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Using discrete uniformity property in a mixed algebraic method
Applied Numerical Mathematics 49 (2004) 355–366 www.elsevier.com/locate/apnum
Using discrete uniformity property in a mixed algebraic method ✩ C. Conti ∗ , R. Morandi, D. Scaramelli Dipartimento di Energetica “Sergio Stecco”, Università di Firenze, Florence, Italy
Abstract Given a mixed algebraic method based on a transfinite Hermite-type interpolant scheme and on a B-spline tensor product method (see [C. Conti, R. Morandi, D. Scaramelli, An automatic control point choice in algebraic numerical grid generation, in: J. Levesly, I.J. Anderson, J.C. Mason (Eds.), Proceedings of Algorithms for Approximation IV, 2002, pp. 2–8] and references quoted therein), in this paper the choice of the B-spline knots and of the control points used in the tensor product is discussed. Both choices are performed by taking into account a discrete “uniformity” property. 2004 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Numerical grid generation; Uniformity property; Grid spacing; Control points
1. Introduction The Hermite-type transfinite interpolating operators allow us the generation of grids with boundary conformity and with orthogonal grid lines emanating from the boundary (see, for example, [5,6,8]). This can be very important, for example, when a domain decomposition is necessary. Furthermore, mixed methods obtained through the combination of transfinite interpolation methods and tensor product schemes give us the opportunity to control the coordinate curves, for instance, via a particular choice of the B-spline knots and/or of the control points appearing in the tensor product. When using an algebraic method, the control of the grid spacing is a crucial point in order to have a uniform/non-uniform distribution of the coordinate curves in the physical domain. Thus, dealing with a “pseudo-rectangular” physical domain, in this paper we propose sufficient conditions based on a discrete “uniformity” property [3,7], v to control the grid spacing when using the Hermite-type mixed algebraic method presented in [1] coupling transfinite and tensor product schemes. In particular, the mentioned conditions involve the B-spline knots or the control points in the tensor product. ✩
Authors gratefully thank GNCS-CNR for supporting the research.
* Corresponding author.
E-mail addresses: [email protected] (C. Conti), [email protected] (R. Morandi), [email protected] (D. Scaramelli). 0168-9274/$30.00 2004 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2003.12.013
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The rest of the paper is organized as follows. In Section 2 the Hermite-type mixed scheme and its properties are recalled. In Section 3 a discrete uniformity property is defined and used to derive a strategy for choosing the B-spline knots and the control points in the tensor product. Finally, in Section 4 some numerical results are presented to illustrate the features of the proposed strategy.
2. A Hermite-type algebraic method Let Ω ⊂ R2 be such that ∂Ω = 4i=1 ∂Ωi , with ∂Ω1 ∩ ∂Ω3 = ∅ ∂Ω2 ∩ ∂Ω4 = ∅ where ∂Ω1 , ∂Ω2 , ∂Ω3 , ∂Ω4 are the supports of four regular curves γi : [0, 1] → ∂Ωi , i = 1, . . . , 4, taken counterclockwise. Furthermore, we assume that the curve intersections occur only at the end points of the boundary curves γi , i = 1, . . . , 4, i.e., γ1 (0) = γ4 (1),
γ1 (1) = γ2 (0),
γ2 (1) = γ3 (0),
γ4 (0) = γ3 (1).
We define φ1 (s) := γ1 (s), φ2 (s) := γ3 (1 − s), s ∈ [0, 1] and ψ1 (t) := γ4 (1 − t), ψ2 (t) := γ2 (t), t ∈ [0, 1] and four additional curves by computing the derivatives of the φ- and ψ -curves, i.e., Ki y x − φi (s) , φi (s) , i = 1, 2, φi 2 Kj +2 y x − ψj (t) , ψj (t) , j = 1, 2, ψj +2 (t) = ψj 2
φi+2 (s) =
(1)
denoting by φ x , φ y and ψ x , ψ y the components of the φ-curves and ψ -curves, respectively. The symbol · 2 stands for the Euclidean norm and Kl , l = 1, . . . , 4 are constant values, also depending on the curve orientations, whose choice will be specified in Section 4. As we are going to deal with orthogonal lines emanating from the boundary of the domain, we assume the following conditions on the boundary curves φi+2 (0) = ψ1 (ui ),
φi+2 (1) = ψ2 (ui ),
ψi+2 (0) = φ1 (ui ),
ψi+2 (1) = φ2 (ui ),
φi (0) = ψ1 (ui ),
i = 1, 2,
φi (1) = ψ2 (ui ),
(2)
where u1 = 0, u2 = 1. Then, we introduce the linear operators P1 [φ](s, t) :=
4 i=1
αi (t)φi (s),
P2 [ψ](s, t) :=
4
αj (s)ψj (t),
j =1
2 ∂P2 [ψ](s, ui ) . αi (t)P2 [ψ](s, ui ) + αi+2 (t) P1 P2 [φ, ψ](s, t) := ∂t i=1
(3)
The functions αj (s), j = 1, . . . , 4, in (3) are the so-called blending functions. Here, they are defined as the dilated versions of the classical Hermite bases with support on Is0 := [0, u¯ s ] and on Is1 := [1 − u˜ s , 1] being 0 < u¯ s < 1 and 0 < u˜ s < 1, i.e., s s 2 s 2 , α3 (s) := s 1 − , s ∈ Is0 1− α1 (s) := 1 + 2 u¯ s u¯ s u¯ s
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s + u˜ s − 1 s + u˜ s − 1 2 α2 (s) := 3 − 2 , u˜ s u˜ s
s + u˜ s − 1 α4 (s) := (s − 1) u˜ s
357
2 ,
s ∈ Is1 .
(4)
The blending functions αi (t), i = 1, . . . , 4, are analogously defined with support on [0, u¯ t ] and on [1 − u˜ t , 1]. The Hermite-type blending function surface is (P1 ⊕ P2 )[φ, ψ](s, t) = P1 [φ](s, t) + P2 [ψ](s, t) − P1 P2 [φ, ψ](s, t).
