Least-squares approximation of affine mappings for sweep mesh generation. Functional analysis and applications

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1 Least-squares approxiation of affine appings for sweep esh generation. Functional analysis and applications evi Roca Josep Sarrate Abstract Sweep ethods are one of the ost robust techniques to generate hexahedral eshes in extrusion volues. The ain issue in sweep algoriths is the projection of cap surface eshes along the sweep path. The ost copetitive technique to deterine this projection is to find a leastsquares approxiation of an affine apping. Several functional forulations have been defined to carry out this leastsquares approxiation. However, these functionals generate unacceptable eshes for several coon geoetries in CAD odels. In this paper we present a new coparative analysis between these classical functional forulations and a new functional presented by the authors. In particular, we prove under which conditions the iniization of the analyzed functionals leads to a full rank linear syste. Moreover, we also prove the equivalences between these forulations. These allow us to point out the advantages of the proposed functional. Finally, fro this analysis we outline an autoatic algorith to copute the nodes location in the inner layers. Keywords Finite eleent ethod, esh generation, hexahedral eleents, sweep, affine apping 1 Introduction The finite eleent ethod has becoe one of the ost iportant tools in applied sciences and engineering. However, in large industrial applications, it is often hapered by the conversion of the CAD odel into a finite eleent odel This work was partially sponsored by the Spanish Ministerio de Ciencia e Innovación under grants DPI , BIA and CGL C03-02/CLI. Laboratori de Càlcul Nuèric (LaCàN), Departaent de Mateàtica Aplicada III, Universitat Politècnica de Catalunya, Jordi Girona 1-3, E Barcelona, Spain. www-lacan.upc.edu E-ail: {xevi.roca,jose.sarrate}@upc.edu adapted to the details of the geoetry and to the prescribed distribution of the eleent sizes. Several fast and robust algoriths have been developed to generate triangular and tetrahedral eshes, see [1, 2]. Quadrilateral and hexahedral eshes are ore constrained, and therefore uch ore difficult to generate. Nevertheless, the use of ixed forulations in incopressible fluid and solid echanics, where quadrilateral and hexahedral eleents are preferred by several authors, have increased the general interest in unstructured quadrilateral and hexahedral discretizations. Several algoriths have been devised in order to generate hexahedral eshes for any geoetry, see [3] for a detailed survey. However, a fully autoatic hexahedral esh generation algorith for any arbitrary geoetry is still not available. Therefore, special attention has been focused on existing algoriths that decopose the entire geoetry into several sipler pieces. These saller volues can be easily eshed by well-known ethods that exhibit an outstanding perforance in these sipler volues, such as apping [4], subapping [5], and sweeping [6 14]. Most of the CAD coercial packages allow to odel volues by extruding, or sweeping, a surface along a curved path. These one-to-one sweep volues are defined by a source surface, a target surface and a series of linking sides, see Figure 1. Reference [15] presents a detailed analysis on the constraints that a given volue ust verify to be sweepable. Based on the definition of an extrusion geoetry, the traditional procedure to generate an all hexahedral esh by sweeping consists in the following five steps: (i) to generate a structured or unstructured quadrilateral esh on the source surface; (ii) to ap the source esh into the target surface; (iii) to generate a structured quadrilateral esh over the linking sides. This esh defines the ribs that join each node on the boundary of the source surface with its corresponding node on the boundary of the target esh. Moreover, it also defines the boundary of the inner layers of nodes; (iv) to generate the inner layers of nodes; and (v) to generate the

2 2 sweep direction linking sides target surface source surface (c) Fig. 1 Sweeping process of an extrusion volue geoetry, available data, i.e. surface eshes, and (c) shrunk hexahedral esh. hexahedral eleents by carefully connecting the nodes of two consecutive layers. Several robust quadrilateral surface esh generation algoriths have been developed which greatly siplify the surface eshing process involved in the first step [16, 18, 19]. The apping of the source surface esh to the target surface can be perfored by any ethod based on orthogonal projections of nodes onto the target surface [17,7], or by ethods based on a least-squares approxiation of an affine apping defined between the paraetric representations of the loops of boundary nodes of the cap surfaces [13]. The gridding of the linking sides involved in the third step ust be generated using any standard structured quadrilateral surface esh generator [1]. Hence, the ain issue to be dealt with by any sweep algorith is the fourth step. Inner layers are deliited by several loops of nodes that belong to the structured eshes of the linking sides. In fact, for every layer there is one outer loop, and one inner loop for each hole in the sweep volue. Several algoriths have been developed in order to generate the inner layer of nodes. Mingwu and Benzley [9] presented a ethod that geoetrically deterines the position of the interior nodes. Staten and co-workers [10] developed the BMSweep algorith which uses barycentric co-ordinates in a background esh to locate the interior nodes. Blacker [8] first developed a projection algorith based on a least-squares weighted residual functional which does not require the creation of a background esh. Knupp [6] detailed a projection ethod in which the inner nodes are located using a least-squares approxiation of a linear transforation (the hoogeneous part of an affine apping) between the boundary nodes of the source surfaces and the boundary nodes of the inner layer. Although all the previous algoriths to generate the inner layer of nodes have their strengths and weaknesses, projection algoriths based on a least-squares approxiation of an affine apping are the fastest option, see [11] for a coparative analysis. Moreover, they generate high-quality eshes, specially in translatable, rotatable, scalable, and shearable geoetries (that is, between loops of points that are affine). If the loops of nodes are not affine, then an additional soothing step ay be needed. However, this soothing step is also needed in other ethods, such as [9,17]. The standard procedure of a projection algorith is coposed by two steps. First, starting at the eshes of the cap surfaces, the position of the layer is coputed fro the previous one in an advancing front anner, see [6, 9]. Second, to increase the robustness of the projection algorith, the final location of the inner nodes is coputed averaging the nodal position obtained starting fro the eshes of both cap surfaces. Several functionals have been defined to copute this least-squares approxiation, see section 2. However, the iniization of soe of these functionals does not always generate an acceptable projected esh. For several usual geoetric configurations the iniization of the proposed functionals ay lead to a set of noral equations with a singular syste atrix or ay introduce several undesired geoetrical effects on the final esh, see [12,14]. In this paper we present an in depth theoretical study on the least-squares fitting of affine transforations defined between two finite sets of points. This analysis provides the necessary theoretical background for the projection algoriths presented in [6, 12, 13]. Moreover, we propose a new functional that overcoes the drawbacks of the previous forulations and aintains their perforance. 