J S I A M Vol.3 (2011) pp.1-100 The Japan Society for Industrial and Applied Mathematics Vol.3 (2011) pp.1-100 The Japan Society for Industrial and Applied Mathematics Editorial Board Chief Editor Yoshimasa Nakamura (Kyoto University) Vice-Chief Editor Kazuo Kishimoto (Tsukuba University) Associate Editors Reiji Suda (University of Tokyo) Satoshi Tsujimoto (Kyoto University) Masashi Iwasaki (Kyoto Prefectural University) Norikazu Saito (University of Tokyo) Koh-ichi Nagao (Kanto Gakuin University) Koichi Kato (Japan Institute for Pacific Studies) Nagai Atsushi (Nihon University) Takeshi Mandai (Osaka Electro-Communication University) Ryuichi Ashino (Osaka Kyoiku University) Tamotu Kinoshita (University of Tsukuba) Yuzuru Sato (Hokkaido University) Ken Umeno (NiCT) Katsuhiro Nishinari (University of Tokyo) Tetsu Yajima (Utsunomiya University) Narimasa Sasa (Japan Atomic Energy Agency) Fumiko Sugiyama (Kyoto University) Hiroko Kitaoka (JSOL) Hitoshi Imai (University of Tokushima) Nobito Yamamoto (University of Electro-Communications) Daisuke Furihata (Osaka University) Takahiro Katagiri (The University of Tokyo) Tetsuya Sakurai (University of Tsukuba) Takayasu Matsuo (Tokyo University) Tomohiro Sogabe (Aichi Prefectural University) Yoshitaka Watanabe (Kyushu University) Katsuhisa Ozaki (Shibaura Institute of Technology) Kenta Kobayashi (Kanazawa University) Takaaki Nara (The University of Electro-Communications) Takashi Suzuki (Osaka University) Tetsuo Ichimori (Osaka Institute of Technology) Tatsuo Oyama (National Graduate Institute for Policy Studies) Hideyuki Azegami (Nagoya University) Kenji Shirota (Aichi Prefectural University) Eiji Katamine (Gifu National College of Technology) Masami Hagiya (University of Tokyo) Toru Fujiwara (Osaka University) Yasuyuki Tsukada (NTT Communication Science Laboratories) Naoyuki Ishimura (Hitotsubashi University) Jiro Akahori (Ritsumeikan University) Kiyomasa Narita (Kanagawa University) Ken Nakamula (Tokyo Metropolitan University) Miho Aoki (Shimane University) Kazuto Matsuo (Institute of Information Security) Keiko Imai (Chuo University) Ichiro Kataoka (HITACHI) Naoshi Nishimura (Kyoto University) Hiromichi Itou (Gunma University) Shin-Ichi Nakano (Gunma University) Akiyoshi Shioura (Tohoku University) Contents Regular solution to topology optimization problems of continua ・・・ 1-4 Hideyuki Azegami, Satoshi Kaizu and Kenzen Takeuchi A convergence improvement of the BSAIC preconditioner by deflation ・・・ 5-8 Ikuro Yamazaki, Hiroto Tadano, Tetsuya Sakurai and Keita Teranishi Cache optimization of a non-orthogonal joint diagonalization method ・・・ 9-12 Yusuke Hirota, Yusaku Yamamoto and Shao-Liang Zhang Quasi-minimal residual smoothing technique for the IDR(s) method ・・・ 13-16 Lei Du, Tomohiro Sogabe and Shao-Liang Zhang A new approach to find a saddle point efficiently based on the Davidson method ・・・ 17-20 Akitaka Sawamura On rounding off quotas to the nearest integers in the problem of apportionment ・・・ 21-24 Tetsuo Ichimori Traveling wave solutions to the nonlinear evolution equation for the risk preference ・・・ 25-28 Naoyuki Ishimura and Sakkakom Maneenop Approximation algorithms for a winner determination problem of single-item multi-unit ・・・ 29-32 auctions Satoshi Takahashi and Maiko Shigeno On the new family of wavelets interpolating to the Shannon wavelet ・・・ 33-36 Naohiro Fukuda and Tamotu Kinoshita Conservative finite difference schemes for the modified Camassa-Holm equation ・・・ 37-40 Yuto Miyatake, Takayasu Matsuo and Daisuke Furihata A multi-symplectic integration of the Ostrovsky equation ・・・ 41-44 Yuto Miyatake, Takaharu Yaguchi and Takayasu Matsuo Solutions of Sakaki-Kakei equations of type 1, 2, 7 and 12 ・・・ 45-48 Koichi Kondo Analysis of credit event impact with self-exciting intensity model ・・・ 49-52 Suguru Yamanaka, Masaaki Sugihara and Hidetoshi Nakagawa On the reduction attack against the algebraic surface public-key cryptosystem(ASC04) ・・・ 53-56 Satoshi Harada, Yuichi Wada, Shigenori Uchiyama and Hiro-o Tokunaga Deterministic volatility models and dynamics of option returns ・・・ 57-60 Takahiro Yamamoto and Koichi Miyazaki Stochastic estimation method of eigenvalue density for nonlinear eigenvalue problem on ・・・ 61-64 the complex plane Yasuyuki Maeda, Yasunori Futamura and Tetsuya Sakurai Computation of multipole moments from incomplete boundary data for ・・・ 65-68 Magnetoencphalography inverse problem Hiroyuki Aoshika, Takaaki Nara, Kaoru Amano and Tsunehiro Takeda An alternative implementation of the IDRstab method saving vector updates ・・・ 69-72 Kensuke Aihara, Kuniyoshi Abe and Emiko Ishiwata Error analysis of H1 gradient method for topology optimization problems of continua ・・・ 73-76 Daisuke Murai and Hideyuki Azegami Evolution of bivariate copulas in discrete processes ・・・ 77-80 Yasukazu Yoshizawa and Naoyuki Ishimura On boundedness of the condition number of the coefficient matrices appearing in ・・・ 81-84 Sinc-Nyström methods for Fredholm integral equations of the second kind Tomoaki Okayama, Takayasu Matsuo and Masaaki Sugihara A modified Calogero-Bogoyavlenskii-Schiff equation with variable coefficients and its ・・・ 85-88 non-isospectral Lax pair Tadashi Kobayashi and Kouichi Toda A parallel algorithm for incremental orthogonalization based on the compact WY ・・・ 89-92 representation Yusaku Yamamoto and Yusuke Hirota Analysis of downgrade risk in credit portfolios with self-exciting intensity model ・・・ 93-96 Suguru Yamanaka, Masaaki Sugihara and Hidetoshi Nakagawa Automatic verication of anonymity of protocols ・・・ 97-100 Hideki Sakurada JSIAM Letters Vol.3 (2011) pp.1{4 ⃝c 2011 Japan Society for Industrial and Applied Mathematics Regular solution to topology optimization problems of continua Hideyuki Azegami1, Satoshi Kaizu2 and Kenzen Takeuchi3 1 Graduate School of Information Science, Nagoya University, A4-2 (780) Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan 2 College of Science and Technology, Nihon University, 7-24-1 Narashinodai, Funabashi, Chiba 274-8501, Japan 3 Quint Corporation, 1-14-1 Fuchu-cho, Fuchu, Tokyo 183-0055, Japan E-mail azegami is.nagoya-u.ac.jp Received September 30, 2010, Accepted November 1, 2010 Abstract The present paper describes a numerical solution to topology optimization problems of do- mains in which boundary value problems of partial differential equations are defined. Density raised to a power is used instead of the characteristic function of the domain. A design vari- able is set by a function on a fixed domain which is converted to the density by a sigmoidal function. Evaluation of derivatives of cost functions with respect to the design variable appear as stationary conditions of the Lagrangians. A numerical solution is constructed by a gradient method in a design space for the design variable. Keywords calculus of variations, boundary value problem, topology optimization, density method, H1 gradient method Research Activity Group Mathematical Design 1. Introduction χΩ is substituted by a function evaluated by homoge- A problem finding the optimum layout of holes in nization theory. A numerical scheme was demonstrated domain in which boundary value problem is defined is using the finite element method [4]. called the topology optimization problem of continua [1]. Moreover, it has been found that introducing a den- ! In the present paper, the Poisson problem is considered sity ϕ : D [0; 1] and a constant α > 1, and replacing α as a boundary value problem for the simplicity. χΩ by ϕ obtains a similar result to that from the micro- One of the most natural expressions of a topology opti- structure model. This method is called the SIMP (solid mization problem uses the characteristic function of the isotropic material with penalization) method [1, 5]. The domain as a design variable. Let D be a fixed domain in meaning of the penalization is that the intermediate den- α d sity is weakened by the nonlinear function ϕ . R , d 2 f2; 3g,ΓD ⊂ @D be a fixed subboundary, define However, numerical instabilities such as checkerboard ΓN = @D n Γ¯D, and let functions f, p, and uD be fixed functions on D. Denoting the characteristic function for patterns or mesh-dependencies are observed if the pa- 1 rameters of micro-structure or the density is constructed Ω ⊆ D by χΩ 2 X = fχ 2 L (D; R) j 0 ≤ χ ≤ 1 a.e. in by a constant function in each finite element and they Dg, the normal by ν, and @ν = ν · r, we can write the topology optimization problem as follows. are varied using a gradient method [6,7]. If the design pa- rameters are approximated by continuous functions [8], Problem 1 (Topology optimization problem) it is known that a numerical instability, such as the so- 2 2 1 R For each χΩ X, let u H (D; ) satisfy called island phenomena, is observed [9]. In addition, al- − r · (χΩru) = f in D; though many numerical schemes have been proposed to overcome such numerical instabilities [10,11], regularity χΩ@ν u = p on ΓN; u = uD on ΓD: in the sense of functional analysis has not been shown. Find χΩ such that In the present paper, a regular solution which is free 0 of numerical instability is presented, where the meaning min fJ (χΩ; u) j J(χΩ; u) ≤ 0g; χΩ2X of regular is as follows. First, the admissible set of a design variable is defined. Then, a solution is regular if 0 1 m > l 2 0 × 1 where J and J = (J ;:::;J ) , J C (X H (D; any point obtained by the solution from a point in the R R ); ) are cost functions. admissible set also belongs to the admissible set. However, it has been shown that Problem 1 does not always have a solution [2]. 2. Admissible set of