(5)
Now, in order to define a grid orthogonal at the boundary, we define the linear transformation G : [0, 1]2 → R2 G(s, t) := TP (s, t) + (P1 ⊕ P2 ) [φ, ψ] − TP (s, t), (6) m n where TP (s, t) := i=1 j =1 Qij Bi,3 (s)Bj,3 (t) with Bi,3 (s) and Bj,3 (t) denoting the usual cubic B-splines with knots {si−2 , si−1 , si , si+1 , si+2 } and {tj −2 , tj −1 , tj , tj +1 , tj +2 }, respectively. The knots are such that 0 = s−1 = s0 = s1 = s2 < · · · < sm−1 = sm = sm+1 = sm+2 = 1 and 0 = t−1 = t0 = t1 < t2 < · · · < tn−1 = tn = tn+1 = tn+2 = 1. {Qij }m,n i,j =1 is a set of control points suitably chosen. The Boolean sum operator (P1 ⊕ P2 ) in (6) is also acting on the surface TP (s, t) taking into account the eight boundary curves TP (0, t), TP (1, t), TP (s, 0), TP (s, 1), ∂TP∂s(0,t ) , ∂TP∂s(1,t ) , ∂TP∂t(s,0) , ∂TP∂t(s,1) . As discussed in [1], the transformation G is such that G(s, ui ) = φi (s), i = 1, 2, G(uj , t) = ψj (t), j = 1, 2, ∂G(s, ui ) ∂G(uj , t) = ψj (t), j = 3, 4, = φi (s), i = 3, 4. (7) ∂s ∂t Moreover, setting u¯ s = s3 − s2 , u¯ t = t3 − t2 , u˜ s = sm−1 − sm−2 , u˜ t = tn−1 − tn−2 , because of the blending function locality, it holds G(s, t) = TP (s, t) for all (s, t) ∈ [s3 , sm−2 ] × [t3 , tn−2 ]. The grid G is then obtained by sampling G at a given set of parameter values {(σi , τj )}M,N i,j =1 in M,N [0, 1] × [0, 1], i.e., G := {G(σi , τj )}i,j =1 . 3. Discrete “uniformity” property to control the grid spacing In this section we discuss a discrete approach to control the grid spacing. In particular, let Ω be a “pseudo-rectangular” domain, that is for each pair of opposite boundaries it is possible to find two favourite directions, η, θ ∈ R2 such that φ1 (s) − φ1 (0) · η = s φ1 (1) − φ1 (0) , s ∈ [0, 1], φ2 (s) − φ2 (0) · η = s φ2 (1) − φ2 (0) , s ∈ [0, 1], ψ1 (t) − ψ1 (0) · θ = t ψ1 (1) − ψ1 (0) , ψ2 (t) − ψ2 (0) · θ = t ψ2 (1) − ψ2 (0) ,
t ∈ [0, 1], t ∈ [0, 1].
The control of the grid spacing is performed in the sense of the projections of the vectors G(s, t) − G(0, t) for all t ∈ [0, 1] and G(s, t) − G(s, 0) for all s ∈ [0, 1] on assigned directions η and θ . The idea of looking for projections linearly increasing with s and t, respectively, has been introduced in [2] for the univariate case and generalized in [7] for the bivariate case.
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Definition 3.1 (Uniformity property I). Let η, θ ∈ R2 be two linearly independent directions. A map F : [0, 1]2 → R2 such that F s, t¯ − F 0, t¯ · η = s F 1, t¯ − F 0, t¯ · η, t¯ = 0, 1, s ∈ [0, 1], F s¯ , t − F s¯ , 0 · θ = t F s¯, 1 − F s¯ , 0 · θ, s¯ = 0, 1, t ∈ [0, 1] (8) is said to satisfy the uniformity property I with respect to the projections on η, θ if F (s, t) − F (0, t) · η = s F (1, t) − F (0, t) · η, s, t ∈ (0, 1), F (s, t) − F (s, 0) · θ = t F (s, 1) − F (s, 0) · θ, s, t ∈ (0, 1). For practical purposes it is more convenient to consider an alternative definition of “uniformity” (implied by the previous one). Definition 3.2 (Uniformity property II). Let η, θ ∈ R2 be two linearly independent directions. A map F : [0, 1]2 → R2 for which (8) holds is said to satisfy the uniformity property II with respect to the projections on η, θ if ∂F (s, t) · η = F (1, t) − F (0, t) · η, ∂s ∂F (s, t) · θ = F (s, 1) − F (s, 0) · θ, ∂t
s, t ∈ (0, 1), s, t ∈ (0, 1).
(9)
Next, we consider a discrete version of the second definition. Definition 3.3 (Discrete uniformity property). Let η, θ ∈ R2 be two linearly independent directions. A map F : [0, 1]2 → R2 for which (8) holds is said to satisfy the discrete uniformity property with respect to the projections on η, θ if n−2 ∂F (s, tj ) j =3 m−2 i=3
n−2
F (1, tj ) − F (0, tj ) · η,
s ∈ (0, 1),
∂F (si , t) ·θ = F (si , 1) − F (si , 0) · θ, ∂t i=3
t ∈ (0, 1),
∂s
·η=
j =3 m−2
(10)
where 0 < s3 < · · · < sm−2 < 1 and 0 < t3 < · · · < tn−2 < 1. 3.1. Choosing the B-spline knots n−2 In this section we are going to discuss how to choose the B-spline knots {si }m−2 i=3 , {tj }j =3 so that the transformation G in (6) satisfies the discrete uniformity property. It should be noted that because of (7) and the assumptions on Ω, G satisfies conditions (8). As regards to (10), after little algebra and by the B-spline properties we can write, for l = 3, . . . , n − 2, m Bi−1,2 (s) t ∂G(s, tl ) t =3 (tl ) + f s (s, tl ), Ci (tl ) − Ci−1 ∂s s − s i+1 i−2 i=2
(11)
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where
∂T (0,t ) α1 (s) ψ1 (tl ) − TP (0, tl ) + α3 (s) ψ3 (tl ) − P∂s l , s ∈ [s2 , s3 ), f s (s, tl ) = 0, s ∈ [s3 , sm−2 ), α2 (s) ψ2 (tl ) − TP (1, tl ) + α4 (s) ψ4 (tl ) − ∂TP∂s(1,tl ) , s ∈ [sm−2 , sm−1 ], where Cit (t) = nj=1 Qi,j Bj,3 (t) and Bi−1,2 (s) are the quadratic B-splines with knots {si−2 , si−1 , si , si+1 }, for i = 2, . . . , m. Thus, the left side of the first of (10) becomes n−2 ∂G(s, tl )
∂s
l=3
=3
n−2 m
t (tl ) Cit (tl ) − Ci−1
l=3 i=2
n−2 Bi−1,2 (s) + f s (s, tl ). si+1 − si−2 l=3
Similarly, we find the left side of the second of (10) as m−2
m−2 m−2 n Bj −1,2 (t) s ∂G(sr , t) s =3 + f t (sr , t), Cj (sr ) − Cj −1 (sr ) ∂t t − t j +1 j −2 r=3 r=3 j =2 r=3 m where Cjs (s) = i=1 Qi,j Bi,3 (s) and Bj −1,2 (t) are the quadratic B-splines with knots {tj −2 , tj −1 , tj , tj +1 }, for j = 2, . . . , n and ∂T (s ,0) α1 (t) φ1 (sr ) − TP (sr , 0) + α3 (t) φ3 (sr ) − P ∂t r , t ∈ [t2 , t3 ), f t (sr , t) = 0, t ∈ [t3 , tn−2 ), α2 (t) φ2 (sr ) − TP (sr , 1) + α4 (t) φ4 (sr ) − ∂TP ∂t(sr ,1) , t ∈ [tn−2 , tn−1 ].