2 Proble stateent 2.1 The original forulation Let = {x i },..., R n be a set of source points, and = {y i },..., R n be a set of target points with n. Fro the practical point of view, two cases are iportant: n = 2

3 3 used to sweep curves on a plane, and n = 3 used to project eshes along a given path in a sweep algorith. In a 3D sweep application the source layer is a quadrilateral esh. This esh is coposed by a set of boundary points,, and a set of inner nodes, see figure 2. The initial layer is the source surface esh. Our goal is to ap this quadrilateral esh onto a target layer bounded by the set of points. Note that there does not exist an underlying surface defining this target layer. That is, the set of boundary points is the only available data for generating the new inner nodes of the quadrilateral esh of the target layer. Therefore, we are looking for a apping φ : R n R n such that y i = φ(x i ) for i = 1,...,. Once this apping is obtained, we will use it to ap the inner nodes of the source quadrilateral esh to the target layer. The fastest ethod to sweep a esh is to approxiate φ by an affine apping ϕ fro R n to R n. This affine apping is deterined by a least-squares fitting of the given data. Thus, we want to find ϕ such that it iniizes the functional E(ϕ) = y i ϕ(x i ) 2. (1) Fro the geoetrical point of view, the iniization of (1) eans that the optial affine apping is such that the su of the squares of the distances between the target points and iages of the source points is iniized, see figure 2. We also define c := 1 x i and c := 1 y i, (2) as the geoetric centers of the sets and, respectively. Reark 1 Any affine apping ϕ fro R n to R n can be written as ϕ(x) = Ax + b, where A L (R n ) is a linear transforation, and b R n is the affine part. If we define b := b + Ac, (3) then we can write ϕ as ϕ(x) = A(x c ) + b. (4) Taking into account Reark 1 we can write, without loss of generality, the initial least-squares proble (1) as the iniization of the functional E(A,b) := y i A(x i c ) b 2, (5) where A L (R n ) and b R n. Proposition 1 If (A E,b E ) L (R n ) R n is such that E(A E,b E ) = in (A,b) L (R n ) R ne(a,b), then b E = c. O φ c y Fig. 2 The inner nodes of the source quadrilateral esh (arked with ) are apped according to ap φ. Available data is: the set (arked with ) and the set (arked with ). The iage of set by ϕ, arked with, approxiates the set according to E. Proof The iniization of functional E is equivalent to solving the overdeterined linear syste A(x i c ) + b = y i, i = 1,,. (6) Note that the unknowns of the previous overdeterined linear syste are the coefficients of the n n atrix, A, and the coefficients of the n-diensional vector, b. According to Equation (3) we set b = b Ac. Hence, Equation (6) is equivalent to Ax i +b = y i, for i = 1,,. By applying the well known noral equations to the previous least-squares proble we obtain A x i + b = y i. Taking into account the definitions of c and c we obtain b = c Ac. Finally, according to Equation (3) we conclude that b = c. c x ϕ

4 4 c x c y the affine apping between the source and target points of a sweep volue. However, functional F has an iportant drawback: if the set of source points deterines a hyperplane (for instance, a plane for 3D probles or a straight line for 2D probles), then the iniization of (7) induces a set of noral equations with a singular syste atrix. Note that this geoetrical configuration is usual in practical CAD odels. In order to avoid this drawback, the functional O G(A) := = y i c A(x i c + c c ) 2 y i Ax i 2 (8) O c y - c x c x c y c y - c x Fig. 3 Geoetric representation of the translation of the sets of points and to the origin and to c c. Reark 2 Proposition 1 has a geoetrical eaning. It states that the optial solution (A E,b E ) aps the center c into the center c. This result follows fro Proposition 1 and Equation (4). 2.2 Alternative forulations According to Reark 2, the new coordinates x = x c and y = y c are introduced, see [6] for details. These new coordinates can be interpreted as translating the sets of point and to the origin, see Figure 3. Using these new coordinates the following functional is also introduced in [6]: F(A) := y i c A(x i c ) 2 = y i Ax i 2. (7) Therefore, we are looking for a linear apping A such that it transfors, in the least-squares sense, = {x i },..., into = {y i },...,. Functional (7) is used in [6] in order to siplify the stateent of the least-squares approxiation of is also proposed in [6], where the new coordinates x = x c + c c and y = y c are introduced. These new coordinates have a clear geoetric interpretation: the sets of points and are translated to c c, see Figure 3. Therefore, by iniizing functional G we are looking for a linear apping A such that it approxiately transfors, in the least-squares sense, = {x i },..., into = {y i },...,. Fro the practical point of view, the iniization of functional G is only used for source surfaces with planar boundaries. However, we will prove that functional (8) also leads to a set of noral equations with singular atrix if the vector c c lies in the hyperplane deterined by the source points, see Figure 4. In order to overcoe the shortcoings presented by functionals (7) and (8), while aintaining the perforance of these ethods we propose the functional H(A;u,u ) := = y i c A(x i c ) 2 + u Au 2 y i Ax i 2 + u Au 2, (9) where u and u are two vector paraeters that belong to R n. Note that vectors u and u in (9) can be properly selected in order to obtain several desired properties of functional H. Therefore, we are looking for a linear apping A such that it approxiately transfors, in the least-squares sense, = {x i },..., into = {y i },...,, and u to u, see Figure 4. To suarize, functionals (5), (7), (8), and (9) have been defined in order to find an optial affine apping, in the least-squares sense, between the set of points and. The ai of this work is to analyze their relationships and to prove that functional H is preferable.

