distributions of the structures that reduces frequency response and improves the numerical stabilities of the bi-directional evolutionary structural optimization (BESO). Then, to take into account uncertainties in practical engineering problems, a hybrid interval uncertainty model is employed to efficiently model uncertainties in dynamic structural optimization. A perturbation method is developed to implement an uncertainty-insensitive robust dynamic topology optimization in a form that greatly reduces computational costs. In addition, we introduce a model of interval field uncertainty into dynamic topology optimization. The approach is applied to single material, composites and multi-scale structures topology optimization. Finally, we develop a topology optimization for dynamic brittle fracture structural resistance, by combining topology optimization with dynamic phase field fracture simulations. This framework is extended to design impact-resistant structures. In contrast to stress-based approaches, the whole crack propagation is taken into account into the optimization process.

... Besides, penalty variables are often used as degrees of freedom of the polynomial function. Although the BESO method does not need material interpolations, it has been shown that such a procedure contributes to the avoidance of singularities, as well as to the reduction of computational costs involved in multiphysics problems [10,11]. ...

... The numerical results agree well with the experimental data. Vicente et al. developed an extended optimization method for the structural problem of artificial fluid loading and applied it to the frequency optimization problem [9] . ...

A wet model topology optimization method based on added mass method and bidirectional evolutionary structure optimization algorithm is proposed to maximize the natural frequency of fluid-structure coupling system. The fluid-structure coupling equation of the finite element method and the sensitivity formula of the element are analyzed simultaneously, and then the coupling optimization model is established. Under the constraints of structure volume, the weighted natural frequencies of single-objective and multi-objective of the coupled system are maximized to optimize the underwater coupled structure. The numerical results show that the evolutionary methods can be applied to this kind of problem effectively and efficiently. Besides, the optimization results of cases have smooth boundaries and the numerical stability of the optimization algorithm is good. Under constant volume constraints, the single-order natural frequency is increased by more than 380%. The results of multi-objective weighted optimization show that the weighted coefficients are final. The larger the weighting coefficient is, the closer the topological form is to the optimal result of the model.

... With these adjustments, topology optimization algorithms proved to be robust enough for several problems. Among others, it has been implemented for fluid-structure interaction [12,13], multiscale analyses [14,15], design of piezoelectric harvesters [16] and acoustic insulators [17]. ...

Minimizing vibration levels of dynamic components at their operating frequency range has been a widely studied topic in engineering. However, the design of structures that satisfy geometric constraints and technical performances is an ongoing challenge. In this work, a topology optimization procedure based on the Bi-directional Evolutionary Structural Optimization (BESO) algorithm is performed to maximize the natural frequency separation interval of an elongated structure. The issues of disconnected and trivial solutions are solved using a connectivity constraint. It is imposed by a proposed procedure based on the heat flux solution of an auxiliary system. An assessment of the feasibility of the structure is done by verifying its accordance with manufacturing and design constraints. The optimized structure was manufactured and validated experimentally. The implemented process produces topologies that maximize the natural frequency separation and reduce the mass of the structure. The obtained results demonstrate the effectiveness of the proposed procedure at satisfying geometric design constraints and technical performances.

... Furthermore, whether it is a large-scale structure to an airplane, a high-speed rail, or a small-scale structure to a micro-electromechanical system, the natural working environment may involve the interaction of multiple physical fields, such as thermo-structure coupling (Deaton and Grandhi 2015), acoustic-structure coupling (Picelli et al. 2015) or fluid-structure coupling (Andreasen and Optimization 2020). Among them, the thermal-structure coupling is a typical application scenario, and many scholars have conducted considerable research on topology optimization under the action of the thermal-structure coupling field. ...

With the tremendous development of additive manufacturing technology in recent years, porous infill structures with well-designed topology configurations have been widely used in various physical fields. The porous infill structure may be prone to thermo-mechanical buckling failure under certain extreme thermal conditions due to temperature gradient effects and delicate local details of the porous infill structure.Therefore, a topological optimization design method, which considers the influence of the thermo-solid coupling field on the buckling performance of the porous infill structure, is proposed by using the projection approach merged with the Solid Isotropic Material with Penalization (SIMP) method. Critical buckling load factors obtained with thermal-elastic equilibrium and linear buckling analysis are employed to measure the buckling performance of the structure. Numerical examples show that the proposed method can effectively improve the buckling performance of the porous infill structure under the thermo-mechanical environment.

... In order to solve the eigenvalue topology optimization several approaches such as optimality criteria (OC) method [14], the method of moving asymptotes (MMA) [29], mathematical programming (MP) [31] and less mathematically rigoros approaches such as evolutionary method [32] can be used. In the present paper to solve variable bound optimization of eigenvalue problem we use MMA [29] which has been proven to be amongst the most effective methods [33]. ...

Keeping the eigenfrequencies of a structure away from the external excitation frequencies is one of the main goals in the design of vibrating structures in order to avoid risk of resonance. This paper is devoted to the topological design of freely vibrating continuum structures with the aim of maximizing the fundamental eigenfrequency. Since in the process of topology optimization some areas of domain can potentially be removed, it is quite possible to encounter the problem of localized modes. Hence, the modified Solid Isotropic Material with Penalization (SIMP) model is here used to avoid artificial modes in low density areas. As during the optimization process, the first natural frequency increases, it may become close to the second natural frequency. Due to lack of the usual differentiability of the multiple eigenfrequencies, their sensitivity are calculated by the mathematical perturbation analysis. The optimization problem is formulated by a variable bound formulation and it is solved by the Method of Moving Asymptotes (MMA). Two dimensional plane elasticity problems with different sets of boundary conditions and attachment of a concentrated nonstructural mass are considered. Numerical results show the validity and supremacy of this approach.

... Based on the ESO method, Huang and Xie [11] developed a more reasonable method, termed as bi-directional evolutionary structural optimization (BESO), which can remove and add material in each iteration. Due to its high efficiency and robustness [12], [13], BESO has been used extensively, including functionally graded material design [14], biomechanical design [15], nonlinear structural design [15], [16], fracture resistance [17], photonic bandgap material design [18], [19], frequency design of fluid-structure [20] and acoustic structural design [21]. ...

... Based on the ESO method, Huang and Xie [11] developed a more reasonable method, termed as bi-directional evolutionary structural optimization (BESO), which can remove and add material in each iteration. Due to its high efficiency and robustness [12], [13], BESO has been used extensively, including functionally graded material design [14], biomechanical design [15], nonlinear structural design [15], [16], fracture resistance [17], photonic bandgap material design [18], [19], frequency design of fluid-structure [20] and acoustic structural design [21]. ...

