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.     

Evolutionary topology optimization of continuum structures with an additional displacement constraint

Evolutionary structural optimization (ESO) and its later version Bi-directional ESO (BESO) have been successfully applied to optimum material distribution problems for continuum structures. However, the existing ESO/BESO methods are limited to the topology optimization of an objective function such as mean compliance with a single constraint e.g. structural volume. The present work extends the BESO method to the stiffness optimization with a material volume constraint and a local displacement constraint. As a result, one will obtain a structure with the highest stiffness for a given volume while the displacement of a certain node does not exceed a prescribed limit. Several examples are presented to demonstrate the effectiveness of the proposed method.     

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.     

Multidiscipline topology optimization

Topology optimization is used for determining the best layout of structural components to achieve predetermined performance goals. The density method which uses material density of each finite element as the design variable, is employed. Unlike the most common approach which uses the optimality criteria methods, the topology design problem is formulated as a general optimization problem and is solved by the mathematical programming method. One of the major advantages of this approach is its generality; thus it can solve various problems, e.g. multi-objective and multi-constraint problems. In this study, the structural weight is chosen as the objective function and structural responses such as the compliances, displacements and the natural frequencies, are treated as the constraints. The MSC/NASTRAN finite element code is employed for response analyses. One example with four different optimization formulations was used to demonstrate this approach.     

Structural design for desired eigenfrequencies and mode shapes using topology optimization

A technique is proposed for determining the material distribution of a structure to obtain desired eigenmode shapes for problems of maximizing the fundamental eigenfrequency. The design objective is achieved using the solid isotropic method with penalization (SIMP) for topology optimization. Weighted constraints added in bound formulation are proposed to maximize the fundamental natural frequency, which provides an easy and straightforward way to prevent mode switching in the optimization process. Aside from maximizing the fundamental frequency, a method to modify existing eigenmodes to continuously evolve and assume the same shapes as the desired modes within the optimization process is proposed. The topology layout of a structure with desired eigenmodes is obtained by adding the modal assurance criterion (MAC) as additional constraints in the bound formulation optimization. Examples are presented to illustrate the proposed method, and a potential application of the proposed technique in decoupling a mechanical system is demonstrated.     

Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods

Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch’s combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.     

The evolutionary structural optimization method: theoretical aspects

In 1993, Y.M. Xie and G.P. Steven introduced an approach called evolutionary structural optimization (ESO). ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material from the design domain. The design domain is constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. Additionally, the ESO topologies have been compared with analytical ones, e.g. Michell trusses, and so far the results are quite promising. On the other hand, ESO does not have a solid theoretical basis, and consequently, the ESO minimization problem is still unresolved. Since the good agreement between the results cannot be just a coincidence, in this paper, we will study whether the gradual removal of material can be explained mathematically and whether the theoretical basis of ESO can be outlined.First, a minimization problem solved by ESO is examined. Based on the results of earlier publications, it was assumed that the ESO method minimizes the compliance-volume product of a structure or a finite element model. It was noted that the sequential linear programming (SLP)-based approximate optimization method followed by the Simplex algorithm is equivalent to ESO if the strain energy rejection criterion is utilized. However, ESO should be applied so that the elements corresponding to the design domain are equally sized. If this requirement is not met, the rejection criterion, which also considers the varying sizes of the elements, should be used. Additionally, the element stiffness matrices and element volumes should be linearly dependent on the design variables. Also linearly elastic material is assumed. At each iteration the rejected elements should be removed completely. Most often only element removal is allowed in ESO. If the design variables are initially assigned values other than the maximum value, however, the elements should be allowed to reenter the design domain. This subject, obviously, needs further study. Typically, ESO is applied to problems having a planar design domain with in-plane forces only. In these cases, ESO produces truss-like, equally stressed and maximum-stiffness topologies. It is often recommended that, based on the topology optimization, a new finite element discretization should be employed. After that, the sizing optimization procedure can be performed. Since ESO seems to be producing truss-like topologies, ESO should be applied to structural problems having pin-jointed connections. For other types of structures ESO should be studied further.Finally, it can be concluded that ESO is not just an intuitive method, as it has a very distinct theoretical basis. It is also very simple to employ in engineering design problems. For this reason, ESO has potential to become a tool for design engineers.     

Stress-based topology optimization

Previous research on topology optimization focussed primarily on global structural behaviour such as stiffness and frequencies. However, to obtain a true optimum design of a vehicle structure, stresses must be considered. The major difficulties in stress based topology optimization problems are two-fold. First, a large number of constraints must be considered, since unlike stiffness, stress is a local quantity. This problem increases the computational complexity of both the optimization and sensitivity analysis associated with the conventional topology optimization problem. The other difficulty is that since stress is highly nonlinear with respect to design variables, the move limit is essential for convergence in the optimization process. In this research, global stress functions are used to approximate local stresses. The density method is employed for solving the topology optimization problems. Three numerical examples are used for this investigation. The results show that a minimum stress design can be achieved and that a maximum stiffness design is not necessarily equivalent to a minimum stress design.     

Evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure

This paper deals with the evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure. Two new criteria, namely the material efficiency criterion and the smooth change criterion, are derived for solving this kind of evolutionary optimization problem. Using these two new criteria, the evolutionary optimization method has been further extended and applied to maximize the difference between the fundamental and the second natural frequencies of a structure under both plane stress and thin plate flexural bending conditions. The related results demonstrated that the extended evolutionary structural optimization method is useful in and applicable to dealing with the evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure. Moreover, the results also indicated that owing to the different mechanism between plane stress and thin plate flexural bending conditions, the optimal topologies, the normalized difference between two natural frequencies and the normalized material efficiency are different for a vibrating structure under these two different situations.