5 5 Target loop c y c y - c x c x Source loop u y vector n R n with n = 1 such that < n,x i 0 >= c, for i = 1,..., and c R. Definition 3 (Hoogeneous hyperplane) Let be a hyperplanar set of points. The hoogeneous hyperplane of is the subspace of vectors H = {v R n < n,v >= 0}, where n R n is a unitary noral vector to. The next three leas are basic linear algebra results and will be used in the following sections. Lea 1 Let = {x i },..., R n be a hyperplanar set of points. There exists a set of indices {i 1,...,i n } {1,...,} such that H = span(x i 1 x i n,...,x i n 1 x i n ) = span(x i 1 c,...,x i n 1 c ). u x u y u x c x c y Lea 2 Let U = {u i },...,n be a set of vectors in R n such that: di{span(u 1,...,u n 1 )} = di{span(u 1,...,u n 1,u n )} = di{span(u 1 u n,...,u n 1 u n )} = n 1. If v R n is such that v / span(u 1,...,u n 1 ) then O span(u 1,...,u n 1,v) = span(u 1 + v,...,u n + v). Fig. 4 Exaple of a geoetry where c c lies in the sae plane than the source surface and the boundary of the inner layer. Geoetric representation of the translation of the sets of points and to the origin and additional vectors u and u. 3 Preliinary definitions and results 3.1 Linear algebra In this analysis we only consider sets of points of diension n 1 and n. That is, we do not consider sets of points that generate linear varieties of diension less than n 1. For instance, in R 3 we do not consider source surfaces which degenerate to lines or points, because it does not ake sense to sweep the in practical applications. Definition 1 (Hyperplanar set) A set of points = {x i },..., is hyperplanar if there exists only one hyperplane through all the points in. In other words, if we take any point of, the differences between the rest of points of and the selected point deterine a subspace of diension n 1. Definition 2 (Unitary noral vector) Let be a set of points and 0 the origin. A unitary noral vector to is a Reark 3 If is a hyperplanar set of points and u / H, then R n = span(u ) H. Thus, for every v R n there exist λ R and v H H such that v = v H + λu. Lea 3 Let be a hyperplanar set of points. Assue that u / H, u R n, and A L (R n ) are given. Then, the apping Θ[A,u,u ] : R n R n defined by Θ[A,u,u ](v) := Av H + λu. (10) is such that: (i) Θ[A,u,u ] L (R n ) (ii) Θ[A,u,u ](v H ) = Av H, (iii) Θ[A,u,u ](u ) = u. v H H Reark 4 Fro the geoetrical point of view Lea 3 states that the linear apping Θ[A,u,u ] can be used to take into account the offset data of a non-planar surface esh deliited by a planar boundary, i.e. planar loop of nodes. On the one hand it states that the linear apping Θ[A,u,u ] aps any vector that belongs to the hoogeneous hyperplane according to A. On the other hand it aps the first paraeter vector u (that does not belong to H) onto the second paraeter vector u. Reark 5 Given a atrix Z it is well known that Z Z T has full rank if and only if Z has full rank, see [20]. On the contrary, we can prove that Z Z T is rank deficient if we prove that Z is rank deficient, too.

