A genetic algorithm of optimal design of beam at restricted sagging

There are many recent advances in mesh deformation methods for computational fluid dynamics simulation in deforming geometries. We present a method of constructing dynamic mesh around deforming objects by solving the bi-elliptic equation, an extension of the biharmonic equation. We show that introducing a stiffness coefficient fielda(x) in the bi-elliptic equation can enable mesh deformation for very large boundary movements. An indicator of the mesh quality is constructed as an objective function of a numerical optimization procedure to find the best stiffness coefficient field a(x). The optimization is efficiently solved using steepest descent along adjoint based, integrated Sobolev gradients. A multi-scenario optimization procedure is performed to calculate the optimal stiffness coefficient fielda � (x) for a priori unpredictable boundary movements. Simulation of fluid structure interaction problems requires solving the governing equation of the unsteady flow inside a deforming physical domain, whose geometry is determined by the deformation of the structure. In numerical methods for solving such problems, Arbitrary Lagrangian Eulerian (ALE) 1 method is one of the most popular types of approaches. It formulates the governing equation in a coordinate system that moves with the boundary, and discretizes the deforming domain with a deformable, body conforming mesh. ALE methods are capable of using anisotropic mesh to efficiently resolve boundary layers. However, these methods requires an algorithm to deform the mesh so that it conforms to the boundary movement while maintaining good mesh quality. There has been many recent advances in algorithms for computing mesh deformation in the context of computational fluid dynamics simulations in moving domains. Two of the most popular categories of such algorithms are interpolation based methods and structure analogy methods. Interpolation based methods determines the position or displacement of mesh points in the interior of the domain by interpolating the position or displacement of the mesh points on the boundary of the domain. An example of interpolation based methods is the radial basis function interpolation method 23 . Structural analogy methods include methods based on structural models of the discrete computational mesh and methods based on modeling the computational domain as a continuous elastic media. Recent developments in discrete mesh structural model includes the torsional spring method 45 . Recent developments in the continuous elasticity model include the biharmonic equation method 6 and the quasi-elliptic transformation method 7 . In this paper, we present a mesh deformation method based on continuous elasticity model. Our contribution is introducing a variable stiffness coefficient in t he biharmonic elastic model, and developing a method to efficiently obtain the optimal stiffness coefficient d istribution. We show that the variable stiffness coefficient algorithm can produce a good deformed mesh, whilethe original scheme with constant coefficient produces unacceptable mesh. To obtain a suitable stiffness coefficient, we define a simple global mesh

Under addressing global competition, manufacturing companies strive to produce better and cheaper products more quickly. For a complex production system, the design problem is intrinsically a daunting optimization task often involving multiple disciplines, nonlinear mathematical model, and computation-intensive processes during manufacturing process. Here is a reason to develop a high performance algorithm for finding an optimal solution to the engineering design and/or optimization problems. In this paper, a hybrid metaheuristic approach is proposed for solving engineering optimization problems. A genetic algorithm (GA), particle swarm optimization (PSO), and teaching and learning-based optimization (TLBO), called the GA-PSO-TLBO approach, is used and demonstrated for the proposed hybrid metaheuristic approach. Since each approach has its strengths and weaknesses, the GA-PSO-TLBO approach provides an optimal strategy that maintains the strengths as well as mitigates the weaknesses, as needed. The performance of the GA-PSO-TLBO approach is compared with those of conventional approaches such as single metaheuristic approaches (GA, PSO and TLBO) and hybrid metaheuristic approaches (GA-PSO and GA-TLBO) using various types of engineering optimization problems. An additional analysis for reinforcing the performance of the GA-PSO-TLBO approach was also carried out. Experimental results proved that the GA-PSO-TLBO approach outperforms conventional competing approaches and demonstrates high flexibility and efficiency.

Fundamental advantage of using transversely stiffened plate girders (TSPG) rather than plane web plate girders is in achieving increased strength and stiffness of the web panel. As these girders are made of steel plates, they can be custom designed to suit consumer's requirements. Weight minimization of TSPG by satisfying strength and serviceability conditions, further enables in achieving higher strength to weight ratios. The utilisation of these girders in the building sector in India is growing, and optimising these girders can lead to more cost-effective and structurally sound designs. No significant research has been carried out in the field of optimization of plate girders based on specifications according to IS 800 (2007): General Construction in Steel - Code of Practice (Bureau of Indian Standards). Therefore, in the present study, the main goal is to develop a formulation for optimum weight design of TSPG using genetic algorithm (GA) with weight minimization as objective function. The breadth of flange (bf), thickness of flange (t_f), depth of web (d), thickness of web (t_w), breadth of stiffener (b_s), thickness of stiffener (t_s) and spacing between stiffeners(c) are adopted as the design variables. The optimum girder weight is compared with weight obtained through conventional design using IS 800 (2007). The results demonstrate that conventional design overestimates the girder weight by about 30% on an average and therefore the developed formulation helps in attaining economical girder weight. The viability of the GA formulation is verified by comparing the optimized results with existing experimental and numerical data from literature. Parametric studies are also carried out to understand the behaviour of design variables. Results from these studies show that deeper web reduces flange thickness but requires thicker web in order to ensure ample shear capacity and to prevent shear buckling.

