On The Convergence of Domain Decomposition Algorithm for The Body with Thin Inclusion

Styahar, Andriy; Savula, Yarema

doi:10.1515/ama-2015-0006

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.

References

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.
Search in Google Scholar
2. Dyyak I., Savula Ya. (1997), D-Adaptive mathematical model of solid body with thin coating, Mathematical Studies, 7 (1), 103-109.
Search in Google Scholar
3. Hsiao G.C., Wendland W.L. (2008), Boundary integral equations, Springer.10.1007/978-3-540-68545-6
Search in Google Scholar
4. Nazarov S. (2005), Asymptotic analysis and modeling of a jointing of a massive body with thin rods, Journal of Mathematical Sciences, 127 (5), 2192-2262.10.1007/s10958-005-0177-0
Search in Google Scholar
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.
Search in Google Scholar
6. Pelekh B. (1978), Generalized shell theory, Lviv (in Russian).
Search in Google Scholar
7. Quarteroni A., Valli. A. (1999), Domain decomposition methods for partial differential equations, Oxford.10.1007/978-94-011-4647-0_11
Search in Google Scholar
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), 157-165.10.1016/S0045-7949(99)00091-7
Search in Google Scholar
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).
Search in Google Scholar
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).
Search in Google Scholar
11. Vynnytska L., Savula Ya. (2012), Mathematical modeling and numerical analysis of elastic body with thin inclusion, Computational Mechanics, 50 (5), 533-542.10.1007/s00466-012-0687-9
Search in Google Scholar

On The Convergence of Domain Decomposition Algorithm for The Body with Thin Inclusion

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