On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation

This work deals with the topological design of vibrating continuum structures. The vibration of continuum structure is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. In comparison with well-known compliance minimization in static topology optimization, various objective functions are proposed in literature to minimize the response of vibrating structures, such as power flow, vibration transmission, and dynamic compliances, etc. Even for the dynamic compliance, different definitions are found in literature, which have quite different formulations and influences on the optimization results. The aim of this paper is to provide a comparison of these different objective functions and propose reference forms of objective functions for design optimization of vibration problems. Analytical solutions for two degrees of freedom system and topological design of plane structures in numerical examples are compared using different optimization formulations for given various excitation frequencies. The results are obtained by the finite element method and gradient based optimization using analytical sensitivity analysis. The optimized topologies and vibration response of the optimized structures are presented. The influence of excitation frequencies and the eigenfrequencies of the structure are discussed in the numerical examples.     

Design sensitivity analysis of acoustical damping and its application to design of musical bells

Damping characteristics of a musical bell plays an important role in characterizing the musical sound. The total damping consists of acoustical damping and internal damping. Acoustical damping depends upon resonating frequencies and vibration patterns while internal damping is a material property. The acoustical damping of a vibrating structure is formulated via boundary element method and finite element method using eigenmode decomposition. The design sensitivity of acoustical damping is derived using an adjoint variable method of the eigenvalue problem. Design optimization of a musical bell is then performed in terms of acoustical parameters. The goal of the optimization problem is to design a harmonically tuned bell with given acoustical damping values. The proposed automated design process integrates finite element analysis, boundary element analysis, design sensitivity analysis, mode-tracking algorithm and optimization module, seamlessly. It is demonstrated by numerical examples to show practical applications.     

Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

Topology optimization of electrode coverage of piezoelectric thin-walled structures with CGVF control for minimizing sound radiation

It is impractical to implement arbitrary-shaped piezoelectric patches from the view point of manufacturability of fragile piezoelectric ceramics, thus using designable electrode layers to deliver desired actuation forces provides a more realistic option in engineering applications. This study develops a topological design method of surface electrode distribution over piezoelectric sensors/actuators attached to a thin-walled shell structure for reducing the sound radiation in an unbounded acoustic domain. In the optimization model, the sound pressure norm at specific reference points under excitations at a certain excitation frequency or in a given frequency range is taken as the objective function. The pseudo densities for indicating absence and presence of surface electrodes at each element are taken as the design variables, and a penalized relationship between the densities and the active damping effect is employed. The vibrating structure is discretized with finite element model for the frequency response analysis and the sound radiation analysis in the unbounded acoustic domain is treated by boundary element method. The applied voltage on each actuator is determined by the constant gain velocity feedback (CGVF) control law. The technique of the complex mode superposition in the state space, in conjunction with a model reduction transformation, is adopted in the response analysis of the system characterized by a non-proportional active damping property. In this context, the adjoint-variable sensitivity analysis scheme is derived. The effectiveness and efficiency of the proposed method are demonstrated by numerical examples, and several key factors on the optimal designs are also discussed.     

Shape preserving design of vibrating structures using topology optimization

In several engineering components, the shape of some functional surfaces needs to be preserved in order to avoid losing performance or even its functionality when subjected to loads. This is particularly important when tight tolerances are required for operational conditions in some regions. If the deformation significantly affects product functionality, it is interesting to use a shape preserving design technique. This will often reduce deformation in a local region. To achieve that, we deal with topology optimization of elastic, continuum structures with Rayleigh damping, subjected to time-harmonic, design-independent external dynamic loading with prescribed excitation frequency, amplitude and spatial distribution. In topology optimization for vibrating structures, the obtained design should often have its resonance frequencies driven far away from the given excitation frequency in order to avoid resonance and to reduce vibration levels. In this work, we explore harmonic vibration problems with the excitation frequency lower than the first resonance frequency of the initial structure. Dynamic compliance minimization is used to improve dynamic response of the structure. An additional local dynamic compliance constraint is used to define the shape preserving problem, thus, reducing deformation in specific regions of a part named shape preserving region (SPR). A commercial FE code (ANSYS^?) is used to solve the finite element problem. The optimization Method of Moving Asymptotes (MMA) is used with the modified Solid Isotropic Material with Penalization (SIMP) material interpolation scheme. The effectiveness of this technique is presented using 2D plane structures. Coherent results were achieved using the proposed optimization formulation. It is possible to observe significant decrease on local deformation, at expense of little increase on global dynamic compliance.     

