Eigenvalue topology optimization of structures using a parameterized level set method

Preventing a structure from resonance is important in many real-world applications. Because an external excitation frequency can be lower than the fundamental eigenfrequency or between the eigenfrequencies of a structure, there is a strong need for eigenfrequency optimization technology to optimize the fundamental eigenfrequency and, in addition, the k-th eigenfrequency and to maximize the gap between eigenfrequencies. However, previous optimization studies on vibrating elastic structures that used the level set method have been devoted to the optimization of the fundamental eigenfrequency, whereas the higher-order eigenfrequencies optimization problem has seldom been considered. This paper presents an eigenfrequency optimization technology that is based on the compactly supported radial basis functions (CS-RBFs) parameterized level-set method, using the fundamental eigenfrequency, the eigenfrequency of a given higher-order, and the gap between two consecutive eigenfrequencies as the optimization objectives. Furthermore, to address the oscillation problem of the objective function, we adopt an exponential weighted optimization model of a number of the lower eigenfrequencies for multiple eigenvalue optimizations, and we utilize mode-tracking technology for the single eigenvalue optimization.In addition, we further extend the CS-RBFs parameterized level-set method to an optimization that is performed with geometric constraints, which means that the size and position of the regular holes in the structure can be optimized with the shape and topology. This approach is useful in real-world applications. The effectiveness of this method is demonstrated by several widely investigated examples that have various objectives.     

Topology optimization of freely vibrating continuum structures based on nonsmooth optimization

The non-differentiability of repeated eigenvalues is one of the key difficulties to obtain the optimal solution in the topology optimization of freely vibrating continuum structures. In this paper, the bundle method, which is a very promising one in the nonsmooth optimization algorithm family, is proposed and implemented to solve the problem of eigenfrequency optimization of continuum. The bundle method is well-known in the mathematical programming community, but has never been used to solve the problems of topology optimization of continuum structures with respect to simple or multiple eigenfrequencies. The advantage of this method is that the specified information of iteration history may be collected and utilized in a very efficient manner to ensure that the next stability center is closer to the optimal solution, so as to avoid the numerical oscillation in the iteration history. Moreover, in the present method, both the simple and multiple eigenfrequencies can be managed within a unified computational scheme. Several numerical examples are tested to validate the proposed method. Comparisons with nonlinear semidefinite programming method and 0–1 formulation based heuristic method show the advantages of the proposed method. It is showed that, the method can deal with the nonsmoothness of the repeated eigenvalues in topology optimization in a very effective and efficient manner without evaluating the multiplicity of the eigenvalues.     

Designing plates for minimum internal resonances

Internal resonance is a nonlinear phenomenon for a structure when the eigenfrequencies of the structure are commensurable or close to being commensurable. Using optimization we have the possibility to control the eigenfrequencies, i.e., move an eigenfrequency, maximize a given eigenfrequency, or maximize the gap between eigenfrequencies. It is therefore also possible to design a structure that is as free as possible of internal resonance up to mode of order n.We consider plates made of two materials. The designs depend on the boundary conditions and on the frequency range within which the plate should be as free of internal resonance as possible. The two materials can either be two physical materials, or one can be a physical material and the other a weakening of the first material. By doing this we are in principle solving three different problems: a reinforcement problem, a problem of where to put holes in the structure, and, finally the more involved case (from a manufacturing point of view), of two different materials. The optimizations are performed using the finite element method for analysis and the topology optimization approach for design. The optimization problem is formulated using a bound formulation where the objective is to maximize a minimum detuning parameter. Special attention is given to the formulation of the conditions for internal resonance. Using the method presented in this paper it is possible to remove an unwanted nonlinear phenomenon without the use of a nonlinear model and without knowledge of the nonlinearities present in the system.     

GA topology optimization using random keys for tree encoding of structures

Topology optimization consists in finding the spatial distribution of a given total volume of material for the resulting structure to have some optimal property, for instance, maximization of structural stiffness or maximization of the fundamental eigenfrequency. In this paper a Genetic Algorithm (GA) employing a representation method based on trees is developed to generate initial feasible individuals that remain feasible upon crossover and mutation and as such do not require any repairing operator to ensure feasibility. Several application examples are studied involving the topology optimization of structures where the objective functions is the maximization of the stiffness and the maximization of the first and the second eigenfrequencies of a plate, all cases having a prescribed material volume constraint.     

