## Mesh size functions for implicit geometries and pde-based gradient limiting

Venue: | Eng. with Comput |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Persson_meshsize,

author = {Per-olof Persson},

title = {Mesh size functions for implicit geometries and pde-based gradient limiting},

journal = {Eng. with Comput},

year = {},

volume = {22},

pages = {2006}

}

### OpenURL

### Abstract

Mesh generation and mesh enhancement algorithms often require a mesh size function to specify the desired size of the elements. We present algorithms for automatic generation of a size function, discretized on a background grid, by using distance functions and numerical PDE solvers. The size function is adapted to the geometry, taking into account the local feature size and the boundary curvature. It also obeys a grading constraint that limits the size ratio of neighboring elements. We formulate the feature size in terms of the medial axis transform, and show how to compute it accurately from a distance function. We propose a new Gradient Limiting Equation for the mesh grading requirement, and we show how to solve it numerically with Hamilton-Jacobi solvers. We show examples of the techniques using Cartesian and unstructured background grids in 2-D and 3-D, and applications with numerical adaptation and mesh generation for images. Keywords: mesh generation, size function, background grid, Hamilton-Jacobi, gradation control 1.

### Citations

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Citation Context ...background mesh, we compute the distances to the geometry boundary for the nodes in a narrow band around the boundary (typically a few node points wide). We then use the Fast Marching Method (Sethian =-=[11]-=-, see also Tsitsiklis [12]) to calculate the distances at all the remaining node points. The computed values are considered “known values”, and their neighbors can be updated and inserted into a prior... |

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Citation Context ... algorithms it is advantageous if an appropriate mesh size function h(x) is known prior to computing the mesh. This includes the advancing front method [1], the paving method for quadrilateral meshes =-=[2]-=-, and smoothing-based mesh generators such as the one we proposed in [3],[4]. The popular Delaunay refinement algorithm [5], [6] typically does not need an explicit size function since good element si... |

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Citation Context ...s. The identification of the medial axis is often referred to as skeletonization, and a large number of algorithms have been proposed. Many of them, including the original Grassfire algorithm by Blum =-=[16]-=-, are based on explicit representations of the geometry. Kimmel et al [17] described an algorithm for finding the medial axis from a distance function in two dimensions, by segmenting the boundary cur... |

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Citation Context ... is known prior to computing the mesh. This includes the advancing front method [1], the paving method for quadrilateral meshes [2], and smoothing-based mesh generators such as the one we proposed in =-=[3]-=-,[4]. The popular Delaunay refinement algorithm [5], [6] typically does not need an explicit size function since good element sizing is implied from the quality bound, but higher quality meshes can be... |

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Citation Context ...nds on the mesh. Another method is to build a balanced octree, and let the size function be related to the size of the octree cells [21]. This data structure is used in the quadtree meshing algorithm =-=[22]-=-, and the balancing guarantees a limited variation in element sizes, by a maximum factor of two between neighboring cells. However, when used as a size function for other meshing algorithms it provide... |

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Citation Context ...mplied from the quality bound, but higher quality meshes can be obtained with good apriori size functions. Many techniques have been proposed for automatic generation of mesh size functions, see [7], =-=[8]-=-, [9]. A common solution is to represent the size function in a discretized form on a background grid and obtain the actual values of h(x) by interpolation, as described in Section 2.1. We present sev... |

8 |
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Citation Context ... the magnitude of ∇h(x). In many mesh generation algorithms it is advantageous if an appropriate mesh size function h(x) is known prior to computing the mesh. This includes the advancing front method =-=[1]-=-, the paving method for quadrilateral meshes [2], and smoothing-based mesh generators such as the one we proposed in [3],[4]. The popular Delaunay refinement algorithm [5], [6] typically does not need... |

