A new surface parameterization method based on one-step inverse ...

Int J Adv Manuf Technol (2013) 65:1215–1227 DOI 10.1007/s00170-012-4251-8

ORIGINAL ARTICLE

A new surface parameterization method based on one-step inverse forming for isogeometric analysis-suited geometry Xue-Feng Zhu & Ping Hu & Zheng-Dong Ma & Xiangkui Zhang & Weidong Li & Jingru Bao & Mingzeng Liu

Received: 3 December 2011 / Accepted: 21 May 2012 / Published online: 9 June 2012 # Springer-Verlag London Limited 2012

Abstract In an attempt to construct an isogeometric analysissuited geometry for isogeometric analysis, a new surface parameterization method using the one-step inverse forming (SPIA) is proposed. Initial generation of watertight analysissuitable geometry (NURBS surfaces) with complex shapes can be a significant bottleneck for isogeometric analysis because computer-aided design models often include ambiguities such as gaps and overlaps. Most of traditional surface parameterization techniques are based on geometric method and limited to finite meshes, while SPIA is a physics-based method using sheet metal forming technique with large elastic–plastic deformation and robust enough and rapid to deal with the finite elements mesh with over 100,000 nodes within 2 min without the necessity to simplify the meshes. Using Coons surface parameterization, global mesh parameterization, and NURBS reconstruction, we can rebuild new computer-aided design models with errors under any tolerance to which isogeometric analysis can be applied. The NURBS surfaces after reconstruction are also used for computer-aided manufacturing. Keywords Isogeometric analysis . Surface parameterization . One-step inverse forming . Finite The first author is a visiting Ph.D. student at the University of Michigan. X.-F. Zhu : P. Hu (*) : X. Zhang : W. Li : J. Bao : M. Liu State Key Laboratory of Structural Analysis for Industrial Equipment, School of Automotive Engineering, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, People’s Republic of China e-mail: [email protected] Z.-D. Ma Department of Mechanical Engineering, The University of Michigan, Ann Arbor 48103, USA

element . Isogeometric analysis-suitable geometry . Surface reconstruction

1 Introduction The concept of isogeometric analysis, introduced by Hughes, Cottrell, and Bazilevs [1] and further developed by Cottrell [2] and Bazilevs et al. [3], was initially motivated by the gap existing between computer-aided design (CAD) and finite element analysis. The idea is to have one and only one representation of geometry (initial control mesh) that exactly encapsulates the analysis-suitable geometry (ASG). The fields in questions (including displacement, velocity, temperature, etc.) are represented in terms of the same NURBS basis functions that describe the geometry. Coefficients of the basis functions, or control variables, are the degrees of freedom (DOF) of the discrete system [4]. Cottrell et al. [2] mentioned that the ASG must be constructed but that this could be a timeconsuming process as the mesh generation. The main cause for this is that isogeometric analysis demands that the NURBS surface composed of NURBS patches be C0-continuous so that the displacement field is at least C0 continuous across patch interfaces. Most geometric models, however, are composed of trimmed NURBS patches whose control nets have no topological relationship, which causes the initial control net to be discontinuous. Additionally, the boundary of each trimmed NURBS patch cannot be accurately represented by the control point; this prevents the boundary conditions from being applied to the control variables. Therefore, the control nets belonging to different patches must be merged into a watertight control net. Most of stamping parts of the car body are composed of trimmed NURBS patches that contain many overlaps and gaps, as shown in Fig. 1. Therefore, constructing isogeometric ASG models for sheet metal stamping can be

1216

Int J Adv Manuf Technol (2013) 65:1215–1227

parameterization; Section 3 gives the basic principles of one-step inverse approach (OSIA); Section 4 reveals how to parameterize the mesh data into high continuous surfaces by using Coons surface parameterization and NURBS reconstruction. Finally, we provide summary and concluding remarks in Sections 5 and 6.

2 Related works and surface parameterization—a brief review

Fig. 1 Overlap and gap among the trimmed NURBS patches of a car body

difficult. This has limited further development of isogeometric analysis. NURBS is the standard freedom surface representation in CAD/computer-aided manufacturing (CAM). However, it still exhibits some limitations due to its tensor-product forms and rectangle topology. Many state-of-the-art CAD systems have to cut off the superfluous regions of a NURBS surface and then to tessellate those trimmed NURBS patches into complex CAD surfaces. It is this process that produces a great deal of overlaps and gaps. The overlap and the gap have been the most serious impediments to interoperability between CAD, CAM, and CAE systems. Software for isogeometric analysis will not work properly if the model contains unresolved gaps. It has been estimated that to close the gaps and overlaps would cost the US automotive industry over 600 million annually [1]. Even if the gaps and overlaps are eliminated and all patches are merged together, the model still fails to fulfill isogeometric analysis because almost all of the patches are trimmed NURBS surfaces. It is common that a complicated CAD model contains thousands of trimmed NURBS patches. As shown in Fig. 7, there are more than 900 NURBS patches composing the inner door model. Though isogeometric analysis for trimmed CAD surfaces has been researched, it is limited to a single trimmed NURBS patch. To the best of our knowledge, how to apply isogeometric analysis to two or more trimmed NURBS patches remains a challenge. Surface parameterization based on one-step inverse method (SPIA) provides another alternative: reparameterizing the CAD model and reconstructing it into a watertight NURBS surface, thus keeping the error under tolerance, or segmenting the surface into numerous compatible NURBS patches. SPIA is capable to introduce less angular distortion and unfold complex surfaces step by step. The rest of the article is organized as follows: Section 2 reviews the recent developments in the field of global