... The other classification directly optimizes the structural global [32][33][34][35][36] or local responses [37][38][39] in target frequencies [40]. The BESO studies for this problem were mainly focused on optimizing the structural dynamic characteristics, [14,[41][42][43], which have been proven to be effective. However, the more direct method for this problem is seldom studied, and only a few works focus on the optimization of structural global [44][45][46][47] and local [48] responses were reported. ...

The bi-directional evolutionary structural optimization (BESO) method has been widely studied and applied due to its efficient iteration and clear boundaries. However, due to the use of the discrete design variable, numerical difficulties are more likely to occur with this method, especially in cases with strong nonlinearity. This limits the application of the BESO method in certain cases, such as the suppression of structural dynamic frequency response under high-frequency excitation. In this work, a normalization strategy is proposed for the BESO-based topology optimization, by which the magnitude of the sensitivities can be efficiently unified to the same order to avoid the possible numerical instabilities caused by the nonlinearity. To validate its merit in applications, the normalization-based BESO (NBESO) method is proposed for minimizing the structural frequency response. By means of the weighted sum method, a normalized weighted sum method is also proposed for multi-frequency involved problems. A series of 2D and 3D numerical examples is presented to illustrate the advantages of the NBESO. The effectiveness of the NBESO for multi-frequency response suppression is also demonstrated, in which the frequency ranges below and above the eigenfrequency are involved, respectively.

... The last approaches through their wide applicability. This is shown by their employment in thermal expansion (Sigmund and Torquato 1997), auxetic materials (Kaminakis and Stavroulakis 2012), piezoelectric harvesters (Sigmund et al. 1998), acoustic-structure interaction (Yoon et al. 2007;Picelli et al. 2015), multiscale analysis (Huang et al. 2013;Vicente et al. 2016), compliant mechanism design (Ansola et al. 2007;Ansola et al. 2010), and design-dependent loads problems . The advantages of the BESO algorithm are its simplicity for implementation, the clearly defined boundaries of its results (Munk et al. 2018), and that it does not require geometric features. ...

This work aims to perform the topology optimizationof frequency separation interval of continuous elastic bi-dimensional structures in the high-frequency domain. The studied structures are composed of two materials. The proposed algorithm is an adaptation of the Bidirectional Evolutionary Structural Optimization (BESO). As the modal density is high in this frequency domain, the objective function, based on the weighted natural frequency, is formulated to consider an important number of modes. To implement the algorithm, a mode tracking method is necessary to avoid problems stemming from mode-shifting and local modes. As the obtained results by using structural dynamics analysis present quasi-periodic topology, further calculations are done to compare the results with and without imposed periodicity. A dispersion analysis based on wave propagation theory is performed by using the unit cell previously obtained from the structural optimization to investigate the band gap phenomenon. The resulting band gaps from the dispersion analysis are compared with respect to the dynamic behavior of the structure. The topology optimization methodology and the wave propagation analysis are assessed for different boundary conditions and geometries. Comparison between both analyses shows that the influence of the boundary conditions on the frequency separation interval is small. However, the influence from the geometry is more pronounced. The optimization procedure does not present significant numerical instability. The obtained topologies are well-defined and easily manufacturable, and the obtained natural frequency separation intervals are satisfactory.

... This creates opportunities to customize the mechanical, thermal, and electrical properties of the object, leading to material design freedom and dramatically reformed engineering design processes. Although different material properties were targeted, in specific, the optimization of the frequency responses has been attempted by a variety of methods such as the density-based solid isotropic material with penalization (SIMP), 12-14 the hard-kill evolutionary structural optimization (ESO), its later version, bi-directional evolutionary structural optimization (BESO), [15][16][17][18] and genetic algorithms (GAs). [19][20][21] The basic idea of SIMP was proposed by Bendsoe in 1989. ...

Limitations of the traditional manufacturing methods often force engineered components to be made of single material systems. However, this is going through changes due to the advent of additive manufacturing (AM) methods, as the point-by-point consolidation allows for a possible change of the material constitution within a given part domain. This will give rise to a plethora of new material and property options for the designers, where just human perception may fail to realize the full benefits. Automated design tools integrating material choice, dispersion, analysis, and optimization algorithms need to be developed to assist in finding the optimal multi-material dispersion solutions achieving given performance criteria sets. Considering the fact that the multi-material manufacturing systems are only recently coming into use, design solutions targeting optimal placement of multiple materials are not common. This article addresses this gap, evaluating a numerical model integrated with different optimization schemes to find the optimal material solutions achieving certain preset performance criteria such as combinations of natural frequencies in different degrees of freedom. A case study of three different metaheuristic optimization schemes based on genetic algorithms indicates, first, that it is possible to create a beam with six uniformly spaced natural frequencies and to change these frequencies without modifying the structural geometry; and second that the basic genetic algorithm generally outperforms neural net-based alternatives for this problem. This tailoring of the structural resonance spectrum demonstrates that evolutionary computing combined with multi-material AM can be used to unlock previously unavailable structural functionality.

... The element-based topology optimization defines the design variables within the whole design domain and the optimized solution is therefore less dependent on an initial guess. A segregated finite element model combined with the BESO method has been used to solve the acoustic-mechanical interaction problems [27][28][29], where the acoustic-structural interface during the optimization process is naturally defined and heuristically updated by discrete 0/1 design variables. In their works, the modified isotropic material penalization (SIMP) model [30] has been adopted to overcome artificially local vibration modes at weak elements. ...

Topology optimization of dynamic acoustic-mechanical structures is challenging due to the interaction between the acoustic and structural domains and artificial localized vibration modes of structures. This paper presents a floating projection topology optimization (FPTO) method based on the mixed displacement/pressure (u/p) finite element formulation and the ersatz material model. The former is able to release the need for tracking the interface boundaries explicitly between the structural and acoustic domains during the optimization process. The ersatz material model enables us to entirely avoid artificial localized vibration modes caused by the extremely high ratio between mass and stiffness. The floating projection simulates the original 0/1 constraints, and it gradually pushes the design variables toward 0 or 1 at the desired level so that the optimized element-based design can be accurately represented by a smooth design. Some 2D and 3D numerical examples, including minimizing sound pressure at the designated domain, restraining structural vibration, and maximizing sound transmission loss, are presented to demonstrate the effectiveness of the proposed topology optimization algorithm. The optimized solutions achieve the consistency of the objective function between the element-based design using the mixed formulation and the smooth design using the segregated formulation. The study suggests that the FPTO method using the ersatz material model is a promising approach for optimizing dynamic acoustic-mechanical structures.c

... In 2015, Picelli et al. [16][17][18][19] and Vicente et al. [20] used the bi-directional evolutionary structural optimization (BESO) method to solve problems with fluid-related design-dependent loads, acoustics, and structural topology optimization of the fluid-solid coupling design. Because the design variable of the BESO method is 0 or 1, this variable can be used to represent fluids and solids. ...