6 6 3.2 Properties of functionals Lea 4 states several basic relationships between the previous functionals. The proof follows fro the definitions of the functionals E,F, G, and H. Lea 4 For every A L (R n ): (i) E(A,c ) = F(A) (ii) E ( A,c + A(c c ) ) = G(A) (iii) E(A,c ) = H(A;u,Au ) The following three leas prove that functionals F, G and H can be written in ters of linear apping Θ. Lea 5 Let be a hyperplanar set of points, and assue that u / H and u R n. If A L (R n ) then F ( Θ[A,u,u ] ) = F(A). Proof Since is hyperplanar, x i c H, for i = 1,...,. Therefore, by property (ii) of Lea 3 we have that Θ[A,u,u ](x i c ) = A(x i c ), for i = 1,...,. The proof follows fro the definition of the functional F, see Equation (7). Lea 6 Let be a hyperplanar set of points, and assue that c c / H. If A L (R n ) then: (i) F ( Θ[A,c c,c c ] ) = F ( A ) (ii) G ( Θ[A,c c,c c ] ) = F ( Θ[A,c c,c c ] ) Proof Assue that u = c c and u = c c. Then, property (i) is a particular case of Lea 5. Property (ii) follows fro the definition of functional G, see Equation (8), and Lea 3. Lea 7 Let be a hyperplanar set of points, and assue that u / H and u R n. Then H ( Θ[A,u,u ];u,u ) = F ( Θ[A,u,u ] ). Proof This result follows fro the definitions of functionals F and H, and Lea 3. 4 Rank analysis In this section we prove that the iniization of functionals F and G could lead to a rank deficient set of noral equations. On the contrary, the iniization of functional H can always lead to a full rank set of noral equations. Proposition 2 Let = {x i },..., R n be a hyperplanar set of points. Then, the iniization of functional F is equivalent to solving n uncoupled overdeterined linear systes of rank n 1. Proof The iniization of functional F is equivalent to iposing the following constraints: A(x i c ) = y i c, i = 1,,. (11) Defining the atrices a 1,1... a 1,n A :=.., a n,1... a n,n x1 1 c 1... x 1 c 1 :=.., xn 1 c n... xn c n y 1 1 c 1... y 1 c 1 :=.., y 1 n c n... y n c n we can write (11) as A =. Hence, the iniization of F is equivalent to solving the following n uncoupled overdeterined linear systes T a k = y k, k = 1,,n, (12) where a k := (a k, j ) for j = 1,,n and y k = (y l k c k ), for l = 1,,. To conclude we have to prove that T has rank n 1. By Lea 1, and taking into account that dih = n 1, we have that rank T = di { span(x 1 c,...,x c ) } = dih = n 1. Reark 6 The set of noral equations corresponding to the iniization of functional F can be obtained fro equation (12) as T a k = y k, k = 1,,n, (13) According to Reark 5 the syste atrix T is singular if the set of points is hyperplanar, and it is regular if the set of points spans R n.

7 7 Proposition 3 Let be a hyperplanar set of points. If c c H then the iniization of functional G is equivalent to solving n uncoupled overdeterined linear systes of rank n 1. Otherwise, the rank is n. Proof Siilar to Proposition 2, if we define x1 1 c 1 + c 1 c 1... x 1 c 1 + c 1 c 1 :=...., xn 1 c n + c n c n... xn c n + c n c n then the iniization of functional G is equivalent to solving the following n uncoupled overdeterined linear systes T a k = y k, k = 1,,n, (14) where a k := (a k, j ) for j = 1,,n and y k = (y l k c k ), for l = 1,,. Since the set of points is hyperplanar, by Lea 1, we can assue that we have already reordered the points in in such a way that di{span(x 1 c,...,x n 1 c )} = di{span(x 1 x n,...,x n 1 x n )} = n 1. In these conditions, if we apply Lea 2 considering v = c c, then rank() = di { span(x 1 c + c c,...,x n c + c c ) } = di { span(x 1 c,...,x n 1 c,c c ) }. Therefore, if c c H then rank() = n 1. Otherwise, if c c / H we have rank() = n. Reark 7 If is hyperplanar and c c H then the iniization of functional G leads to a set of noral equations with singular syste atrix. However, if is hyperplanar and c c / H then the syste atrix is regular. Proposition 4 Let be a hyperplanar set of points and assue that u R n. If u H then the iniization of functional H is equivalent to solving n uncoupled overdeterined linear systes of rank n 1. Otherwise, the rank is n. Proof Siilar to Proposition 2, if we define x1 1 c 1... x 1 c 1 u 1 ˆ :=... xn 1 c n... xn c n u n and y 1 1 c 1... y 1 c 1 u 1 Ŷ :=..., y 1 n c n... y n c n u n then the iniization of functional H is equivalent to solving the following n uncoupled overdeterined linear systes ˆ T a k = ŷ k, k = 1,,n, (15) where a k := (a k, j ) for j = 1,,n and ŷ k = (y 1 k c k,,y k c k,u k ). Since is hyperplanar, by Lea 1, we can assue that we have already reordered the points in in such a way that di{span(x 1 c,...,x n 1 c )} = di{span(x 1 x n,...,x n 1 x n )} = n 1. If u H then rank ˆ = di{span(x 1 c,...,x n 1 c,u )} = n 1. Otherwise, u / H, we have that rank ˆ = di{span(x 1 c,...,x n 1 c,u )} = n. Reark 8 In sweeping applications if is hyperplanar we can always select u / H. Therefore, the iniization of H leads to a set of noral equations with regular syste atrix. 5 Equivalences between functionals In this section we prove under which conditions the iniization of functionals F, G and H are equivalent to iniizing functional E. Proposition 5 Let A E L (R n ), b E R n and A F L (R n ) be such that E(A E,b E ) = in (A,b) L (R n ) R ne(a,b) and F(A F ) = in F(A). Then: (i) in (A,b) L (R n ) R ne(a,b) = in F(A) (ii) E(A F,c ) = E(A E,c ) (iii) F(A E ) = F(A F ) Proof We have that F(A F ) F(A E ) since A F iniizes F = E(A E,c ) by Lea 4 E(A F,c ) = F(A F ) by Lea 4. since (A E,c ) iniizes E Note that the first and the last ters in the previous expression are the sae. Then, the inequalities are in fact equalities. All properties follow fro rewriting and reordering the chain of equalities.

