International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2020.13.2.476 Vol. 13, No. 2, April-June 2020 ISSN (Online): 1882-9554
Original Paper
Surface Mesh Generation in Parametric Space Using a Riemannian Surface Definition
Cui Dai1, Zhaoxue Wang 1, Liang Dong2, Yiping Chen1, Junfeng Qiu1
1School of Energy & Power Engineering, Jiangsu University, Zhenjiang 212013, Jiangsu, China, [email protected], [email protected], [email protected], [email protected] 2Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, Jiangsu, China, [email protected]
Abstract
In order to solve the problem of generating distortion elements in the mapping from parameter space to real space, and the boundary coincidence of the mesh generated by the software quality, an approach for parametric surface mesh generation based on Riemannian metric, combined with Delaunay triangulation and AFT is proposed. In our algorithm, the boundary curves are discretized based on the proximity and curvature of the curves in the model after derivation the correlation of curve length between parametric space and real space. Background meshes of parametric space were generated by using improved AFT, and could improve the efficient of the algorithm and control element sizing and metric values. When background mesh of parametric space were refined, to counteract mapping distortion, the traditional Delaunay incremental insertion kernel is replaced by inserting the center of triangle circumscribed ellipse, and the algorithm for locating ellipse center and judging whether nodes within ellipse. In this paper, the details of the surface mesh generated by the algorithm are introduced in detail. The algorithm proposed in this paper has the characteristics of reliable algorithm, high mesh generation efficiency and mesh quality. Finally, the reliability of the proposed algorithm is verified by an example of surface mesh generation. Key words: Surface mesh generation; Riemannian metric; advancing front method; Delaunay triangulation; Background mesh
1. Introduction
Surface mesh generation is one of the most difficult and yet important prerequisites of mesh generation in three dimensions. The surface mesh influences the volume mesh close to the boundary of the domain. The smoothness of the surface mesh, and the conformity of this mesh to the domain boundary, directly affects the quality of any further finite element simulation based on the mesh. The accuracy of the numerical solutions and the convergence of the computation scheme are strongly related to the quality of the underlying surface meshes [1] (Since surface meshes define external
Received March 3 2020; accepted for publication April 9 2020: Review conducted by Xuelin Tang. (Paper number O20005k) Corresponding author: Dong Liang, [email protected]
476 and internal boundaries of computational domains where boundary conditions are imposed). There are two popular surface mesh generation methods [2]: direct method and indirect method. The indirect method is conceptually straightforward as a two-dimensional mesh is generated in the parametric domain and thus it is expected to be faster than the direct approach. In addition, Indirect method is strong the adaptiveness for complex curved-surface, so indirect method are among the most commonly used surface mesh generation. The problem with the methods is the generation of a mesh which conforms to the metric of the surface. In fact, they aimed to minimize the error in the polyhedral approximation of the surface indirectly in the parametric space without paying attention to the quality of the resulting mesh. The mesh in the parametric surface is usually anisotropic, due to the metric deformation from the surface to its parametric domain. Thus, for people in finite element computation, the problem is reduced to the generation of an anisotropic mesh in the parametric domain. Cuillière[3] developed automatic mesh generation procedures based on advancing front methods (AFT) and featuring a priori nodal density calculations, and focused on the discretization of 3D parametric surfaces with strong variations of curvature with respect to a nodal density function with steep gradients. This method can be more accurately calculate element distortion, but may not be so general. In [4], Delaunay method was extended to anisotropic context of 2D domains, and a Riemannian metric map is introduced to remedy the mapping distortion from object space to parametric domain. The algorithm is easier to implemented and efficient, but it doesn’t suit to the surface that differs very much from interior to the boundary. In [5], Guan presented a new mesh generation procedure which is suggested for the triangulation of general combined parametric surfaces using an advancing front approach and metric tensor. The calculation and interpolation method of arbitrary points in surface’s parametric space are detailed. Tristano [6] presented a method for meshing 3D parametric surfaces using the advancing front method with a Riemannian surface definition. The details of the creation of the metric map used to determine the amount of distortion of the elements in parametric space are given along with the details of an advancing front algorithm that utilizes the metric map. The method overcomes anomalies found in the warped parametric space and direct 3D methods. Borouchaki[7]presented an indirect method for meshing parametric surfaces conforming to a user-special size map. First, from the size specification, a Riemannian metric is defined so that the desired mesh is one with unit length edges with respect to the related Riemannian space (the so-called ‘unit mesh’). Then, based on the intrinsic properties of the surface, the Riemannian structure is induced into the parametric space. Finally, a unit mesh is generated completely inside the parametric space so that it conforms to the metric of the induced Riemannian structure. Experiments [8] prove that Riemannian metric has well effect on eliminating mapping distortion. If only Riemannian metric is reasonably defined, the quality of surface mesh in 2D parametric domain can be controlled. A significant amount of research has gone into meshing surfaces using a metric map with Delaunay triangulation or Advancing Front Technique independently. However, those mesh generation methods have their own disadvantages (for example, AFT cannot assure the quality of the generated mesh to meet the standards, Delaunay triangulation needs to look for the triangle with the biggest dimensionless radius in generating internal nodes, so can be time-consuming. In addition, they calculate the size and metric of the surface by calling related functions in the surface mesh generating process and tend to be slower and less robust. Little attention has been paid to the automatic generation of mesh size and metric functions on a background mesh simultaneously. Most this work is only devoted to the automatic generation of mesh size functions [9-11]. In order to strikes a balance between surface mesh quality and computational cost, this paper proposes an approach for parametric surface mesh generation based on Riemannian metric, combined with Delaunay triangulation and AFT. The control element sizing and metric values are provided in a set of discrete points (vertices of a background triangular mesh) covering the parametric domain. Those discrete points are generated by AFT. The interpolation on individual triangles of background control mesh is then used to approximate the relevant mesh control function in the rest of the parametric space. Delaunay triangulation is then used to refine the mesh of parametric domain.
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2. Overall Algorithm
In this section, the overall surface mesh generation scheme is presented. Algorithmic details for each of the major steps in this scheme will be presented later in the paper. More specifically, the proposed algorithm involves the following steps: Read geometry information on parametric surfaces, and determine the corresponding boundary curves. According to the boundary curves on discrete curves with given metrics, make out the coordinate of all discrete points（section 3.1）. According to the discrete results of parameter curves, generate background mesh by using improved AFT and store the metrics of all nodes（section 3.2）. Refine the parameter surfaces using Delaunay until the meshes meet the required quality（section 3.3）. Map the meshes onto surface physical spaces. The proposed algorithm can be used for both flat and curved surfaces. While curved surfaces require the use of a metric map, flat surfaces can use the identity matrix for the metric, provided an initial transformation of the three dimensional boundary nodes to the x-y plane is first accomplished.
3. Details of the Algorithm
This section will define the metric and its uses. It will also describe the creation of the background mesh and the surface mesh that utilize the metric.
3.1 Definition of the metric Metric at a point: the metric tensor at a certain point P in 3D space that defines element size characteristics is a 3×3 matrix, M3p, which has the form [12]
a b c 1 M== b d f e e e e e e T 3p 1 2 3 2 1 2 3 c f e 3 (1) where λi, ei(i=1, 2, 3) is the corresponding eigenvalue and eigenvector of Matrix M3p, and det(M3p)>0, λi>0. Metric specified on the surface: when parametric surface r (u, v) is meshed, the metric tensor of the surface itself and 3D points specialized by users need to be taken into account simultaneously. Then, we get the metric, Msp, at a certain point P on the surface, as follows
τ1 τ ττν MMsp= 2 3 p () 1 2 ν 2 (2)
rr ν = uv τ = r τ = r rr where 1 u , 2 v , uv and the notation of 2 indicates that only the first two lines and two columns of the matrix are considered.