Based on the parameterized level-set method using radial basis functions, a topology optimization method is proposed that can account for design-dependent loads. First, a mathematical model of the optimization problem is established with the structural compliance minimization as the objective function and the structural volume fraction as the design constraint. By converting the line integrals into volume integrals, the design-dependent loads imposed on the structure outline described by the zero-level set are converted, and a relatively easy boundary detection method is used. Subsequently, an updating strategy of the Lagrange multiplier is proposed to make the transition of the objective function smooth during the convergence process. Finally, the effectiveness and efficiency of the proposed optimization method in solving design-dependent load problems can be shown through several classic numerical examples.

... Topology optimization formulations have undergone rapid development since their introduction by Bendsøe and Kikuchi (1988), and they have been applied to a large range of engineering problems. For acoustic-structure interaction (ASI) problems, different design objectives, e.g., maximization or minimization of eigen frequencies (or eigenvalue gap) (Casas and Pavanello 2017;Picelli et al. 2015), structural compliance (Picelli et al. 2017), and radiated sound power , have been investigated. ...

This research contributes to the topology optimization of acoustic–structure interaction problems with infinite acoustic domain by focusing on two issues, namely, the variation of the acoustic–structure interface and free-floating elements. To handle the variation of the coupling interface, we divide the analyzed domain into the bounded and unbounded domains by using a virtual surface. For the bounded domain where the coupling interface may appear, we model it with the mixed displacement–pressure (u/p) finite element method (mixed FEM). Then, the boundary element method (BEM) is utilized to model the non-reflecting boundary conditions on the truncated surface. The mixed FEM enables the explicit representation of the acoustic–solid interface in the optimized domain to be circumvented. By spatial variation of the mass density, shear and bulk moduli in the mixed FEM, the topology optimization procedure can be easily implemented similar to standard density approaches. With regard to the free–floating elements, we establish a connected component filter to delete these useless elements. This treatment avoids unreasonable but possibly optimized results. The proposed method is verified by several optimization examples.

... Okamoto et al. [112] enhanced genetic algorithm, immune algorithm, additional search in the restricted design space with enabling island, and void distribution during FEM analysis to solve a typical magnetic circuit problem. Picelli et al. [113] used BESO to free vibration problems of acoustic-structure systems. Riehl and Steinmann [114] employed the traction method to define descent directions for shape variation. ...

... Since the resonance phenomenon is one of the main causes of failures in structures subject to periodic loads, such as submerged structures and components of rotor designs, several researchers have sought techniques to increase the natural frequency of these structures. An interesting result was found by Picelli et al. [1] who used the bi-directional evolutionary structural optimization (BESO) to maximize the first natural frequency of acoustic-structure systems. A different approach was proposed by Wang et al. [2], who used a homogenization-based topology optimization method for natural frequency optimization in a cantilevered plate with a honeycomb structure increasing the first natural frequency and reducing weight and, using the differential evolution optimization, Roque and Martins [3] maximized the first natural frequency for a functionally graded beam. ...

Various types of engineering structures are subject to periodic loading such as offshore platform parts and wind turbine blades. One of the main causes of failure in these structures is due to the resonance effect, when the frequency of external loading coincides with some natural frequency of the structure. Therefore, the maximization of natural frequencies is an increasingly sought-after topic in the design of these components. In this paper a genetic algorithm is developed to maximize natural frequencies of Euler-Bernoulli beams. Genetic algorithms are stochastic search methods, which are based on biological concepts of adaptation, natural selection, fitness and evolution, to solve optimization problems. A beam population is created, each of them discretized in a mesh of cylindrical elements with different diameters, initially random. The natural frequencies of the beam are found by the Finite Element Method, and the one with the highest natural frequency creates a new generation of offsprings. In each offspring is applied a mutation scheme that changes the diameter of any random element, making the entire population change. So over the generations the algorithm finds out the best diameter combination that maximizes the natural frequency of the beam. Results present different shapes are obtained for several boundary conditions and different natural frequencies maximized.

... To deal with this problem, Querin and Xie [29] proposed bi-directional evolutionary structural optimization, which is capable of adding and deleting material elements simultaneously. The latest version of the BESO method, i.e., the convergent and mesh-independent BESO method [25], has been applied for various structural design problems, such as the design of structural natural frequency [13,30] and the design of nonlinear structures [31]. ...

Constrained layer damping (CLD) is an effective method for suppressing the vibration and sound radiation of lightweight structures. In this article, a two-level optimization approach is presented as a systematic methodology to design position layouts and thickness configurations of CLD materials for suppressing the sound power of vibrating structures. A two-level optimization model for the CLD structure is developed, considering sound radiation power as the objective function and different additional mass fractions as constraints. The proposed approach applies a modified bi-directional evolutionary structural optimization (BESO) method to obtain several optimal position layouts of CLD materials pasted on the base structure, and sound power sensitivity analysis is formulated based on sound radiation modes for the position optimization of CLD materials. Two strategies based on the distributions of average normalized elemental kinetic energy and strain energy of the base plate are proposed to divide optimal position layouts of CLD materials into several subareas, and a genetic algorithm (GA) is employed to optimally reconfigure the thicknesses of CLD materials in the subareas. Numerical examples are provided to illustrate the validity and efficiency of this approach. The sound radiation power radiated from the vibrating plate, which is treated with multiple position layouts and thickness reconfigurations of CLD materials, is emphatically discussed.

... In addition to these main methods, we will briefly review acoustic-structure interaction problems solved by the bi-directional evolutionary topology optimization (BESO) method as presented in Picelli et al. (2015), Vicente et al. (2015), and Chen et al. (2017). It should be noted also that a number of topology optimization problems have been studied for which the topological changes do not involve a change in the boundary between the structural and acoustic domains. ...

The pursuit for design improvements by geometry modifications can easily become prohibitive using a trial and error process. This holds especially when dealing with multi-physics problems—such as acoustic-structure interaction—where it is difficult to realize design improvements intuitively due to the complexity of the coupled physics. Compared to classical shape optimization, where a near optimal shape has to be supplied as an initial guess, topology optimization allows for innovative designs through a completely free material distribution, such that the topology can change during the optimization process. The goal of this article is to provide a comprehensive critical review of the proposed strategies for topology optimization of coupled acoustic-structure interaction problems. The work includes a comparison of topology optimization formulations with density, level set, and evolutionary-based methods and discusses the corresponding strengths and weaknesses through the considered application examples. The review concludes with recommendations for future research directions.