For shortness, we now introduce the quantities Mηψ
:=
n−2
ψ2 (tl ) − ψ1 (tl ) · η,
l=3
MηG,t (s) :=
φ Mθ
:=
m−2
φ2 (sr ) − φ1 (sr ) · θ,
r=3 n−2 l=3
∂G(s, tl ) · η, ∂s
MθG,s (t) :=
m−2 r=3
∂G(sr , t) · θ. ∂t
(12)
The transformation G in (6) satisfies the discrete uniformity property if and only if MηG,t (s) = Mηψ ,
s ∈ (0, 1)
and
MθG,s (t) = Mθφ ,
t ∈ (0, 1).
As a first step for choosing the B-splines knots so that G satisfies the discrete uniformity property, the following proposition can be easily proved. n−2 Proposition 3.1. Let the knots {sr }m−2 r=3 , {tl }l=3 be such that m−2
m−2
r=3
r=3
n−2
n−2
l=3
and
φ3 (sr ) · θ = Mθφ , ψ3 (tl ) · η = Mηψ ,
l=3
φ4 (sr ) · θ = Mθφ , ψ4 (tl ) · η = Mηψ ,
(13)
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3 t (tl ) · η = Mηψ , Crt (tl ) − Cr−1 sr+1 − sr−2 l=3
r = 2, . . . , m,
m−2 3 φ s (sr ) · θ = Mθ , Cls (sr ) − Cl−1 tl+1 − tl−2 r=3
l = 2, . . . , n.
n−2
(14)
φ
Thus, MηG,t (si ) = Mηψ , i = 2, . . . , m − 1 and MθG,s (tj ) = Mθ , j = 2, . . . , n − 1. Proposition 3.1 allows us to formulate the following theorem. n−2 Theorem 3.1. Let G be the transformation defined in (6). Let {sr }m−2 r=3 , {tl }l=3 satisfy conditions (13) and (14) and be such that m−2
m−2
φ1 (sr ) − C1s (sr ) · θ = 0,
r=3
φ2 (sr ) − Cns (sr ) · θ = 0,
r=3
n−2
n−2
ψ1 (tl ) − C1t (tl ) · η = 0,
l=3
ψ2 (tl ) − Cmt (tl ) · η = 0.
(15)
l=3
Then, G satisfies the discrete uniformity property. Proof. Because of the B-spline support, it holds n−2 4 Bi−1,2 (s) t 3 l=3 i=2 Cit (tl ) − Ci−1 (tl ) si+1 ·η −si−2 n−2 + l=3 α1 (s) ψ1 (tl ) − C1t (tl ) · η, s ∈ [s2 , s3 ), Bi−1,2 (s) n−2 n−2 j +2 t t ∂G(s, tl ) (tl ) si+1 · η, 3 l=3 i=j Ci (tl ) − Ci−1 −si−2 ·η= ∂s s ∈ [sj , sj +1 ), j = 3, . . . , m − 3, l=3 Bi−1,2 (s) t n−2 m t (t ) − C (t ) ·η C 3 l l i l=3 i=m−2 i−1 si+1 −si−2 n−2 t + l=3 α2 (s)(ψ2 (tl ) − Cm (tl )) · η, s ∈ [sm−2 , sm−1 ]. Thus, by the assumptions, it holds 4 ψ M s ∈ [s2 , s3 ), η i=2 Bi−1,2 (s) + 0, n−2 ∂G(s, tl ) +2 · η = Mηψ ji=j Bi−1,2 (s), s ∈ [sj , sj +1 ), j = 3, . . . , m − 3, ∂s l=3 Mψ m B (s) + 0, s ∈ [s , s ], η
i=m−2
n−2
i−1,2
m−2
(16)
(17)
m−1
l) · η = Mηψ for s ∈ (0, 1). In an analogous way we show that allowing us to conclude that l=3 ∂G(s,t ∂s m−2 ∂G(sr ,t ) φ · θ = Mθ for t ∈ (0, 1), thus completing the proof. 2 r=3 ∂t
3.2. A practical way for choosing the control points The aim of this section is the discussion of a practical way for choosing the control points in the tensor product. It is derived from the condition (10) on the map TP under the assumption of uniformly distributed knots and repeated end knots. In fact, the analysis of (14) easily yields the proposition below
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m,n n−2 Proposition 3.2. If the knots {sr }m−2 r=3 , {tl }l=3 are uniformly distributed and the control points {Qij }i,j =1 are such that the following equalities hold
hs t (tl ) · η Mη = C2t (tl ) − C1t (tl ) · η = Cmt (tl ) − Cm−1 3 l=3 l=3 n−2
n−2
1 t 1 t t (tl ) · η C3 (tl ) − C2t (tl ) · η = Cm−1 (tl ) − Cm−2 2 l=3 2 l=3 n−2
=
n−2
1 t Ci+1 (tl ) − Cit (tl ) · η, = 3 l=3 n−2
i = 3, . . . , m − 3,
(18)
ht s Mθ = (sr ) · θ C2s (sr ) − C1s (sr ) · θ = Cns (sr ) − Cn−1 3 r=3 r=3 m−2
m−2
1 s 1 s s (sr ) · θ C3 (sr ) − C2s (sr ) · θ = Cn−1 (sr ) − Cn−2 = 2 r=3 2 r=3 m−2
m−2
1 s Cj +1 (sr ) − Cjs (sr ) · θ, = 3 r=3 m−2
j = 3, . . . , n − 3,
(19)
then (10) holds for F ≡ TP . Next, a theorem gives sufficient conditions for using Proposition 3.2. Theorem 3.2. If the control points {Qij }m,n i,j =1 are such that 1 (Q2,j − Q1,j ) · η = (Qm,j − Qm−1,j ) · η = (Q3,j − Q2,j ) · η 2 1 1 = (Qm−1,j − Qm−2,j ) · η = (Qi+1,j − Qi,j ) · η, 2 3 i = 3, . . . , m − 3, j = 1, . . . , n,
(20)
1 (Qi,2 − Qi,1 ) · θ = (Qi,n − Qi,n−1 ) · θ = (Qi,3 − Qi,2 ) · θ 2 1 1 = (Qi,n−1 − Qi,n−2 ) · θ = (Qi,j +1 − Qi,j ) · θ 2 3 i = 1, . . . , m, j = 3, . . . , n − 3,
(21)
where (Q2,j − Q1,j ) · η = and (19) are verified.
