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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: M. Papadrakakis and B.H.V. Topping
Paper 51

Alignment of Surface Triangulations for Approximating Interior Curves

J.M. Escobar1,2, E. Rodríguez2, R. Montenegro2 and G. Montero2

1Department of Signal and Communications,
2University Institute for Intelligent Systems and Numerical Applications in Engineering,
University of Las Palmas de Gran Canaria, Spain

Full Bibliographic Reference for this paper
, "Alignment of Surface Triangulations for Approximating Interior Curves", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 51, 2008. doi:10.4203/ccp.89.51
Keywords: mesh alignment, moving meshes, mesh adaptation, surface mesh smoothing, node movement, R-adaptivity.

Summary
The numerical simulation of physical problems requires the internal boundaries and discontinuities be properly represented. Usually, the largest errors are introduced in a neighborhood of such discontinuities. These errors are often greatly reduced if the mesh is aligned with the discontinuities. That is why it is desirable to have a procedure capable of deforming a given triangulation to get its alignment with a curve. Although there are numerous works dealing with r-adaptivity, that is, mesh adaption allowing only changes in the position of the nodes, only a few of them consider the problem of the exact mesh alignment with interior curves [1].

The procedure that we describe in this paper is to align a given surface triangulation with an arbitrary curve. The method moves the nodes of the mesh, maintaining its topology, in order to achieve two objectives: the piecewise approximation of the curve by edges of the mesh, and the optimization of the deformed mesh resulting from the previous process. The overall method, which we will designate as projecting/smoothing, is based on the surface mesh smoothing technique proposed in [2] where the quality improvement of the mesh is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes a certain objective function. The objective function is derived from the algebraic quality measure mean ratio [3] extended to the set of triangles connected to the free node. The projecting/smoothing method allows us to track an object moving through the reference mesh without the necessity of remeshing.

Usually we do not have an analytical representation of the curve. Instead, it is approximately known by a sequence of interpolating data points. We have chosen a parametric cubic spline as the interpolating curve as it is C2 continuous and it has others interesting properties that will be used later. Obviously, the grade of approximation of the curve depends on the element sizes, therefore, a good strategy is to combine the projecting/smoothing technique with a local mesh refinement [4]. Our procedure is specially indicated for evolutionary problems in which the boundaries change their shape or position with time. For example the ones related to fluid-structure interaction involving large displacement (see, for example [5]), or crack modeling. The projecting/smoothing technique could be also applied to domain decomposition, definition of material interfaces, free boundary problems, etc.

References
1
J.M. Hyman, S. Li, P.M. Knupp, M. Shashkov, "An algorithm to align a quadrilateral grid with internal boundaries", Journal of Computational Physics, 163, 133-149, 2000. doi:10.1006/jcph.2000.6560
2
J.M. Escobar, G. Montero, R. Montenegro, E. Rodríguez, "An algebraic method for smoothing surface triangulations on a local parametric space", Int. J. Numer. Methods. Engng., 66, 740-760, 2006. doi:10.1002/nme.1584
3
P.M. Knupp, "Algebraic mesh quality metrics", SIAM J. Sci. Comput., 23, 193-218, 2001. doi:10.1137/S1064827500371499
4
J.M. González-Yuste, R. Montenegro, J.M. Escobar, G. Montero, E. Rodríguez, "Local refinement of 3-D triangulations using object-oriented methods", Advances in Engineering Software, 35, 693-702, 2004. doi:10.1016/j.advengsoft.2003.07.003
5
K. Stein, T.E. Tezduyar, R. Benney, "Automatic mesh update with the solid-extension mesh moving technique", Comput. Methods Appl. Mech. Engng., 193, 2019-2032, 2004. doi:10.1016/j.cma.2003.12.046

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