8 8 Reark 9 The nuber of degrees of freedo involved in the iniization of F is saller than in the iniization of E. Hence, it is preferable to iniize F because it siplifies the projection algorith. Proposition 6 Let be a hyperplanar set of points, and assue that c c / H. If A F L (R n ) and A G L (R n ) are such that F(A F ) = in F(A) and G(A G ) = in G(A). Then: (i) in G(A) has one and only one solution (ii) in F(A) = in G(A) (iii) A G = Θ[A F,c c,c c ] (iv) F(A G ) = F(A F ) Proof Property (i) follows fro Proposition 3 and Reark 7. Assue we have (A E,b E ) L (R n ) R n such that E(A E,b E ) = in (A,b) L (R n ) R ne(a,b). Then: Proposition 7 Let be a hyperplanar set of points, and assue that u / H and u R n. If A F L (R n ) and A H L (R n ) are such that F(A F ) = in F(A) and H(A H ;u,u ) = in H(A;u,u ). Then: (i) in H(A;u,u ) has one and only one solution (ii) in F(A) = in H(A;u,u ) (iii) A H = Θ[A F,u,u ] (iv) F(A H ) = F(A F ) Proof Property (i) follows fro Proposition 4 and Reark 8. We define R(A;u,u ) := u Au 2. Hence, H(A;u,u ) = F(A) + R(A;u,u ). (16) We consider the following sequence of equalities and inequalities: F(A F ) = E ( A E,b E) by Proposition 5 E ( A G,c + A G (c c ) ) (A E,b E ) iniizes E = G ( A G) by Lea 4 G ( Θ[A F,c c,c c ] ) A G iniizes G = F ( Θ[A F,c c,c c ] ) by Lea 6 = F(A F ) by Lea 6. Note that the first and the last ters in the previous expression are the sae. Then, the inequalities are in fact equalities, and property (ii) follows. Fro the previous sequence of equalities, we have proved that Θ[A F,c c,c c ] and A G iniize functional G. Thus, property (iii) is also proved since G has a unique solution. Property (iv) follows fro Lea 6 and property (iii) of this Proposition. Reark 10 Property (iii) of Proposition 6 states that the optial solution of the iniization of functional G, A G, can be coputed fro the optial solution of the iniization of functional F, A F, when c c / H (note that this property is not used in the original work of [6]). In this case we have that A G = Θ[A F,c c,c c ]. Moreover, by Lea 3 if c c / H and taking u = u = c c, we have that c c = A G (c c ). That is, c c is a fixed vector of A G. F(A F ) F(A H ) A F iniizes F F(A H ) + R(A H ;u,u ) since R 0 = H(A H ;u,u ) by Equation (16) H ( Θ[A F,u,u ];u,u ) A H iniizes H = F ( Θ[A F,u,u ] ) by Lea 7 = F(A F ) by Lea 5 Note that the first and the last ters are the sae. Thus, all the inequalities are in fact equalities. Therefore, properties (ii) and (iv) hold. Fro the previous sequence of equalities, we have also proved that Θ[A F,u,u ] and A H iniize the functional H. Since the iniization of H has a unique solution we have that A H = Θ[A F,u,u ], and property (iii) holds. Reark 11 Property (iii) of Proposition 7 states how to copute the optial solution of the iniization of functional H, A H, fro the optial solution of the iniization of functional F, A F, when is hyper-planar. In this case, A H = Θ[A F,u,u ]. Taking into account the basic properties of Θ, see Lea 3, we have that u = A H u and A H v H = A F v H for v H H. In other words, the optial solution of the iniization of H aps u into u. Moreover, it is equal to the optial solution of the iniization of F over H.

9 9 Sweep direction Sweep direction Sweep direction (c) Fig. 5 Illustration of the flattening effect: projecting a non-planar esh with planar boundary into a planar boundary loop; projecting a non-planar esh with planar boundary into a non-planar boundary loop; and (c) projecting a non-planar esh into a planar boundary loop. Proposition 8 Let be a non-hyperplanar set of points and assue that u R n. If A F L (R n ) and A H L (R n ) are such that F(A F ) = in F(A) and H(A H ;0,u ) = in H(A;0,u ). Then: (i) in F(A) = in H(A;0,u ) u 2 (ii) F(A H ) = F(A F ) Proof Observe that, fro the definitions of functionals F and H we have H(A;0,u ) = F(A)+ u 2. In other words, H differs fro F only by a constant that does not depend on A. Hence, both properties follow. Reark 12 Proposition 8 extends the equivalence between the iniization of F and H to non-hyperplanar sets of points. The requireent is that u = 0. 6 Exact apping characterization When the least-squares fitting of affine appings is applied in a sweep algorith, it is iportant to characterize under which conditions we can exactly ap the set of points to the set of points by eans of an affine apping. To this end, the three following propositions are introduced. Proposition 9 There exists an affine apping ϕ such that ϕ(x i ) = y i, for i = 1,...,, if and only if in F(A) = 0. Proof Assue that there exists an affine apping ϕ such that ϕ(x i ) = y i for i = 1,...,. Taking into account Equation (4) and Proposition 1, this is equivalent to iposing y i c A(x i c ) = 0 for i = 1,...,. Thus F(A) = 0. On the other hand, assue that in F(A) = 0. In this case there exists a linear transforation A F L (R n ) such that F(A F ) = 0. Fro the definition of F we have that y i c A F (x i c ) 2 = 0. Since all the suation ebers are non-negative, and defining ϕ(x) := A F (x c ) + c, we have that ϕ(x i ) = y i, for i = 1,...,. Proposition 10 Assue that is hyperplanar and c c / H. Then, there exists an affine apping ϕ such that ϕ(x i ) = y i, for i = 1,...,, if and only if in G(A) = 0. Proof This result follows directly fro property (ii) of Proposition 6 and Proposition 9. Proposition 11 There exists an affine apping ϕ such that ϕ(x i ) = y i, for i = 1,...,, if and only if there exist u,u R n such that in H(A;u,u ) = 0. Proof It is proved analogously to Proposition 9. 7 Shortcoings of functionals F and G In this section we suarize the shortcoings of functionals F and G. In addition, we highlight that it is always possible to select a vector u such that the iniization of H overcoes these drawbacks. Flattening. We have seen that the iniization of functional F has two ain shortcoings:

10 10 = = A G c x c y - c x c y Fig. 7 Least-squares approxiation of an affine apping between a circular shaped set of points and an elliptical shaped set of points. c y - c x c x Fig. 6 A non-planar surface with planar boundary; cross-section view of the surface and its iage iniizing G (dotted line) and H (thin solid line). c y If the set of source points,, is hyperplanar (for instance a source surface esh with planar boundary), then the iniization of F leads to a set of noral equations with singular syste atrix, see Proposition 2. In practice a singular value decoposition, SVD, ay be used to solve the set of noral equations. In this case, the inner part of the projected esh will be planar. Hence, a flattening effect will be introduced and the shape of the inner part of the source surface esh will be lost, see Figures 5 and 5. If a given esh is projected over an inner layer with a hyperplanar boundary by iniizing F, then the projected esh will always have a planar inner part, see Figure 5(c). Thus, a flattening effect is also introduced. Functional G was introduced to overcoe the first drawback of F. In particular, Proposition 6 states the equivalence between both functionals. However, if is hyperplanar and c c H then the iniization of G also leads to a set of noral equations with a singular syste atrix, see Proposition 3. Hence, the iniization of G also introduces the flattening effect. Note that, on the one hand Proposition 7 holds in the case that the source points are hyperplanar. On the other hand, Proposition 7 holds even in the case of c c H. Recall that in these cases the iniization of F and G lead to a set noral equations with a singular syste atrix. Thus, the iniization of H is always preferable in these two cases because it is always possible to choose a vector u / H such that a set noral equations with a regular syste atrix is obtained. Skewing. Functional G has an additional drawback when the projection algorith is applied to planar sets of points, even in the case of c c / H. Consider a non-planar source surface with planar boundary, see Figure 6. Assue that we want to project this source surface esh into an inner layer (of a sweep volue) defined by a planar boundary, but non-parallel to the source surface. Figure 6 shows a crosssection of the source surface (thick solid line), the crosssection of the obtained solution iniizing functionals G (dotted line) and the cross-section of the obtained solution iniizing functionals H (thin solid line). Since c c is a fixed vector of A G, see Reark 10, the cross-section obtained with A G (dotted line in the top cross section of Figure 6) does not preserve the shape of the original surface. On the contrary, Reark 11 states the optial solution of the iniization of H, A H, aps u into u. If we select u = n and u = n, then the noral of the source boundary is apped into the noral of the target boundary. Therefore, its shape is preserved (thin solid line in the top cross section of Figure 6). This exaple illustrates that the iniization of functional H is preferable since its optial solution is not affected by the skewing introduced by the iniization of functional G and tends to preserve the shape of the original surface. This property was first reported in [12]. Not capturing affinities. If is hyperplanar and c c H then Proposition 10 does not hold. In this case, if the set and are affine, then the iniization of functional G could provide an affine apping that does not exactly ap to. To illustrate this we consider two affine and coplanar sets of points = {x i := (cost i,sint i,0)} i=0,...,11 R 3 and = {y i := (5 + 2cost i,sint i,0)} i=0,...,11 R 3, where t i = i2π/12 for i = 0,...,11, see Figure 7. In this case c = (0,0,0) and c = c c = (5,0,0). Note that in this exa-

11 11 ple c c H. We solve the corresponding sets of noral equations by eans of the SVD which supplies the solution with the sallest nor, and we obtain the following linear transforations: A F = , A G = 0 1 0, A H = A Θ[A,u,u ] u c λu Θ[A,u,u ](v) := λu + Av H Av H AH These linear transforations verify that F(A F ) = 0, G(A G ) = 100/17 and H(A H ) = 0. Thus, A F and A H exactly ap the circular shaped set (black solid line in Figure 7) into an elliptical shaped set (black dotted line in Figure 7). On the contrary, A G aps the circular shaped set (grey solid line in Figure 7) into an alost circular set of points A G (grey dotted line in Figure 7), whereas the set of target points is elliptical. Note that in this exaple both solutions, A F and A G, ap the noral coponent of H into zero. Thus, all the offset inforation that the inner part of the source loop of points ay contain will be lost (voluetric distributions of points are apped into planar distributions). Hence, the iniization of H is preferable. 