主动约束层阻尼梁有限元建模与动态特性研究

基于弹性、粘弹性和压电材料的本构关系,利用Hamilton原理,推导了主动约束层阻尼梁的有限元动力学模型．结合压电材料的机电耦合特性,采用自感电压的位移反馈,研究了主动约束层阻尼梁的闭环控制特性．求解了主动约束层阻尼简支梁的动态特性如固有频率、模态损耗因子及频率响应特性等．对被动控制、主动控制和主被动混合控制的控制效果进行了分析比较．研究了粘弹性层与约束层厚度等参数对减振控制效果的影响．     

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.     

Design of materials using hybrid cellular automata

This paper presents a new approach for the topological design of materials with extreme properties. The method is based on hybrid cellular automaton (HCA), which is an implicit optimization technique that uses local rules to update design variables iteratively until meeting the described optimality conditions. By means of an energy-based homogenization approach, the effective properties of the considered material are calculated in terms of element mutual energies. By this method, no sensitivity information is required to find the optimal topology for the considered design objectives: bulk modulus, shear modulus, and negative Poisson’s ratio. The proposed method is validated by a series of numerical examples.     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.     

Optimal structural damping of skis using a genetic algorithm

Two problems of the optimal damping of skis with the use of an improved genetic algorithm (GA) are presented. The first problem is the optimal location of viscoelastic constrained layers; the second problem is the optimization of the stacking sequence with or without variable layer thicknesses. It is demonstrated that the GA is a very attractive and powerful method for these problems in which the objective function has no derivatives and several optimal solutions simultaneously exist.     

Multi-objective optimal design of hybrid viscoelastic/composite sandwich beams by using the generalized differential quadrature method and the non-dominated sorting genetic algorithm II

In this study, the multi-objective optimal design of hybrid viscoelastic/composite sandwich beams for minimum weight and minimum vibration response is aimed. The equation of motion for linear vibrations of a multi-layer beam is derived by using the principle of virtual work in the most general form. These governing equations together with the boundary conditions are discretized by the generalized differential quadrature method (GDQM) in the frequency domain for the first time. Also, the time and temperature dependent properties of the viscoelastic materials are taken into consideration by a novel ten-parameter fractional derivative model that can realistically capture the response of these materials. The material variability is accounted for by letting an optimization algorithm choose a material freely out of four fiber-reinforced composite materials and five viscoelastic damping polymers for each layer. The design parameters, i.e., the orientation angles of the composites, layer thicknesses and the layer materials that give the set of optimal solutions, namely the Pareto frontier, is obtained for the three and nine-layered clamped-free sandwich beams by using a variant of the non-dominated sorting genetic algorithms (NSGA II).     

Topology optimization of periodic microstructures for enhanced dynamic properties of viscoelastic composite materials

We present a topology optimization method for the design of periodic composites with dissipative materials for maximizing the loss/attenuation of propagating waves. The computational model is based on a finite element discretization of the periodic unit cell and a complex eigenvalue problem with a prescribed wave frequency. The attenuation in the material is described by its complex wavenumber, and we demonstrate in several examples optimized distributions of a stiff low loss and a soft lossy material in order to maximize the attenuation. In the examples we cover different frequency ranges and relate the results to previous studies on composites with high damping and stiffness based on quasi-static conditions for low frequencies and the bandgap phenomenon for high frequencies. Additionally, we consider the issues of stiffness and connectivity constraints and finally present optimized composites with direction dependent loss properties.     

Optimal design of periodic functionally graded composites with prescribed properties

The computational design of a composite where the properties of its constituents change gradually within a unit cell can be successfully achieved by means of a material design method that combines topology optimization with homogenization. This is an iterative numerical method, which leads to changes in the composite material unit cell until desired properties (or performance) are obtained. Such method has been applied to several types of materials in the last few years. In this work, the objective is to extend the material design method to obtain functionally graded material architectures, i.e. materials that are graded at the local level (e.g. microstructural level). Consistent with this goal, a continuum distribution of the design variable inside the finite element domain is considered to represent a fully continuous material variation during the design process. Thus the topology optimization naturally leads to a smoothly graded material system. To illustrate the theoretical and numerical approaches, numerical examples are provided. The homogenization method is verified by considering one-dimensional material gradation profiles for which analytical solutions for the effective elastic properties are available. The verification of the homogenization method is extended to two dimensions considering a trigonometric material gradation, and a material variation with discontinuous derivatives. These are also used as benchmark examples to verify the optimization method for functionally graded material cell design. Finally the influence of material gradation on extreme materials is investigated, which includes materials with near-zero shear modulus, and materials with negative Poisson’s ratio.     