Several approaches have been put forward to construct geometry suitable for analysis for isogeometric analysis. None of them, however, resolves it perfectly. As a superset of NURBS, T-spline is a helpful tool for the integration of CAD and CAE. T-spline can give local refinement and use fewer control points than NURBS does to depict same CAD model. Bazilevs et al. [5] developed isogeometric analysis using T-spline and proved its powerful analysis capability. As an open standard, NURBS has been widely adopted by many CAD systems. Although T-spline is more powerful than NURBS, it still takes a long time to replace NURBS. The Boolean/feature-based trimming technique is still widely adopted. Regarding a complex surface with many NURBS patches, the creator of T-spline also stated that two NURBS patches can be merged into a single Tspline patch easily but that a solution for how to handle three or more intersecting surfaces is not available and is a topic for further study [6]. Kim et al. [7] tried to apply isogeometric analysis directly to the trimmed NURBS, but until now, to our knowledge, their methods could address only a single patch with multiple trimming curves. How to analyze two or more intersecting trimmed CAD surfaces also has not been researched. Mesh parameterization [8] provides an alternative choice. Mesh parameterization has been a very active research topic for the last several years. Its applications include texture mapping, geometry processing, remeshing, mesh simplicity [9], reverse engineering [10], pasting, and reconstruction [11]. Surface parameterization [12], as a branch of mesh parameterization, is suitable for constructing the NURBS surface that is used for isogeometric shell analysis. The main idea is this: Given a surface (mesh) S in 3D, find a mapping function to enable the adjustment from 2D to 3D as shown in Fig. 2. In the past, many surface parameterization methods have been developed. The mostly isometric parameterization of surfaces method [13] was, to our knowledge, the first mesh parameterization method to compute a natural boundary. All methods can be separated into two methods: linear methods and nonlinear methods. Linear methods are faster but fail to produce natural

Int J Adv Manuf Technol (2013) 65:1215–1227

1217

Fig. 2 Mapping functions of surface parameterization

boundaries while nonlinear methods are slower than linear methods but can produce more accurate boundaries. Additionally, all methods can be characterized as angle-preservation methods or area-preservation methods. In the field of computational mathematics, there is always controversy about the criteria standard. Kai Hormann et al. [14] provided four standards to evaluate those mesh parameterization methods:

OSIA has the potential to become a new geometric modeling technique for reverse engineering. SPIA may produce a watertight C2 NURBS surface with a watertight NURBS control net. Moreover, the SPIA and SPIA-based isogeometric analysis will allow for a seamless integration between CAD and CAE.

1. 2. 3. 4.

3 Introduction to NURBS and Coons surface parameterization

Minimal distortion: as close to isometry as possible Global optimization: boundaries develop naturally Bijectivity: no fold-overs of parameter triangles Fast enough

Isometry is an ideal objective but is impossible in most cases. Many mathematicians focus on the methods providing angle preservation. ABF++ and Global Conformal Mapping based on Ricci flow are two of the best methods for angle preservation. ABF++ is one of the fastest and robust methods [15], but it fails to give symmetric boundaries. Global conformal parameterization based on Ricci Flow [16] gives the best conformal mapping. Though it almost eliminates all the angle distortion, it becomes extremely slow when dealing with a large amount of mesh data or meshes with complex boundaries. SPIA is a physics-based method based on shell theory with large elastoplastic deformation. SPIA is almost as fast as ABF++ and is more robust than ABF++ methods; this is because the present methods can introduce less angular distortions. An inverse finite element method named the OSIA, which is based on large deformation theory, is employed to construct the isogeometric analysis-suited geometry. OSIA is used mainly in the unfolding of complicated 3D surface and successfully applied in the field of automobile body manufacturing, including sheet metal forming, the unfolding of auto-body parts with flanges, the prediction of the flanging lines in progressive die design, and so on. Considering there are a great deal of 3D open surfaces in CAD/CAM, especially in the automobile body field,

