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In this paper nonlinear analysis of structures are performed considering material and geometric nonlinearity using force method and energy concepts. For this purpose, the complementary energy of the structure is minimized using ant colony algorithms. Considering the energy term next to the weight of the structure, optimal design of structures is performed. The first part of this paper contains the formulation of the complementary energy of truss and frame structures for the purpose of linear analysis. In the second part material and geometric nonlinearity of structure is considered using Ramberg-Osgood relationships. In the last part optimal simultaneous analysis and design of structure is studied. In each part, the efficiency of the methods is illustrated by means simple examples.

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The Isogeometric Analysis (IA) method is applied for structural topology optimization instead of the finite element method. For this purpose, the material density is considered as a continuous function throughout the design domain and approximated by the Non-Uniform Rational B-Spline (NURBS) basis functions. The coordinates of control points which are also used for constructing the density function, are considered as design variables of the optimization problem. In order to change the design variables towards optimum, the Method of Moving Asymptotes (MMA) is used. To alleviate the formation of layouts with porous media, the density function is penalized during the optimization process. A few examples are presented to demonstrate the performance of the method.

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A new method for structural topology optimization is introduced which employs the Isogeometric Analysis (IA) method. In this approach, an implicit function is constructed over the whole domain by Non-Uniform Rational B-Spline (NURBS) basis functions which are also used for creating the geometry and the surface of solution of the elasticity problem. Inspiration of the level set method zero level of the function describes the boundary of the structure. An optimality criterion is derived to improve the implicit function towards the optimum boundaries. The last section of this paper is devoted to some numerical examples in order to demonstrate the performance of the method as well as the concluding remarks.

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Keeping the eigenfrequencies of a structure away from the external excitation frequencies is one of the main goals in the design of vibrating structures in order to avoid risk of resonance. This paper is devoted to the topological design of freely vibrating continuum structures with the aim of maximizing the fundamental eigenfrequency. Since in the process of topology optimization some areas of domain can potentially be removed, it is quite possible to encounter the problem of localized modes. Hence, the modified Solid Isotropic Material with Penalization (SIMP) model is here used to avoid artificial modes in low density areas. As during the optimization process, the first natural frequency increases, it may become close to the second natural frequency. Due to lack of the usual differentiability of the multiple eigenfrequencies, their sensitivity are calculated by the mathematical perturbation analysis. The optimization problem is formulated by a variable bound formulation and it is solved by the Method of Moving Asymptotes (MMA). Two dimensional plane elasticity problems with different sets of boundary conditions and attachment of a concentrated nonstructural mass are considered. Numerical results show the validity and supremacy of this approach.