3.2 Discretizing Boundary Curve Curve meshing is one of the main steps in the meshing process of planes, curved surface and volumes. In fact, most of the automatic mesh generation methods for domain in R2 or R3 build the desired covering up from the data of the 478 boundary meshing delimiting the domain considered. The mesh of a domain is strictly dependent on the mesh of its boundary. Thus, the properties of the latter are one of the parameters influencing the quality of the final mesh. The curve discretization is controlled by checking the curve deviation from the discrete segments. If all the deviation values are smaller than a specified value, the curve discretization is considered to be finished. Euclidean geometric length of a curve: the physical space surface (X, Y, Z) is denoted by Σ, whose corresponding parametric surface (u, v) is denoted by Ω. Moreover, we consider an interval [a, b] in R. The curve PQ on Σ is denoted by a function γ in the interval [a, b] in R. This curve is also the image by the function σ of a curve AB which comes
(tt )= ( ( )) from a function ω of the interval [a, b]. Therefore, we have .The length of the curve PQ in Σ can be defined as follows:
bb L( PQ )== '( t ) dt '( t ), '( t ) dt aa (3) τ ==r as τ1 ==ruu , 2 vv, hence ,we find
b T ` L( PQ )= ' ( t ) M3 p '( t ) dt a (4) Through eq. (4), we have found the link between the length of curve AB in parametric space and curve PQ in Σ.
Also, we have found the matrix Msp corresponding to the given matrix M3p (see eq. (2) and eq. (3)). The length of curve
PQ for the metric M3p is then:
b b T T LMp( PQ )= '() t M3 (()'()) t t dt LMp( PQ )= w '() t M3 (()'()) w t w t dt 3 p a (5) or 3 p a (6) Discretizing boundary curve algorithm is generally part of a surface mesh generation procedure including: Firstly, we get geometry information on parametric surfaces and metric. Then, Romberg formula is used to subdivide curve into a set of curve segments whose length size is less than a given threshold value (for example equal to 0.1). Calculate the length L of the curve for eq. (6), and compute the nearest integer n to L. Finally, we determine the real values ti by (7). We can get the coordinate of discrete points by the values ti.
L ti+1 = w'T ( t ) M ( w ( t ) w '( t )) dt t sp n i (7)
where, 0≤i≤n-1，t0=a，tn=b. Figure1 shows the discrete results of blade boundary curves. We can see that the points in the section of large curvature are more than that in the section of small curvature. That is because large curvature equals to large metric of
MSP（w(ti)). So the less the distance of discrete points is, the more the points in the unit length are. So, this method allows the mesher to capture the curvature of the surface by putting more element divisions on highly curved boundaries.
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Fig. 1 boundary curves of a blade
3.3 Background mesh generation For the sake of efficiency, a background mesh is defined to control element sizing and metric values in this section. The background mesh consists of selected points in the parametric domain where both size and metric are known exactly. The mesher can utilize this information to interpolate local size and metric data as a function of the parametric u, v coordinates. One of the benefits of computing the metric map and size map using the same background mesh is that all sizing information and metric information is stored in one place. A second benefit is that the metric of a point in parametric space can be determined by a simple interpolation rather than a costly evaluation of the surface followed by the computation of the metric. An additional benefit is the ability to refine the background mesh based on size and metric values at the same time on the same mesh. This is a fast and convenient way to compute essentially two different maps on one mesh. The main steps of the background mesh generation algorithm can be summarized as follows: (1) Set AB as the initial front .Compute Pa（resp. Pb）, so that the triangle ABPa（resp. ABPb） is equilateral with respect to the metric .
P'=+ A ( AP / L ( AP )) (2) Set P=(Pa + Pb）/2, and P is the candidate point. Iteratively adjust P’s position. Set a and
P'=+ B ( BP / L ( BP )) b ,where L(AP）and L(BP) is the corresponding Riemannian metric, and
1 1 T T L(AP) = AP M sp (A + t AP)APdt L(BP) = BP M sp (B + tBP)BPdt 0 0 .