Damping optimization of viscoelastic laminated sandwich composite structures

Recent developments on the optimization of passive damping for vibration reduction in sandwich structures are presented in this paper, showing the importance of appropriate finite element models associated with gradient based optimizers for computationally efficient damping maximization programs. A new finite element model for anisotropic laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach. The complex modulus approach is used for the viscoelastic material behavior, and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for the maximization of modal loss factors, using gradient based optimization associated with the developed model, and single and multiobjective optimization based on genetic algorithms using an alternative ABAQUS finite element model. The model has been applied successfully and comparative optimal design applications in sandwich structures are presented and discussed.     

Viscoelastic material design with negative stiffness components using topology optimization

An application of topology optimization to design viscoelastic composite materials with elastic moduli that soften with frequency is presented. The material is a two-phase composite whose first constituent is isotropic and viscoelastic while the other is an orthotropic material with negative stiffness but stable. A concept for this material based on a lumped parameter model is used. The performance of the topology optimization approach in this context is illustrated using three examples.     

The gradient projection method for structural topology optimization including density-dependent force

This paper proposes a modified gradient projection method (GPM) that can solve the structural topology optimization problem including density-dependent force efficiently. The particular difficulty of the considered problem is the non-monotonicity of the objective function and consequently the optimization problem is not definitely constrained. Transformation of variables technique is used to eliminate the constraints of the design variables, and thus the volume is the only possible constraint. The negative gradient of the objective function is adopted as the most promising search direction when the point is inside the feasible domain, while the projected negative gradient is used instead on condition that the point is on the hypersurface of the constraint. A rational step size is given via a self-adjustment mechanism that ensures the step size is a good compromising between efficiency and reliability. Furthermore, some image processing techniques are employed to improve the layouts. Numerical examples with different prescribed volume fractions and different load ratios are tested respectively to illustrate the characteristics of the topology optimization with density-dependent load.     

Adaptive topology optimization of elastoplastic structures

Material topology optimization is applied to determine the basic layout of a structure. The nonlinear structural response, e.g. buckling or plasticity, must be considered in order to generate a reliable design by structural optimization. In the present paper adaptive material topology optimization is extended to elastoplasticity. The objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement. The mass in the design space is prescribed. The design variables are the densities of the finite elements. The optimization problem is solved by a gradient based OC algorithm. An elastoplastic von Mises material with linear, isotropic work-hardening/softening for small strains is used. A geometrically adaptive optimization procedure is applied in order to avoid artificial stress singularities and to increase the numerical efficiency of the optimization process. The geometric parametrization of the design model is adapted during the optimization process. Elastoplastic structural analysis is outlined. An efficient algorithm is introduced to determine the gradient of the ductility with respect to the densities of the finite elements. The overall optimization procedure is presented and verified with design problems for plane stress conditions.     

Optimal riser design in sand casting process by topology optimization with SIMP method I: Poisson approximation of nonlinear heat transfer equation

The optimal design of a casting feeding system is considered. The problem is formulated as the volume constrained topology optimization and is solved with the finite element analysis, explicit design sensitivity, and numerical optimization. In contrast to the traditional topology optimization where the objective function is defined on the design space, in the presented method, the design space is a subset of the complement of the objective function space. To accelerate optimization procedure, the nonlinear unsteady heat transfer equation is approximated with a Poisson-like equation. The feasibility of the presented method is supported with illustrative examples.     

Design of laminated composite plates for optimal dynamic characteristics using a constrained global optimization technique

The lamination arrangements of moderately thick laminated composite plates for optimal dynamic characteristics are studied via a constrained multi-start global optimization technique. In the optimization process, the dynamical analysis of laminated composite plates is accomplished by utilizing a shear deformable laminated composite finite element, in which the exact expressions for determining shear correction factors were adopted and the modal damping model constructed based on an energy concept. The optimal layups of laminated composite plates with maximum fundamental frequency or modal damping are then designed by maximizing the frequency or modal damping capacity of the plate via the multi-start global optimization technique. The effects of length-to-thickness ratio, aspect ratio and number of layer groups upon the optimum fiber orientations or layer group thicknesses are investigated by means of a number of examples of the design of symmetrically laminated composite plates.     