... Recently, Picelli et al. (2015a) extended this method to the application of general movable fluid-structure interfaces with design-dependent pressure loads. Later, Picelli et al. (2015b) applied this method to topology optimisation problems for frequency maximisation considering acoustic-structure interactions. Most recently, Munk et al. (2017b) coupled a BESO algorithm to a LBM for the design of micro fluidic mixers with the fluid-structural coupling present. ...

This article presents an optimisation framework for the compliance minimisation of structures subjected to design-dependent pressure loads. A finite element solver coupled to a Lattice Boltzmann method is employed, such that the effect of the fluid-structure interactions on the optimised design can be considered. It is noted that the main computational expense of the algorithm is the Lattice Boltzmann method. Therefore, to improve the computational efficiency and to assess the effect of the fluid-structure interactions on the fi nal optimised design, the degree of coupling is changed. Several successful topology optimisation algorithms exist with thousands of associated publications in the literature. However, only a small portion of these are applied to real-world problems, with even fewer offering a comparison of methodologies. This is especially important for problems involving fluid-structure interactions, where discrete and continuous methods can provide different advantages. The goal of this research is to couple two key disciplines, fluids and structures, into a topology optimisation framework, which shows fast convergence for multi-physics optimisation problems. This is achieved by offering a comparison of three popular, but competing, optimisation methodologies. The needs for the exploration of larger design spaces and to produce innovative designs make meta-heuristic algorithms less efficient for this task. A coupled analysis, where the fluid and structural mechanics are updated, provides superior results compared with an uncoupled analysis approach, however at some computational expense. The results in this article show that the method is sensitive to whether fluid-structure coupling is included, i.e. if the fluid mechanics are updated with design changes, but not to the degree of the coupling, i.e. how regularly the fluid mechanics are updated, up to a certain limit. Therefore, the computational efficiency of the algorithm can be considerably increased with small penalties in the quality of the objective by relaxing the coupling.

... Lee and Kim (2009) also studied the muffler structure optimization but focusing on rigid wall partitions, several cases were presented in order to show that acoustic topology optimization (ATO) can produce better results than conventional methods for instance. In works Yoon (2013), Lee and Kim (2009) the SIMP method is used, in the same way BESO was also adapted and used in acoustic-structural problems in Picelli et al. (2015), Vicente et al. (2016a, b). ...

This article proposes an acoustic muffler design procedure based on finite element models and a Bi-directional Evolutionary Acoustic Topology Optimization. The main goal is to find the best configuration of barriers inside acoustic mufflers used in the automotive industry that reduces sound pressure level in the outlet of the muffler. The acoustic medium is governed by Helmholtz equation and rigid wall boundary conditions are introduced to represent acoustic barriers. The continuum problem is written in the frequency domain and it is discretized using the finite element method. The adopted objective function is Transmission Loss (TL). Increasing TL guarantees that the sound pressure level ratio between outlet and inlet of the muffler is reduced. To find the configuration of acoustic barriers that increases the Transmission Loss function of the muffler an adaptation of the Bi-directional Evolutionary Structural Optimization (BESO) method is used. Applying the proposed design procedure topologies in 2D models are reached, which raises the Transmission Loss function for one or multiple frequencies. Three examples are presented to show the efficiency of the proposed procedure.

... Recently, Picelli et al. [7] extended this method to the application of general movable fluid-structure interfaces with design-dependent pressure loads. Later, Picelli et al. [29] applied this method to topology optimisation problems for frequency maximisation considering acoustic-structure interactions. Most recently, Munk et al. [6] coupled a BESO algorithm to a LBM for the design of micro fluidic mixers with the fluid-structural coupling present. ...

This article presents an optimisation framework for the compliance minimisation
of structures subjected to design-dependent pressure loads. A finite element solver coupled to a Lattice Boltzmann method is employed, such that the effect of the fluid-structure interactions on the optimised design can be considered. It is noted that the main computational expense of the algorithm is the Lattice Boltzmann method. Therefore, to improve the computational efficiency and to assess the effect of the fluid-structure interactions on the final optimised design, the degree of coupling is changed.
Several successful topology optimisation algorithms exist with thousands
of associated publications in the literature. However, only a small portion of
these are applied to real-world problems, with even fewer offering a comparison
of methodologies. This is especially important for problems involving fluid-structure interactions, where discrete and continuous methods can provide different advantages.
The goal of this research is to couple two key disciplines, fluids and structures,
into a topology optimisation framework, which shows fast convergence
for multi-physics optimisation problems. This is achieved by offering a comparison
of three popular, but competing, optimisation methodologies. The
needs for the exploration of larger design spaces and to produce innovative
designs make meta-heuristic algorithms less efficient for this task. A coupled
analysis, where the fluid and structural mechanics are updated, provides
superior results compared with an uncoupled analysis approach, however at some computational expense. The results in this article show that the method is sensitive to whether fluid-structure coupling is included, i.e. if the fluid mechanics are updated with design changes, but not to the degree of the coupling, i.e. how regularly the fluid mechanics are updated, up to a certain limit. Therefore, the computational efficiency of the algorithm can be considerably increased with small penalties in the quality of the objective by relaxing the coupling.

... In the current work, the dynamic approach is developed for the maximization of the fundamental natural frequency of cyclic systems. In the context of the BESO method, many works have already been done in the study of the natural frequency optimization and in the minimization of dynamic response of the structures ( Xie and Steven, 1996), ( ), (Zuo et al., 2013), ( Vicente et al., 2015), ( Picelli et al., 2015), ( Vicente et al., 2016a), ( Vicente et al., 2016b), ( Picelli et al., 2017). Herein, it is extended these applications to the case of cyclic symmetric structures. ...

Jack-up platforms are widely used for the offshore oil and gas exploration, constructions and offshore wind farms. The jack-up platform legs are import component, which not only support the whole weight but also withstands various external loads, such as wind, current and wave load. A topology optimization algorithm-BESO (bi-directional evolutionary structural optimization) is proposed in this paper for the jack-up leg optimization. Based on the sensitivity analysis, the optimization strategy with sensitivity filtering and updated have been applied. And a simple numerical example is used to verify the proposed optimization method. A case jack-up platform is introduced to implement the leg structure optimization. After a series of iterations, a new leg structure is obtained. And the ultimate analysis, Eigen analysis and buckling of optimized leg structure optimized structure are all discussed compared with the traditional structures (K-type, the reverse(rev) K-type and X-type). To further evaluate the performance of the optimized leg structures, an experiment test is designed in the wave tank considering the current and wave load. Also, the optimized leg structure and the traditional ones are all manufactured to perform the experiment. The results show that the optimized leg structure has better ultimate bearing capacity, Eigen analysis and dynamic response with lighter weight under the same environmental loads. It is indicated that the BESO topology method is superior in finding best structure of the jack-up platform leg.