hs (Qm,j 3
− Q1,j ) · η and (Qi,2 − Qi,1 ) · θ =
ht (Qi,n 3
− Qi,1 ) · θ , then (18)
The proofs of the proposition and of the theorem are tedious but straightforward.
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Last, we discuss how to choose the control points satisfying the hypothesis of the previous theorem. This can be realized by using a Lagrange transfinite scheme L(s, t) (see [4] for details) interpolating the n four boundary curves and sampling it at parameter values {σi }m i=1 and {τj }j =1 in [0, 1] such that 1 1 σ2 − σ1 = σm − σm−1 = (σ3 − σ2 ) = (σm−1 − σm−2 ) 2 2 1 = (σi+1 − σi ), i = 3, . . . , m − 3, 3 1 1 τ2 − τ1 = τn − τn−1 = (τ3 − τ2 ) = (τn−1 − τn−2 ) 2 2 1 (22) = (τj +1 − τj ), j = 3, . . . , n − 3. 3 The strategy for choosing the parameter values (here referred as NUSLB strategy) is motivated by the proposition below whose proof is trivial due to the linearity of the blending functions used in L(s, t). Proposition 3.3. If, for any positive real number ρ, the four boundary curves satisfy φi (ρt) − φi (0) · θ = ρ φi (s, t) − φi (0) · θ, i = 1, 2, ψj (ρs) − ψj (0) · η = ρ ψj (s) − ψj (0) · η, j = 1, 2, L(s, t) satisfies L(s, ρt) − L(s, 0) · θ = ρ L(s, t) − L(s, 0) · θ, L(ρs, t) − L(0, t) · η = ρ L(s, t) − L(0, t) · η,
∀ s ∈ [0, 1], ∀ t ∈ [0, 1].
4. Examples In this section we present some numerical results to put into evidence how the grid spacing can be n−2 controlled by means of the discrete uniformity property. In particular, first the knots {si }m−2 i=3 , {tj }j =3 used for the B-splines in the tensor product are chosen by solving the constrained non-linear system we get from (14) and (15), that is n+1 2 m+1 2 min s ,...,s ,t ,...,t 3 m−2 3 n−2 l=1 Ul + r=1 Wr , m−2 (23) sr < sr+1 , r=2 (sr+1 − sr ) = 1, n−2 tl < tl+1 , l=2 (tl+1 − tl ) = 1, where Ul−1 = Wr−1 =
m−2 3 φ s (sr ) · θ − Mθ , Cls (sr ) − Cl−1 tl+1 − tl−2 r=3 n−2 t 3 t (tl ) · η − Mηψ , Cr (tl ) − Cr−1 sr+1 − sr−2 l=3
l = 2, . . . , n, r = 2, . . . , m,
C. Conti et al. / Applied Numerical Mathematics 49 (2004) 355–366
Un =
m−2
363
φ1 (sr ) − C1s (sr ) · θ,
r=3
Wm =
n−2 ψ1 (tl ) − C1t (tl ) · η, l=3
Un+1 =
m−2
φ2 (sr ) − Cns (sr ) · θ,
r=3
Wm+1 =
n−2 ψ2 (tl ) − Cmt (tl ) · η.
(24)
l=3
We note that the conditions (13) are not included in the minimization procedure since the constants Kl , l = 1, . . . , 4 in (1) are set so that (13) holds. The set of control points used in the tensor product is simply obtained by uniformly sampling a Lagrange blending surface interpolating the four given boundary curves φ1 , φ2 , ψ1 , ψ2 on a coarse parameter set. We refer to this way of getting the control points as USLB strategy. In case of uniformly distributed knots, we use the procedure NUSBL for choosing the control points. We conclude the section showing four “pseudo-rectangular” domains. For each of them we present the initial grid performed by using uniform parameter values with Kl = 1, 1 l 4 and control points obtained via USLB strategy. Then (on the left of each picture) we show the grid obtained by using nonuniform B-spline knots solution of the minimization problem and with Kl , 1 l 4 satisfying (13). The non-linear problem has been solved by using the routine constr of the optimization toolbox of the Matlab package. Finally, (on the right of each picture) we show the grid obtained applying the NUSLB strategy to get the control points for the tensor product when using uniform B-spline knots. The values of Kl , for 1 l 4 still satisfy (13). All the examples show the effectiveness of the proposed strategies.
Fig. 1. Initial grid.
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C. Conti et al. / Applied Numerical Mathematics 49 (2004) 355–366
Fig. 2. Final grids.
Fig. 3. Initial grid.
Fig. 4. Final grids.
C. Conti et al. / Applied Numerical Mathematics 49 (2004) 355–366
Fig. 5. Initial grid.
Fig. 6. Final grids.
Fig. 7. Initial grid.
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Fig. 8. Final grids.