Damping of beams. Optimal distribution of cuts in the viscoelastic constrained layer

To damp the flexural vibrations of homogeneous beams or plates in a large frequency range, one of the most efficient methods is the use of constrained viscoelastic layers. Since most of the damping ability is due to shearing stresses in the viscoelastic layer, it is interesting to determine the most appropriate distribution of the shear in the layers. This paper presents the influence of a new parameter involved in this repartition, i.e. the distribution of cuts in the elastic constraining layer. It will be demonstrated that modal damping may be significantly modified in this way. The number and the locations of the cuts may vary and are determined to optimize the damping. The vibrating beam modal analysis is performed by a finite element analysis using special finite elements which have variable d.o.f. in order to take into account the lack of continuity of the viscoelastic constrained displacement field. Using a genetic algorithm, an optimal distribution of the cuts has been determined for a maximum damping of one or serval flexural modes.     

A discrete layer beam finite element for the dynamic analysis of composite sandwich beams with integral damping layers

A discrete layer finite element is presented for the dynamic analysis of laminated beams. The element uses C⁰ continuous linear and quadratic polynominals to interpolate the in-plane and transverse displacement field, respectively, and is free from the effects of shear locking. Modal frequencies and damping are estimated using both the modal strain energy method and the complex modulus method. A forced response version of the model is also presented. The model predictions are compared with experimental data for composite sandwich beams with integral damping layers. Four damping configurations are considered, a constrained layer treatment, a segmented constrained layer treatment and two internal treatments.     

Design of periodic elastoplastic energy dissipating microstructures

The design of periodic elastoplastic microstructures for maximum energy dissipation is carried out using topology optimization. While the topology optimization of elastic microstructures has been performed in numerous studies, microstructural design considering inelastic behavior is relatively untouched due to a number of reasons which are addressed in this study. An RVE-based multiscale model is employed for computational homogenization with periodic boundary constraints, satisfying the Hill-Mandel principle. The plastic anisotropy which may be prevalent in materials fabricated through additive manufacturing processes is considered by modeling the constitutive behavior at the microscale with Hoffman plasticity. Discretization is done using enhanced assumed strain elements to avoid locking from incompressible plastic flow under plane strain conditions and a Lagrange multiplier approach is used to enforce periodic boundary constraints in the discrete system. The design problem is formulated using a density-based parameterization in conjunction with a SIMP-like material interpolation scheme. Attention is devoted to issues such as dependence on initial design and enforcement of microstructural connectivity, and a number of optimized microstructural designs are obtained under different prescribed deformation modes.

     

Study on energy dissipation pattern in vibrating fluid filled cylindrical shells with a constrained viscoelastic layer

The modal strain energy method is used to study the energy dissipation pattern in vibrating cylindrical shells with a viscoelastic damping layer, for various circumferential and axial modes. The effect of tank size, boundary condition and height of contained fluid on the distribution pattern is also investigated. The regions of energy dissipation concentration are identified. The effect of redistribution of the (initially uniformly distributed) constrained viscoelastic material by concentrating it in the above regions is studied.     

Topological design of vibrating structures with respect to optimum sound pressure characteristics in a surrounding acoustic medium

This paper deals with topological design optimization of vibrating bi-material elastic structures placed in an acoustic medium. The structural vibrations are excited by a time-harmonic external mechanical surface loading with prescribed excitation frequency, amplitude and spatial distribution. The design objective is minimization of the sound pressure generated by the vibrating structures on a prescribed reference plane or surface in the acoustic medium. The design variables are the volumetric densities of material in the admissible design domain for the structure. A high frequency boundary integral equation is employed to calculate the sound pressure in the acoustic field. This way the acoustic analysis and the corresponding sensitivity analysis can be carried out in a very efficient manner. The structural damping is considered as Rayleigh damping. Penalization models with respect to the acoustic transformation matrix and/or the damping matrix are proposed in order to eliminate intermediate material volume densities, which have been found to appear obstinately in some of the high frequency designs. The influences of the excitation frequency and the structural damping on optimum topologies are investigated by numerical examples. Also, the problem of maximizing (rather than minimizing) sound pressures in points on a reference plane in the acoustic medium is treated. Many interesting features of the examples are revealed and discussed.     

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.