This work details a new acoustic topology optimization methodology with applications on the design of systems composed of rigid and porous materials. The Bi-directional Evolutionary Structural Optimization (BESO) algorithm is combined with the Virtual Temperature Method to minimize the occurrence of air cavities and rough surfaces inside rigid and porous domains, hence configuring a multiconstrained optimization approach. The modeling of porous materials is done by the Johnson–Champoux–Allard (JCA) formulation, while the Finite Element Method is used to approximate the governing equations of the physical system. Two different optimization problems are considered separately: first, a rigid–acoustic metasurface is optimized to reduce regional sound pressure levels (SPL) in a set of observed frequencies, while also considering wind permeability through the structure. Secondly, a coupled poro-acoustic absorptive system is treated in order to enhance the sound absorption coefficient in the low frequency range. Both problems are systematized by the implementation of acoustic–rigid and acoustic–porous material interpolation schemes, respectively. The effectiveness of the proposed approach is explored through numerical examples. Here, it is remarked that the methodology maintains the uniformity of rigid barriers, by guaranteeing the absence of internal holes to them. In addition, well-defined cavities are formed in porous domains, increasing their absorption coefficients, but without inflicting macroscopic closed spaces within such structures. In these cases, comparison with appropriate literature is also provided.

A concurrent topology optimization for thermoelastic structures with random and interval hybrid uncertainties is discussed in this work. A robust topology optimization method is proposed for structures composed of periodic microstructures under thermal and mechanical coupled loads. The robust objective function is defined as a linear combination of the mean and standard variance under the worst case for the robust optimization model. An efficient hybrid orthogonal polynomial expansion (HOPE) method is developed to evaluate the robust objective function. The sensitivities for the robust topology optimization are then calculated based on the uncertainty analysis. Three numerical examples are provided to verify the effectiveness of the proposed method, and the Monte-Carlo-Scanning (MCS) test is used to validate the numerical accuracy of our proposed method. For comparison purpose, the topology optimizations under deterministic assumptions are also provided for these examples to show the importance of considering hybrid uncertainties.

An effective and straightforward method to implement topology optimization using high-level programming is presented. The method uses the LiveLink for MATLAB, which couples the commercial COMSOL Multiphysics software with MATLAB programming environment via COMSOL Application Programming Interface (API). The integrated environment allows one to implement advanced and customized functions and methods from scratch easily. Topology optimization of an acoustic-structure interaction problem with a mixed displacement–pressure (u/p) formulation is employed to demonstrate the effective-ness of the presented implementation method to design multiphysics problems systematically. The governing equations of the system are derived in a weak form, which is inserted directly in equation-based modeling in COMSOL Multiphysics via MATLAB programming environment. The tight integration of MATLAB and COMSOL Multiphysics allows one to easily pass the matrices and derivatives to perform design sensitivity analysis. A comprehensive code to perform the optimization of the acoustic-structure interaction problem is provided in Appendix. The well-structured code can be used as a platform for educational and research purposes, and it can be extended to other topology optimization applications involving various types of physical problems that use the equation-based modeling functionality ofCOMSOL.

In view of the general inertia and damping features as well as the inevitable uncertainty factors in engineering structures, a novel dynamic reliability-based topology optimization (DRBTO) strategy is investigated for time-variant mechanical systems with overall consideration of material dispersion and loading deviation effects. The static interval-set model is first utilized to quantify multi-source uncertainty inputs and the transient interval-process model is then established to characterize unknown-but-bounded response results, which can be readily solved through the proposed interval-process collocation approach combined with a classical Newmark difference scheme. Different from the traditional deterministic design framework, the present DRBTO scheme will directly consider new reliability constraints, for which the non-probabilistic time-variant reliability (NTR) index is mathematically deduced using the first-passage principle. In addition, the issues related to uncertainty-oriented design sensitivity and filtering method are discussed. The usage and effectiveness of DRBTO are demonstrated with three numerical examples.

The demand for advanced manufacturing is gradually turning to the multi-functional integrated structure, so the effective and innovative functional structure topology optimization design method can maximize the performance of the structure. This paper proposes a parallel topology optimization design method for dynamic and static characteristics of multiphase materials. The inherent discrete design variables of bidirectional evolutionary structure optimization (BESO) can clearly distinguish the topological boundaries of multiphase materials without the intermediate density materials. The pre-processing density filtering scheme is introduced into the BESO method to eliminate the numerical calculation instability problem. Then, the improved multiphase materials interpolation scheme can overcome the local modal phenomenon of vibration problems. Finally, a series of typical numerical examples show that the method is effective, efficient and easy to implement in multiphase materials topology optimization with dynamic and static characteristics.

The bi-directional evolutionary structural optimization (BESO) method has been widely studied and applied for its efficient iteration and clear boundaries. However, due to the use of the discrete design variable, numerical difficulties are more likely to occur to this method, especially in the cases with strong nonlinearity. This limits the application of the BESO method in certain topics, such as the suppression of structural dynamic frequency response under high-frequency excitation. In this work, a normalization strategy is proposed for the BESO based topology optimization, by which the magnitude of the sensitivities can be efficiently unified to the same order to avoid the possible numerical instabilities caused by the nonlinearity. To validate its merit in applications, the normalization based BESO (NBESO) method is proposed for minimizing the structural frequency response. By means of the weight sum method, a normalized weight sum (NWS) method is also proposed for the multi-frequency involved problems. A series of 2D and 3D numerical examples are presented to illustrate the advantages of the NBESO. The effectiveness of the NBESO for the multi-frequency response suppression is also demonstrated, in which the frequency ranges below and above the eigenfrequency are involved, respectively.

