Quadrilateral Mesh Generation: Meromorphic Quartic Differential and Abel-Jacobi Condition Na Lei 1 1 Dalian University of Technology GAMES Webinar 2020-02-27 Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 1 / 82
Thanks This work is collaborated with David Xianfeng Gu , Stony Brook University Feng Luo , Rutgers University Zhongxuan Luo , Dalian University of Technology Xiaopeng Zheng , Dalian University of Technology Wei Chen , Dalian University of Technology Jingyao Ke , University of Science and Technology of China and many students. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 2 / 82
References W. Chen, X. Zheng, J. Ke, N. Lei, Z. Luo, X. Gu. “Quadrilateral Mesh Generation I : Metric Based Method”, Computer Methods in Applied Mechanics and Engineering , V356:652-668, 2019. N. Lei, X. Zheng, Z. Luo, F. Luo, X. Gu “Quadrilateral Mesh Generation II : Meromorphic Quartic Differentials and Abel-Jacobi Condition”, https://arxiv.org/pdf/1907.00216.pdf Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 3 / 82
Motivation Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 4 / 82
Simulation Numerical simulation is very important in flight vehicle design and engineering. After we have a designed CAD model, the first step is to convert the designed models and the external flow fields into meshes and then to use computational fluid dynamics software to simulate. In the whole procedure, meshing step cost 70% time and human power for manufacture industry, such as Boeing. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 5 / 82
Simulation Triangular/Tetrahedral meshes and quadrilateral/hexahedral meshes have been widely used in simulation. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 6 / 82
Simulation Comparing to triangular meshes, quadrilateral meshes have many advantages. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 7 / 82
Advantages Advantages of Quad-mesh Quad-mesh can better capture the local principle curvature directions or sharp features, as well as the semantics of modeled objects, therefore it is widely used in animation industry. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 8 / 82
Advantages Advantages of Quad-mesh Quad-mesh has tensor product structure, it is suitable for fitting splines or NURBS. Therefore it is applied for high-order surface modeling, such as CAD/CAM for Splines and NURBS, and the entertainment industry for subdivision surfaces. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 9 / 82
Quad-Mesh Generation Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 10 / 82
Categories Categories of Quad-meshes Regular quad-mesh : all the interior vertices are with topological 1 valence 4, there are no singularities, such as torus. Semi-regular quad-mesh : The separatrices divide the 2 quad-mesh into several topological rectangles, the interior of each topological rectangle is regular grids. Unstructured quad-mesh : A large fraction of its vertices are 3 irregular. h k / 2 h j / 2 t i l k / 2 l j / 2 t j t k h j / 2 h k / 2 h i / 2 h i / 2 l i / 2 Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 11 / 82
Regular vs Semi-regular Quad-mesh Regular vs Semi-regular Quad-mesh Regular quad-meshes have strong topological requirements for the surfaces, such as topological torus or annulus. Semi-regular quad-meshes can be realized for surfaces with any topologies, but the number of singularities, the behavior of separatrices are difficult to control. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 12 / 82
Quad-Mesh Figure: Quad-meshes with different number of singularities. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 13 / 82
Quad-Mesh Figure: Quad-meshes with different number of singularities. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 14 / 82
Quad-Mesh Figure: Quad-meshes with different number of singularities. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 15 / 82
Generic Surface Model - Triangular Mesh Surfaces are represented as polyhedron triangular meshes. Isometric gluing of triangles in E 2 . Isometric gluing of triangles in H 2 , S 2 . Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 16 / 82