NURBS is built from B-splines, which are piecewise polynomial curves or surfaces composed of a linear combination of B-spline basic functions. The B-spline parametric space is specific to “patches” as opposed to finite elements in which each element carries its own parameterization. Patches may be thought of as subdomains. A NURBS surface formulation can be generally expressed as S ðx; ηÞ ¼

n X m X i¼1 j¼1

ðpÞ

ðqÞ

Ni ðxÞMj ðηÞwij Pn Pm ðpÞ ðqÞ i¼1 j¼1 Ni ðxÞMj ðηÞwij |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Rij

Pij ¼

n X m X

Rij Pij

ð1Þ

i¼1 j¼1

It can be defined on a knot interval called a knot vector. A knot vector in one dimension is a set of coordinates in the parametric space written as   ð2Þ Ξ p ¼ x1 ; . . . ; xnþpþ1 where ξi∈R is the ith knot, i is the knot index, i ¼ 1; 2; . . . ; p þ n þ 1 is the polynomial order of the B-spline, and n is the number of basic functions corresponding to it. The knots partition the parameter space into elements, and the interval [ξ1, ξn+p+1] constitutes a patch. A knot vector is said to be uniform if its knots are uniformly spaced and are non-uniform

1218

Int J Adv Manuf Technol (2013) 65:1215–1227

otherwise. Knot values may be repeated; that is, more than one knot may assume a given value. For example: 9 8 > > = < ð3Þ Ξ p ¼ 0; 0;    ; 0 ; Ξ int ; 1; 1;    ; 1 |fflfflfflfflfflffl{zfflfflfflfflfflffl} > > ; :|fflfflfflfflfflffl{zfflfflfflfflfflffl} pþ1copies

pþ1copies

A trimmed NURBS is defined by a NURBS and a trimmed curve. The trimmed curve is not really on the NURBS surface. However, when OPENGL renders the trimmed NURBS patch, the surface external to the trimming curve is abandoned. That is why so many gaps exist in the CAD data of aerospace or automotive parts. As shown in Fig. 3, a trimmed surface is depicted, which is defined by a NURBS control net and a trimming NURBS curve. In CAD files, the design is encapsulated in some types of CAD model. These models often include ambiguities such as gaps and overlaps that make it inappropriate for the finite element analysis. It should be mentioned that the ambiguities must be removed and that the defection must be corrected to arrive at an ASG. Creating mesh and merging these gaps in trimmed NURBS models can be one of the most timeconsuming steps in the traditional finite element analysis process. The same is true for the isogeometric analysis process.

4 SPIA: surface parameterization based on one-step inverse method 4.1 Introduction to one-step inverse approach The OSIA for sheet metal forming analysis predicts an initial blank shape based on a final deformed shape in one-step calculation with the aid of the large deformation theory. Due to the fact that it is much less time-consuming, OSIA is

Fig. 4 General description of the inverse approach

efficient and well suited for the initial body part design, compared with incremental approaches like PAM-STAMP or LS-DYNA. The first work related to the simulation of sheet metal forming using IA (which is also called the “one-step approach”) was carried out in the early 1980s, simultaneously in Europe and in the USA. Some scholars, including Majlessi and Lee [17], Batoz et al. [18], Liu and Assempoor [19], Lee and Huh [20], Chung and Richmond [21], as well as Zhang et al. [22], had made excellent contributions to the research. The starting point of the inverse simulation is the finite element model of the stamped part depicted in Fig. 4. The algorithm of inverse simulation enables the user to find the position of the stamped part’s nodes on the blank’s initial geometry. The calculated displacements field ensures the equilibrium of the stamped part, through the calculations of the deformation (thinning, elongation), and of the constrained boundary conditions. It may be said that the IA is built on the final shape, whereas the unknowns arise between the flat sheet metal (result of the calculation) and the stamped part (starting point of the calculation). Table 1 gives the scheme of the IA. For satisfying the force equilibrium conditions with the PVW in C, the nodal positions in C0 are updated in an iterative manner. In addition, in order to avoid the pathdependent incremental procedure of calculating plastic deformation and associated contact problems, the following assumptions are made: 1. Large elastoplastic strains with full incompressibility Table 1 Known and unknown quantities of IA

Fig. 3 A trimmed NURBS patch

Known quantities

Unknown quantities

C0 initial shape (flatten roughcast)

Thickness h Initial stress 0 Initial strain 0

C entire shape (part)

Geometry shape Vertical displacements of nodes Resistance and friction

Horizontal Displacements of nodes Roughcast’s shape and size Thickness h Stress and strain Die force

Int J Adv Manuf Technol (2013) 65:1215–1227

1219

deformation. A fundamental concept of the shell theory with large elastic–plastic deformation is that all distortion measures for a given deformation, including length, angle, and area distortion, can be expressed in terms of the inverse deformation gradient tensor and the left Cauchy-green tensor describing the length and angle variations. A shell is a three-dimensional body with one of the shell dimensions much smaller than the other two and nonzero curvature of the shell’s mid-surface. As shown in Fig. 5, in the planar parametric domain and space surface, the position of node p on middle surface can be expressed as follows: Fig. 5 Kinematic description of One-step inverse method

