A Construction of Injective Parameterizations of Domainsfor Isogeometric Applications

[Extended Abstract]

Thien NguyenInst. of Applied Geometry

JKU Linz,Austria

Bernard MourrainGALAAD

INRIA Sophia AntipolisFrance

André GalligoNice University

NiceFrance

Gang XuComp. Sci. Dep.

Hangzhou Dianzi Univ.China

ABSTRACTWe present a new method to construct a B-spline param-eterization of a domain defined by its boundary curves orsurfaces. The method is based on solving Laplace equationson the physical domain. The equations are then pulled backto the parameter domain to deduce an elliptic system of Par-tial Differential Equations with boundary conditions. Thissystem, solved by relaxation techniques, yields the requiredparameterization.

Categories and Subject DescriptorsI.3.5 [Computational Geometry and Object Model-ing]: Curve, surface, solid, and object representations, Splines

General TermsAlgorithms

KeywordsIsogeometric analysis, Surface parameterization, elliptic PDE

1. INTRODUCTIONThe problem of parameterizing a domain defined by its

boundary curves (or surfaces) has long been discussed incomputer graphics and geometric modeling due to its im-portant role in many applications. The well-known Gordons-Coons method, which is represented in almost all standardliterature, see [3], is a simple but suitable to solve this prob-lem for many cases. Later, Farin and Hansford [2] proposeddiscrete Coons patches which minimizes a surface energy toproduce good shape Coons patches. With the recent de-velopment of isogeometric analysis (IGA for short) (see [4]),solving this problem once again is a big challenge to produceprerequisite geometry for IGA. In IGA, the computationaldomains are solid objects which are represented by tensorproduct B-splines or NURBS and the discretization space isspanned by B-splines or NURBS basis. However, in practice,the NURBS representations of solid objects are not always

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available since CAD systems usually only provide param-eterization of the boundary, i.e. mappings from the para-metric domains to the boundary of the physical domain, asdescribed in [1].

Motivated by IGA, Xu et al. [8] have used the discreteCoons method as the initial solution for the iterative methodminimizing a functional arising from shape optimization toobtain a better parameterization of a planar shape. Basi-cally, Xu et al.’s method consists of finding a mapping fromthe parametric domain (usually the unit square) to the phys-ical domain which is initially injective via suited inequalitieson the coordinates of inner control points and then of mini-mizing some quality functionals. There the important pointis to achieve injectivity in the initialization and keep it dur-ing the optimization process.

On the other way around, especially for volume parame-terization, there are several methods for finding a mappingfrom the physical domain to the parametric domain by solv-ing Laplace equations or minimizing some energy function-als. Then the physical domain is reconstructed by the ap-proximation of the inverse mapping.

In [6], a parameterization method for a generalized cylinder-type volume defined by a tetrahedral mesh is proposed basedon the investigation of discrete volumetric harmonic func-tions. In another paper [5], the authors also solve Laplaceequations on the physical domain by the fundamental solu-tion method to find the mapping between solid objects giventhat the surface mapping is available to be the boundaryconditions for the PDEs.

In the previous works, Laplace equations are solved withrespect to the coordinate functions separately. A differentapproach is given in [7]: the first coordinate function is foundby minimizing a Dirichlet energy, then the second coordi-nate function is found by minimizing its harmonic energy onthe level sets of previously found coordinate function. Theimportant point is that this method naturally guaranteesthe injectivity of the mapping from the physical domain tothe parameter domain. Once this mapping is obtained, thespline approximation for the inverse of the mapping is foundby the least squares fitting method.

Inspired by these works, we propose a new iterative algo-rithm to obtain an injective parameterization of the physicaldomain from the initial parameterization by solving Laplaceequations on the physical domain, via splines defined on theparameter domain.

2. PARAMETERIZATION CONSTRUCTIONWe describe this method for a planar simply connected

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domain bounded by B-spline curves, but it can be extendedstraighforwardly to volumes bounded by B-spline surfaces.Our objective is to construct a parametrisation σ from asquare domain S = [0, 1] × [0, 1] into the domain Ω whichboundary is formed by the curves Γ1, . . . ,Γ4. We search amap σ in the B-spline basis associated to the parametricdomain S of the form:

σ(ξ, η) = (x(ξ, η), y(ξ, η)) =

d1Xi=0

d2Xj=0

Nd1i (ξ)Nd2j (η)pij , (1)

where (Nd1i (ξ))i=0,...,d1 , (Nd2j (η))j=0,...,d2 , are the bases of

our spline spaces in ξ and η. In order to construct an injec-tive map from S to Ω, we solve Laplace equations

