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徐岗, Ran Ling, Ming Li, Jinlan Xu, Qing Wu, Bernard Morrain, Timon Rabczuk, Stéphane P.A. Bordas

hDp://xugang.hdu.edu.cn gxu@hdu.edu.cn

G

1

construction of parameterization with unstructured Bézier elements

2017.4.14

(2)

Outline

• Parameterization in isogeometric analysis

• Analysis-suitable G1 planar parameterization from complex boundary

(3)

IGA-meshing

(4)

Parameterization of computational domain

•  Open problem

(5)

计算域参数化质量对分析结果影响 CAD2013

(6)

计算域参数化质量对分析结果影响 CAD2013

(7)

Main difficulties

•  Trimmed surface

•  Complex topology

•  Analysis-suitable

(8)

Related work on parameterization for IGA

Ø  Analysis-aware optimal parameterization

E. Cohen et al.(CMAME, 2010) , Xu et al.(CMAME,2011), Pilgerstorfer et al ( CMAME, 2013)

Ø  Volumetric spline parameterization from boundary triangulation

T. Martin et al.(CMAME, 2009), Zhang et al.(CMAME, 2012).

Ø  Analysis-suitable planar parameterization from spline boundary

Xu et al.(CAD 2013), Gravessen et al.(CMAME, 2014）, Xu et al. (CMAME 2015)

Ø  Analysis-suitable volume parameterization from spline boundary

Xu et al.(JCP2013), , Zhang et al.(CM, 2012)，Chan et al (CAD 2017)

Most of the current methods with spline boundary on ly focus on the computational domain with simple bo undaries.

(9)

Problem statement

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Analysis-suitable parameterization

(11)

CMAME 2013, CAD 2013

(12)

Multi-block case

(13)

Variational harmonic method

( Journal of Computational Physics, 2013)

(14)

Two examples （ 1/2 ）

(15)

Two examples （ 2/2 ）

(16)

3D example I

(17)

Boundary reparameterization for volumetric parameterization (Computational Mechanics, 2014)

Goal: construct opSmal Möbius reparameterizaSon of boundary surfaces to achieve high-quality isoparametric structure without changing the boundary geometry

(18)

Möbius reparameterization

New NURBS surface with the same control points

but different weights and knot vectors

(19)

Optimization method

•  Find the optimal

such that the reparameterized NURBS surface minimizes the following objective function

(20)

Reparameterization for VP problem

(21)
(22)

Outline

• Parameterization in isogeometric analysis

• Analysis-suitable G1 planar parameterization from complex boundary

(23)

Planar domain with arbitrary topology

•  Given the boundary spline curves of a planar domain with arbitrary topology, construct the patch structure and

control points to obtain IGA-suitable parameterization

(24)

Desired parameterization method

•  Boundary-preserving

•  Automatic continuity imposition (≠high-order meshing with C

0

)

•  Automatic construction of segmentation curve

•  Injective

•  Uniform patch size

•  Orthogonal iso-parametric structure

(25)

Proposed framework

1.  Pre-processing for high-quality parameterization

2.  Topology information generation of quadrilateral decomposition 3.  Quadrilateral patch partition by global optimization

4.  High-quality patch parameterization by local optimization

Reference:

Gang Xu, Ming Li, Bernard Mourrain, Timon Rabczuk, Jinlan Xu, Stephane P.A. Bordas.

ConstrucSng IGA-suitable planar parameterizaSon from complex CAD boundary by domain parSSon and global/local opSmizaSon,CMAME, in revision

(26)

Framework Overview

(27)

Pre-processing of input boundary curves

•  Bézier extraction

•  Bézier subdivision

(28)

Pre-processing of input boundary curves

•  Subdivision of a Bézier curve with concave shape

(29)

Proposed framework

1.  Pre-processing for high-quality parameterization

2.  Topology information generation of quadrilateral decomposition 3.  Quadrilateral patch partition by global optimization

4.  High-quality patch parameterization by local optimization

Ref:

Gang Xu, Ming Li, Bernard Mourrain, Timon Rabczuk, Jinlan Xu, Stephane P.A. Bordas.

ConstrucSng IGA-suitable planar parameterizaSon from complex CAD boundary by domain parSSon and global/local opSmizaSon,CMAME, in revision

(30)

Global optimization method

•  Propose a global optimization method to construct the four- sided curved partition of the computational domain

Topology generation of

quadrilateral decomposition

Laplacian smoothing

method

Construction of segmentation

curves

(31)

Topology generation of quadrilateral decomposition

(a)input discrete boundary

•  Step. 1: Construct the discrete boundary by connecting the endpoints of the extracted Bézier curves.