Xp0 ¼ Xp  Up 2. Proportional loading, viz. deformation theory of plasticity, without unloading 3. Material hardening law with isotropy or normal anisotropy

ð4Þ

where Up is the displacement vector of node p. It can be described in the global coordinate system (i, j, k) as well as the local coordinate system (s1, s2, n) at node p on the space surface as follows:

These assumptions lead one-step IA to a total and direct method with high efficiency and low memory cost because only two DOF per node are considered.

Up ¼ Ui þ Vj þ wk ¼ us1 þ vs2 þ wn

4.2 Mathematical description of one-step inverse finite element method

According to the Kirchhoff straight-normal assumption, as shown in Fig. 1, the position vector of node q located on the normal line from node p is written as

In this paper, we think of the surface meshes as the thin shell meshes of sheet metal that can endure large plastic

Xq ¼ Xp þ zn

ð5Þ

ð6Þ

Fig. 6 Flow chart of local unfolding for complex model (an application in UG NX 5 of the present method. Courtesy of Mr. Jerry Hu in UG Co.)

1220

Int J Adv Manuf Technol (2013) 65:1215–1227

where Xp ¼ x1p s1 þ y2p s2 þ z3p s3 and z stands for the position in thickness direction of space surface. In the planar parametric domain, we have x0q ¼ x0p þ z0 n0

ð7Þ

where n0 ¼ kx s1 þ ky s2 þ kz n. Thus, the position of node q in the planar parametric domain can be obtained by Eqs. (4) and (7) as follows: x0q ¼ xp  up þ z0 n0 It can also be written in local coordinate system:     Xq0 ¼ x1p  u þ kx s1 þ y1p  v þ ky s2   þ z1p  w þ kz n

ð8Þ

If we only consider the membrane component on the middle surface, the inverse deformation gradient tensor, an important index to evaluate the distortion, can be obtained as follows: ½ F 1 ¼

@Xq0 @Xq

@Xq0

ð10Þ

@Xp

By Eqs. (9) and (10), we can obtain the inverse deformation gradient tensor in matrix form 2

ð9Þ



1  @u @x

 @u @x

6 @u ½ F 1 ¼ 4  @v @x 1 @x @w @u  @x  @x

Fig. 7 a One-step unfolding. b Local unfolding algorithm step by step for complex surfaces

kx l3 ky l3 kz l3

3 7 5

ð11Þ

Int J Adv Manuf Technol (2013) 65:1215–1227

1221

Then the left Cauchy-green tensor {B} describing the length and angle distortion is defined as 2 3 a b 0 0 5 ½ B1 ¼ ½ F T ½ F 1 4 b c ð12Þ 0 0 l2 3 where the variation l3 can be obtained by following two eigenvalues controlling the stretching of the plastic metal surfaces: ( )  1=2 1=2 l1 1 1 ð a þ cÞ  ða  cÞ2 þ 4b2 ð13Þ ¼ 2 2 l2 where a, b, and c is as follows:

2 a ¼ 1  u;x þ v2;x þ w2;x



b ¼ u;y 1  u;x  v;x 1  v;y þ w;x w;y

2 c ¼ u2;y þ 1  v;y þ w2;y

ð14Þ

l1 and l2 are very important indices that enable verification of the distortion of the surface. According to two eigenvalues, we get the principle strain as follows: 2

3 2 ln l1 0 "1 0 0 ln l1 ½"1  ¼ 4 0 "2 0 5 ¼ 4 0 0 0 2 0 0 "3 3 "x "xy 0 ½"1  ¼ 4 "xy "y 0 5 ¼ ½m½"1 ½mT 0 0 "z

3 0 0 5 ln l1

ð15Þ

Table 2 Unfolding speed for different amounts of meshes Unfolding speed

Amount of meshes (elements)

Speed (s)

20,150 50,290 83,889 104,889

21.6 49.2 89.8 109

Inner door with holes (Intel Double Core CPU 1.83 GHz, 1-G Memory)

In the OIFEM, the principle of virtual work is established on the known final configuration C as shown in Fig. 1 with respect to itself, as a reference that gives a very simple expression of PVW:

W ¼

n X

0 @

i¼1

Z

 * " fσgdV 

Ve

Z

1  * u f f gdVA ¼ 0

ð16Þ

Ve

Assuming that virtual displacements of nodes under the initial coordinate (OXYZ) are U*, V*, W*, one has W ¼

X     

e e Fint  Fext ¼0 Un*

ð17Þ

e

Fig. 8 Flowchart of SPIA

Finite element mesh generation

NURBS reconstruction FT1 ,Q2 ( x, y, z )