Design optimization of vibro-acoustic systems over a wide frequency band is challenging. It does not only require a computationally efficient numerical prediction model of sufficient accuracy, but the optimization scheme itself should also be computationally efficient and the design space should be limited by all relevant manufacturing and performance constraints. In this paper, a methodology is presented for the shape optimization of components in a complex wall system, with the aim of achieving an optimized sound insulation of the overall system across the entire building acoustics frequency range. As an example, the cross-sectional shape of studs in a double-leaf wall is first parameterized and subsequently optimized for broadband sound insulation with a gradient-based optimization scheme. A recently developed sound insulation prediction model that has the required balance between accuracy and computational efficiency is adopted and validated for a range of plasterboard walls with acoustic studs. The model is further complemented with a novel sensitivity analysis, such that the sensitivities of the predicted sound insulation to the cross-sectional stud shape parameters can be obtained in a semi-analytic way. This approach reduces the computation cost related to broadband acoustic design optimization significantly. Furthermore, inequality constraints that are necessary for obtaining a feasible design in terms of material usage and manufacturing limitations are identified and incorporated in the optimization procedure. The relevant constraints related to strength and stiffness of the wall are very mild and therefore verified after optimization. As an example of the proposed methodology, the cross-sectional shape of flexible metal studs in double-leaf plasterboard walls is optimized for the overall A-weighted sound reduction under pink noise excitation. A range of combinations in stud depths and number of sheets is analyzed. Walls containing the optimized studs have an airborne sound insulation that is close to that of walls with fully decoupled leafs. Their sound insulation is on average 11.8 dB higher than when they would contain conventional C-shaped studs, and 5.1 dB higher than when they would contain acoustic studs that are presently available.

Topology optimization is an effective method in the design of acoustic media. This article presents optimization for graded damping materials to minimize sound pressure at a reference point or sound power radiation under harmonic excitation. The Helmholtz integral equation is used to calculate an acoustic field to satisfy the Sommerfeld conditions. The equation of motion is solved using a unit dynamic load method. Formulations for the sound pressure or sound power radiation in an integral form are derived in terms of mutual kinetic and strain energy densities. These integrals lead to novel physical response functions for solving the proposed optimization problem to design graded damping materials. The response function derived for individual frequency is utilized to solve the multi-objective optimization problem of minimizing sound pressure at the reference point for excitations with a range of frequencies. Numerical examples are presented to verify the efficiency of the present formulations.

Determining an optimized solution by means of topology optimization in vibroacoustic problems often requires a high computational cost. Conventional density-based topology optimization using the finite element method is a time-consuming approach, owing to the large model size and repeated function evaluations involved in the frequency response. To address this issue, an efficient topology optimization method that uses the hybrid finite element–wave based method is proposed in this paper. In this method, the entire problem domain is divided into design and non-design domains. The mixed displacement–pressure finite element method is applied to the design domain for material interpolation in the topology optimization. Moreover, the wave based method, which is an efficient numerical scheme for acoustic problems, is applied to the non-design domain to reduce the computational cost. A direct coupling approach is proposed to construct hybrid models for vibroacoustic problems. The adjoint variable method is also presented to compute the design sensitivities efficiently. The effectiveness of the proposed method is demonstrated by benchmark problems. The optimization results indicate that the proposed method can significantly reduce the computational cost, while maintaining almost identical optimized layouts to those obtained using the conventional approach.

This study proposes a systematic inverse design method of a novel thermal metastructure that improves the thermal dynamic characteristics by considering thermal eigenvalues for the first time. To this end, we apply topology optimization and introduce a relaxation scheme to address the design-dependent heat convection boundary on the implicit interfaces between solid and (ambient) fluid phases updated during the optimization process. With the proposed inverse design method, optimization of various thermal metastructures with enhanced thermal dynamic characteristics is performed. A physical interpretation of the heat transfer mechanism for each optimized thermal metastructure is also carried out by employing two dimensionless numbers, in this case Biot and Fourier numbers. Through this interpretation, it is demonstrated that among many design factors, the heat convection effect with an ambient fluid is the most important factor when seeking to improve the thermal dynamic characteristics. Also, performance of the optimized thermal metastructures is physically analyzed through thermal energy. Moreover, we find that the thermal eigenvalue physically corresponds to a time constant related to the response rate in a first-order linear system. The thermal eigenvalue-based inverse design method to improve the thermal dynamic characteristics is computationally very efficient, as it predicts the dynamic characteristics from an eigenvalue analysis instead of a full time-dependent analysis. It also has the advantage of providing more physically quantifiable insight regarding optimized thermal metastructures. In this regard, the proposed inverse design method can be an important and critically useful design tool for improving the dynamic characteristic of various thermal actuator and sensor systems.

In this work, we present a topology optimization method for acoustic-structure interaction problems, which combines bi-directional evolutionary structural optimization (BESO) with a mixed displacement-pressure (u/p) formulation as an effective and straightforward design method for a multi-physics system involving acoustic-structure interactions. Due to the binary characteristics of the BESO and the multi-physics modeling approach of the mixed formulation, the proposed optimization procedure could benefit from high computational efficiency and high-quality design in acoustic-structure interaction problems. Several topology optimization problems for vibro-acoustic systems are carried out, in order to demonstrate the effectiveness of the presented method.

Numerical modelling is a key point for vibro-acoustic analysis and optimization of hearing aids. The great number of small components constituting the devices, and the strong structure-acoustic coupling of the system make it a challenge to obtain accurate and computationally efficient models. In this thesis, several challenges encountered in the process of modelling and optimizing hearing aids are addressed. Firstly, a strategy for modelling the contacts between plastic parts for harmonic analysis is developed. Irregularities in the contact surfaces, inherent to the manufacturing process of the parts, introduce variations on the final contact points in practice, making the contact properties unknown. The suggested technique aims at characterising the contact in terms of distributed stiffness values, which are identified by means of a model updating method that matches simulation to experimental data. Secondly, the applicability of Model Order Reduction (MOR) techniques to lower the computational complexity of hearing aid vibro-acoustic models is studied. For fine frequency response calculation and optimization, which require solving the numerical model repeatedly, a computational challenge is encountered due to the large number of Degrees of Freedom (DOFs) needed to represent the complexity of the hearing aid system
accurately. In this context, several MOR techniques are discussed, and an adaptive reduction method for vibro-acoustic optimization problems is developed as a main contribution. Lastly, topology optimization techniques for structure acoustic interaction problems are investigated with the aim of evaluating their applicability to the design of hearing aid parts. The strong fluid-structure interaction between the air and some of the thin, soft parts makes it necessary to include the effects of the interface variations in the optimization, which poses a challenge due to the need of interpolating between solid and fluid elements. Two techniques are compared in this context for a 2D hearing aid suspension design problem.

This article presents an evolutionary topology optimization method for compliance minimization of structures under design-dependent pressure loads. In traditional density based topology optimization methods, intermediate values of densities for the solid elements arise along the iterations. Extra boundary parametrization schemes are demanded when these methods are applied to pressure loading problems. An alternative methodology is suggested in this article for handling this type of load. With an extended bi-directional evolutionary structural optimization method associated with a partially coupled fluid-structure formulation, pressure loads are modelled with hydrostatic fluid finite elements. Due to the discrete nature of the method, the problem is solved without any need of pressure load surfaces parametrization. Furthermore, the introduction of a separate fluid domain allows the algorithm to model non-constant pressure fields with Laplace's equation. Three benchmark examples are explored in order to show the achievements of the proposed method.

IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion
Appendix 4.1References

Describes development work to combine the basic ESO with the additive evolutionary structural optimisation (AESO) to produce bidirectional ESO whereby material can be added and can be removed. It will be shown that this provides the same results as the traditional ESO. This has two benefits, it validates the whole ESO concept and there is a significant time saving since the structure grows from a small initial one rather than contracting from a sometimes huge initial one where 90 per cent of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Presents a brief background to the current state of Structural Optimisation research. This is followed by a discussion of the strategies for the bidirectional ESO (BESO) algorithm and two examples are presented.

A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external
excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher
order, or the gap between two consecutive eigenfrequencies of given order. This problem is often complicated by the fact that
the eigenfrequencies in question may be multiple, and this is particularly the case in topology optimization. In the present
paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies
of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and
several illustrative results are presented.

This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

Topology optimization of steady heat conduction problem under both design-independent and design-dependent heat loads is studied by means of a modified bidirectional evolutionary structural optimization (BESO) method. Two types of problems are distinguished by their physical meanings and particularly design-dependent load effect is highlighted in the following two points. At the stage of sensitivity analysis, both the heat conductivity matrix and the design-dependent heat generation load associated with the void element are penalized in the same manner. The rationality is illustrated based on numerical tests. Furthermore, as the sensitivity of the objective function changes its sign during the iteration, a modified BESO procedure is presented to deal with the non-monotonicity of the objective function defined by the heat potential capacity. Detailed steps of the BESO procedure are presented for the element removal and growth while the inequality volume constraint is imposed. To conclude the work, numerical results and the element sensitivity obtained are discussed to show the effect of design-dependent load.

Computational methods within structural acoustics, vibration and fluid-structure interaction are powerful tools for investigating acoustic and structural-acoustic problems in many sectors of industry; in the building industry regarding room acoustics, in the car industry and aeronautical industry for optimizing structural components with regard to vibrations characteristics etc. It is on the verge of becoming a common tool for noise characterization and design for optimizing structural properties and geometries in order to accomplish a desired acoustic environment. The book covers the field of computational mechanics, and then moved into the field of formulations of multiphysics and multiscale.
The book is addressed to graduate level, PhD students and young researchers interested in structural dynamics, vibrations and acoustics. It is also suitable for industrial researchers in mechanical, aeronautical and civil engineering with a professional interest in structural dynamics, vibrations and acoustics or involved in questions regarding noise characterization and reduction in building, car, plane, space, train, industries by means of computer simulations.

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method’s focuses on new developments and novel applications.

The current techniques for topology optimization of material microstructure are typically based on infinitely small and periodically repeating base cells. These base cells have no actual size. It is uncertain whether the topology of the microstructure obtained from such a material design approach could be translated into real structures of macroscale. In this work we have carried out a first systematic study on the convergence of topological patterns of optimal periodic structures, the extreme case of which is a material microstructure with infinitesimal base cells. In a series of numerical experiments, periodic structures under various loading and boundary conditions are optimized for stiffness and frequency. By increasing the number of unit cells, we have found that the topologies of the unit cells converge rapidly to certain patterns. It is envisaged that if we continue to increase the number of unit cells and thus reduce the size of each unit cell until it becomes the infinitesimal material base cell, the optimal topology of the unit cell would remain the same. The finding from this work is of significant practical importance and theoretical implication because the same topological pattern designed for given loading and boundary conditions could be used as the optimal solution for the periodic structure of vastly different scales, from a structure with a few (e.g. 20) repetitive modules to a material microstructure with an infinite number of base cells.

This paper presents a method for applying topology optimization to fluid–structure interaction problems in saturated poroelastic media. The method relies on a multiple-scale method applied to periodic media. The resulting model couples the Stokes flow in the pores of the structure with the deformation of the elastic skeleton through a macroscopic Darcy-type flow law. The method allows to impose pressure loads for static problems through a one way coupling, while transient problems are fully coupled modeling the interaction between fluid and solid. The material distribution is determined by topology optimization in order to optimize the performance of a shock absorber and test the pressure loading capabilities and optimization of an internally pressurized lid.

This research details a new acoustic topology optimization (ATO) framework with an empirical material formulation for fibrous material. Despite the importance of considering pressure attenuation not only by internal solid structures but also by fibrous (porous) structures in acoustic design, a systematic ATO approach with an empirical material formulation has not yet been proposed. Thus, in this paper, an empirical material formulation called the Delany–Bazley model is implemented for the development of an ATO framework for fibrous material with porosity close to 1. By means of the SIMP (solid isotropic material with penalization) interpolation functions developed for multiple structural materials, ATO processes for fibrous structures as well as internal solid structures are carried out. In addition, a heuristic filter method that allows fibrous material to emerge only at the boundaries or rims of an internal solid structure is presented. Finally, the effect of the pressure attenuation on the topological layout for fibrous materials is investigated by solving several illustrative topology optimization examples.

Frequency optimization is of great importance in the design of machines and structures subjected to dynamic loading. When the natural frequencies of considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this paper, a modified SIMP model is developed to effectively avoid the artificial modes. Based on this model, a new bi-directional evolutionary structural optimization (BESO) method combining with rigorous optimality criteria is developed for topology frequency optimization problems. Numerical results show that the proposed BESO method is efficient, and convergent solid-void or bi-material optimal solutions can be achieved for a variety of frequency optimization problems of continuum structures.

Topology optimization is frequently used to design structures and acoustic systems in a large range of engineering applications. In this work, a method is proposed for maximizing the absorbing performance of acoustic panels by using a coupled finite element model and evolutionary strategies. The goal is to find the best distribution of porous material for sound absorbing panels. The absorbing performance of the porous material samples in a Kundt tube is simulated using a coupled porous–acoustic finite element model. The equivalent fluid model is used to represent the foam material. The porous material model is coupled to a wave guide using a modal superposition technique. A sensitivity number indicating the optimum locations for porous material to be removed is derived and used in a numerical hard kill scheme. The sensitivity number is used to form an evolutionary porous material optimization algorithm which is verified through examples.