Displacement-amplifying mechanisms can be systematically designed using topology optimization. Due to the special need of large displacement amplification for piezoelectric stacked actuators, the objective function should be formulated properly. Among output stroke, magnification factor and work ratio, magnification factor better describes the design goal of displacement amplifier and is thus adopted in this work. To depict the dynamic operation of displacement amplifier, undamped harmonic response is used in the formulation. The design problem is thus posed as a material distribution problem, which maximize the dynamic magnification factor by varying the thickness of 2-D domain. Plane stress solid is assumed for the design domain. Stiffness of the actuator and the workpiece is included in the analysis. The design problem is solved by Method of Moving Asymptotes or MMA. To show the viability of this design methodology, two examples of magnification mechanism for printer head driven at different excitation frequency are presented.

Engineers design systems by searching through the large number of possible solutions to discover the best specific solution. The search process is often time consuming and expensive. But by exploiting the natural processes that biological systems use to evolve and adapt, design engineers can often quickly solve otherwise difficult design problems that resist solution by traditional optimization methods. This paper explains the basic technique of the genetic algorithm and shows how design engineers can use a genetic algorithm to solve real design engineering problems. This paper focuses on explaining how genetic algorithms work. A brief example at the end demonstrates how the practicing engineer can use this powerful technique to solve real world problems in engineering design. The example of a structural design problem uses a genetic algorithm to minimize the weight of a pin jointed frame, but the genetic algorithm can be applied to almost any type of design problem.

This paper proposes a topology optimization method to optimize the material distribution of electrical machines using Genetic Algorithm (GA). In the method, a cluster of material is taken into account to form a group of iron. Moreover, the GA is iterated with the newly increased length of genes. The proposed method is applied to the topology optimization of the stator teeth in a Brushless DC motor.

A model-based parameter estimation (MBPE) technique is introduced in this paper for efficiently interpolating periodic moment method (PMM) impedance matrices over a wide frequency band. In the model, only the Floquet harmonics that strongly affect the frequency band of interest are employed to approximate the matrix elements, while the contributions from all other higher-order harmonics are compactly represented by two additional terms. The derivation of the model is physics-based, and the objective is to find the coefficients of the terms in the model by utilizing the values of the impedance matrix elements calculated via PMM at only a few frequency points. The number and position of these fitting points can be pre-determined from the cutoff frequencies of the Floquet harmonics, which allows the MBPE interpolation process in this case to be completely automated. In other words, the number and position of the sampling points are only dictated by the periodicity of the frequency selective surface (FSS) structure and the frequency range of interest. Unlike many of the other scattering parameter-based techniques, the shape and the resonances in the response of the FSS do not have any impact on the construction of the interpolation model. This makes it particularly useful in genetic algorithm (GA) aided FSS design, since for a fixed periodicity and frequency range the MBPE interpolation is independent of the scattering response of candidate FSS designs. Several examples of the new PMM-MBPE approach are presented including one in which it is used to considerably speed up the GA-based design process for a reconfigurable FSS.

Multi-objective optimization of functionally graded materials with temperature-dependent material properties

Here is solved the optimization problem for the longitudinal depth distribution in the beam with a limitation on the maximum value of deflection. A review of the references is done, and it is shown that the known solutions are either erroneous, because they are based on false hypotheses, or have a narrow field of application, limited only to symmetrical constructions for which the point of the maximum deflection is known a priori. The paper considers a beam of the rectangular cross-section of constant width. The beam is assumed to be statically determinate, and the load is arbitrary and asymmetric and multidirectional as well. The points (or point) of the beam maximum deflections are unknown in advance and would be determined in the problem-solution procedure. A linear problem is considered. The optimization criterion is the mass of the beam. To find the deflections of the beam, i.e. to solve the differential equation of a variable cross-section beam bending the finite difference method is used. The design problem is reduced to the required beam depths obtaining in the system of nodal points. In this case, the desired solution must satisfy the restriction system for the nodal points shift and the sign of variables as well. Since the restrictions of the shift of each node are considered separately and independently, so the proposed method allows flexible control of the beam shift restrictions. Using the change of variables proposed in the paper, the problem to be solved is reduced to a nonlinear programming problem where the criterion function is separable and restrictions are linear functions. Using linearization, this problem can be reduced to the linear programming problem relatively to new variables. The model problem is solved, and it is shown that the proposed algorithm efficiently allows us to solve the problems of the beam optimal design with the restrictions of the maximally allowed deflection. The proposed approach can be spread for the strength limitations, for beams of variable width, I-beam cross-section, etc.