ΨT1 ,T2 ( x, y, z ) Onestep

inverse forming

Φ Q1 ,Q2 ( x, y, z )

fT2 ,Q1 ( x, y, z )

∂T2

Coons parameterization

Mesh mapping

1222

Int J Adv Manuf Technol (2013) 65:1215–1227

Fig. 9 Local unfolding for an inner door which is composed of more than 900 trimmed NURBS patches. a The CAD model of an inner door. b The unfolded result applying SPIA

in which  *  * Un ¼ Ui

Vi*

*

Wi ; i ¼ 1; 2; 3; 4 and Wi* ¼ 0

ð18Þ

All forces satisfy the following nonlinear equilibrium system. X    e 

e Fext  Fintt ¼ 0 ð19Þ fRðU ; V Þg ¼

Now, it is possible to define the tangent stiffness matrix  

i @ fRðU Þg ; ð23Þ KT ¼  @ fU g U ¼U i At that point, the matrix [KT] can be solved by using finite center difference algorithm involving applying a perturbation analysis scheme to the residual force vector {R(Ui)}, i.e.,

e

The equilibrium equation is solved in general by using the Newton–Raphson iterative algorithm due to strong nonlinear characteristics involving with geometry, physics, and boundary friction. Assembling all nodal internal forces and external forces, one can obtain a residual force vector corresponding to total coordinate (OXYZ) X    

e e  Fint ð20Þ Fext fRðU Þg ¼

K ði; jÞ ¼



 1 i Ri U þ dej  Ri U i  dej 2d

 d ¼ 106 : 1010

ð24Þ

where δ is a title perturbation amount of nodal displacement and ej is the unit vector under the total coordinate (OXYZ). 4.3 Local one-step spatial unfolding algorithm

e

Assuming that in the ith iterative step the approximate solution of {R(Ui)}is detached {Ui}, one has  i   i   i  ¼ Fext U  Fint U 6¼ f0g ð21Þ R U From the N–R scheme, the {R(Ui)} is detached by Taylor expandedness and kept as only a linear term. One has i (h Þg i i  @ f@RðU fU g U ¼U i fΔU g ¼ fRðU Þg ð22Þ  iþ1  U ¼ fU i g þ fΔU i g Fig. 10 a NURBS patches after reconstruction. b Control net

Most of one-step methods fail to handle models with complex morphologies (surfaces with high curvature or roughness). In the community of surface parameterization, those surfaces or point cloud often are segmented into many charts [23] and unfolded, respectively. However, parameterization methods coupling with mesh segmentation are likely to encounter difficulties near the patch boundaries [24]. Furthermore, performing the unfolding from a nontrivial 3D surface with arbitrary genus and holes to another 3D surface is also a question which is never completely settled.

Int J Adv Manuf Technol (2013) 65:1215–1227

1223

Fig. 11 The mesh of a carbody model before starting the unfolding

In this section, we introduce “local one-step unfolding method” and attempt to address those issues. The codes based on this method have been integrated into UG NX 6.0 and higher version. The idea of the algorithm was coming from the flanging simulation of sheet metal. Figure 6 illustrates a flowchart of the present method for a complex model. First, we fix a part of the surfaces (yellow area); second, intermediate faces (also called binder face) are automatically generated as objective surfaces as shown in third picture of Fig. 6. Usually this procedure is settled manually by users, which often leads to severe error. We revamped this procedure and enable the automatic generation of the intermediate face. Third, we employed a similar technique with flanging simulation of sheet metal forming to enforce the sheet metal flow naturally on the intermediate faces. Finally, we can obtain a new surface illustrated in the fourth picture of Fig. 6. Repeating this circle, the initial surface can be unfolded into a planar face shown in last picture of Fig. 6. In Fig. 7, another example, a geometrically complicated surface of a car-body model, is also used to verify the efficient of the new method. The relatively flat area (yellow zone as shown in Fig. 7a) is selected to generate the intermediate faces. The green domain is projected into the intermediate domain. Figure 7b shows the stress contours and the meshes after unfolding in detail. Local one-step method allows mapping arbitrary 2D manifold to another 2D manifold, which does not rely on mesh segmentation. Automatic generation of the binder Fig. 12 The improved mesh of a car-body model after finishing the unfolding

