Shape optimization in laminated composite plates

A review on optimization of composite structures Part I: Laminated composites

Compared with traditional metal-materials, fiber-reinforced composite materials have better performance in many aspects such as strength, stiffness, and fracture resistance. At present, fiber-reinforced composite materials have been widely used in automotive, aerospace, and other industrial fields. This paper proposes a topology optimization method for solving the fundamental frequency maximization of undamped free vibration of continuous fiber-reinforced composite structures. To achieve the simultaneous optimization of the structural topological configuration and the fiber angle. A dynamic topological optimization model is established with the permitted material usage as the constraint and the structure's first-order eigenvalue as the objective function. The model includes density design variables that characterizes the topological configuration of the structure and angular design variables that characterizes the fiber orientation. The analytical sensitivity formulas of the objective function of eigenvalue with respect to density design variables and angle design variables are derived in detail, and the method of moving asymptotes (MMA) is used to solve the optimization problem. Finally, three numerical examples are performed to verify the effectiveness of the proposed method, which includes a static optimization example with the stiffness maximization as the goal and two dynamic optimization examples with the first-order eigenvalue maximization as the goal. The results show that the proposed method can achieve a stable iterative history and fast convergence, and can effectively improve the structural frequency while achieving the integrated optimization of the structural topological configuration and the fiber angle.

Optimization of composite I-sections using fiber angles as design variables

Design candidates obtained from optimization techniques may have meaningful information, which provides not only the best solution, but also a relationship between object functions and design variables. In particular, trade-off studies for optimum airfoil shape design involving various objectives and design variables require the effective analysis tool to take into account a complexity between objectives and design variables. In this study, for the multiple-conflicting objectives that need to be simultaneously fulfilled, the real-coded Adaptive Range Multi-Objective Genetic Algorithm code, which represents the global and stochastic multi-objective evolutionary algorithm, was developed for an airfoil shape design. Furthermore, the PARSEC method reflecting geometrical properties of airfoil is adopted to generate airfoil shapes. In addition, the Self-Organizing Maps, based on the neural network, are used to visualize trade-offs of a relationship between the objective function space and the design variable space obtained by evolutionary computation. The Self-Organizing Maps that can be considered as data mining of the engineering design generate clusters of object functions and design variables as an essential role of trade-off studies. The aerodynamic data for all candidate airfoils is obtained through Computational Fluid Dynamics. Lastly, the relationship between the maximum lift coefficient and maximum lift-to-drag ratio as object functions and 12 airfoil design parameters based on the PARSEC method is investigated using the Self-Organizing Maps method.

Recently, research and practical application of free-form shells are very active owing to development of computer software tools as well as progress of technology of construction and material. Design problem of free-form shells can be naturally formulated as an optimization problem considering mechanical performances. Parametric surfaces such as Bézier surfaces are effectively used for generating smooth and complex surfaces of continuum and latticed shells. However, to design a practically acceptable design, non-structural performances such as cost of construction should be taken into account. In this study, a shape optimization approach is presented for free-form shells using ruled surface. Latticed shells consisting of ruled surface have high constructability, because members on generating lines require no torsion. Boundary shape of the latticed shell is defined using a pair of Bézier curves to reduce the number of variables. The points with the same parameter value of the two curves are connected by a line to model a ruled surface. The locations of control points of Bézier curves are chosen as design variables. A latticed shell modeled by ruled surface consisting of two parabolic boundary curves is selected as the initial shape for optimization. Since a simple ruled surface does not have high degree of freedom for shape representation, two ruled surfaces are connected to model a roof with a ridge line. The members of latticed shell is modeled using beam elements, and the four corners are fixed. The vertical self-weight and live loads as well as horizontal loads representing seismic loads are considered in the process of optimization. First, we minimize the strain energy under volume constraint to obtain a stiff structure. It is confirmed that the strain energy due to vertical and horizontal loads is drastically reduced through shape optimization. Constructability is maintained, because the stiff beams along the generating lines can be manufactured without torsion. Although a stiff structure is obtained through strain energy minimization, the local stress may not be reduced. Therefore, we solve the additional optimization problem, which minimizes the material volume under stiffness and stress constraint. The parameter defining the cross-sectional property of each member is considered as design variables of this optimization problem, which is carried out after shape optimization. By solving this problem, the member stresses are reduced through the process of minimizing the material volume, while maintaining the stiffness.