Evolutionary natural frequency optimization of thin plate bending vibration problems

This paper extends the evolutionary structural optimization method to the solution for maximizing the natural frequencies of bending vibration thin plates. Two kinds of constraint conditions are considered in the evolutionary structural optimization method. If the weight of a target structure is set as a constraint condition during the natural frequency optimization, the optimal structural topology can be found by removing the most ineffectively used material gradually from the initial design domain of a structure until the weight requirement is met for the target structure. However, if the specific value of a particular natural frequency is set as a constraint condition for a target structure, the optimal structural topology can be found by using a design chart. This design chart describes the evolutionary process of the structure and can be generated by the information associated with removing the most inefficiently used material gradually from the initial design domain of a structure until the minimum weight is met for maintaining the integrity of a structure. The main advantage in using the evolutionary structural optimization method lies in the fact that it is simple in concept and easy to be included into existing finite element codes. Through applying the extended evolutionary structural optimization method to the solution for the natural frequency optimization of a thin plate bending vibration problem, it has been demonstrated that the extended evolutionary structural optimization method is very useful in dealing with structural topology optimization problems.     

Topology optimization of piezoelectric smart structures for minimum energy consumption under active control

This paper investigates topology optimization of the electrode coverage over piezoelectric patches attached to a thin-shell structure to reduce the energy consumption of active vibration control under harmonic excitations. The constant gain velocity feedback control method is employed, and the structural frequency response under control is analyzed with the finite element method. In the mathematical formulation of the proposed topology optimization model, the total energy consumption of the control system is taken as the objective function, and a constraint of the maximum allowable dynamic compliance is considered. The pseudo-densities indicating the distribution of surface electrode coverage over the piezoelectric layers are chosen as the design variables, and a penalized model is employed to relate the active damping effect and these design variables. The sensitivity analysis scheme of the control energy consumption with respect to the design variables is derived with the adjoint-variable method. Numerical examples demonstrate that the proposed optimization model is able to generate optimal topologies of electrode coverage over the piezoelectric layers, which can effectively reduce the energy consumption of the control system. Also, numerical comparisons with a minimum-volume optimization model show the advantage of the proposed method with respect to energy consumption. The proposed method may provide useful guidance to the layout optimization of piezoelectric smart structures where the energy supply is limited, such as miniature vibration control systems.     

Microstructural topology optimization with respect to sound power radiation

The paper deals with the problem of topological design of microstructure with respect to minimization of the sound power radiation from a vibrating macrostructure. The macrostructure is excited at a single or a band of excitation frequencies by a time-harmonic mechanical loading with prescribed amplitude and spatial distribution. The structural damping is considered to be proportional damping. The sound power is calculated using a high frequency approximation formulation and thus the sensitivity analysis may be performed in a very efficient manner. The microstructure composed of two different solid isotropic materials is assumed to be identical from point to point at the macro-level which implies that the interface between the structure and the acoustic medium is unchanged during the design process. The equivalent material properties of the macrostructure are calculated using homogenization method and the bi-material SIMP model is employed to achieve zero-one design at the micro-scale. Numerical examples are given to validate the model developed. Some interesting features of acoustic microstructure topology optimization are revealed and discussed.     