faces also has this unfolding process improvement. Therefore, the whole process is automatically accomplished. 4.4 Basic process of SPIA The main idea of our algorithm is to reconstruct a threedimensional NURBS surface suited for isogeometric analysis in R3. This is done by flattening the initial finite element mesh of original surface using OSIA and solving the two-dimensional problem in R2 instead as shown in Fig. 8. Taking into consideration only mesh T1 with its boundary, it is flattened into a twodimensional mesh T2. The function y T1 ;Q1 ðx; y; zÞ : T1 ! T2 is then defined by linearly mapping each triangle of T1 to the corresponding triangle in T2, while the inverse function fQ1 ;Q2 ðx; y; zÞ : Q1 ! Q2 enables us to get back from Q1 to Q2. 4.4.1 Unfolding meshes via SPIA In SPIA, the mesh T1 was treated as a stamping part. Our program will automatically calculate the stamping direction. Using a multi-step iteration based on local unfolding algorithm, the metal of stamping part flows naturally until it is stable. After the initial triangle mesh is flattened to a planar domain, the remeshing procedure is reduced to the two-dimensional problem T2, which is much easier to solve than the original problem T1. For the mesh parameterization, the unfolding speed and stability are significant factors that influence the results because the approximation error will be reduced if the CAD model is

1224

Int J Adv Manuf Technol (2013) 65:1215–1227

Fig. 13 The process of constructing the fender with 302 trimmed NURBS patches

refined adequately. Many nonlinear algorithms of surface parameterizations become time-consuming and unstable when the amount of meshes reaches a certain size. Our solution speed appears to increase linearly as the mesh is refined. Table 2 and Fig. 9 depict the statistical speeds for different meshes which imply the ability of SPIA to handle large amounts of meshes. 4.4.2 Coons surface parameterization to obtain mesh Q2 All that is necessary is to create a planar regular quadrilateral mesh Q1 whose boundary ∂Q1 geometrically coincides with ∂T1. The easiest way of finding such a Q1 is to force the boundary vertices of the parameterization T2 to form a rectangle. Then the boundary can be detected and four vertices taken on Q2. We can get four boundaries such as ∂2,1, ∂2,2, ∂2,3, ∂2,4,which can be looked as a coarse remesh Q2 of the parameterization T2, which can now be refined to obtain more detailed meshes. We use Coons surface parameterization to refine the initial profile and obtain the inner points. Figure 8 depicts the four edges separated by the four vertices of T2. A Coons surface can be reconstructed using those edges. For four edges, we can describe the surface as NURBS curves ordered one by chord parameterization. The boundary AB, BC, CD, DA corresponds to NURBS c1 (u), d1 (v), c2 (u), and d2 (u) defined over u ∈ [0,1] and v ∈ [0,1], respectively. We can find a surface X (u, v) that has these four curves as its boundary curves. The equation of the Coons surface after reconstruction is:

It is defined over [0,1]×[0,1]. A regular quadrilateral mesh Q2 can be obtained by refining the parametric domain of X (u, v). 4.4.3 Mapping from Q1 to Q2 Once this planar mesh is obtained, we can use the function FT1 ;T2 ðx; y; zÞ to lift it back into T1 in order to get the quadrilateral mesh Q2 of T1. The vertices Pi of Q2 are determined by analyzing the corresponding vertices Pi of Q1, detecting the surrounding triangle of T2 for each vertex of Q2, computing the barycentric coordinates with respect to this triangle, and linearly interpolating the corresponding triangle of T3 with these barycentric coordinates. Thus, a 3D rectangular quadrilateral mesh is approached. 4.4.4 Reconstruction of isogeometric analysis-suited geometry After remeshing T1, we have a set of ðn þ 1Þ  ðm þ 1Þ data   points Qk;l ; k ¼ 0; . . . ; n and l00,…, m in Q2, and we want to construct a (p, q)-degree NURBS surface by interpolating these points. Again, the first task is to compute

   1v X ð0; vÞ þ ½ X ðu; 0Þ X ðu; 1Þ  v X  ð1; vÞ   X ð0; 0Þ X ð0; 1Þ 1  v u X ð1; 0Þ X ð1; 1Þ v 

X ðu; vÞ ¼ ½1  u þ ½1  u

u

ð25Þ

Fig. 14 The control nets of two NURBS patches for isogeometric analysis

Int J Adv Manuf Technol (2013) 65:1215–1227 Table 3 Comparison of different parameterization methods

Models

1225

Number of faces

Methods

Runtime (s)

Area (rms)

Angular (rms)

Car body (Fig. 8)

207,239

Inner door (Fig. 9)

104,889

SPIA ABF++ ARAP SPIA ABF++ ARAP SPIA ABF++ ARAP

252 348 391 109 91 102 4 2 1

0.13604 0.46743 0.20382 0.03248 0.22971 0.08923 0.09556 0.30384 0.08631

0.000545343 0.000837923 0.002853432 0.000134787 0.000912236 0.000457398 0.000846713 0.001024391 0.003536823

Face (Fig. 15)

2,134

reasonable values for the (ui, vj) and the knot vectors U and V. A common method is to use chord parameterization. Secondly, to complete the computation of the control points, define a linear equation in the unknown Pi, j. S ðx; ηÞ ¼

n X m X i¼1 j¼1

ðpÞ

ðqÞ

Ni ðxÞMj ðηÞwij Pn Pm ðpÞ ðqÞ i¼1 j¼1 Ni ðxÞMj ðηÞwij |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Rij