PPP=+( ' ') / 2 (3) Set ab , and P is the optimal candidate point. Msp(P)is computed by the coordinate of P. The
PxT M Px =1 ellipse ( sp ) of which the center is P determined by the metric of Msp(P). Adjacent points (not including A and B) are searched in the ellipse, and store those points. (4)Those Adjacent points and P are connected with edge AB, and some triangles are created. The highest-quality triangle is stored, and the corresponding point is the best optimal inserting point. The metric and size of the point is stored. (5) Update the initial front and the current mesh. (6) If the front is not empty, return to Step1. In order to reduce the numbers of inserting points, adjacent points are preferred to use in Step 4. When the quality of
480 the triangle (made up by P and AB) is computed, the quality is multiplied by factor τ, τ with (0.7~0.9). The element size and metric values is determined anywhere in the domain through linear interpolation inside the corresponding triangle. Once the background mesh is generated, the 3D sizes and metrics are stored at the nodes of the 2D background mesh for later reference.
3.4 Delaunay refinement algorithm with Riemannian metric considered A new point is inserted in the center of the triangle’ circumcircle for the traditional Delaunay refinement algorithm and there doesn’t exist any nodes within the triangle’ circumcircle. While in the cases with Riemannian metric considered, the circumcenter of the elements is obtained by solving many complex nonlinear equations. Its computing process is very difficult and time-consuming. And, the ellipse in parametric space is needed to be used to replace the triangle circumcircle. In this paper, a simplified parameter space method is applied. The method can avoid solving a large number of nonlinear equations. Specific process is as follows The center of the triangle K, circumcircle OK is obtained by solving equations (8). Where Vi(i=1,…,3) is the vertex of K , the vertex coordinate is [xi, yi], the OK coordinates is[Ox，Oy]，and Msp=[mij]1≤i，j≤2. Combining eq. (6) and eq. (8), we can obtain Linear eq. (9).
LOVLOVM()() k12= M k a bO c 33PP 2 2x = 2 LOVLOV()()= a bO c M33PP k13 M k (8) 3 3y 3 (9) where ai=2(m11(xi-x1)+m12(yi-y1)),bi=2(m22(yi-y1)+m12(xi-x1))and
c = m x2 + 2m x y + m y2 −(m x2 + 2m x y + m y2 ) i 11 i 12 i i 22 i 11 1 12 1 1 22 1 So we can get Coordinates OK, OKE is the distance from OK to any vertex of triangle K. OKP is the distance from P to the center of the triangle K circumcircle. So, if OKE>OKP, P is considered to be inside the ellipse. If OKE=OKP, P is considered to be on the ellipse. If OKE CA CB K =23 i CA2++ AB 2 BC 2 ,where AB，BC，CA is respectively the three edges of the triangular. 4. Results and Comparisons Figure2 (a)-(c) show the parametric surface mesh generated by a mesh generation software and the proposed algorithm in the paper. We can see the isotropy of the mesh generated by the software is better, but its boundary coincidence is poor (see red arrows in Fig. 2 (b)). And because the proposed algorithm considers the Riemannian metric specified on surfaces and employs AFT, it makes the meshes be good coincident with the original model. In Table 1, the mesh quality for the two different algorithms are computed using the mesh quality metric Ki. It can be seen that fewer elements are generated by the proposed algorithm, and the quality of the worst mesh and the average mesh is higher. 481 (a) Model (b) mesh generation software (c) The proposed algorithms Fig. 2 Mesh generation of parametric surface Table 1 Comparison of the quality of the mesh for different algorithms The distribution of surface mesh quality Max Min Mean 0.0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 mesh generation 0 0 41 248 407 0.9999 0.5754 0.9175 software The proposed 0 0 16 187 449 0.9999 0.5951 0.9362 algorithms Figure3 (a) is a compound surface mesh. Fig. 3(b) shows the relation of the number of elements and the time, which is linear. The whole process includes the discretization of boundary curves and generation of triangular elements, etc. It is finished in PC with frequency 20GHz and memory 1GB. 100,000 triangular elements are generated in one minute. Fig. 3(c) is the distribution of surface mesh quality, the minimum of the quality coefficient is 0.254, the maximum is 0.999, and the mean value is 0.908(the quality measurement criterion for surface mesh is Ki). 