Topology optimization of rubber isolators considering static and dynamic behaviours

A topology optimization for the design of rubber vibration isolators is proposed. Many vibration isolators are made of rubbers and they operate under small oscillatory load superimposed on large static deformation. Vibration isolators must have a certain degree of static stiffness in order to endure the static loading due to large gravitational and inertial forces. On the other hand, isolators must have a small dynamic stiffness in order to reduce the force transmission from vibrating systems to base structures. Therefore both the static and dynamic behaviours of rubber should be simultaneously considered in the design process. The static behaviours of rubber under large and slow loads are generally treated with hyperelastic constitutive models. Rubber under fast dynamic loads can be modelled as a viscoelastic material. In this paper, the steady state viscoelastic model, which is suggested by Kim and Youn and correctly predicts the influence of the pre-strain on the relaxation function, is applied for the dynamic analysis. The continuum-based design sensitivity analyses (DSA) of both the static hyperelastic model and dynamic viscoelastic model are developed. The topology optimization formulation is proposed in order to generate the system layouts considering both the static and dynamic performance. The density distribution approach and sequentially linear programming (SLP) are used as the optimization algorithms. Some design examples are presented in order to verify the proposed approach.     

Minimization of sound radiation from vibrating bi-material structures using topology optimization

Up to now, work on topological design optimization of vibrating structures against noise radiation has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, and minimization of the dynamic compliance subject to harmonic loading on the structure. In this paper, we deal with topology optimization problems formulated directly with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. It is assumed that air is the acoustic medium and that a feedback coupling to the structure can be neglected. Certain conditions are assumed that imply that the sound power emission from the structural surface can be obtained in a simpler way than by solving Helmholz’ integral equation. Hereby, the computational cost of the structural-acoustical analysis is substantially reduced. Several numerical results are presented and discussed for plate- and pipe-like structures with different sets of boundary and loading conditions.     

Topology optimization of sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems

Due to the significant difference of dynamic properties between the fluid medium and the structure, when a vibro-acoustic system is subjected to a higher frequency excitation, it may typically exhibit mid-frequency behavior which involves different wavelength deformations and is very sensitive to the uncertainties of the system. This paper deals with optimized distribution of a sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems by using hybrid boundary element analysis and statistical energy analysis. Based on the solid isotropic material with penalization approach, an artificial sound absorbing material model is suggested and the relative densities of the sound absorbing material are taken as design variables. The sound pressure level at a specified point in the acoustic cavity is to be minimized by distributing a given amount of sound absorbing material. An efficient direct differentiation scheme for the response sensitivity analysis is proposed. Then, the optimization problem is solved by using the method of moving asymptotes. A numerical example illustrates the validity and effectiveness of the present optimization model. Impact of the excitation frequency on optimized topology is also discussed.

     

Multi-material structural topology optimization with decision making of stiffness design criteria

A new methodology for making design decisions of structures using multi-material optimum topology information is presented. Multi-material analysis contributes significant applications to enhance the bearing capacity and performance of structures. A method that chooses an appropriate material combination satisfying design stiffness requirement economically is currently needed. An alternative method of making design-decision is to utilize a multi-material topology optimization (MMTO) approach. This study provides a new computational design optimization procedure as a guideline to find the optimal multi-material design by considering structure strain energy and material cost. The MMTO problem is analyzed using an alternative active-phase approach. The procedure consists of three design steps. First, steel grid configurations and composite with material properties are defined as a given structure for automatic design decision-making (DDM). And then design criteria of the steel composites structure is given to be limited strain energy by designers and engineers. Second, topology changes in the automatic distribution of multi-steel materials combination and volume control of each material during optimization procedures are achieved and at the same time, their converged minimal strain energy is produced for each material combination. And third, the strain energy and material cost which is computed based on the material ratio in the combinations are used as design decision parameters. A study in constructional steel composites to produce optimal and economical multi-material designs demonstrates the efficiency of the present DDM methodology.     

Structural optimization in magnetic fields using the homogenization design method — Part II: Reduction of vibration caused by magnetic forces

Summary The homogenization design method has been expanded to obtain the optimal topology of a structure in magnetic fields to maximize the magnetic energy. In this study, the homogenization design method (HDM) is applied to a structure in magnetic fields to reduce the vibration level of a structure excited by the magnetic forces, especially by the magnetic harmonic forces. This is accomplished by obtaining the optimal material distribution of the structure to minimize the frequency response. The Maxwell stress method is used to compute the magnetic force and the HDM is applied for the optimization. It is verified that the HDM is useful to minimize the frequency response by some actual applications. The effects of mesh density of the design domain and the rotor-stator position are also examined.     

