• International Journal of Control, Automation, and Systems
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• v.6 no.3
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• pp.364-377
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• 2008

Due to the difficulty in solving the Hamilton-Jacobi-Isaacs equation, the nonlinear optimal control approach is not very practical in general. To overcome this problem, Ezal et al. (2000) first solved a linear optimal control problem for the linearized model of a nonlinear system given in the strict-feedback form. Then, using the backstepping procedure, a nonlinear feedback controller was designed where the linear part is same as the linear feedback obtained from the linear optimal control design. However, their construction is based on the cancellation of the high order nonlinearity, which limits the application to the smooth ($C^{\infty}$) vector fields. In this paper, we develop an alternative method for backstepping procedure, so that the vector field can be just $C^1$, which allows this approach to be applicable to much larger class of nonlinear systems.

• Journal of the korean Society of Automotive Engineers
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• v.8 no.2
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• pp.51-64
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• 1986

There has been considerable interest over the last twenty years in the subject of the elastic properties of the cord-rubber laminate. This has been due to the rather intensive study of the composites materials characteristics brought about by the increased use of rigid composites materials characteristics brought about by the increased use of rigid composites in many structural applications. The object of this study is to obtain the natural frequencies and modes of the simply supported cord-rubber laminate plates prior to the study on the analysis of the dynamic properties of the pneumatic tire. To obtain these natural frequencies and modes, the 12 degrees of freedom orthotropic rectangular plate finite elements are developed. By using classical lamination theory, the stress-strain relations are represented. The governing equation for the finite element is derived by energy method. To find the natural frequencies and modes, he eigenvalues and corresponding eigenvectors are computed by the well known Jacobi power method. In order to verify the capability of this present finite element, the results of the specially orthotropic plate and the angle-ply laminate plate are compared with the analytical solution. The analytical and numberical results are in good agreement. The following problems of the simply supported plate are analyzed by the present finite element. a) the natural frequencies and mode shapes of the cord-rubber laminate plate for various aspect ratio. b) The natural frequencies and mode shapes of the orthotropic plate with the rectangular hole in its center.

• Journal of the Korean Society of Safety
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• v.7 no.2
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• pp.30-41
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• 1992

Analysis of Dynamic Characterisocs of Rectangular Plate by Finite Element Method. Dynamic characteristics of a rectangular plate with opening in it is studied by finite element method. To investigate these characteristics 12 degrees of freedom membrane finite element in used. The rectangular membrane finite elements are defined by specifying geometry, internal displacement functions and strain-displacement relations. Then, the governing equation for the finite element is derived by energy method. To derive the mass matrix and stiffness matrix of the element, expressions for strain and kineic energy in terms of the node displacement are generated. In constructing the overall structure matrix, the matrix of each elements are superposed and partitioned by applying the given boundary condition to obtain a nonslngular matrix. To find the natural freguencies and viration modes, the eigen values and the corresponding eigen vectors are computed by the computer using well known Jacobi power method. In order to verify the capability of the membrane finite element, a flat rectangular plate is analyzed first, and the result is compared with well known analytical results to show the good agreement. A rectangular plate with opening in It is analyzed with the same finite element. The results are presented in this paper. Unfortunately, the literature study could not provide with some results to compare, but the results reveal that the output of this research is phlslcally reasonable. And the results of this research are useful not only in practice but also for the future experimental research in comparison purpose.

• Journal of Electrical Engineering and Technology
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• v.13 no.4
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• pp.1740-1751
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• 2018

This paper proposes a novel fault tolerant tracking control (FTTC) scheme for a class of nonlinear systems with actuator failures based on the policy iteration (PI) algorithm and the adaptive fault observer. The estimated actuator failure from an adaptive fault observer is utilized to construct an improved performance index function that reflects the failure, regulation and control simultaneously. With the help of the proper performance index function, the FTTC problem can be transformed into an optimal control problem. The fault tolerant tracking controller is composed of the desired controller and the approximated optimal feedback one. The desired controller is developed to maintain the desired tracking performance at the steady-state, and the approximated optimal feedback controller is designed to stabilize the tracking error dynamics in an optimal manner. By establishing a critic neural network, the PI algorithm is utilized to solve the Hamilton-Jacobi-Bellman equation, and then the approximated optimal feedback controller can be derived. Based on Lyapunov technique, the uniform ultimate boundedness of the closed-loop system is proven. The proposed FTTC scheme is applied to reconfigurable manipulators with two degree of freedoms in order to test the effectiveness via numerical simulation.

• Steel and Composite Structures
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• v.47 no.5
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• pp.569-585
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• 2023

This paper proposes an efficient approach for the structural topology optimization of bi-directional functionally graded structures by incorporating popular radial basis functions (RBFs) into an implicit level set (ILS) method. Compared to traditional element density-based methods, a level set (LS) description of material boundaries produces a smoother boundary description of the design. The paper develops RBF implicit modeling with multiquadric (MQ) splines, thin-plate spline (TPS), exponential spline (ES), and Gaussians (GS) to define the ILS function with high accuracy and smoothness. The optimization problem is formulated by considering RBF-based nodal densities as design variables and minimizing the compliance objective function. A LS-RBF optimization method is proposed to transform a Hamilton-Jacobi partial differential equation (PDE) into a system of coupled non-linear ordinary differential equations (ODEs) over the entire design domain using a collocation formulation of the method of lines design variables. The paper presents detailed mathematical expressions for BiDFG beams topology optimization with two different material models: continuum functionally graded (CFG) and mechanical functionally graded (MFG). Several numerical examples are presented to verify the method's efficiency, reliability, and success in accuracy, convergence speed, and insensitivity to initial designs in the topology optimization of two-dimensional (2D) structures. Overall, the paper presents a novel and efficient approach to topology optimization that can handle bi-directional functionally graded structures with complex geometries.

