Shape Optimization Approach by Traction Method to Inverse Free Boundary Problems
The importance of the optimal shape design has been increasing in the present industrial design due to the request to make their production more efficient.
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1 Introduction The importance of the optimal shape design has been increasing in the present industrial design due to the request to make their production more efficient. Actually, a number of techniques and algorithms for shape optimization have been proposed in engineering and industry [1, 11, 12]. Mathematical analysis is playing an essential role there, mainly in the following three processes: (1) choice of a cost functional, (2) derivation of the shape derivative of the cost functional, (3) shape deformation to reduce the cost functional. Concerning the choice of the cost functional, mathematical study for the existence and uniqueness of the minimizer is important. For the second process which is often called the shape sensitivity analysis, there have been many mathematical studies. However, not so many mathematical studies have dealt with the shape deformation process. In this paper, we focus on the traction method which was proposed by one of the authors S. Shioda · A.U. Maharani · M. Kimura (B) Kanazawa University, Kanazawa, Japan e-mail: [email protected] S. Shioda e-mail: [email protected] A.U. Maharani Bandung Institute of Technology, Bandung, Indonesia e-mail: [email protected] H. Azegami Nagoya University, Nagoya, Japan e-mail: [email protected] K. Ohtsuka Hiroshima Kokusai Gakuin University, Hiroshima, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 H. Itou et al. (eds.), Mathematical Analysis of Continuum Mechanics and Industrial Applications, Mathematics for Industry 26, DOI 10.1007/978-981-10-2633-1_8
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for stable shape deformation (see detail in [2]). The traction method and its variants are widely used in the optimal shape design in engineering and industry. We consider an optimal shape design approach to a free boundary problem (1) arising from inverse potential problems, where the cost functional is defined by a boundary integral. We derive its shape derivative and give a weak formulation of the traction method. The geometrical difficulty caused by the presence of a curvature term can be dissolved by introducing the weak form. For a special case of the free boundary problem (1) in plane which will be considered in Sect. 5, all the solutions are classified and described by using the conformal mapping method [4, 8]. We give several numerical examples and compare them with the exact solutions. Often in industrial examples, because of many constraints and complicated geometry, precise property or even existence and uniqueness of the exact solution is not clear in most cases. In this sense, our target problem (1) is appropriate for the mathematical study of the optimal shape design problem. The outline of this paper is as follows. In Sect. 2, the free boundary problem and our optimal shape design approach to it are described. We derive the shape derivative of the cost functional in Sect. 3, and give a weak formulation of the traction method in Sect. 4. In Sect. 5, we introduce some numerical examples and s
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