Topological material layout in plates for vibration suppression and wave propagation control

We propose a topological material layout method to design elastic plates with optimized properties for vibration suppression and guided transport of vibration energy. The gradient-based optimization algorithm is based on a finite element model of the plate vibrations obtained using the Mindlin plate theory coupled with analytical sensitivity analysis using the adjoint method and an iterative design update procedure based on a mathematical programming tool. We demonstrate the capability of the method by designing bi-material plates that, when subjected to harmonic excitation, either effectively suppress the overall vibration level or alternatively transport energy in predefined paths in the plates, including the realization of a ring wave device. Most of this work was performed while AAL was employed at the Department of Mechanical Engineering.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

Sensitivity analysis of vibro-acoustic systems in statistical energy analysis framework

The main objective of this paper is to present first and second-order sensitivity analysis of vibro-acoustic systems in the statistical energy analysis (SEA) frame work. Equations for computing these sensitivities for a general SEA model are obtained from two different approaches: (1) direct method and (2) adjoint method. The above equations are applied to a simple model of three plates, joined in the form of a ‘Z’, to minimize the total energy of one of the plates. It has been verified that these approaches lead to the same results and the difference between them is only with respect to the computational efficiency. The design sensitivity results calculated from the proposed analytical methods are compared with those obtained from the finite difference method, which show good agreement. The results of this paper can be useful to optimization of vibro-acoustic systems at the drawing board stage in the SEA framework.     

Analysis of the deflection of a circular plate with an annular piezoelectric actuator

A new analytical model for predicting the deflection of a circular plate with an annular piezoelectric actuator is presented. The plate and actuator are treated as a mechanically over-constrained system and a structural mechanics approach is applied to establish the relevant equations of geometrical compatibility and static equilibrium, assuming that the interaction forces between the actuator and plate are concentrated at the edges of the actuator annulus. These equations can be solved analytically or numerically to determine the interaction forces. Analytical expressions for plate deflection in terms of the interaction forces are then presented for three sets of plate boundary conditions. The analytical results are shown to be in good agreement with finite element simulations and provide an efficient alternative to finite element analysis for design and optimization studies.     

Developments in structural-acoustic optimization for passive noise control

Summary  Low noise constructions receive more and more attention in highly industrialized countries. Consequently, decrease of noise radiation challenges a growing community of engineers. One of the most efficient techniques for finding quiet structures consists in numerical optimization. Herein, we consider structural-acoustic optimization understood as an (iterative) minimum search of a specified objective (or cost) function by modifying certain design variables. Obviously, a coupled problem must be solved to evaluate the objective function. In this paper, we will start with a review of structural and acoustic analysis techniques using numerical methods like the finite- and/or the boundary-element method. This is followed by a survey of techniques for structural-acoustic coupling. We will then discuss objective functions. Often, the average sound pressure at one or a few points in a frequency interval accounts for the objective function for interior problems, wheareas the average sound power is mostly used for external problems. The analysis part will be completed by review of sensitivity analysis and special techniques. We will then discuss applications of structural-acoustic optimization. Starting with a review of related work in pure structural optimization and in pure acoustic optimization, we will categorize the problems of optimization in structural acoustics. A suitable distinction consists in academic and more applied examples. Academic examples iclude simple structures like beams, rectangular or circular plates and boxes; real industrial applications consider problems like that of a fuselage, bells, loudspeaker diaphragms and components of vehicle structures. Various different types of variables are used as design parameters. Quite often, locally defined plate or shell thickness or discrete point masses are chosen. Furthermore, all kinds of structural material parameters, beam cross sections, spring characteristics and shell geometry account for suitable design modifications. This is followed by a listing of constraints that have been applied. After that, we will discuss strategies of optimization. Starting with a formulation of the optimization problem we review aspects of multiobjective optimization, approximation concepts and optimization methods in general. In a final chapter, results are categorized and discussed. Very often, quite large decreases of noise radiation have been reported. However, even small gains should be highly appreciated in some cases of certain support conditions, complexity of simulation, model and large frequency ranges. Optimization outcomes are categorized with respect to objective functions, optimization methods, variables and groups of problems, the latter with particular focus on industrial applications. More specifically, a close-up look at vehicle panel shell geometry optimization is presented. Review of results is completed with a section on experimental validation of optimization gains. The conclusions bring together a number of open problems in the field.     

