The impact of the Dirichlet boundary conditions on the convergence of the discretized system of nonlinear equations for potential problems

Jan Kucwaj

doi:10.24423/cames.6

Jan Kucwaj Institute of Computer Science, Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, Kraków

Abstract

The purpose of this paper is the analysis of numerical approaches obtained by describing the Dirichlet boundary conditions on different connected components of the computational domain boundary for potential flow, provided that the domain is a rectangle. The considered problem is a potential flow around an airfoil profile. It is shown that in the case of a rectangular computational domain with two sides perpendicular to the speed direction, the potential function is constant on the connected components of these sides. This allows to state the Dirichlet conditions on the considered parts of the boundary instead of the potential jump on the slice connecting the trail edge with the external boundary. Furthermore, the adaptive remeshing method is applied to the solution of the considered problem.

Keywords

adaptation, rate of convergence, remeshing, Delaunay triangulation, finite element method, potential flow, Kutta-Joukovsky condition, Dirichlet condition,

References

[1] H. Berger, G. Warnecke, W.L. Wendland. Analysis of a FEM/BEM coupling method for transonic flow computations. Mathematics of Computation, 66(220): 1407–1440, 1997.
[2] B. Bojarski, Subsonic flow of compressible fluid, Arch. Mech., 18(4): 497–520, 1966.
[3] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam, New York-Oxford, Studies in Mathematics and its Application, 1978.
[4] L. Demkowicz, J.T. Oden, W. Rachowicz, O. Hardy. Towards a universal h-p adaptive finite element strategy. Part 1: Constrained approximation and data structure. Comp. Meth. Appl. Mech. Engrg., 77: 79–112, 1989.
[5] R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, J. Periaux. A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies: application to particulate flow. Comput. Methods Appl. Mech. Engrg., 184: 241–267, 2000.
[6] J. Kucwaj. The algorithm of adaptation by using graded meshes generator. Computer Assisted Mechanics and Engineering Sciences, 7: 615–624, 2000.
[7] J. Kucwaj. Adaptive unstructured solution to the problem of elastic-plastic hardening twist of prismatic bars. Technical Transactions. Fundamental Sciences [in Polish: Czasopismo Techniczne. Nauki Podstawowe], 2-NP: 63–79, 2014.
[8] J. Kucwaj. Numerical investigations of the covergence of a remeshing algorithm on an example of subsonic flow. Computer Assisted Mechanics and Engineering Sciences, 17: 147–160, 2010.
[9] J.F. Thompson, B.K. Soni, N.P. Weatherwill. Handbook of Grid Generation. CRC Press, Boca Raton, 1999.
[10] S.H. Lo. Finite element mesh generation and adaptive meshing. Progress in Structural Engineering and Materials, 4(4): 381–399, 2002.
[11] J.T. Oden. h-p adaptive finite element methods for compressible and incompressible flows. Computing Systems in Engineering, 1(2–4): 523–534, 1990.
[12] A. Zdunek, W. Rachowicz. hp-Adaptive CEM in practical applications, Lecture Notes in Computational Science and Engineering, 76: 339–346, 2011.
[13] A. Safjan, L. Demkowicz, D.P. Young. Compressible flow hp-adaptivity and Kutta-Joukovsky condition. The Boeing Company Project, 2006.
[14] D. Wang, O. Hassan, K. Morgan, N. Weatherill. Enhanced remeshing from STL files with applications to surface grid generation. Comm. in Num. Meth. in Engrg., 65: 734–751, 2006.
[15] O.C. Zienkiewicz, J.Z. Zhu. Adaptivity and mesh generation, Int. J. Num. Meth. Engng., 32: 783–810, 1991.
[16] MAdLib: an open source Mesh Adaptation Library, http://sites.uclouvain.be/madlib/, 2010.