References [1] C. Conti, R. Morandi, D. Scaramelli, An automatic control point choice in algebraic numerical grid generation, in: J. Levesly, I.J. Anderson, J.C. Mason (Eds.), Proceedings of Algorithms for Approximation IV, 2002, pp. 2–8. [2] P.R. Eiseman, High level continuity for coordinate generation with precise controls, J. Comput. Phys. 47 (1982) 352–374. [3] P.R. Eiseman, Control point grid generation, Comput. Math. Appl. 24 (5–6) (1992) 57–67. [4] W.J. Gordon, L.C. Thiel, Transfinite mappings and their application to grid generation, in: J.F. Thompson (Ed.), Numerical Grid Generation, Appl. Math. Comput., vol. 10–11, Elsevier, Amsterdam, 1982, pp. 171–192. [5] P. Knupp, S. Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Raton, FL, 1994. [6] V.D. Liseikin, Grid Generation Methods, Springer, Berlin, 1999. [7] R. Morandi, A. Sestini, Precise controls in numerical grid generation, in: F. Fontanella, K. Jetter, P.J. Laurent (Eds.), Advanced Topics in Multivariate Approximation, World Scientific, Singapore, 1996, pp. 243–258. [8] J. Thompson, B.K. Soni, N.P. Weatherill (Eds.), Handbook of Grid Generation, CRC Press, Boca Raton, FL, 1999.
Using discrete uniformity property in a mixed algebraic method ✩ C. Conti ∗ , R. Morandi, D. Scaramelli Dipartimento di Energetica “Sergio Stecco”, Università di Firenze, Florence, Italy
Abstract Given a mixed algebraic method based on a transfinite Hermite-type interpolant scheme and on a B-spline tensor product method (see [C. Conti, R. Morandi, D. Scaramelli, An automatic control point choice in algebraic numerical grid generation, in: J. Levesly, I.J. Anderson, J.C. Mason (Eds.), Proceedings of Algorithms for Approximation IV, 2002, pp. 2–8] and references quoted therein), in this paper the choice of the B-spline knots and of the control points used in the tensor product is discussed. Both choices are performed by taking into account a discrete “uniformity” property. 2004 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Numerical grid generation; Uniformity property; Grid spacing; Control points
1. Introduction The Hermite-type transfinite interpolating operators allow us the generation of grids with boundary conformity and with orthogonal grid lines emanating from the boundary (see, for example, [5,6,8]). This can be very important, for example, when a domain decomposition is necessary. Furthermore, mixed methods obtained through the combination of transfinite interpolation methods and tensor product schemes give us the opportunity to control the coordinate curves, for instance, via a particular choice of the B-spline knots and/or of the control points appearing in the tensor product. When using an algebraic method, the control of the grid spacing is a crucial point in order to have a uniform/non-uniform distribution of the coordinate curves in the physical domain. Thus, dealing with a “pseudo-rectangular” physical domain, in this paper we propose sufficient conditions based on a discrete “uniformity” property [3,7], v to control the grid spacing when using the Hermite-type mixed algebraic method presented in [1] coupling transfinite and tensor product schemes. In particular, the mentioned conditions involve the B-spline knots or the control points in the tensor product. ✩
Authors gratefully thank GNCS-CNR for supporting the research.
* Corresponding author.
E-mail addresses: [email protected] (C. Conti), [email protected] (R. Morandi), [email protected] (D. Scaramelli). 0168-9274/$30.00 2004 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2003.12.013
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The rest of the paper is organized as follows. In Section 2 the Hermite-type mixed scheme and its properties are recalled. In Section 3 a discrete uniformity property is defined and used to derive a strategy for choosing the B-spline knots and the control points in the tensor product. Finally, in Section 4 some numerical results are presented to illustrate the features of the proposed strategy.
2. A Hermite-type algebraic method Let Ω ⊂ R2 be such that ∂Ω = 4i=1 ∂Ωi , with ∂Ω1 ∩ ∂Ω3 = ∅ ∂Ω2 ∩ ∂Ω4 = ∅ where ∂Ω1 , ∂Ω2 , ∂Ω3 , ∂Ω4 are the supports of four regular curves γi : [0, 1] → ∂Ωi , i = 1, . . . , 4, taken counterclockwise. Furthermore, we assume that the curve intersections occur only at the end points of the boundary curves γi , i = 1, . . . , 4, i.e., γ1 (0) = γ4 (1),
γ1 (1) = γ2 (0),
γ2 (1) = γ3 (0),
γ4 (0) = γ3 (1).
We define φ1 (s) := γ1 (s), φ2 (s) := γ3 (1 − s), s ∈ [0, 1] and ψ1 (t) := γ4 (1 − t), ψ2 (t) := γ2 (t), t ∈ [0, 1] and four additional curves by computing the derivatives of the φ- and ψ -curves, i.e., Ki y x − φi (s) , φi (s) , i = 1, 2, φi 2 Kj +2 y x − ψj (t) , ψj (t) , j = 1, 2, ψj +2 (t) = ψj 2
φi+2 (s) =
(1)
denoting by φ x , φ y and ψ x , ψ y the components of the φ-curves and ψ -curves, respectively. The symbol · 2 stands for the Euclidean norm and Kl , l = 1, . . . , 4 are constant values, also depending on the curve orientations, whose choice will be specified in Section 4. As we are going to deal with orthogonal lines emanating from the boundary of the domain, we assume the following conditions on the boundary curves φi+2 (0) = ψ1 (ui ),
φi+2 (1) = ψ2 (ui ),
ψi+2 (0) = φ1 (ui ),
ψi+2 (1) = φ2 (ui ),
φi (0) = ψ1 (ui ),
i = 1, 2,
φi (1) = ψ2 (ui ),
(2)
where u1 = 0, u2 = 1. Then, we introduce the linear operators P1 [φ](s, t) :=
4 i=1
αi (t)φi (s),
P2 [ψ](s, t) :=
4
αj (s)ψj (t),
j =1
2 ∂P2 [ψ](s, ui ) . αi (t)P2 [ψ](s, ui ) + αi+2 (t) P1 P2 [φ, ψ](s, t) := ∂t i=1
(3)
The functions αj (s), j = 1, . . . , 4, in (3) are the so-called blending functions. Here, they are defined as the dilated versions of the classical Hermite bases with support on Is0 := [0, u¯ s ] and on Is1 := [1 − u˜ s , 1] being 0 < u¯ s < 1 and 0 < u˜ s < 1, i.e., s s 2 s 2 , α3 (s) := s 1 − , s ∈ Is0 1− α1 (s) := 1 + 2 u¯ s u¯ s u¯ s
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s + u˜ s − 1 s + u˜ s − 1 2 α2 (s) := 3 − 2 , u˜ s u˜ s
s + u˜ s − 1 α4 (s) := (s − 1) u˜ s
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2 ,
s ∈ Is1 .