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.     

Dynamic problems of shape optimal design for shallow curved plates

The problem of optimal structural design of shallow thin-walled elements such as curved rectangular plates are formulated and solved for dynamic conditions. The distribution of the initial curvature of shallow plates in a nonstrained state is taken as the control function. Dynamic compliance is considered as the minimized performance functional. Optimality conditions are derived for the distributed parameter system considered and applied for the construction of the analytical solution. The rigorous analysis of extremum conditions and behavioural equations shows that the initial optimization problem is decomposed into several problems of classical structural analysis, which can be successfully solved analytically. Some optimal designs obtained for rectangular plates under stretching and bending, and a plate lying on an elastic foundation and subjected to lateral forces are presented. Received: November 27, 1998     

Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

Maximum energy dissipation for elasto-plastic plates via isogeometric shape optimization

In this research, optimum shape of plate structures is sought to maximize the energy dissipation via structural shape optimization. To achieve this, isogeometric analysis (IGA) is utilized for structural analysis of plates considering elasto-plastic behavior of materials. The von Mises material model is employed for this purpose. Non-uniform rational B-splines basis functions are used for both geometry definition and approximating the unknown deformation field. The optimization problem is to maximize the structural dissipated energy until a prescribed displacement is reached and a fixed amount of material is considered in the design domain. A direct shape sensitivity analysis is performed and a mathematical based approach is employed for the optimization process. To demonstrate the efficiency of the proposed algorithm three examples are illustrated. Using the IGA prevents adjusting analysis model during the optimization process, which is time-consuming especially when iterative nonlinear analysis is performed. The results also show that large geometry modifications can be properly managed by the proposed algorithm.

     

Vibration sensitivity analysis of the ‘Butterfly-gyro’ structure

The ‘Butterfly-gyro’ is simple to manufacture with single sided electrostatic excitation and capacitive detection, and it is considered as one kind of the microgyroscope with high sensitivity due to its unique structure. This paper provides the sensitivity analytical model by solving the dynamic equations of motion and the design guidelines for microgyroscope with high sensitivity. Using Coriolis Effect and Newton’s second law, the dynamic equations are built. The sensitivity analytical model, including the denotations of Q factors and the resonant frequencies, is built. The approximate analytical expressions of Q factors and the resonant frequencies are derived by rational assumptions. Based on the sensitivity analytical model, the parametric analysis is carried out, and the design guidelines of high sensitivity are also deduced. Finally, Q factor, frequency split and other factors influencing the sensitivity are discussed in details to enhance its sensitivity. Results presented are valuable in the design and parameters optimization of the microgyroscope with high sensitivity.     

Exact analytical solutions for frequency sensitivity of flat plates with respect to boundary shape change

Exact analytical solutions for the natural frequency sensitivity of flat plates subjected to prescribed shape design changes are obtained. The results are based on the domain mapping idea and are restricted to certain simple changes in domain geometry. Since the approach is exact and analytical it may be used as a reference for a simple yet precise accuracy assessment of more sophisticated and general numerical calculations of boundary shape eigen value sensitivity.     

A stabilized MITC element for accurate wave response in Reissner–Mindlin plates

Residual based finite element methods are developed for accurate time-harmonic wave response of the Reissner–Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variational form for the finite element approximation. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used to develop a new mixed least-squares (MLS4) four-node quadrilateral plate element with improved wave accuracy. Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle, match both propagating and evanescent analytical wave numbers for Reissner–Mindlin plates. Numerical results demonstrates the significantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4 element.     