A level-set method for vibration and multiple loads structural optimization

We extend the level-set method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads. This method is based on a combination of the classical shape derivative and of the Osher–Sethian level-set algorithm for front propagation. In two and three space dimensions we maximize the first eigenfrequency or we minimize a weighted sum of compliances associated to different loading configurations. The shape derivative is used as an advection velocity in a Hamilton–Jacobi equation for changing the shape. This level-set method is a low-cost shape capturing algorithm working on a fixed Eulerian mesh and it can easily handle topology changes.     

Topology optimization for femto suspension design

The ongoing increase of track density requirements of hard disk drives (HDDs) and decrease of flying height of sliders have brought along formidable challenges to suspension design. Conventional design processes are quite tedious and inefficient. This paper presents a HDD suspension design process by using topology optimization. An efficient structural topology optimization method, based on the second derivatives information, is proposed to generate structures which satisfy multiple design objectives including both compliance and eigenfrequencies. This topology optimization approach is successfully applied in the HDD suspension design. The design begins with a very simple initial draft, and the design objectives are defined to minimize the spring rate and maximize the resonant frequencies of first bending, first torsion and sway modes of a suspension. An optimal design concept can be generated from the topology optimization. Then the design is further tuned by using the shape optimization. Finally, an optimal suspension design for femto sliders with much better dynamic characteristics is presented.     

Stability optimization of beams conveying fluid or carrying other axially moving materials

Optimal designs, and sensitivities of such designs, are calculated for transversely vibrating structures carrying an axially moving material. The structures studied consist of piecewise uniform and initially straight beam elements conveying a piecewise constant-speed plug flow of material along their deflected axes. The beams can be supported by a distributed Winkler-type ambient medium. Viscous damping in the beam material and the ambient medium is considered. Large static axial loads may act on the beam and on the moving material. The beams are modelled with a generalized second-order Rayleigh-Timoshenko theory including the Euler-Bernoulli theory as a special case. The structures investigated may also contain taut strings and rigid bodies. Considering a given subcritical material speed, the aim of the present study is to modify the initial design of a given beam structure in such a way that the transient vibrational motion following a transverse disturbance will die out as quickly as possible. To this end, complex eigenfrequencies pertaining to transverse vibration are calculated, and design parameters are changed so as to maximally raise the imaginary part of that eigenfrequency which has the smallest such part. In one of the examples, the objective is to maximally raise the product of the real and imaginary parts of the eigenfrequency which has the smallest such product.     

Topology optimization of actuator arms in hard disk drives for reducing bending resonance-induced off-tracks

The objective of this work is to find an optimal actuator arm configuration in hard disk drives (HDD’s) for sufficiently small arm bending-induced off-track error by maximizing the fundamental eigenfrequency of arm bending modes subject to a constraint on the mass moment of inertia of arms. By applying the topology optimization method for the purpose, an arm configuration having two balancing holes instead of a single balancing hole in conventional designs was obtained. It is numerically shown that an optimized arm configuration substantially increases the arm bending resonant frequency with little change in the mass moment of inertia of arms in comparison with conventional designs having a single balancing hole. Finally, the performance of an optimized actuator arm is verified by showing that the eigenfrequencies associated with arm bending modes are increased by about 100?Hz and the off-track error by measuring the position error signal (PES) from actual sample drives can be reduced by as much?as?25%.     

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.     

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.     

2D shape optimization with static and dynamic constraints

The paper presents an approach that allows us to consider in the shape optimization several static loading conditions together with constraints imposed on eigenfrequencies. The idea of the method is based upon simultaneous solutions of equations and inequalities arising from the Kuhn-Tucker necessary conditions for an optimum problem. The paper is illustrated with four examples in which stress and eigenfrequency are active constraints.     