(4)
The blending functions αi (t), i = 1, . . . , 4, are analogously defined with support on [0, u¯ t ] and on [1 − u˜ t , 1]. The Hermite-type blending function surface is (P1 ⊕ P2 )[φ, ψ](s, t) = P1 [φ](s, t) + P2 [ψ](s, t) − P1 P2 [φ, ψ](s, t).
(5)
Now, in order to define a grid orthogonal at the boundary, we define the linear transformation G : [0, 1]2 → R2 G(s, t) := TP (s, t) + (P1 ⊕ P2 ) [φ, ψ] − TP (s, t), (6) m n where TP (s, t) := i=1 j =1 Qij Bi,3 (s)Bj,3 (t) with Bi,3 (s) and Bj,3 (t) denoting the usual cubic B-splines with knots {si−2 , si−1 , si , si+1 , si+2 } and {tj −2 , tj −1 , tj , tj +1 , tj +2 }, respectively. The knots are such that 0 = s−1 = s0 = s1 = s2 < · · · < sm−1 = sm = sm+1 = sm+2 = 1 and 0 = t−1 = t0 = t1 < t2 < · · · < tn−1 = tn = tn+1 = tn+2 = 1. {Qij }m,n i,j =1 is a set of control points suitably chosen. The Boolean sum operator (P1 ⊕ P2 ) in (6) is also acting on the surface TP (s, t) taking into account the eight boundary curves TP (0, t), TP (1, t), TP (s, 0), TP (s, 1), ∂TP∂s(0,t ) , ∂TP∂s(1,t ) , ∂TP∂t(s,0) , ∂TP∂t(s,1) . As discussed in [1], the transformation G is such that G(s, ui ) = φi (s), i = 1, 2, G(uj , t) = ψj (t), j = 1, 2, ∂G(s, ui ) ∂G(uj , t) = ψj (t), j = 3, 4, = φi (s), i = 3, 4. (7) ∂s ∂t Moreover, setting u¯ s = s3 − s2 , u¯ t = t3 − t2 , u˜ s = sm−1 − sm−2 , u˜ t = tn−1 − tn−2 , because of the blending function locality, it holds G(s, t) = TP (s, t) for all (s, t) ∈ [s3 , sm−2 ] × [t3 , tn−2 ]. The grid G is then obtained by sampling G at a given set of parameter values {(σi , τj )}M,N i,j =1 in M,N [0, 1] × [0, 1], i.e., G := {G(σi , τj )}i,j =1 . 3. Discrete “uniformity” property to control the grid spacing In this section we discuss a discrete approach to control the grid spacing. In particular, let Ω be a “pseudo-rectangular” domain, that is for each pair of opposite boundaries it is possible to find two favourite directions, η, θ ∈ R2 such that φ1 (s) − φ1 (0) · η = s φ1 (1) − φ1 (0) , s ∈ [0, 1], φ2 (s) − φ2 (0) · η = s φ2 (1) − φ2 (0) , s ∈ [0, 1], ψ1 (t) − ψ1 (0) · θ = t ψ1 (1) − ψ1 (0) , ψ2 (t) − ψ2 (0) · θ = t ψ2 (1) − ψ2 (0) ,
t ∈ [0, 1], t ∈ [0, 1].
The control of the grid spacing is performed in the sense of the projections of the vectors G(s, t) − G(0, t) for all t ∈ [0, 1] and G(s, t) − G(s, 0) for all s ∈ [0, 1] on assigned directions η and θ . The idea of looking for projections linearly increasing with s and t, respectively, has been introduced in [2] for the univariate case and generalized in [7] for the bivariate case.
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Definition 3.1 (Uniformity property I). Let η, θ ∈ R2 be two linearly independent directions. A map F : [0, 1]2 → R2 such that F s, t¯ − F 0, t¯ · η = s F 1, t¯ − F 0, t¯ · η, t¯ = 0, 1, s ∈ [0, 1], F s¯ , t − F s¯ , 0 · θ = t F s¯, 1 − F s¯ , 0 · θ, s¯ = 0, 1, t ∈ [0, 1] (8) is said to satisfy the uniformity property I with respect to the projections on η, θ if F (s, t) − F (0, t) · η = s F (1, t) − F (0, t) · η, s, t ∈ (0, 1), F (s, t) − F (s, 0) · θ = t F (s, 1) − F (s, 0) · θ, s, t ∈ (0, 1). For practical purposes it is more convenient to consider an alternative definition of “uniformity” (implied by the previous one). Definition 3.2 (Uniformity property II). Let η, θ ∈ R2 be two linearly independent directions. A map F : [0, 1]2 → R2 for which (8) holds is said to satisfy the uniformity property II with respect to the projections on η, θ if ∂F (s, t) · η = F (1, t) − F (0, t) · η, ∂s ∂F (s, t) · θ = F (s, 1) − F (s, 0) · θ, ∂t
s, t ∈ (0, 1), s, t ∈ (0, 1).