7 |
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Citation Context ...egmenting the boundary curve with respect to curvature extrema. Siddiqi et al [18] used a divergence based formulation combined with a thinning process to guarantee a correct topology. Telea and Wijk =-=[19]-=- showed how to use the fast marching method for skeletonization and centerline extraction. Although in principle we could use any existing algorithm for skeletonization using distance functions, we ha... |

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Citation Context ...olution satisfies |∇h(x)| ≤ g only approximately, in a way that depends on the mesh. Another method is to build a balanced octree, and let the size function be related to the size of the octree cells =-=[21]-=-. This data structure is used in the quadtree meshing algorithm [22], and the balancing guarantees a limited variation in element sizes, by a maximum factor of two between neighboring cells. However, ... |

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Citation Context ...ant g (Figures 14 and 15). One way to limit the gradients of a discretized size function is to iterate over the edges of the background mesh and update the size function locally for neighboring nodes =-=[20]-=-. When the iterations converge, the solution satisfies |∇h(x)| ≤ g only approximately, in a way that depends on the mesh. Another method is to build a balanced octree, and let the size function be rel... |

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Citation Context ...known prior to computing the mesh. This includes the advancing front method [1], the paving method for quadrilateral meshes [2], and smoothing-based mesh generators such as the one we proposed in [3],=-=[4]-=-. The popular Delaunay refinement algorithm [5], [6] typically does not need an explicit size function since good element sizing is implied from the quality bound, but higher quality meshes can be obt... |

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Citation Context ... few additional refinements. Several methods have been developed to solve Hamilton-Jacobi equations on unstructured grids, and we have implemented the positive coefficient scheme by Barth and Sethian =-=[25]-=-. The solver is slightly more complicated than the Cartesian variants, but the numerical schemes can essentially be used as black-boxes. A triangulated version of the fast marching method was given in... |

3 |
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Citation Context ...s slightly more complicated than the Cartesian variants, but the numerical schemes can essentially be used as black-boxes. A triangulated version of the fast marching method was given in [26], and in =-=[27]-=- the algorithm was generalized to arbitrary node locations. One particular unstructured background grid is the octree representation, and the Cartesian methods extend naturally to this case (both the ... |

2 |
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Citation Context ...d from the quality bound, but higher quality meshes can be obtained with good apriori size functions. Many techniques have been proposed for automatic generation of mesh size functions, see [7], [8], =-=[9]-=-. A common solution is to represent the size function in a discretized form on a background grid and obtain the actual values of h(x) by interpolation, as described in Section 2.1. We present several ... |

2 |
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Citation Context ... The solver is slightly more complicated than the Cartesian variants, but the numerical schemes can essentially be used as black-boxes. A triangulated version of the fast marching method was given in =-=[26]-=-, and in [27] the algorithm was generalized to arbitrary node locations. One particular unstructured background grid is the octree representation, and the Cartesian methods extend naturally to this ca... |

1 |
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Citation Context ... is implied from the quality bound, but higher quality meshes can be obtained with good apriori size functions. Many techniques have been proposed for automatic generation of mesh size functions, see =-=[7]-=-, [8], [9]. A common solution is to represent the size function in a discretized form on a background grid and obtain the actual values of h(x) by interpolation, as described in Section 2.1. We presen... |

1 |
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Citation Context ...zation, and a large number of algorithms have been proposed. Many of them, including the original Grassfire algorithm by Blum [16], are based on explicit representations of the geometry. Kimmel et al =-=[17]-=- described an algorithm for finding the medial axis from a distance function in two dimensions, by segmenting the boundary curve with respect to curvature extrema. Siddiqi et al [18] used a divergence... |

1 |
Size Functions and Mesh Generation for High-Quality Adaptive Remeshing
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(Show Context)
Citation Context ...ns from the original size function, even where the gradient is small. A better method is to use gradient limiting and solve (8) on the same unstructured mesh that the size function is defined on, see =-=[28]-=- for further details. Figure 12 shows an example of adaptive meshing for a compressible flow simulation over a bump at Mach 0.95. We solve the Euler equations with a finite volume solver, and use a si... |