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Keywords: mesh generation, mesh smoothing, surface meshes, surface mesh optimization, finite element, adaptive meshes.

Summary

In this work we present a procedure to smooth meshes defined on surfaces. The construction of the objective function is done in the framework of theory of algebraic quality measures developed in [1]. For 2D or 3D meshes the quality improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes certain objective function [2]. This function is derived from a quality measure of the local mesh, that is, the set of tetrahedra connected to the adjustable or free node. We have chosen, as a starting point, a 2D objective function that presents a barrier in the boundary of the feasible region (set of points where the free node could be placed to get a valid local mesh, that is, without inverted elements). This barrier has an important role because it avoids the optimization algorithm to create a tangled mesh when it starts with a valid one.

Nevertheless, objective functions constructed by algebraic quality measures are only directly applicable to 2D or 3D meshes, but they are not in surface meshes. To overcome this problem, the local mesh, , sited on a surface , is orthogonally projected on a plane (if this exists) in such a way that it performs a valid local mesh . Here is the free node on and is its projection on . The optimization of is got by the appropriated optimization of . To do this we search ideal triangles in that become equilateral ones in .

In general, when the local mesh is on a curved surface, each triangle is placed on a different plane and it is not possible to define a feasible region. Indeed, it is not clear the concept of valid mesh in this case. In order to make sense of it, we say that is acceptable if is valid. Note that the feasible region is always perfectly defined in .

To construct the objective function in , it is first necessary to define the objective function in and, afterwards, to establish the connection between them. A crucial aspect for this construction is to keep the barrier of the 2D objective function. This is got by doing a suitable approximation in the process that transforms the original problem on into an entirely two-dimensional one.

Several examples and applications presented in this work show how this technique is capable to improve the quality of surface meshes. In particular, the algorithm is specially easy to implement when the plane is the same for all the mesh. This is the case of meshes adapted to the terrain used for the simulation of environmental problems [3].

References

1: P.M. Knupp, "Algebraic Mesh Quality Metrics", SIAM J. Sci. Comput., 23, 193-218, 2001.
2: L.A. Freitag and P.M. Knupp, "Tetrahedral Mesh Improvement via Optimization of the Element Condition Number", Int. J. Numer. Meth., 53, 1377-1391, 2002.
3: R. Montenegro, G. Montero, J. M. Escobar, E. Rodríguez and J. M. González-Yuste, "Tetrahedral Mesh Generation for Environmental Problems over Complex Terrains", Lecture Notes in Computer Science, 2329, 335-344, 2002.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 80 PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares Paper 12 Optimization of Surface Meshes by Projections on the Plane: Applications to Environmental Problems J.M. Escobar, G. Montero, R. Montenegro, E. Rodríguez and J.M. González-Yuste Institute of Intelligent Systems and Numerical Applications in Engineering, University of Las Palmas de Gran Canaria, Spain doi:10.4203/ccp.80.12 purchase the full-text of this paper Full Bibliographic Reference for this paper , "Optimization of Surface Meshes by Projections on the Plane: Applications to Environmental Problems", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Fourth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 12, 2004. doi:10.4203/ccp.80.12 Keywords: mesh generation, mesh smoothing, surface meshes, surface mesh optimization, finite element, adaptive meshes. Summary In this work we present a procedure to smooth meshes defined on surfaces. The construction of the objective function is done in the framework of theory of algebraic quality measures developed in [1]. For 2D or 3D meshes the quality improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes certain objective function [2]. This function is derived from a quality measure of the local mesh, that is, the set of tetrahedra connected to the adjustable or free node. We have chosen, as a starting point, a 2D objective function that presents a barrier in the boundary of the feasible region (set of points where the free node could be placed to get a valid local mesh, that is, without inverted elements). This barrier has an important role because it avoids the optimization algorithm to create a tangled mesh when it starts with a valid one. Nevertheless, objective functions constructed by algebraic quality measures are only directly applicable to 2D or 3D meshes, but they are not in surface meshes. To overcome this problem, the local mesh, , sited on a surface , is orthogonally projected on a plane (if this exists) in such a way that it performs a valid local mesh . Here is the free node on and is its projection on . The optimization of is got by the appropriated optimization of . To do this we search ideal triangles in that become equilateral ones in . In general, when the local mesh is on a curved surface, each triangle is placed on a different plane and it is not possible to define a feasible region. Indeed, it is not clear the concept of valid mesh in this case. In order to make sense of it, we say that is acceptable if is valid. Note that the feasible region is always perfectly defined in . To construct the objective function in , it is first necessary to define the objective function in and, afterwards, to establish the connection between them. A crucial aspect for this construction is to keep the barrier of the 2D objective function. This is got by doing a suitable approximation in the process that transforms the original problem on into an entirely two-dimensional one. Several examples and applications presented in this work show how this technique is capable to improve the quality of surface meshes. In particular, the algorithm is specially easy to implement when the plane is the same for all the mesh. This is the case of meshes adapted to the terrain used for the simulation of environmental problems [3]. References 1 P.M. Knupp, "Algebraic Mesh Quality Metrics", SIAM J. Sci. Comput., 23, 193-218, 2001. 2 L.A. Freitag and P.M. Knupp, "Tetrahedral Mesh Improvement via Optimization of the Element Condition Number", Int. J. Numer. Meth., 53, 1377-1391, 2002. 3 R. Montenegro, G. Montero, J. M. Escobar, E. Rodríguez and J. M. González-Yuste, "Tetrahedral Mesh Generation for Environmental Problems over Complex Terrains", Lecture Notes in Computer Science, 2329, 335-344, 2002. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £95 +P&P)
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