(9)
Next, we consider a discrete version of the second definition. Definition 3.3 (Discrete uniformity property). Let η, θ ∈ R2 be two linearly independent directions. A map F : [0, 1]2 → R2 for which (8) holds is said to satisfy the discrete uniformity property with respect to the projections on η, θ if n−2 ∂F (s, tj ) j =3 m−2 i=3
n−2
F (1, tj ) − F (0, tj ) · η,
s ∈ (0, 1),
∂F (si , t) ·θ = F (si , 1) − F (si , 0) · θ, ∂t i=3
t ∈ (0, 1),
∂s
·η=
j =3 m−2
(10)
where 0 < s3 < · · · < sm−2 < 1 and 0 < t3 < · · · < tn−2 < 1. 3.1. Choosing the B-spline knots n−2 In this section we are going to discuss how to choose the B-spline knots {si }m−2 i=3 , {tj }j =3 so that the transformation G in (6) satisfies the discrete uniformity property. It should be noted that because of (7) and the assumptions on Ω, G satisfies conditions (8). As regards to (10), after little algebra and by the B-spline properties we can write, for l = 3, . . . , n − 2, m Bi−1,2 (s) t ∂G(s, tl ) t =3 (tl ) + f s (s, tl ), Ci (tl ) − Ci−1 ∂s s − s i+1 i−2 i=2
(11)
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where
∂T (0,t ) α1 (s) ψ1 (tl ) − TP (0, tl ) + α3 (s) ψ3 (tl ) − P∂s l , s ∈ [s2 , s3 ), f s (s, tl ) = 0, s ∈ [s3 , sm−2 ), α2 (s) ψ2 (tl ) − TP (1, tl ) + α4 (s) ψ4 (tl ) − ∂TP∂s(1,tl ) , s ∈ [sm−2 , sm−1 ], where Cit (t) = nj=1 Qi,j Bj,3 (t) and Bi−1,2 (s) are the quadratic B-splines with knots {si−2 , si−1 , si , si+1 }, for i = 2, . . . , m. Thus, the left side of the first of (10) becomes n−2 ∂G(s, tl )
∂s
l=3
=3
n−2 m
t (tl ) Cit (tl ) − Ci−1
l=3 i=2
n−2 Bi−1,2 (s) + f s (s, tl ). si+1 − si−2 l=3
Similarly, we find the left side of the second of (10) as m−2
m−2 m−2 n Bj −1,2 (t) s ∂G(sr , t) s =3 + f t (sr , t), Cj (sr ) − Cj −1 (sr ) ∂t t − t j +1 j −2 r=3 r=3 j =2 r=3 m where Cjs (s) = i=1 Qi,j Bi,3 (s) and Bj −1,2 (t) are the quadratic B-splines with knots {tj −2 , tj −1 , tj , tj +1 }, for j = 2, . . . , n and ∂T (s ,0) α1 (t) φ1 (sr ) − TP (sr , 0) + α3 (t) φ3 (sr ) − P ∂t r , t ∈ [t2 , t3 ), f t (sr , t) = 0, t ∈ [t3 , tn−2 ), α2 (t) φ2 (sr ) − TP (sr , 1) + α4 (t) φ4 (sr ) − ∂TP ∂t(sr ,1) , t ∈ [tn−2 , tn−1 ].
For shortness, we now introduce the quantities Mηψ
:=
n−2
ψ2 (tl ) − ψ1 (tl ) · η,
l=3
MηG,t (s) :=
φ Mθ
:=
m−2
φ2 (sr ) − φ1 (sr ) · θ,
r=3 n−2 l=3
∂G(s, tl ) · η, ∂s
MθG,s (t) :=
m−2 r=3
∂G(sr , t) · θ. ∂t
(12)
The transformation G in (6) satisfies the discrete uniformity property if and only if MηG,t (s) = Mηψ ,
s ∈ (0, 1)
and
MθG,s (t) = Mθφ ,
t ∈ (0, 1).
As a first step for choosing the B-splines knots so that G satisfies the discrete uniformity property, the following proposition can be easily proved. n−2 Proposition 3.1. Let the knots {sr }m−2 r=3 , {tl }l=3 be such that m−2
m−2
r=3
r=3
n−2
n−2
l=3
and
φ3 (sr ) · θ = Mθφ , ψ3 (tl ) · η = Mηψ ,
l=3
φ4 (sr ) · θ = Mθφ , ψ4 (tl ) · η = Mηψ ,
(13)
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3 t (tl ) · η = Mηψ , Crt (tl ) − Cr−1 sr+1 − sr−2 l=3
r = 2, . . . , m,
m−2 3 φ s (sr ) · θ = Mθ , Cls (sr ) − Cl−1 tl+1 − tl−2 r=3
l = 2, . . . , n.
n−2
(14)
φ
Thus, MηG,t (si ) = Mηψ , i = 2, . . . , m − 1 and MθG,s (tj ) = Mθ , j = 2, . . . , n − 1. Proposition 3.1 allows us to formulate the following theorem. n−2 Theorem 3.1. Let G be the transformation defined in (6). Let {sr }m−2 r=3 , {tl }l=3 satisfy conditions (13) and (14) and be such that m−2
m−2
φ1 (sr ) − C1s (sr ) · θ = 0,
r=3
φ2 (sr ) − Cns (sr ) · θ = 0,
r=3
n−2
n−2
ψ1 (tl ) − C1t (tl ) · η = 0,
l=3
ψ2 (tl ) − Cmt (tl ) · η = 0.
(15)
l=3
Then, G satisfies the discrete uniformity property. Proof. Because of the B-spline support, it holds n−2 4 Bi−1,2 (s) t 3 l=3 i=2 Cit (tl ) − Ci−1 (tl ) si+1 ·η −si−2 n−2 + l=3 α1 (s) ψ1 (tl ) − C1t (tl ) · η, s ∈ [s2 , s3 ), Bi−1,2 (s) n−2 n−2 j +2 t t ∂G(s, tl ) (tl ) si+1 · η, 3 l=3 i=j Ci (tl ) − Ci−1 −si−2 ·η= ∂s s ∈ [sj , sj +1 ), j = 3, . . . , m − 3, l=3 Bi−1,2 (s) t n−2 m t (t ) − C (t ) ·η C 3 l l i l=3 i=m−2 i−1 si+1 −si−2 n−2 t + l=3 α2 (s)(ψ2 (tl ) − Cm (tl )) · η, s ∈ [sm−2 , sm−1 ]. Thus, by the assumptions, it holds 4 ψ M s ∈ [s2 , s3 ), η i=2 Bi−1,2 (s) + 0, n−2 ∂G(s, tl ) +2 · η = Mηψ ji=j Bi−1,2 (s), s ∈ [sj , sj +1 ), j = 3, . . . , m − 3, ∂s l=3 Mψ m B (s) + 0, s ∈ [s , s ], η
i=m−2
n−2
i−1,2
m−2
(16)
(17)
m−1
l) · η = Mηψ for s ∈ (0, 1). In an analogous way we show that allowing us to conclude that l=3 ∂G(s,t ∂s m−2 ∂G(sr ,t ) φ · θ = Mθ for t ∈ (0, 1), thus completing the proof. 2 r=3 ∂t
3.2. A practical way for choosing the control points The aim of this section is the discussion of a practical way for choosing the control points in the tensor product. It is derived from the condition (10) on the map TP under the assumption of uniformly distributed knots and repeated end knots. In fact, the analysis of (14) easily yields the proposition below
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361
m,n n−2 Proposition 3.2. If the knots {sr }m−2 r=3 , {tl }l=3 are uniformly distributed and the control points {Qij }i,j =1 are such that the following equalities hold
hs t (tl ) · η Mη = C2t (tl ) − C1t (tl ) · η = Cmt (tl ) − Cm−1 3 l=3 l=3 n−2
n−2
1 t 1 t t (tl ) · η C3 (tl ) − C2t (tl ) · η = Cm−1 (tl ) − Cm−2 2 l=3 2 l=3 n−2
=
n−2
1 t Ci+1 (tl ) − Cit (tl ) · η, = 3 l=3 n−2
i = 3, . . . , m − 3,
(18)
ht s Mθ = (sr ) · θ C2s (sr ) − C1s (sr ) · θ = Cns (sr ) − Cn−1 3 r=3 r=3 m−2
m−2
1 s 1 s s (sr ) · θ C3 (sr ) − C2s (sr ) · θ = Cn−1 (sr ) − Cn−2 = 2 r=3 2 r=3 m−2
m−2
1 s Cj +1 (sr ) − Cjs (sr ) · θ, = 3 r=3 m−2
j = 3, . . . , n − 3,
(19)
then (10) holds for F ≡ TP . Next, a theorem gives sufficient conditions for using Proposition 3.2. Theorem 3.2. If the control points {Qij }m,n i,j =1 are such that 1 (Q2,j − Q1,j ) · η = (Qm,j − Qm−1,j ) · η = (Q3,j − Q2,j ) · η 2 1 1 = (Qm−1,j − Qm−2,j ) · η = (Qi+1,j − Qi,j ) · η, 2 3 i = 3, . . . , m − 3, j = 1, . . . , n,
(20)
1 (Qi,2 − Qi,1 ) · θ = (Qi,n − Qi,n−1 ) · θ = (Qi,3 − Qi,2 ) · θ 2 1 1 = (Qi,n−1 − Qi,n−2 ) · θ = (Qi,j +1 − Qi,j ) · θ 2 3 i = 1, . . . , m, j = 3, . . . , n − 3,
(21)
where (Q2,j − Q1,j ) · η = and (19) are verified.