• Structural Engineering and Mechanics
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• v.78 no.5
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• pp.585-597
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• 2021

This paper presents a comparative study of analytical method, finite element method (FEM) and Multilayer Perceptron (MLP) for analysis of a contact problem. The problem consists of a functionally graded (FG) layer resting on a half plane and pressed with distributed load from the top. Firstly, analytical solution of the problem is obtained by using theory of elasticity and integral transform techniques. The problem is reduced a system of integral equation in which the contact pressure are unknown functions. The numerical solution of the integral equation was carried out with Gauss-Jacobi integration formulation. Secondly, finite element model of the problem is constituted using ANSYS software and the two-dimensional analysis of the problem is carried out. The results show that contact areas and the contact stresses obtained from FEM provide boundary conditions of the problem as well as analytical results. Thirdly, the contact problem has been extended based on the MLP. The MLP with three-layer was used to calculate the contact distances. Material properties and loading states were created by giving examples of different values were used at the training and test stages of MLP. Program code was rewritten in C++. As a result, average deviation values such as 0.375 and 1.465 was obtained for FEM and MLP respectively. The contact areas and contact stresses obtained from FEM and MLP are very close to results obtained from analytical method. Finally, this study provides evidence that there is a good agreement between three methods and the stiffness parameters has an important effect on the contact stresses and contact areas.

• Journal of the Korea Society of Computer and Information
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• v.17 no.9
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• pp.9-16
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• 2012

In this paper, we propose the two dimensional variable-length vector storage format which can be used for efficient storage of sparse matrix in the FEM (finite element method). The proposed storage format is the method storing only actual needed non-zero values of each row on upper triangular matrix with the total rows N, by using two dimensional variable-length vector instead of $N{\times}N$ large sparse matrix of entire equation of finite elements. This method only needs storage spaces of the number of minimum 1 to maximum 5 in 2D grid structure and the number of minimum 1 to maximum 14 in 3D grid structure of analysis target. The number doesn't excess two times although involving index number. From the experimental result, we can find out that the proposed storage format can reduce the memory space more effectively, as the total number of nodes increases, than the existing skyline storage format storing maximum column height.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.21 no.3
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• pp.239-245
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• 2008

A level set based topological shape optimization using extended B-spline basis functions is developed for steady-state heat conduction problems. The only inside of complicated domain identified by the level set functions is taken into account in computation, so we can remove the effects of domain outside parts in heat conduction problem. The solution of Hamilton-Jacobi equation leads to an optimal shape according to the normal velocity field determined from the sensitivity analysis, minimizing a thermal compliance while satisfying a volume constraint. To obtain exact shape sensitivity, the precise normal and curvature of geometry need to be determined using the level set and B-spline basis functions. Using topological derivative concept, the nucleation of holes for topological changes can be made whenever and wherever necessary during the optimization.

• Structural Engineering and Mechanics
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• v.19 no.4
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• pp.425-440
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• 2005

In this paper, the behavior of a crack between two half-planes of functionally graded materials subjected to arbitrary tractions is resolved using a somewhat different approach, named the Schmidt method. To make the analysis tractable, it is assumed that the Poisson's ratios of the mediums are constants and the shear modulus vary exponentially with coordinate parallel to the crack. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. This process is quite different from those adopted in previous works. Numerical examples are provided to show the effect of the crack length and the parameters describing the functionally graded materials upon the stress intensity factor of the crack. It can be shown that the results of the present paper are the same as ones of the same problem that was solved by the singular integral equation method. As a special case, when the material properties are not continuous through the crack line, an approximate solution of the interface crack problem is also given under the assumption that the effect of the crack surface interference very near the crack tips is negligible. It is found that the stress singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.27 no.1
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• pp.9-16
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• 2014

Using a level set method and topological derivatives, a topological shape optimization method that is independent of an initial design is developed for linearly elastic structures. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in the level set function, which facilitates to handle complicated topological shape changes. The "Hamilton-Jacobi(H-J)" equation and computationally robust numerical technique of "up-wind scheme" lead the initial implicit boundary to an optimal one according to the normal velocity field while minimizing the objective function of compliance and satisfying the constraint of allowable volume. Based on the asymptotic regularization concept, the topological derivative is considered as the limit of shape derivative as the radius of hole approaches to zero. The required velocity field to update the H-J equation is determined from the descent direction of Lagrangian derived from optimality conditions. It turns out that the initial holes are not required to get the optimal result since the developed method can create holes whenever and wherever necessary using indicators obtained from the topological derivatives. It is demonstrated that the proper choice of control parameters for nucleation is crucial for efficient optimization process.