(32)

Step. 2

•  Multiply-connected region à simply-connected region.

(a)multiply-connected domain (b) simply-connected domain

(33)

Step. 3

•  Approximate convex decomposition of the simply- connected regions

(b) quasi-convex polygon decomposition

(34)

Step. 4

•  For each quasi-convex polygon obtained in Step.3, generate the quadrangulation topology information

•  Only introduce irregular vertices with valence 3 or 5, which guarantees the solution existence for G1 planar parameterization around the

irregular vertex

Reference:

K.Takayama,D.Panozzo,O.

Sorkine-Hornung Pattern-based quadrangulation for N-sided patches. CGF, 2015

(c) quad-meshing result by our method with 147 elements and 14 irregular vertices

(35)

Laplacian smoothing

•  We adapt an iterative Laplacian smoothing method to improve the quality of the quad mesh.

Termination rules:

(36)

Construction of segmentation curves

•  The segmentation curves should interpolate two vertices on the quad mesh Q(V,E).

•  Global optimization method to construct the optimal shape of segmentation curves.

Fig.5(a) segmentation curves(red) and quad edges( black)

(37)

Desired parameterization method

•  Boundary-preserving

•  Automatic continuity imposition (≠high-order meshing with C

0

)

•  Automatic construction of segmentation curve

•  Injective

•  Uniform patch size

•  Orthogonal iso-parametric structure

(38)

Uniform patch size

Computing the area of planar region bounded by B-spline curves?

(39)

Computing the area of planar region with Bézier boundary

Then the area A(Ω) of the planar region is

•  For the planar region bounded by N pieces of Bézier curves

(40)

Global optimization method

•  Objective functions:

.

(41)

An example

(42)

Proposed framework

1.  Pre-processing for high-quality parameterization

2.  Topology information generation of quadrilateral decomposition 3.  Quadrilateral patch partition by global optimization

4.  High-quality patch parameterization by local optimization

Ref:

Gang Xu, Ming Li, Bernard Mourrain, Timon Rabczuk, Jinlan Xu, Stephane P.A. Bordas.

ConstrucSng IGA-suitable planar parameterizaSon from complex CAD boundary by domain parSSon and global/local opSmizaSon,CMAME, in revision

(43)

High-quality patch parameterization

Step 1: Construction of boundary second-layer control points with

orthogonality optimization and continuity constraints.

Step 2: Local C1 linear-energy-minimizing method for constructing inner control points.

Step 3: Find out the invalid patches on the parameterization, then recover patch validity.

(44)

Step. 1: Construction of boundary control points

Initial construction

Imposition of C1-continuity

at regular region

Imposition of G1-continuity

around the irregular

vertices

(45)

Initial construction

•  Firstly, we will describe the initial construction by orthogonality optimization.

(46)

the Lagrange function:

Minimize the change of related control points along the segmentaSon curves such that they saSsfy the C1-constraints

Imposition of C1-continuity by Lagrange Multiplier method

(47)

Imposition of G1-continuity around irregular vertex

•  Some special treatments should be done achieve G1-continuity at the irregular vertices.

G1-continuity

(48)

Imposition of G

1

-continuity

(Mourrain et al，CAGD 2016)

•  The G1-continuity constraints around the irregular vertex can be described as follows:

There exists unique solution for

M=3 and M=5

(49)

High-quality patch parameterization

Step 1: Construction of boundary second-layer control points with

orthogonality optimization and continuity constraints.

Step 2: Local C1 linear-energy-minimizing method for constructing inner control points.

Step 3: Find out the invalid patches on the parameterization, then recover patch validity.

(50)

Local C

1

linear-energy-minimizing m

•  Constructing interior (n−3)×(m−3) control points of each patch

(51)

Inner control points construction

•  A tensor product Bézier surface r(u,v) has minimal energy E(r) if and only if remaining inner control points satisfy

•  A linear system with (n−3)×(n−3) equations and (n−3)×(m−3) variables

.

(52)

High-quality patch parameterization

Step 1: Construction of boundary second-layer control points with

orthogonality optimization and continuity constraints.

Step 2: Local C1 linear-energy-minimizing method for constructing inner control points.

Step 3: Find out the invalid patches on the parameterization, then recover patch validity.

(53)

Step 3: Injective parameterization

•  Jacobian

•  Logarithmic-barrier method for invalid patch Valid patch

(54)

Example I

(55)

Example II

(56)

Example III

(57)

Example III with the skeleton-based decomposition

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Quality comparison

(59)

Parameters and computing time

Références

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