8 Outline of the algorith The previous analysis provides the theoretical background to develop a general and autoatic algorith to generate eshes by sweep. The new algorith is designed to attain two ain goals. On the one hand, it has to lead to a set of noral equations with full rank syste atrix. Note that by iniizing H with u / H we eet this condition, see Reark 8. On the other hand, the algorith has to preserve the shape of the eshes of the cap surfaces on the obtained inner layers. To this end we have to specify how to select u and u. In [14] we detail and prove an algorith ipleentation that eets these requireents. It is outlined here in three steps for clarity: First. We copute the SVD of the optial solution A F = UWV T, where U and V are two n n orthogonal atrices, and W is a n n diagonal atrix with positive or zero entries, the singular values, diag(w 1,...,w n ) such that w 1 w 2 w n 1 w n 0. We denote by u i R n and v i R n, for i = 1,...,n, the coluns of atrices U and V respectively. Second. We set u and u as: hyperplanar and hyperplanar: u = n and u = n, hyperplanar and non-hyperplanar: u = n and u = u n, non-hyperplanar and hyperplanar: u = v n and u = n, non-hyperplanar and non-hyperplanar: u = n pseudo and u = n pseudo, λu u c v = λu + v H Fig. 8 Transforation of a given vector v by apping Θ when is a planar set. where n pseudo and n pseudo are the pseudo-noral vectors to sets and, respectively: n pseudo := 1 2 x i x i+1 and n pseudo := 1 2 v H H y i y i+1, being x +1 := x 1 and y +1 := y 1. Third. Proposition 7 states that if is a planar loop of nodes, like the one depicted in Figure 8, the unique solution of the iniization of functional H can be coputed fro one of the optial solutions of the iniization of F. Using the linear apping Θ introduced in Lea 3, A H can be coputed as the linear transforation that aps u into u, and any vector v H H to Av H. Once we have selected vectors u and u, for any centered vector x R n we copute the linear part of the affine projection as A(x) := A F (x < x,u > u )+ < x,u > u. Finally, we copute the desired affine apping according to Equation (4) ϕ(x) = A(x) + c = A(x c ) + c. That is, to obtain the optial solution A H, we first find the optial solution A F, and based on its SVD and the geoetric configuration of sets and we select the vectors u and u. 9 Exaples and discussion In this section we present five exaples in order to assess the capabilities of the proposed functional. In all the exaples the set of points corresponds to the boundary nodes of the source esh, and the set of points corresponds to the boundary nodes of the current inner layer. To highlight the analyzed issues, in all the exaples the source surface has

12 12 (c) Fig. 10 Projection of a non-planar surface esh with planar boundary into planar inner layers along a curved sweep path: iniizing F; iniizing G; and (c) iniizing H. (c) Fig. 9 Projection of a non-planar surface esh with planar boundary into planar inner layers along a straight sweep path: iniizing F; iniizing G; and (c) iniizing H. a planar boundary, with non-planar interior. The first two exaples illustrate the advantages and drawbacks of the analyzed functionals. These two eshes are obtained with a sweeping tool that ipleents the iniizations of functionals F, G, and H. In these two exaples the inner layers are obtained projecting directly the source surface onto the inner layer. That is, we have used neither a weighted interpolation of the projections fro both cap surface eshes nor an additional soothing step to iprove the quality of the final esh. To solve the overdeterined linear systes that do not have full rank we have used a SVD which supplies the solution with the sallest nor, see [20]. In the first exaple we illustrate the flattening effect introduced by the iniization of functional F. In addition, we observe that in this case the iniization of functionals G and H preserve the shape of the cap surface in the inner layers. To this end we consider a non-planar surface esh with planar boundary that is extruded along a straight sweep path, see figure 9. The first row shows a general view of several layers of eleents corresponding to the eshes obtained iniizing each functional. The second row shows a front view of these layers. Since all the boundary loops are planar, the iniization of functional F leads to flattened inner layers despite the cap surfaces are non-planar, see figures 9. Since we have a straight sweep path and c c / H, the obtained eshes iniizing G and H are able to aintain the shape of the cap surfaces in the inner layers, see figures 9 and 9(c). In the second exaple we show that the iniization of functionals F and G ay generate bad eshes due to the flattening and skewing effects, whereas the iniization of the new functional H produces a high quality esh. Figure 10, presents a C-shaped geoetry with square cross sections. The cap surfaces are non-planar but have a planar

13 13 Fig. 11 Mesh of a gear coposed by several one-to-one extrusion volues. boundary. Since the boundary loop of the cap surfaces is planar the iniization of functional F generates planar inner layers, see figures 10. The iniization of functional G generates non-planar inner layers, see figures 10. Since ost of the planar loop of nodes that define the inner layers are non-parallel, and c c is a fixed vector of the optial solution of functional G, the skewing effect appears on the generated esh, see Reark 10. Moreover, there exists an inner layer such that its planar boundary is parallel to the planar boundary of the source surface (c c H). In this case iniizing functional G leads to an overdeterined linear syste that does not have full rank, and a degenerated projection is obtained, see figures 10. Miniizing functional H we are able to properly select vectors u and u and we obtain a esh that preserves the shape of the cap surfaces in each inner layer. The proposed algorith can deal with ore coplicated geoetries, which ay contain curved and twisted paths or ay be defined as an assebly of several volues. The goal of the last three exaples is to illustrate that the proposed algorith can be successfully applied to odels coposed by several sweep volues that ay contain planar and nonplanar cap surfaces. Figure 11 shows the esh generated for a gear odel. A conforal esh is generated over the shared