There are several well-established techniques for the generation of solid-void optimal topologies such as solid isotropic
material with penalization (SIMP) method and evolutionary structural optimization (ESO) and its later version bi-directional
ESO (BESO) methods. Utilizing the material interpolation scheme, a new BESO method with a penalization parameter is developed
in this paper. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent
optimal solutions for structures with one or multiple materials. The results show that the optimal designs from the present
BESO method are independent on the degree of penalization. The resulted optimal topologies and values of the objective function
compare well with those of SIMP method.

A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.

This paper outlines a new procedure for topology optimization in the steady-state fluid–structure interaction (FSI) problem. A review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics (hereafter, the fluid and structural equations, respectively) and in imposing coupling boundary conditions between the separated fluid and solid domains. To overcome these difficulties, we propose an alternative monolithic procedure employing a unified domain rather than separated domains, which is not computationally efficient. In the proposed analysis procedure, the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor. For the coupling boundary conditions, the divergence of the pressure and the Darcy damping force are inserted into the solid and fluid equations, respectively. The proposed method is validated in several benchmark analysis problems. Topology optimization in the FSI problem is then made possible by interpolating Young's modulus, the fluid pressure of the modified solid equation, and the inverse permeability from the damping force with respect to the design variables. Copyright © 2009 John Wiley & Sons, Ltd.

The paper presents an introduction to two general approaches used in the soluation of coupled structures and fluid systems in which effects of large scale flow are excluded. In the first approach, the Lagrangian, the fluid is simply treated as a ‘solid’ with a negligible shear modulus. In the second method, Eulerian, a single pressure variable is used in the fluid.
The numerical problems posed, discretization methods used and possible simplifications are discussed.

The paper presents a gradient-based topology optimization formulation that allows to solve acoustic–structure (vibro-acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface-coupling integrals, however, such a formulation does not allow for free material re-distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p-formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two-dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.

Topology optimization is used to optimize the eigenvalues of plates. The results are intended especially for MicroElectroMechanical Systems (MEMS) but can be seen as more general. The problem is not formulated as a case of reinforcement of an existing structure,
so there is a problem related to localized modes in low density areas. The topology optimization problem is formulated using
the SIMP method. Special attention is paid to a numerical method for removing localized eigenmodes in low density areas. The
method is applied to numerical examples of maximizing the first eigenfrequency. One example is a practical MEMS application;
a probe used in an Atomic Force Microscope (AFM). For the AFM probe the optimization is complicated by a constraint on the stiffness and constraints on higher
order eigenvalues.

A multi-objective topology optimization formulation for the design of dynamically tunable fluidic devices is presented. The
flow is manipulated via external and internal mechanical actuation, leading to elastic deformations of flow channels. The
design objectives characterize the performance in the undeformed and deformed configurations. The layout of fluid channels
is determined by material topology optimization. In addition, the thickness distribution, the distribution of active material
for internal actuation, and the support conditions are optimized. The coupled fluid-structure response is predicted by a non-linear
finite element model and a hydrodynamic lattice Boltzmann method. Focusing on applications with low flow velocities and pressures,
structural deformations due to fluid-forces are neglected. A mapping scheme is presented that couples the material distributions
in the structural and fluid mesh. The governing and the adjoint equations of the resulting fluid-structure interaction problem
are derived. The proposed method is illustrated with the design of tunable manifolds.
KeywordsFluid-structure interaction-Hydrodynamic lattice Boltzmann method-Non-linear elasticity-Adjoint sensitivity analysis

This note considers topology optimization of large scale 2D and 3D Stokes flow problems using parallel computations. We solve
problems with up to 1,125,000 elements in 2D and 128,000 elements in 3D on a shared memory computer consisting of Sun UltraSparc
IV CPUs.

This paper presents a simple method for structural optimization with frequency constraints. The structure is modelled by a fine mesh of finite elements. At the end of each eigenvalue analysis, part of the material is removed from the structure so that the frequencies of the resulting structure will be shifted towards a desired direction. A sensitivity number indicating the optimum locations for such material elimination is derived. This sensitivity number can be easily calculated for each element using the information of the eigenvalue solution. The significance of such an evolutionary structural optimization (ESO) method lies in its simplicity in achieving shape and topology optimization for both static and dynamic problems. In this paper, the ESO method is applied to a wide range of frequency optimization problems, which include maximizing or minimizing a chosen frequency of a structure, keeping a chosen frequency constant, maximizing the gap of arbitrarily given two frequencies, as well as considerations of multiple frequency constraints. The proposed ESO method is verified through several examples whose solutions may be obtained by other methods.

We present a method to maximize the separation of two adjacent eigenfrequencies in structures with two material components. The method is based on finite element analysis and topology optimization in which an iterative algorithm is used to find the optimal distribution of the materials. Results are presented for eigenvalue problems based on the 1D and 2D scalar wave equations. Two different objectives are used in the optimization, the difference between two adjacent eigenfrequencies and the ratio between the squared eigenfrequencies. In the 1D case, we use simple interpolation of material parameters but in the 2D case the use of a more involved interpolation is needed, and results obtained with a new interpolation function are shown. In the 2D case, the objective is reformulated into a double-bound formulation due to the complication from multiple eigenfrequencies. It is shown that some general conclusions can be drawn that relate the material parameters to the obtainable objective values and the optimized designs.

To bring down noise levels in human surroundings is an important issue and a method to reduce noise by means of topology optimization is presented here. The acoustic field is modeled by Helmholtz equation and the topology optimization method is based on continuous material interpolation functions in the density and bulk modulus. The objective function is the squared sound pressure amplitude. First, room acoustic problems are considered and it is shown that the sound level can be reduced in a certain part of the room by an optimized distribution of reflecting material in a design domain along the ceiling or by distribution of absorbing and reflecting material along the walls. We obtain well defined optimized designs for a single frequency or a frequency interval for both 2D and 3D problems when considering low frequencies. Second, it is shown that the method can be applied to design outdoor sound barriers in order to reduce the sound level in the shadow zone behind the barrier. A reduction of up to 10 dB for a single barrier and almost 30 dB when using two barriers are achieved compared to utilizing conventional sound barriers.

A shortcoming of the traditional density based approach to topology optimization is the handling of design dependent loads that relate to boundary data, such as for example pressure loads. Previous works have introduced spline and iso-density curves or alternative parametrization schemes to determine the load surfaces. In this work we suggest a new way to solve pressure load problems in topology optimization. Using a mixed displacement–pressure formulation for the underlying finite element problem, we define the void phase to be an incompressible hydrostatic fluid. In this way we can transfer pressure loads through the fluid without any needs for special load surface parametrizations. The method is easily implemented in the standard density approach and is demonstrated to work efficiently for both 2D and 3D problems. By extending the method to a three phase (solid/fluid/void) design method we also demonstrate design of water containing dams.

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