Journal : Acta Mechanica et Automatica
Article : On the convergence of domain decomposition algorithm for the body with thin inclusion

Authors :
Azimi, A.
Department of Chemical Engineering, College of Chemical Engineering, Islamic Azad University, Mahshahr Branch, Mahshahr, Farhangsara Street, Iran,
Azimi, M.
Faculty of New Sciences and Technologies, Department of Aerospace, University of Tehran, Tehran, North Kargar, Amirabad, Iran,
Javanfar, A.
Faculty of Mechanical Engineering, Babol University of Technology, Shariati Street, Babol, Iran,
Trąbka, A.
*Faculty of Mechanical Engineering and Computer Science, Department of Engineering Fundamentals, University of Bielsko-Biala, ul. Willowa 2, 43-309 Bielsko-Biała, Poland, atrabka@ath.bielsko.pl,
Szpica, D.
Faculty of Mechanical Engineering, Department of Mechanical Engineering, Bialystok University of Technology, ul. Wiejska 45C, 15-351 Bialystok, Poland, d.szpica@pb.edu.pl,
Czaban, J.
Faculty of Mechanical Engineering, Department of Mechanical Engineering, Bialystok University of Technology, ul. Wiejska 45C, 15-351 Bialystok, Poland, j.czaban@pb.edu.pl,
Banaszuk, P.
Faculty of Civil and Environmental Engineering, Department of Environmental Protection and Management, Bialystok University of Technology, ul. Wiejska 45E, 15-351 Bialystok, Poland, p.banaszuk@pb.edu.pl,
Weresa, E.
Faculty of Mechanical Engineering, Department of Mechanical Engineering, Bialystok University of Technology, ul. Wiejska 45C, 15-351 Bialystok, Poland, e.weresa@pb.edu.pl,
Tomaszewski, J.
Faculty of Mechanical Engineering and Computer Science, Department Of Mechanical Engineering Fundamentals, University of Bielsko-Biala, ul. Willowa 2, 43-300 Bielsko-Biala, Poland, jtomaszewski@ath.bielsko.pl,
Rysiński, J.
Faculty of Mechanical Engineering and Computer Science, Department Of Mechanical Engineering Fundamentals, University of Bielsko-Biala, ul. Willowa 2, 43-300 Bielsko-Biala, Poland, jrysinski@ath.bielsko.pl,
Fedorynenko, D.
Mechanical Engineering Department, Chernihiv National University of Technology, 95 Shevchenka Str., 14027 Chernihiv, Ukraine, fdy@mail.ru,
Boyko, S.
Mechanical Engineering Department, Chernihiv National University of Technology, 95 Shevchenka Str., 14027 Chernihiv, Ukraine, svboyko.cstu@gmail.com,
Sapon, S.
Mechanical Engineering Department, Chernihiv National University of Technology, 95 Shevchenka Str., 14027 Chernihiv, Ukraine, s.sapon@gmail.com,
Styahar, A.
Faculty of Applied Mathematics and Informatics, Department of Applied Mathematics, Ivan Franko Lviv National University, Universytetska,1, 79000, Lviv, Ukraine, astyahar@gmail.com,
Savula, Y.
Faculty of Applied Mathematics and Informatics, Department of Applied Mathematics, Ivan Franko Lviv National University, Universytetska,1, 79000, Lviv, Ukraine, savula@franko.lviv.ua,
Abstract : We consider a coupled 3D model that involves computation of the stress-strain state for the body with thin inclusion. For the description of the stress-strain state of the main part, the linear elasticity theory is used. The inclusion is modelled using Timoshenko theory for shells. Therefore, the dimension of the problem inside the inclusion is decreased by one. For the numerical solution of this problem we propose an iterative domain decomposition algorithm (Dirichlet-Neumann scheme). This approach allows us to decouple problems in both parts and preserve the structure of the corresponding matrices. We investigate the convergence of the aforementioned algorithm and prove that the problem is well-posed.

Keywords : teoria sprężystości, teoria Tymoszenki, operator Steklova-Poincare'a, elasticity theory, Timoshenko shell theory, Steklov-Poincare operator, domain decomposition,
Publishing house : Oficyna Wydawnicza Politechniki Białostockiej
Publication date : 2015
Number : Vol. 9, no. 1
Page : 27 – 32

Bibliography
: 1. Dyyak I., Savula Ya., Styahar A. (2012), Numerical investigation of a plain strain state for a body with thin cover using domain decomposition, Journal of Numerical and Applied Mathematics, 3 (109), 23–33.
2. Dyyak I., Savula Ya. (1997), D-Adaptive mathematical model of solid body with thin coating, Mathematical Studies, 7 (1), 103–109.
3. Hsiao G.C., Wendland W.L. (2008), Boundary integral equations, Springer.
4. Nazarov S. (2005), Asymptotic analysis and modeling of a jointing of a massive body with thin rods, Journal of Mathematical Sciences, 127 (5), 2192-2262.
5. Niemi A.H., Babuska I., Pitkaranta J., Demkowicz L. (2010), Finite element analysis of the Girkmann problem using the modern hpversion and the classical h-version, ICES Report, 10-47.
6. Pelekh B. (1978), Generalized shell theory, Lviv (in Russian).
7. Quarteroni A., Valli. A. (1999), Domain decomposition methods for partial differential equations, Oxford.
8. Savula Ya., Mang H., Dyyak I., Pauk N. (2000), Coupled boundary and finite element analysis of a special class of two-dimensional problems of the theory of elasticity, Computers and Structures, 75 (2), 157-165.
9. Sulym H. (2007), Bases of mathematical theory of thermoelastic eqiulibrium of deformable solids with thin inclusions, Research and Publishing Center of Shevchenko Scientific Society, Lviv (in Ukrainian).
10. Vynnytska L., Savula Ya. (2008), The stress-strain state of elastic body with thin inclusion, Ph.-Math. Modeling and Information Techonogies, 7, 21-29 (in Ukrainian).
11. Vynnytska L., Savula Ya. (2012), Mathematical modeling and numerical analysis of elastic body with thin inclusion, Computational Mechanics, 50 (5), 533-542.
DOI :
Qute : Azimi, A. ,Azimi, M. ,Javanfar, A. ,Trąbka, A. ,Szpica, D. ,Czaban, J. ,Banaszuk, P. ,Weresa, E. ,Tomaszewski, J. ,Rysiński, J. ,Fedorynenko, D. ,Boyko, S. ,Sapon, S. ,Styahar, A. ,Savula, Y. ,Savula, Y. , On the convergence of domain decomposition algorithm for the body with thin inclusion. Acta Mechanica et Automatica Vol. 9, no. 1/2015
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