Optimal rib layout design for noise reduction based on topology optimization and acoustic contribution analysis

In this paper, an optimum rib layout design method for reducing radiated noise is proposed based on topology optimization and acoustic contribution analysis. According to radiated noise depends on acoustic transfer vector (ATV) and normal velocity, the influence of rib layout on ATV is analyzed and it is found rib layout has little influence on ATV. Only if a region has maximum acoustic contribution, the normal velocities on this region can have the most remarkable influence on radiated noise. So the determination procedure of region with maximum acoustic contribution is introduced. Based on this, the topology optimization model is established to minimize the normal velocities on this region. Ribs can be arranged according to the optimum topologies to reduce the normal velocities, which in turn results in a reduction of radiated noise. The topology optimization model is used to obtain the optimal rib layout by taking a plate-like structure as an example. The plate is fixed along all side edges and excited by a time-harmonic external point load with different prescribed frequencies. The radiated noise is simulated using the finite element method and boundary element method. Four plates are manufactured according to the optimal rib layouts for different single frequency excitation. Modal test and sound measurement are conducted to validate the proposed method. The influence of loading position on the topology optimization results is also investigated and discussed.     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

Process for design optimization of honeycomb core sandwich panels for blast load mitigation

A general process for optimization of a sandwich panel to minimize the effects of air blast loading is presented here. The panel geometry consists of two metal face plates with a crushable honeycomb or other type of core. Optimization is necessary as there is strong coupling between the several variables and the physics, which makes parametric studies relatively ineffective. Virtual testing is used to develop a homogenized model for the stress–strain curve of the honeycomb core, which can be readily applied to other types of cellular core. The homogenized model has been validated by comparison to existing results as well as to results from detailed finite element (FE) models. A design of experiments (DOE) based response surface optimization method in combination with LS-DYNA is used to minimize dynamic deflection or acceleration of the back face plate. Constraints on total mass and on plastic strain in the face plates are imposed. The mechanism of lowering the backface deflection is by increasing front face plate thickness which effectively distributes the blast load to a larger area of the core and avoids local concave deformation of the front face plate. Further, core depth is increased which increases panel stiffness. For acceleration minimization, results again produce a stiffer front face plate, but accompanied by a sufficiently soft core. The mechanism of lowering the backface acceleration is by absorbing energy with low transmitted stress. A clear cut comparison between monolithic metal plates and sandwich plates, for the same loading and failure criteria, is presented here.     

Design considerations of rectangular electrostatic torsionactuators based on new analytical pull-in expressions

An important design issue of electrostatic torsion actuator is the relative locations of the actuating electrodes, where the bias voltage is applied. These geometrical design parameters affect both the pull-in angle as well as the pull-in voltage. In this paper, a new approximated analytical solution for the pull-in equation of an electrostatic torsion actuator with rectangular plates is derived. The analytical expression is shown to be within 0.1% of the one degree of freedom (1DOF) lumped-element model numerical simulations. Moreover, the analytical expressions are compared with the full coupled-domain finite-elements/boundary-elements (FEM/BEM) simulations provided by MEMCAD4.8 Co-solve tool, showing excellent agreement. The approach presented here provides better physical insight, more rapid simulations and an improved design optimization tool for the actuator     

Optimal design of sandwich and laminated composite plates based on the planning of experiments

Optimal design problems of sandwich plates with soft core and laminated composite face layers, and multilayered composite plates are investigated. The optimal design problems are solved by using the method of the planning of experiments. The optimization procedure is divided into the following stages: choice of control parameters and establishment of the domain of search, elaboration of plans of experiment for the chosen number of reference points, execution of the experiment, determination of simple mathematical models from the experimental data, design of the structure on the basis of the mathematical models discovered, and finally verification experiments at the point of the optimal solution. Vibration and damping analysis is performed by using a sandwich plate finite elements based on a broken line model. Damping properties of the core and face layers of the plate are taken into account in the optimal design. Modal loss factors are computed using the method of complex eigenvalues or the energy method. Frequencies and modal loss factors of the plate are constraints in the optimal design problem. There are also constraints on geometrical parameters and the bending stiffness of the plate. The mass of the plate is the objective function. Design parameters are the thickness of the plate layers. In the points of experiments computer simulation using FEM is carried out. Using this information, simple mathematical models for frequencies and modal loss factors for the plate are determined. These simple mathematical functions are used as constraints in the nonlinear programming problem, which is solved by random search and the penalty function method. Numerical examples of the optimal design of clamped sandwich and simply supported laminated composite plates are presented. A significant improvement of damping properties of a sandwich plate is observed in comparison with a simple plate of equal natural frequencies.