Discrete Metrics Definition (Discrete Metric) A Discrete Metric on a triangular mesh is a function defined on the vertices, l : E = { all edges } → R + , satisfies triangular inequality. A mesh has infinite metrics. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 17 / 82
Discrete Curvature Definition (Discrete Curvature) Discrete curvature: K : V = { vertices } → R 1 . K ( v ) = 2 π − ∑ α i , v �∈ ∂ M ; K ( v ) = π − ∑ α i , v ∈ ∂ M i i Theorem (Discrete Gauss-Bonnet theorem) K ( v )+ ∑ ∑ K ( v ) = 2 πχ ( M ) . v �∈ ∂ M v ∈ ∂ M v α 1 v α 2 α 3 α 1 α 2 Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 18 / 82
Quad-Mesh Metric Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 19 / 82
Quad-Mesh Metric Definition (Quad-Metric) Given a quad-mesh Q , each face is treated as the unit planar square, this will define a Riemannian metric, the so-called quad-mesh metric g Q , which is a flat metric with cone singularities. Theorem (Quad-Mesh Metric Conditions) Given a quad-mesh Q , the induced quad-mesh metric is g Q , which satisfies the following four conditions: Gauss-Bonnet condition; 1 Holonomy condition; 2 Boundary Alignment condition; 3 Finite geodesic lamination condition. 4 Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 20 / 82
Gauss-Bonnet condition Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 21 / 82
Gauss-Bonnet Condition Definition (Curvature) Given a quad-mesh Q , for each vertex v i , the curvature is defined as � π v �∈ ∂ Q 2 ( 4 − k ( v )) K ( v ) = π v ∈ ∂ Q 2 ( 2 − k ( v )) where k ( v ) is the topological valence of v , i.e. the number of faces adjacent to v . k = π / 2 k = − π / 2 k = − π k = − 2 π k = 0 Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 22 / 82
Gauss-Bonnet Condition Theorem (Gauss-Bonnet) Given a quad-mesh Q , the induced metric is g Q , the total curvature satisfies K ( v i )+ ∑ ∑ K ( v i ) = 2 πχ ( Q ) . v i ∈ ∂ Q v i �∈ ∂ Q Namely ( 2 − k ( v i ))+ ∑ ∑ ( 4 − k ( v i )) = 4 χ ( Q ) . v i ∈ ∂ Q v i �∈ ∂ Q k = π / 2 k = − π / 2 k = − π k = − 2 π k = 0 Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 23 / 82
Quad-Mesh Figure: Quad-meshes with different number of singularities. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 24 / 82
Quad-Mesh Figure: Quad-meshes with different number of singularities. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 25 / 82
Holonomy Condition Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 26 / 82
Holonomy Condition Definition (Holonomy) Given a quad-mesh Q , the induced flat metric is g Q , the set of singular vertices is S Q . Suppose γ : [ 0 , 1 ] → Q \ S Q is a closed loop not through singularities, choose a tangent vector v ( 0 ) ∈ T γ ( 0 ) Q , parallel transport v ( 0 ) along γ ( t ) to obtain v ( t ) . The rotation angle from v ( 0 ) to v ( 1 ) in T γ ( 0 ) Q is the holonomy of γ , denoted as ρ ( γ ) . Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 27 / 82
Face Loop Definition (face path) A sequence of faces, { f 0 , f 1 , ··· , f n } , such that f i and f i + 1 share an edge. If f 0 equals to f n , then the face path is called a face loop. Figure: A face path and a face loop. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 28 / 82
Holonomy Definition (Holonomy of a face loop) Given a face loop γ through σ 0 , fix a frame on σ 0 , parallel transport the frame along γ . When we return to σ 0 , the frame is rotated by an angle k π / 2, which is called the holonomy of γ , and denoted as � γ � . π 2 γ σ 0 Figure: Parallel transportation along a face loop. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 29 / 82
Holonomy Theorem (Holonomy Condition) Suppose Q is a closed quad-mesh, then the holonomy group induced by g Q is a subgroup of the rotation group { e i k 2 π , k ∈ Z } . π 2 γ σ 0 Figure: Parallel transportation along a face loop. Na Lei (Dalian University of Technology) Quad Meshing via Conformal Geometry Feb. 27, 2020 30 / 82
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