Topology optimization using a dual method with discrete variables

This paper deals with topology optimization of continuous structures in static linear elasticity. The problem consists in distributing a given amount of material in a specified domain modelled by a fixed finite element mesh in order to minimize the compliance. As the design variables can only take two values indicating the presence or absence of material (1 and 0), this problem is intrinsicallydiscrete. Here, it is solved by a mathematical programming method working in the dual space and specially designed to handle discrete variables. This method is very wellsuited to topology optimization, because it is particularly efficient for problems with a large number of variables and a small number of constraints. To ensure the existence of a solution, the perimeter of the solid parts is bounded. A computer program including analysis and optimization has been developed. As it is specialized for regular meshes, the computational time is drastically reduced. Some classical 2-D and new 3-D problems are solved, with up to 30,000 design variables. Extensions to multiple load cases and to gravity loads are also examined.     

Topological design considering flexibility under periodic loads

Topology optimization has been extensively considered to design the structural configuration for the stiffness maximization and the eigenfrequency maximization. In this paper, we construct a topology optimization method implementing flexibility with the time-periodic loading condition. First, the flexibility in the dynamic periodic loading is formulated using the mutual energy concept. Second, the multi-optimization problem is formulated using a new multi-objective function in order to obtain an optimal solution incorporating both flexibility and stiffness. Next, the topology optimization procedure is developed using the homogenization design method. Finally, some examples are provided to confirm the optimal design method presented here. Received January 18, 1999     

Natural frequencies of tube bundle in an uncompressible fluid

A method of eigenfrequency computation for a cluster of tubes immersed in an uncompressible liquid is described. Several numerical examples are presented, showing that the presence of fluid involves the spreading of the natural frequencies of the tubes in vacuum. A lower bound of these eigenfrequencies can be obtained for a spatially periodic bundle.     

Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and buckling

In this paper we consider the optimization of general 3D truss structures. The design variables are the cross-sections of the truss bars together with the joint coordinates, and are considered to be continuous variables. Using these design variables we simultaneously carry out size optimization (areas) and shape optimization (joint positions). Topology optimization (removal and introduction of bars) is only considered in the sense that bars of minimum cross-sectional area will have a negligible influence on the performance of the structure. The structures are subjected to multiple load cases and the objective of the optimizations is minimum mass with constraints on (possibly multiple) eigenfrequencies, displacements, and stresses. For the case of stress constraints, we deal differently with tensile and compressive stresses, for which we control buckling on the element level. The stress constraints are imposed in correlation with industrial standards, to make the optimized designs valuable from a practical point of view. The optimization problem is solved using SLP (Sequential Linear Programming).     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

Design of multi-component structural systems for optimal layout topology and joint locations

Present day topology optimization techniques for continuum structures consider the design of single structural components, while most real life engineering design problems involve multiple components or structures. It is therefore necessary to have a methodology that can address the design of multi-component systems and generate designs for the optimal layouts of individual structures and locations for interconnections. The interconnections include supports provided by the ground, joints and rigid connections like rivets, bolts and welds between components. While topology optimization of structures has been extensively researched, relatively little work has been done on optimizing the locations of the interconnections. In this research, a method to model and define domains for the interconnections has been developed. The optimization process redistributes material in the component design domains and locates the connections optimally based on an energy criterion. Some practical design examples are used to illustrate the capability of this method.     

Checkerboard and minimum member size control in topology optimization

Checkerboard-like material distributions are frequently encountered in topology optimization of continuum structures, especially when first order finite elements are used. It has been shown that this phenomenon is caused by errors in the finite element formulation. Minimum member size control is closely related to the problem of mesh dependency of solutions in topology optimization. With increasing mesh density, the solution of a broad class of problems tends to form an increasing number of members with decreasing size. Different approaches have been developed in recent years to overcome these numerical difficulties. However, limitations exist for those methods, either in generality or in efficiency. In this paper, a new algorithm for checkerboard and direct minimum member size control has been developed that is applicable to the general problem formulation involving multiple constraints. This method is implemented in the commercial software Altair OptiStruct. March 22, 2000     

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.