Fathi El-Yafi Project and Software Development Manager Engineering Simulation

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1 An Introduction to Mesh Generation Algorithms Part 2 Fathi El-Yafi Project and Software Development Manager Engineering Simulation April

2 Overview Adaptive Meshing: Remeshing Decimation Optimization April

the mesh from a tree Quaternary Nodal generation within each cell of the tree

3 Finite Element Context Objective: Adaptive Meshing 2D Adaptive Mesh according to the specifications sizes imposed by the user Method: Calculating the density of the mesh from a tree Quaternary Nodal generation within each cell of the tree Meshing by advancing front method according to a criterion of Delaunay April

It is a dynamic, multilevel index, with maximum and minimum bounds on the number of keys in each index segment (usually called a block' or

4 Finite Element Context Quaternary Tree In computer science, a B+ tree (also known as a Quaternary Tree) is a type of tree, which represents sorted data in a way that allows for efficient insertion, retrieval and removal of records, each of which is identified by a key. It is a dynamic, multilevel index, with maximum and minimum bounds on the number of keys in each index segment (usually called a block' or node'). In a B+ tree, in contrast to a B+ tree, all records are stored at the lowest level of the tree; only keys are stored in interior blocks April

5 Finite Element Context Remeshing a Polygon? n points Possible Remeshing? April

6 Finite Element Context Principle = = = April

7 Finite Element Context Principle n nodes lead to n-2 triangles = It shows that we can create Tn different triangles n(n 1)(n 2) T n = N=10, 1430 possible triangulations! April

8 State of the Art Methods that use the surface parameterization: Appropriate Plane + transport functions Appropriate Plane + Metric: INRIA Discrete Models: Very Refined Initial Mesh + Simplifications Reconstruction: Direct Method: Frontal Triangulation (R. Löhner) Remeshing: HDI (A. Rassineux) April

9 Typical Examples April

10 Objective: Adaptive Meshing using Diffuse Interpolation Building an adaptive mesh of a 3D surface according to size map, using the initial mesh as only data Initial Mesh Local geometric model Adaptive mesh Diffuse Interpolation Hermite type Remeshing Procedure April

11 Remeshing April

12 Some Remeshing Tools Edge splitting Edge collapsing Node removing Edge collapsing and nodal insertion Swapping Nodal Shifting April

13 Remeshing Remeshing according to a size map Simplification of a mesh (70%) April

14 Remeshing Initial Mesh Intermediate Mesh Optimized Mesh April

15 Surface Remeshing Initial Mesh Optimized Mesh April

16 Surface Remeshing Initial model Reduction: 75% Reduction: 60% Adapted to curvature April

17 Surface Meshing Adapted to curvature Simplified model April

18 Simplified Curvature In a plane θ i = 360 α ι Θ 1 Θ 2 Of all the triangles α i 360 ( α θ ) = θ = i i default angle = f (Gaussian Curvature) April

19 Feature Lines Detection April

20 Feature Lines Detection (a) (b) Curvature using Diffuse Interpolation Model Detecting neighborhood Identification of geometrical parameters (radius, axis) April

21 Segmentation into subsets Surface Identification Detection of Feature lines Meshing Principal Curvatures April

22 Segmentation into subsets Example Surface Identification Detection of Feature lines Meshing Instituto Tecnológico de Veracruz Meshing Generation Tutorial April

23 Adaptive Mesh Main Steps Tecnológico de Veracruz Instituto Meshing Generation Tutorial April

24 21-25 April

25 Local model Diffuse Model z = f ( x, y) =< 1, x, y, x, xy, y > α = 2 2 T P α z y MLS Technique : Diffuse approximation Minimisation of an interpolation criterion x J i n T 1 x ( ) = = α w( xi, x) p ( xi x) i i= 1 ( ) α z April

26 Polynomial Base Moving Least Squares 2 2 T < 1,,,,, >=, = x y x xy y p x y p x ( ) T ( ) n = Total number of points «standard» Least Squares i = n i= 1 ( ) ( T ) 2 i α i Jx( α) = p x x f ( x x) Z Select a neighbor V(x) around a point(x,y) i V ( x) ( ) ( T ) 2 i α i Jx( α) = p x x f ( x x) X R The approximation is local April

27 Weighted Moving Least Squares Weight Function i V ( x) T ( ) ( ) ( ) α 2 Jx( α) = w xi x p xi x f ( xi x) xi x w(x i,x) = wref R x ( ) w(x i,x) = ( 1-l i) avec l i = d i/r 2 ( 1+ 2li) April

28 Moving Least Squares Interpolation i V ( x) T ( ) ( ) ( α ) 2 Jx( α) = w xi x p xi x f ( xi x) If w(x,x i ) interpolation at point x i w( x, ) [ ] i x w( x, x) i 1 w( x i, x) April

29 CAD-STL Face Per Face April

30 Finite Element Context STL/Mesh/Adaptive Mesh STL Mesh Mesh Face per Face Adaptive Mesh April

31 STL/Adaptive Mesh Adaptive Mesh STL Mesh April

32 CAD/Adaptive Meshes April

33 Mesh Details April

34 Feature Lines April

35 Surface Remeshing Initial model Simplified model April

36 Surface Remeshing April

37 Surface Remeshing Adaptive Mesh Initial Mesh April

38 Surface Remeshing April

39 Repairing Surface Mesh April

40 Demonstrations April

Overview. Efficient Simplification of Point-sampled Surfaces. Introduction. Introduction. Neighborhood. Local Surface Analysis

Overview. Efficient Simplification of Point-sampled Surfaces. Introduction. Introduction. Neighborhood. Local Surface Analysis Overview Efficient Simplification of Pointsampled Surfaces Introduction Local surface analysis Simplification methods Error measurement Comparison PointBased Computer Graphics Mark Pauly PointBased Computer