We present a numerical calculation of the frequencies of natural vibrations of beams with a cross section varying linearly in height under different conditions of fixing their ends. The differential equation is derived using the finite difference method. The eigenvalues of the equation and the frequency parameters are determined. The results are obtained in several various steps of nodes of the approximation, with their refinement being performed using the extrapolation method. The accuracy of the results is confirmed by comparison with the analytical solution for the case of a beam with a uniform cross section.

<p>Polymer dispersed liquid crystal (PDLC) films play an important role in liquid crystal display technology. PDLC films are used in devices such as switchable windows, complex billboards, and flat panel televisions. PDLC films consist of nematic bipolar droplets dispersed randomly in a polymer matrix. In this work, the effect of droplet shape, physical properties, and external field strength is examined on the performance of PDLC films. Finite element method (FEM) and finite difference method (FDM) are used to model a single droplet with external field applied parallel to the droplet axis of symmetry. Results of previous simulations with FEM are reproduced and compared with those with FDM. Genetic algorithm is employed to determine the optimum aspect ratio, and elastic constant ratio for the droplet, and external field strength with respect to minimum energy, response time, and relaxation time.</p>

Prediction of spiral groove dry gas seal performance and intelligent optimization of structural parameters

This work proposes an inverse analysis methodology for estimating soil physical-hydraulic parameters, based on the coupling between a Genetic Algorithm and the Finite Difference Method for solving the Richards equation. The Richards equation models water flow in unsaturated porous media, being in this work discretized by the Finite Difference Method with an implicit scheme (Picard), thus ensuring stability and consistency in the presence of strong nonlinearities. The inverse problem is formulated as an optimization problem, in which hydraulic parameters are adjusted to maximize the coefficient of determination between simulated and observed moisture profiles. Numerical simulations were performed for a one-dimensional vertical infiltration problem in a 40 cm deep soil column, using the Haverkamp model to describe the soil constitutive relations. The Genetic Algorithm is configured with a fixed population, tournament selection operators, arithmetic crossover, mutation and elitism. Different scenarios are investigated, considering the simultaneous identification of four and five parameters. Results indicate efficient convergence of the inverse process, with determination coefficients greater than R^2 > 0.98 in most tests, demonstrating the robustness of the proposed coupling.

Due to the development of urban areas, underground structures are increasingly needed. Therefore, not only it is likely to have two tunnels located close to each other, but also it happens frequently.Because of the complexity of realistic problems, closed form analytical solutions are not accessible for them. However, latest advances in computational techniques have made numerical approaches more practical for realistic problems. Boundary element method (BEM) is one of such approaches. This technique formulates the problem in terms of boundary values. The main advantage of BEM, especially in comparison with finite element and finite difference methods, is that the discretization is only applied to the boundary, thus reducing the volume of modeling and the number of unknown variables. Moreover, the radiation condition at infinity is exactly satisfied in this method which is very striking for wave propagation problems. First the formulation of boundary element method in the time domain is presented and then a parametric study on the dynamic response of unlined circular tunnels subjected to internal explosion is done. Depth of tunnels, vertical and horizontal distance between two tunnels, diameter of tunnels, and other parameters related to geometry are important factors which can affect displacements and stresses of tunnels under explosive loading. In the present study, effect of two parameters has been investigated: changing the depth of the tunnel that is subjected to internal explosion (case A) and changing the horizontal distance between two shallow tunnels subjected to explosion (case B). Radial displacements and circumferential stresses are the results obtained. Case A- Tunnel depth: The tunnel behavior in depth of H=50R is alike to behavior of same tunnel in infinite depth. Also, radial displacement in upper point of tunnel in depth of H=2R is about 2.7 times of the same tunnel in infinite depth. Circumferential stress in points placed on horizontal diameter of tunnel in depth of H=2R is about 1.8 times of the same tunnel in infinite depth. Case B- Tunnels distance: Two tunnels with horizontal distance of greater than L=50R don’t affect each other. When horizontal distance parameter of two tunnels is L=2.5R then the maximum normalized displacement of the side point of the loaded tunnel, is approximately 2.15 times of the same tunnels with distance of infinite. Also, when horizontal distance of two tunnels is L=2.5R, circumferential stress in lower point of the loaded tunnel, which is normalized with static stress, is approximately 1.2 times of the same tunnels with distance of infinite. Normalized results calculated here display the importance of the geometric parameters on the behavior of unlined tunnels subjected to internal explosion and provide suitable criteria for displacement design of shallow tunnels that are subjected to internal explosion. The accuracy of the present study for the computation of radial displacements and circumferential stresses is verified by solving several problems.

Application of genetic algorithm for unknown parameter estimations in cylindrical fin