210000 0.6 180000 0.5 150000 0.4 120000 0.3 90000 elements(%) 60000 0.2 The number of elements 30000 0.1 0 0 0 20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 Time（S） Mesh quality (a) The surface mesh generated (b) The relation between the number of elements and the time (c) The distribution of surface mesh quality Fig. 3 Example of surface mesh generation 5. Conclusion 1) We have proposed an effective surface mesh generation algorithm based on Riemannian metric by combining Advancing Front Technique and Delaunay triangulation. A background triangulation is generated to control the distribution of node points generated in the interior by advancing front technique. The background triangulation can be also used to develop local guidelines for the size and metric of elements. 2）Furthermore, our approach ensures the quality of boundary triangular elements, and gives full play to the advantage of Delaunay triangulation by which the quality of surface grid obtained is higher after the mesh generated in parameter domain is mapped back to physical space. 3）Compared with the mesh generation software, the surface mesh generated by the algorithm has better boundary coincidence and maintains good consistency with the original model. The number of meshes can be well controlled, and 482 the efficiency of surface mesh generation is effectively improved. 4）As a tool for generating surface meshes, the algorithm embodies certain advantages in theory. In order to further study this method, the algorithm will be applied to the meshing of the physical model in the future, and then its performance will be verified by numerical simulation. Acknowledgements This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0200901, 2017YFC0804107), National Natural Science Foundation of China (No. 51879122，51779108，51779106, 51509111), the association innovation fund of production, learning, and research (BY2016072-01), Zhenjiang key research and development plan (GY2017001, GY2018025), the Open Research Subject of Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University (szjj2017-094，szjj2016-068), Sichuan Provincial Key Lab of Process Equipment and Control (GK201816), the Advanced Talent Foundation of Jiangsu University (15JDG052) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and Jiangsu top six talent summit project (GDZB-017). References [1] R. V. Garimella , J.Mikhail. and P. M. Knupp, Comput, 2004, Methods Appl. Mech. Eng.. 193, 913. [2] X.D.Huang，W.S.Ding and Q.G.Du , 2010, J J. 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G Paul, 2000, Mesh generation[M]．Oxford：Hermes Science Publishing，2000：489-551 483 Cui Dai, female, born in January 1984, Ph.D., associate professor, master tutor. Her research areas are unstable transient flow characteristics of hydraulic machinery, cavitation dynamics and cavitation mechanism of hydraulic machinery, and flow-induced noise characteristics of hydraulic machinery. Presided over 2 projects of national natural science foundation of China. Presided over the Chinese Doctoral Fund, the Jiangsu Postdoctoral Fund, and the Open Project of the Key Laboratory of the Ministry of Education. The first and corresponding author published more than 20 SCI / EI papers; 2 authorized invention patents. Zhaoxue Wang, female, born in January 1994, postgraduate student of Jiangsu University, mainly engaged in related topics of centrifugal pump cavitation, noise, bionic centrifugal pump blade. Liang Dong, male, born in August 1981, PhD, associate researcher, doctoral tutor, visiting scholar at the University of Minnesota, USA; Currently participated in the National Natural Science Foundation Project, National Science and Technology Support Project, Jiangsu More than 10 projects at and above the provincial and ministerial level, such as special achievement transformation projects, key projects of science and technology in Jiangsu Province, and more than 10 projects commissioned by enterprises; 1 published book, the first and corresponding author published more than 30 academic papers, including more than 20 papers included in SCI/EI; authorized invention More than 10 patents and more than 10 software copyrights. Yiping Chen, male, born in May 1995, postgraduate student of Jiangsu University, mainly engaged in related topics of centrifugal pump noise and bionic blade optimization. Yiping Chen, male, born in August 1995, postgraduate student of Jiangsu University, mainly engaged in related topics of centrifugal pump noise and centrifugal pump fault diagnosis. 484