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Figures


Figure 1: Rational Bezier entities: a) curve, b) surface.


Figure 2: Principal directions at point on a parametric surface.


Figure 3: Location of ideal point on the ellipse of stretches.


Figure 4: Location of ideal point for equal to a) , b) , c) , d) .


Figure 5: Location of ideal point for angle of edge equal to: a) , b) , c) , d) , e) , f) .


Figure 6: Degenerated surfaces: a) an octant of a sphere, b) a rectangle.


Figure 7: Evaluation of the ellipse of stretches at a singular point .


Figure 8: Triangulation of a surface with collapsed side.


Figure 9: Weighting factors for the smoothing in the parametric space.


Figure 10: Model (left) and mesh (right) of a hemisphere modeled by a single patch with two singularities.


Figure 11: Model (left) and mesh (right) of a hemisphere modeled by a single patch without stretch singularity.


Figure 12: Model (left) and mesh (right) of a hemisphere modeled by four patches with a singularity.


Figure 13: Principal stretches (left) and anisotropic mesh (right) in the parametric space of a hemisphere modeled by a single patch with two singularities.


Figure 14: Principal stretches (left) and anisotropic mesh (right) in the parametric space a hemisphere modeled by a single patch without stretch singularity.


Figure 15: Principal stretches (left) and anisotropic mesh (right) of one of the patches in the parametric space of a hemisphere modeled by four patches with a singularity.


Figure 16: Anisotropic mesh of hyperbolic paraboloid in the parametric space.


Figure 17: Anisotropic mesh of twisted plane in the parametric space.


Figure 18: Model (left) and mesh (right) of a hyperbolic paraboloid.


Figure 19: Model (left) and mesh (right) of a twisted plane with a hole.


Figure 20: Octree hierarchy (2D case).


Figure 21: Enforcement of the octree one-level difference (2D case).


Figure 22: Octree built around a mechanical part (left) and corresponding element spacing contours (right).


Figure 23: Octree built around a double bended curve.


Figure 24: Projection of point : a) to a curve, b) to a surface.


Figure 25: Discretization of a curve.


Figure 26: Location of the ``ideal'' point of a tentative triangle.


Figure 27: Correction of the ``ideal'' point due to the surface curvature.


Figure 28: Correction of the ``ideal'' size due to the element size gradation.


Figure 29: Neighbourhood of point (2D case).


Figure 30: Selection of point (2D case).


Figure 31: Uniform mesh of a circle with (left) and without (right) the application of the ellipsoidal neighbourhood .


Figure 32: Intersection check on a poorly parameterized surface.


Figure 33: Diagonal edge swapping.


Figure 34: Model (left) and mesh (right) of a hemisphere modeled by a single patch with two singularities.


Figure 35: Model (left) and mesh (right) of a hemisphere modeled by a single patch without stretch singularity.


Figure 36: Model (left) and mesh (right) of a hemisphere modeled by four patches with a singularity.


Figure 37: Model (left) and mesh (right) of a rotational hyperboloid.


Figure 38: Model (left) and mesh (right) of a chair.


Figure 39: Chair - performance diagram.


Figure 40: Chair - distribution of the element quality.


Figure 41: Chair - distribution of the dihedral angle.


Figure 42: Model and mesh of Utah teapot.


Figure 43: Two levels of hierarchical refinement of a triangular element.


Figure 44: Two stages of the refinement - splitting and averaging.


Figure 45: Averaging mask of the Loop's scheme (after splitting).


Figure 46: Averaging mask of the butterfly scheme (before splitting).


Figure 47: Loop's scheme applied to a star-shaped polyhedron.


Figure 48: Averaging mask of the modified butterfly scheme (n = 7).


Figure 49: Averaging mask of curve interpolating scheme: a) 4-point curve mask, b) 3-point curve mask.


Figure 50: Modified butterfly scheme applied to a star-shaped polyhedron.


Figure 51: Projection to a discrete surface: a) target triangle localization, b) progressive refinement of the target triangle.


Figure 52: Localization of a point into a target triangle: a) initial configuration, b) first refinement, c) second refinement, d) final (limit) configuration.


Figure 53: Correction of the sub-triangle selection: a) backtracking process, b) sub-triangle numbering, c) neighbour numbering.


Figure 54: Unit hemisphere - performance diagram (EP).


Figure 55: Unit hemisphere - performance diagram (AP).


Figure 56: Unit hemisphere - deviation from the ideal shape.


Figure 57: Mesh of a concrete dam with a spillway.


Figure 58: Concrete dam - performance diagram.


Figure 59: Stanford bunny - performance diagram.


Figure 60: Mesh of Stanford bunny.


Figure 61: Location of the ``ideal'' points and of a tentative quadrilateral.


Figure 62: Selection of the base edge for the generation of the second triangle.


Figure 63: Mesh schemes for the determination of smoothing weights.


Figure 64: Topological operations: a) merge operations (left), b) swap operations (middle), c) refine operation (right top), d) coarse operation (right middle), e) split operation (right bottom).


Figure 65: Transform operations.


Figure 66: Mesh generation process on a planar domain with two openings.


Figure 67: Mixed (left) and all-quadrilateral (right) graded mesh of a complex planar domain.


Figure 68: Mixed (left) and all-quadrilateral (right) uniform mesh of a torus.


Figure 69: Mixed (left) and all-quadrilateral (right) graded mesh of a chair.



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Daniel Rypl
2005-12-07