∆ξ(x, y) = 0,∆η(x, y) = 0, (2)

with boundary conditions; namely

• ξ(x, y) = 0 on Γ1, ξ(x, y) = 1 on Γ3 (Dirichlet) and∇ξ(x, y) · t∂Ω(x, y) = 0 on Γ2, Γ4 (Neumann),• η(x, y) = 0 on Γ2, η(x, y) = 1 on Γ4 and (Dirichlet)

and ∇η(x, y) · t∂Ω(x, y) = 0 on Γ1, Γ3 (Neumann),where t∂Ω(x, y) is tangent to the boundary ∂Ω at the point(x, y) ∈ ∂Ω. The equations (2) are pulled back in the para-metric domain and give

(x2η+y2η)∂2U

∂ξ2−2(xξxη+yξyη) ∂

2U

∂ξ∂η+(x2ξ+y

2ξ)∂2U

∂η2= 0 (3)

for U = x(ξ, η) and U = y(ξ, η).Then the equation (3) is solved by a relaxation technique,

which consists at step k to use the computed functions xk(ξ, η),yk(ξ, η) instead of x(ξ, η) and y(ξ, η) in the equation (3), inorder to compute the solutions xk+1(ξ, η), yk+1(ξ, η) at stepk+1. If U is expressed in the B-spline basis associated to theparametric domain S, each iteration requires the solution ofa linear system in the coefficients c and the evaluation ofthe functions xξ, xη, yξ, yη at some well-chosen interpolationpoints in S. The initial parametrisation (x0(ξ, η), y0(ξ, η))is constructed using Coons patches [2].

3. EXPERIMENTATIONWe demonstrate our method by two examples in 2D. The

implementation is conducted in the Axel1 platform. In thefirst example, we start with a shape defined by 4 B-splinecurves. Then we construct the Coons surface showed in theleft picture. We solve the PDE system and check if the re-sulting surface is injective or not by the method in [8]. Ifnot, we do knot insertion and consider the resulting surfaceas the initial solution in this next step. The right pictureis the resulting after 4 steps of knot insertion. The param-eterization has 82 × 82 control points and is injective. Thezoom-in picture (a) is the result for 3 steps and it is still notinjective. We get the better result in (b) after 4 steps.

1http://axel.inria.fr

The second example is a shape with very narrow segments.The left picture shows very tangled initial Coons surface. Af-ter 4 steps of knot insertion, we get the non-self-overlappingshape in the right.

In the comparison with the method described in [7], theadvantages of our method are that the boundary curves arekept exactly as the original ones and the PDEs are solved inthe parameter domain instead of the physical domain. Bothmethods seem to have difficulties with the high curvatureregions where the iso-curves are nearly parallel.

The method is extendible to 3D by solving a PDE systemof three variables. In the full paper, we will report on prac-tical experimentation of this approach in 3D. We will testdifferent boundary conditions and homotopy techniques tomap one solution for a simple domain into a solution for adomain with a more complex boundary.

4. REFERENCES[1] E. Cohen, T. Martin, R.M. Martin, T.Lyche, and R.F.

Riesenfeld. Analysis-aware modeling: Understandingquality considerations in modeling for isogeometricanalysis. Comput.Methods Appl.Mech.Engrg, 2009.

[2] G. Farin and D. Hansford. Discrete Coons patches.Comput. Aided Geom. Des., 16(7):691–700, 1999.

[3] J. Hoschek and D. Lasser. Fundamentals of computeraided geometric design. Wellesley, 1993.

[4] T.J.R Hughes, J.A. Cottrell, and Y. Bazilevs.Isogeometric analysis: CAD, finite elements, NURBS,exact geometry, and mesh refinement. ComputerMethods in Applied Mechanics and Engineering,194(39-41):4135–4195, 2005.

[5] X. Li, H. Xu, S. Wan, Z. Yin, W. Yu. Feature-alignedharmonic volumetric mapping using MFS. Computers& Graphics, 34(3):242-251, 2010.

[6] T. Martin, E. Cohen, and R.M. Kirby. Volumetricparameterization and trivariate B-spline fitting usingharmonic functions. Computer Aided GeometricDesign, 26(6):648-664, 2009.

[7] T. Nguyen, B. Jüttler. Parameterization of SimplyConnected Domains Using Sequences of HarmonicMaps. submitted to the Proceedings of Curves andSurfaces, Avignon, France, 2010.

[8] G. Xu, B. Mourrain, R. Duvigneau, and A. Galligo.Parametrization of computational domain inisogeometric analysis: methods and comparison.Computer Methods in Applied Mechanics andEngineering, In Press, Accepted Manuscript, 2011.

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