surfaces that define this odel. Figure 12 presents two views of the esh generated for a echanical brace decoposed in several sub-volues using a ulti-sweep algorith. Figure 13 shows the CAD odel of a inertia syste coposed by three wheels. Siilar to the previous two exaples we discretize this assebly odel by iniizing functional H. Figure 13 shows a detail of the esh corresponding to a one quarter of the central wheel. In addition, the circular Fig. 12 Mesh of a echanical brace obtained with a ulti-sweep decoposition: top view; and botto view. boundary loops that deliitate the cap surfaces are arked with thick lines. Although each inertia wheel is defined by non-planar cap surfaces deliited by a planar boundary, we are able to generate inner layers that preserve the shape of the cap surfaces. In all cases we obtain hexahedral eleents of high quality. 10 Conclusions In this paper we present a new theoretical and coparative analysis of several functionals that are extensively used in sweeping tools. We prove that iniizing F leads to a set of noral equations with a singular syste atrix if the set of source points is hyperplanar. To avoid this drawback we prove the equivalence between iniizing F and G. However, we prove that iniizing G also leads to a set of noral equations with a singular syste atrix if, in addition, c c lies in the sae hyperplane that. To over-

14 14 to obtain an affine transforation that exactly aps into. In these situations an additional soothing step ay be required in order to iprove the quality of the final esh. We clai that the iniization of functional H provides better initial node location than the iniization of functionals F and G. Moreover, this projection algorith ay provide an excellent initial guess for orphing procedures. References Fig. 13 Discretization of a syste of inertia wheels: CAD odel; and one quarter of the generated esh for the central wheel. coe these drawbacks we introduce functional H and we prove the equivalence between iniizing F and H. Note that this result does not depend on where the vector c c lies. Moreover, we also prove that we can always enforce that iniizing H leads to a set of noral equations with a regular syste atrix. Miniizing H has two additional advantages over iniizing F and G. On the one hand, if the source surface is non-planar but has a planar boundary, then the nuerical solution obtained fro the iniization of F generates planar inner layers of nodes. However, iniizing the new functional H we obtain non planar inner layers of nodes. On the other hand, if the source surface is non-planar but has a planar boundary and c c / H, then the iniization of H avoids the skewing effect introduced by G. Therefore, the iniization of H tends to preserve the shapes of cap surfaces on inner layers. The source and the target loops of nodes are not affine in a wide range of applications. Therefore, it is not possible 1. Thopson JF, Soni B., Weatherill N (1999) Handbook of Grid Generation. CRC Press 2. Frey PJ, George PL (2000) Mesh Generation. Aplication to finite eleents. Heres Science Publishing 3. Tautges TJ (2001) The generation of hexahedral eshes for assebly geoetry: survey and progress. Int. J. Nuer. Meth. Eng. 50, Cook WA, Oakes WR (1982) Mapping ethods for generating three-diensional eshes. Coputers In Mechanical Engineering 1, Whiteley M, White D, Benzley S, Blacker T (1996) Two and threequarter diensional eshing facilitators. Eng. Coput. 12, Knupp PM (1998) Next-generation sweep tool: a ethod for generating all-hex eshes on two-and-one-half diensional geoetries. In Proc. of 7th International Meshing Roundtable, Sandia National Laboratory, Knupp PM (1999) Applications of esh soothing: copy, orph, and sweep on unstructured quadrilateral eshes. Int. J. Nuer. Meth. Eng. 45, Blacker TD (1996) The cooper tool. In Proc. of 5th International Meshing Roundtable, Sandia National Laboratory, Mingwu L, Benzley SE (1996) A ultiple source and target sweeping ethod for generating all-hexahedral finite eleent eshes. In Proc. of 5th International Meshing Roundtable, Sandia National Laboratory, Staten ML, Canann, SA, Owen SJ (1999) BMSweep: locating interior nodes during sweeping. Eng. Coput. 15, Scott MA, Earp MN, Benzley SE, Stephenson MB (2005) Adaptive sweeping techniques. In Proc. of 14th International Meshing Roundtable, Sandia National Laboratory, Roca, Sarrate J, Huert, A (2005) A new least squares approxiation of affine appings for sweep algoriths. In Proc. of 14th International Meshing Roundtable, Sandia National Laboratory, Roca, Sarrate J, Huerta A (2006) Mesh projection between paraetric surfaces. Coun. Nuer. Meth. Eng. 22, Roca, Sarrate J (2006) An autoatic and general least-squares projection procedure for sweep eshing. In Proc. of 15th International Meshing Roundtable, Sandia National Laboratory, White DR, Tautges TJ (2000) Autoatic schee selection for toolkit hex eshing. Int. J. Nuer. Meth. Eng. 49, Cass RJ, Benzley SE, Meyers RJ, Blacker TD (1996) Generalized 3-D paving: an autoated quadrilateral surface esh generation algorith. Int. J. Nuer. Meth. Eng. 39, Goodrich D (1997) Generation of all-quadrilateral surface eshes by esh orphing. Master in Science Thesis, Brigha oung University 18. Sarrate J, Huerta A (2000) Efficient unstructured quadrilateral esh generation. Int. J. Nuer. Meth. Eng. 49, Sarrate J, Huerta A (2000) Autoatic esh generation of nonstructured quadrilateral eshes over curved surfaces in R 3. In Proc. 3th European Congress on Coputational Methods in Applied Sciences and Engineering, ECCOMAS, Barcelona, Spain 20. Gill PE, Murray W, Wright MH (1991) Nuerical Linear Algebra and Optiization. Addison-Wesley