hs (Qm,j 3
− Q1,j ) · η and (Qi,2 − Qi,1 ) · θ =
ht (Qi,n 3
− Qi,1 ) · θ , then (18)
The proofs of the proposition and of the theorem are tedious but straightforward.
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Last, we discuss how to choose the control points satisfying the hypothesis of the previous theorem. This can be realized by using a Lagrange transfinite scheme L(s, t) (see [4] for details) interpolating the n four boundary curves and sampling it at parameter values {σi }m i=1 and {τj }j =1 in [0, 1] such that 1 1 σ2 − σ1 = σm − σm−1 = (σ3 − σ2 ) = (σm−1 − σm−2 ) 2 2 1 = (σi+1 − σi ), i = 3, . . . , m − 3, 3 1 1 τ2 − τ1 = τn − τn−1 = (τ3 − τ2 ) = (τn−1 − τn−2 ) 2 2 1 (22) = (τj +1 − τj ), j = 3, . . . , n − 3. 3 The strategy for choosing the parameter values (here referred as NUSLB strategy) is motivated by the proposition below whose proof is trivial due to the linearity of the blending functions used in L(s, t). Proposition 3.3. If, for any positive real number ρ, the four boundary curves satisfy φi (ρt) − φi (0) · θ = ρ φi (s, t) − φi (0) · θ, i = 1, 2, ψj (ρs) − ψj (0) · η = ρ ψj (s) − ψj (0) · η, j = 1, 2, L(s, t) satisfies L(s, ρt) − L(s, 0) · θ = ρ L(s, t) − L(s, 0) · θ, L(ρs, t) − L(0, t) · η = ρ L(s, t) − L(0, t) · η,
∀ s ∈ [0, 1], ∀ t ∈ [0, 1].
4. Examples In this section we present some numerical results to put into evidence how the grid spacing can be n−2 controlled by means of the discrete uniformity property. In particular, first the knots {si }m−2 i=3 , {tj }j =3 used for the B-splines in the tensor product are chosen by solving the constrained non-linear system we get from (14) and (15), that is n+1 2 m+1 2 min s ,...,s ,t ,...,t 3 m−2 3 n−2 l=1 Ul + r=1 Wr , m−2 (23) sr < sr+1 , r=2 (sr+1 − sr ) = 1, n−2 tl < tl+1 , l=2 (tl+1 − tl ) = 1, where Ul−1 = Wr−1 =
m−2 3 φ s (sr ) · θ − Mθ , Cls (sr ) − Cl−1 tl+1 − tl−2 r=3 n−2 t 3 t (tl ) · η − Mηψ , Cr (tl ) − Cr−1 sr+1 − sr−2 l=3
l = 2, . . . , n, r = 2, . . . , m,
C. Conti et al. / Applied Numerical Mathematics 49 (2004) 355–366
Un =
m−2
363
φ1 (sr ) − C1s (sr ) · θ,
r=3
Wm =
n−2 ψ1 (tl ) − C1t (tl ) · η, l=3
Un+1 =
m−2
φ2 (sr ) − Cns (sr ) · θ,
r=3
Wm+1 =
n−2 ψ2 (tl ) − Cmt (tl ) · η.
(24)
l=3
We note that the conditions (13) are not included in the minimization procedure since the constants Kl , l = 1, . . . , 4 in (1) are set so that (13) holds. The set of control points used in the tensor product is simply obtained by uniformly sampling a Lagrange blending surface interpolating the four given boundary curves φ1 , φ2 , ψ1 , ψ2 on a coarse parameter set. We refer to this way of getting the control points as USLB strategy. In case of uniformly distributed knots, we use the procedure NUSBL for choosing the control points. We conclude the section showing four “pseudo-rectangular” domains. For each of them we present the initial grid performed by using uniform parameter values with Kl = 1, 1 l 4 and control points obtained via USLB strategy. Then (on the left of each picture) we show the grid obtained by using nonuniform B-spline knots solution of the minimization problem and with Kl , 1 l 4 satisfying (13). The non-linear problem has been solved by using the routine constr of the optimization toolbox of the Matlab package. Finally, (on the right of each picture) we show the grid obtained applying the NUSLB strategy to get the control points for the tensor product when using uniform B-spline knots. The values of Kl , for 1 l 4 still satisfy (13). All the examples show the effectiveness of the proposed strategies.
Fig. 1. Initial grid.
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Fig. 2. Final grids.
Fig. 3. Initial grid.
Fig. 4. Final grids.
C. Conti et al. / Applied Numerical Mathematics 49 (2004) 355–366
Fig. 5. Initial grid.
Fig. 6. Final grids.
Fig. 7. Initial grid.
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C. Conti et al. / Applied Numerical Mathematics 49 (2004) 355–366
Fig. 8. Final grids.
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