Pij ¼

n X m X

Rij Pij

Single Coons patch often produces ugly quadrilateral parameterization and leads to folded control nets. Via the Laplacian mesh optimization technique in [25], the quadrilateral meshes after Coons parameterization are significantly improved as shown in Figs. 11 and 12. 4.5 Example with two patches

ð26Þ

i¼1 j¼1

The interpolation problem for the bi-cubic NURBS surface S (ξ, η) leads to the ðn þ 1Þ  ðm þ 1Þ interpolation conditions S (ξi, ηj)0Pi, j. In order to obtain a uniquely solvable problem, we have to add further conditions, i.e., setting all second derivatives along the border and the forth cross derivatives at the corners to zero; other choices, however, are conceivable. The conditions now add up to (m + 3) (n + 3), which matches exactly the unknown control points P; these can be determined by solving m+3 linear systems with the same (m+3) (n+3) matrix and n+3 systems of order (m+3) (n+3). Due to the local support property of the NURBS basis functions, the matrices are diagonal so that the linear problems can each be solved in linear time. Figure 10 depicts the final NURBS surface after reconstruction.

Fig. 15 Unfolding a face model using SPIA

4.4.5 Mesh optimization

Using some simple mesh segment tools, we can divide the finite element meshes into two segments and then carry out SPIA on each one. After that, two control nets, which belong to two patches, respectively, can be merged into a control net. Figure 13a depicts the model of the fender, Fig. 13b depicts finite element meshes, Fig. 13c shows the control points, and Fig. 13d demonstrates the NURBS patches. The yellow control points as shown in Fig. 10 define the red NURBS patch depicted in Fig. 13d, and the blue control points as shown in Fig. 14 generate the yellow patches as shown in Fig. 13d. 4.6 Comparisons We compare the present techniques with the state-of-theart methods such as ABF++ and ARAP. All codes are

1226

Int J Adv Manuf Technol (2013) 65:1215–1227

Table 4 Comparison for SPIA and ABF++ Surface parameterization One-step inverse approach

ABF++

Speed Boundary free Prosperities (global optimization) Mechanics

1~2min for 100K elements More natural physics Boundaries Shell theory with large plastic deformation for sheet metal stamping

1~2 min for 100K elements Fail to guarantee natural boundaries

Distortions Bijective (folder) Area or angle preservation

2 Table 3 Almost having no folder Table 3 Volume preservation

coming from the authors of ABF++ and ARAP. The distortion data shown in Table 3 are derived via Graphite-3D Editor. For the fair comparison, all models are computed on a machine with 1.83-GHz Intel Double Core CPU and 1-G Memory. Table 3 illustrates the distortion comparison for several models as shown in Figs. 8, 9, and 15. For area distortion, we adopt the methods in [26]. According to [27], the angular distortion is measured using

FðaÞ 3nf

Geometry-based Conformal mapping Slightly introduce distortion Folder exist and need to be removed Angle preservation

5 Deformation based on SPIA In this section, we reveal an application of SPIA in the field of deformation. As shown in Fig. 16, the previous car-body model generated by SPIA is deformed using a free-form deformation method. This application suggests potential uses for reverse engineering, morphing, deformation, and analysis in a broader sense.

and F (a) is defined as

follows: F ða Þ ¼

P X 3  X



aij  fij wij

i¼1 j¼1

where fij is the optimal angle and aij is the unknown planar angles. From Tables 1 and 2, SPIA introduces less angular distortion than all of the others. As mentioned in Section 4.3, SPIA allows users to unfold the complicated surfaces step by step. This is a significantly improvement compared with other methods. This advantages benefit from its mechanical background, namely the whole unfolding process is seen as a sheet metal forming process. Free deformation of the sheet metal under punch and other process control techniques does not allow the sheet metal blank overlapping, which reduce the possibility of distortion. Table 4 summarizes those differences.

Fig. 16 Deformation of a car-body model based on SPIA

6 Conclusion We have presented a new method SPIA for reconstructing complex NURBS surfaces well suited for isogeometric analysis. Essential procedure is the construction of a global parameterization of initial triangle meshes, one that minimizes geometric distortion. SPIA enables us to circumvent the three-dimensional surface fitting problem which is timeconsuming and consider a two-dimensional problem instead. SPIA is a physics-based method that introduces less distortion than other method. Shell theory with large plastic deformation for sheet metal forming is introduced into the field of reverse engineering to construct isogeometric analysis-suitable geometry, which can be used for isogeometric shell analysis, including spring back prediction. Our future work will focus on reconstructing multi-patches NURBS surface with holes and developing an automatic algorithm to segment the geometric model into multi

Int J Adv Manuf Technol (2013) 65:1215–1227

NURBS patches compatible with each other. We will also conduct isogeometric shell analysis on the watertight models generated by SPIA. Acknowledgments This work was funded by the Key Project of the National Natural Science Foundation of China (no. 10932003), “973” National Basic Research Project of China (no. 2010CB832700), Project 11102035 supported by National Natural Science Foundation of China and “04” Great Project of Ministry of Industrialization and Information of China (no. 2011ZX04001-21). These supports are gratefully acknowledged. The first author is supported by the China Scholarship Council (CSC). Many thanks are due to the referees for their valuable comments.