Topology optimization of a swing arm type actuator using the response surface method

In the frame of topology optimization, the multi-objective ability has to be considered since structural design is usually required to satisfy more than one requirement. A modified topology optimization method based on the response surface method (RSM) is proposed to generate a structure of a small form factor (SFF) swing arm type actuator satisfying maximum compliance and maximum stiffness at the same time using the multi-objective optimization approach. The multi-objective function is defined to maximize the compliance in the direction of focusing as well as the eigen-frequency of the structure. The design of experiments (DOE) is performed to select sensitive variables. Based on DOE results, the response surface functions are formulated to construct the multi-objective function. The weight factors between conflicting objective functions are determined by the Pareto optimum method. By applying the optimal combination of design variables to the design domain, the optimized topology can be obtained.This work was supported by Korea Research Foundation (KRF) Grant KRF-2004-042-D00004.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps

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. An erratum to this article can be found at     

非经典阻尼系统的精确解

提出了一种求解非经典阻尼系统的通用的解析方法,分别给出了多自由度非比例阻尼系统的自由振动和强迫振动的实数解的表达式.自由振动归结为求解一个线性代数方程组,把系统分为亚临界阻尼情况、退化阻尼情况和部分退化阻尼情况分别进行了讨论.对于强迫振动,研究了系统的稳态解和瞬态解,求解过程也归结为求解代数方程组.在几个算例中,把得到的精确解和数值计算结果进行了比较,两者吻合得很好.该方法为非比例阻尼系统的求解提供了一条有效的途径.     

An integrated approach for shape and topology optimization of shell structures

In this paper an automated approach for simultaneous shape and topology optimization of shell structures is presented. Most research in the last decades considered these optimization techniques separately, seeking an initial optimal material layout and refining the shape of the solution later. The method developed in this work combines both optimization techniques, where the shape of the shell structure and material distribution are optimized simultaneously, with the aim of finding the optimum design that maximizes the stiffness of the shell. This formulation involves a variable ground structure for topology optimization, since the shape of the shell is modified in the course of the process. The method has been implemented into a computational model and the feasibility of the approach is demonstrated using several examples.     

Structural topology optimization for frequency response problem using model reduction schemes

This study uses model reduction (MR) schemes such as the mode superposition (MS), Ritz vector (RV), and quasi-static Ritz vector (QSRV) methods, which reduce the size of the dynamic stiffness matrix of dynamic structures, to calculate dynamic responses and sensitivity values with adequate efficiency and accuracy for topology optimization in the frequency domain. The calculation of structural responses to dynamic excitation using the framework of the finite element (FE) procedure usually requires a significant amount of computation time; that is mainly attributable to repeated inversions of dynamic stiffness matrices depending on time or frequency intervals, which hastens the dissemination of the MR schemes in the analysis. However, using well-established MR schemes in topology optimization has not been prevalent. Therefore, this study conducted a comprehensive investigation to highlight the drawbacks and advantages of these MR schemes for topology optimization. In the results, the MS method, which generates reduction bases by considering some of the lowest eigenmodes, can lose the accuracy in both approximated structural responses and sensitivity values due to locally vibrating eigenmodes and higher mode truncation in the solid isotropic material with penalization (SIMP) approach. In addition, the RV and QSRV methods, which generate reduction bases by considering the external force, mass, and stiffness matrices of a structure, can be used as alterative model reduction schemes for stable optimization. Through several analysis and design examples, the efficiency and reliability of the model reduction schemes for topology optimization are compared and validated.     

Topology optimization of free vibrating continuum structures based on the element free Galerkin method

In this paper, the topology optimization design of the free vibrating continuum structures is formulated based on the element free Galerkin (EFG) method. Considering the relative density of nodes as design variable, and the maximization of the fundamental eigenvalue as an objective function, the mathematical formulation of the topology optimization model is developed using the solid isotropic microstructures with penalization (SIMP) interpolation scheme. The topology optimization problem is solved by the optimality criteria method. Finally, the feasibility and efficiency of the proposed method are illustrated with several 2D examples that are widely used in the topology optimization design.