Various optimization problems have been studied for the purpose of structural morphogenesis by many researchers. Structural optimization problems can be classified into three different sub-problems, namely size, shape and topology optimization problem. Size optimization problem is consisted of small sets of design variables. The shape and topology of a structure are defined by a set of design variables, and these design variables are adjusted to achieve given objectives, such as minimum volume. Such optimization problems can be solved iteratively, using gradient-based techniques. Introducing more design variables increases the complexity of the optimization problem. Therefore, it becomes difficult to solve the optimization problem by using MP techniques with large sets of design variables. Heuristics can be improved the difficulty. There are many studies using heuristics like genetic algorithm and simulated annealing. An effective method for structural morphogenesis inspired by self-organization phenomena is presented in this paper. Self-organization is a phenomenon that an entire structure gradually emerges by interaction between elements of the structure, and the elements are affected by the entire structure. Since the self-organization algorithm consists of simple calculation iteration, it can be applied to problems with large number of design variables. Proposed method is applied to problems of uniform member length, minimization of strain energy and cross-section design for frame structures, and the effectiveness is demonstrated through some examples. On the other hand, the number of projects incorporating computational design has increased in recent year, and designs with complicated forms composed of free surfaces are also increasing. In order to design such a shape, the designer needs to consider the rationality of the structure from the initial stage. So, it seems necessary to develop simple software for the structural morphogenesis. Therefore, we developed components of Grasshopper that works within Rhinoceros, so that it will be possible to reassemble algorithms in an intuitive way for designers who have never experienced programming. Grasshopper is one of Graphical Algorithm Editor (GAE), and can be visually constructed an algorithm by connecting components that is a function of modeling. Analysis by gradient-based MP techniques often involves jumping of solutions and analytical instability peculiar to non-linear problems, and it is difficult to apply to the computer aided systems as described above. Analysis algorithms based on self-organizing algorithms are considered to be suitable for the above system due to their high robustness.

This work presents a parametric study and optimization of a single impinging jet with cross flow to enhance heat transfer with two design variables. The fluid flow and heat transfer have been analyzed using three-dimensional compressible Reynolds-averaged Navier–Stokes equations with a uniform heat flux condition being applied to the impingement plate. The aspect ratio of the elliptic jet hole and the angle of inclination of the jet nozzle are chosen as the two design variables, and the area-averaged Nusselt number on a limited target plate is set as the objective function. The effects of the design variables on the heat transfer performance have been evaluated, and the objective function has been found to be more sensitive to the angle of inclination of the jet nozzle than to the aspect ratio of the elliptic jet hole. The optimization has been performed by using the radial basis neural network model. Through the optimization, the area-averaged Nusselt number increased by 7.89% compared to that under the reference geometry.

Fiber path optimization based on a family of curves in composite laminate with a center hole

A layerwise optimization approach, developed for the design of vibration of laminated rectangular plates, is extended to the symmetrically laminated, composite elliptical plates. There is little information available on the optimum lay-ups of such elliptical plates, including the special case of circular plates. The optimum design problem is formulated to maximize the critical buckling load and the lowest frequency, and fiber orientation angles in the layers are used as the design variables. A Ritz method is used to calculate eigenvalues (buckling load and frequency) of the elliptical plates and the convergence and comparison studies indicate accuracy of the calculated eigenvalues as the object function. A set of numerical results is given in tabular and graphical form, illustrating the optimum lay-ups of the laminated elliptical plates with respect to the fiber orientation angles.

The optimal location of renewable distributed generations (DGs) into a radial distribution system (RDS) has attracted major concerns from power system researchers in the present years. The main target of DG integration is to improve the overall system performance by minimizing power losses and improving the voltage profile. Hence, this paper proposed a hybrid approach between an analytical and metaheuristic optimization technique for the optimal allocation of DG in RDS, considering different types of load. A simple analytical technique was developed in order to determine the sizes of different and multiple DGs, and a new efficient metaheuristic technique known as the Salp Swarm Algorithm (SSA) was suggested in order to choose the best buses in the system, proportionate to the sizes determined by the analytical technique, in order to obtain the minimum losses and the best voltage profile. To verify the power of the proposed hybrid technique on the incorporation of the DGs in RDS, it was applied to different types of static loads; constant power (CP), constant impedance (CZ), and constant current (CI). The performance of the proposed algorithm was validated using two standards RDSs—IEEE 33-bus and IEEE 69-bus systems—and was compared with other optimization techniques.

16 - Captive (Emergency) Power Generation

The study of reactive silencers by shape and parametric optimization techniques

Non-linear analysis of porous elastically supported FGM plate under various loading

Bellows is a component in piping systems which absorbs mechanical deformation with flexibility. Its geometry is an axial symmetric shell which consists of two toroidal shells and one annular plate or conical shell. In order to analyze bellows, this study presents the finite element analysis using a conical frustum shell element. A finite element analysis is developed to analyze various bellows. The validity of the developed program is verified by the experimental results for axial and lateral stiffness. The formula for calculating the natural frequency of bellows is made by the simple beam theory. The formula for fatigue life is also derived by experiments. The shape optimal design problem is formulated using multiple objective optimization. The multiple objective functions are transformed to a scalar function by weighting factors. The stiffness, strength and specified stiffness are considered as the multiple objective function. The formulation has inequality constraints imposed on the fatigue limit, the natural frequencies, and the manufacturing conditions. Geometric parameters of bellows are the design variables. The recursive quadratic programming algorithm is selected to solve the problem. The results are compared to existing bellows, and the characteristics of bellows is investigated through optimal design process. The optimized shape of bellows is expected to give quite a good guideline to practical design.

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