References 1. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135–4195 2. Cottrell JA, Hughes TJR, Reali A (2007) Studies of refinement and continuity in isogeometric structural analysis. Comput Methods Appl Mech Eng 196(41–44):4160–4183 3. Bazilevs Y, Da Veiga LB, Cottrell JA, Hughes TJR, Sangalli G (2006) Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math Models Methods Appl Sci 16 (7):1031 4. Zhang Y, Bazilevs Y, Goswami S, Bajaj CL, Hughes TJR (2007) Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput Methods Appl Mech Eng 196(29– 30):2943–2959 5. Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using Tsplines. Comput Methods Appl Mech Eng 199(5–8):229–263 6. Sederberg TW, Finnigan GT, Li X, Lin H, Ipson H (2008) Watertight trimmed NURBS. ACM, New York, p 79 7. Kim HJ, Seo YD, Youn SK (2009) Isogeometric analysis for trimmed CAD surfaces. Comput Methods Appl Mech Eng 198 (37):2982–2995 8. Sheffer A, Praun E, Rose K (2006) Mesh parameterization methods and their applications. Found Trends® Comput Graph Vis 2 (2):105–171 9. Choi HK, Kim HS, Lee KH (2010) An improved mesh simplification method using additional attributes with optimal positioning. Int J Adv Manuf Technol 50(1):235–252

1227 10. Chang DY, Chang YM (2002) A freeform surface modelling system based on laser scan data for reverse engineering. Int J Adv Manuf Technol 20(1):9–19 11. Yuwen S, Dongming G, Zhenyuan J, Weijun L (2006) B-spline surface reconstruction and direct slicing from point clouds. Int J Adv Manuf Technol 27(9):918–924 12. Floater MS, Hormann K (2005) Surface parameterization: a tutorial and survey. Advances in multiresolution for geometric modelling. Springer, New York, pp 157–186 13. Hormann K, Greiner G (2000) MIPS: an efficient global parameterization method. DTIC document 14. Hormann K, Lévy B, Sheffer A (2007) Mesh parameterization: theory and practice. SIGGRAPH course notes 15. Sheffer A, Lévy B, Mogilnitsky M, Bogomyakov A (2005) ABF++: fast and robust angle based flattening. ACM Trans Graph (TOG) 24(2):311–330 16. Gu X, Yau ST (2003) Global conformal surface parameterization. Eurographics Association, Aire-la-Ville, pp 127–137 17. Majlessi SA, Lee D (1988) Development of multistage sheet metal forming analysis method. J Mater Shap Technol 6(1):41–54 18. Batoz JL, Guo YQ, Detraux JM (1990) An inverse finite element procedure to estimate the large plastic strain in sheet metal forming. Proc Adv Technol Plast 3:1403–1408 19. Liu SD, Assempoor A (2005) Development of FAST 3D, designoriented one step FEM in sheet metal forming. COMPLAS IV:1515–1526 20. Lee CH, Huh H (1997) Blank design and strain prediction of automobile stamping parts by an inverse finite element approach. J Mater Process Technol 63:645–650 21. Chung K, Richmond O (1993) A deformation theory of plasticity based on minimum work paths. Int J Plast Technol 9:907–920 22. Zhang X, Hu S, Lang Z, Guo W, Hu P (2007) Energy-based initial guess estimation method for one-step simulation. Int J Comput Methods Eng Sci Mech 8:411–417 23. Courtial A, Vezzetti E (2008) New 3D segmentation approach for reverse engineering selective sampling acquisition. Int J Adv Manuf Technol 35(9):900–907 24. Kim HC, Hur SM, Lee SH (2002) Segmentation of the measured point data in reverse engineering. Int J Adv Manuf Technol 20 (8):571–580 25. Nealen A, Igarashi T, Sorkine O, Alexa M (2006) Laplacian mesh optimization. ACM, New York, pp 381–389 26. Floater MS (1997) Parametrization and smooth approximation of surface triangulations. Comput Aided Geom D 14(3):231–250 27. Sheffer A, de Sturler E (2001) Parameterization of faceted surfaces for meshing using angle-based flattening. Eng Comput-Germany 17(3):326–337