A C^2 Finite Element and Interpolation

Computing 50, 69-76 (1993)

COITI~[~ 9 Springer-Verlag 1993 Printed in Austria

Short Communications/Kurze Mitteilungen A C 2 Finite Element and Interpolation J. Gao, Wuhan Received April 22, 1992; revised October 18, 1992 Abstract - - Zusammenfassung

A C 2 Finite Element and Interpolation. In this paper, we define so-called QT triangulation of a given partition with quadrilateral, establish a C 2 finite element and interpolation of the space of piecewise bivariate polynomial of total degree 6, and in the process obtain a local bases for the space.

AMS( MOS ) Subject Classification: 41A15 Key words: Bivariate splines, QT-triangulation, interpolation. Ein C2-finites Element and Interpolation. In diesem Artikel wird eine sogenannte QT-Triangulierung einer gegebenen Vierecksunterteilung eingef/ihrt. Ein Finite-Elemente-Unterraum des Raumes C 2 wird konstruiert. Dieser Unterraum besteht aus den Funktionen, die stfickweise Polynome yon zwei Ver~inderlichen des Gesamtgrades 6 sind und gewissen Interpolationsbedingungen Geniigen.

I. Introduction

Let f2 be a simply connected domain in R 2 whose boundary is polygonal and A be any triangulation of I2, and d and r be integers such that 0 < r _< d. Denote by S~(A) the vector space of all function in C'(f2) whose restriction to each triangular domain of A is a polynomial of total degree at most d (cf. [1, 2]). The space S~(A) is called a bivariate spline space. On the finite element or interpolation point of view, we are interested in finding the functions belonging to S~(A) with minimal support. If d is fairly larger than r, it is possible to construct a basis of S~(A) consisting of functions with smaller support. For instance, when d > 4r + 1, there exist some functions being a bases of SS(A) whose supports contain at most one vertex ofA (cf. [2]); when d > 3r + 1, such functions may have large support (cf. [3]). Since lower-degree piecewise polynomials are more desirable on the practical point of view (such as computing efficiency), many authors have been working on developing special methods to consider the spline interpolation element on so-called micro-triangulations such as well-known Clough-Tocher splitting and Powell-Sabin splitting (cf. [6, 7, 8, 9]). The aim of this paper is devoted to constructing a finite element space of C z spline of degree 6 with some interpolation conditions on so-called QT triangulation A. of t2. This paper is organized as follows. In section 2 we will introduce some basic concepts and notations. In section 3, we drive the main results about the dimension

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J. G a o

of space $6z(A,). A local interpolation scheme is defined and proved to be unisolvable in section 4.

2. The Basic Tools and Conclusions In this section we introduce some notation and general results. Given some simply connected domain ~2, let o~ = {f2i: Oi is a convex quadrilateral for each i = 1, 2 . . . . . N} be a partition of f2 in the sense that

i) ~ , n ~ j = ~,i # j ,

ii) f2i n ~j =

c o m m o n vertex L c o m m o n edge

N

iii) U ~i = ~i=l

The vertices (edges) of ~i = v~,,[i]~2"[q"t~]"[i]~3 ~4 (in counterclockwise order) for each i = 1, 2 . . . . . N are called the vertices (edges) of m. The vertices in interior domain of O are called interior vertices, and otherwise boundary vertices of f2. For each interior (boundary) vertex v of e), we say deg(v) = n ( = n + 1), if there exist n quadrilaterals with a vertex at v.

Definition 2.1. The partition ~o of (2 is regular if for any edge of ~o whose two end-points are interior vertices there exists at least one of end-points v with deg(v) > 3. We assume henceforth that e~ is regular. We define a Q T triangulation A, on ~o with connecting vt~i] with vt~1, and v~] with v~] by lines for each (2~ in co. Denote by vt~] the cross-point of ~,,[q,,t~l ,,m,,[q 1 ~ 3 and ~2 v 4 , and Tlt~] = ~o"~i]"UMi]~l ~+~ (I ~ Z4). Given a partition ~o (regular), A, be a Q T triangulation of co, and denote V = n u m b e r of all vertices of w, E = total n u m b e r of edges of ~o, N = n u m b e r of quadrilaterals of Following [2], each spline s ~ S~(A) can be written in the form s(x) = sin(x)

for x ~ T m, 1 = 1, 2 . . . . . M

(2.1)

and d I

sV](x) =

Z

i+j+k=d

C~]k~-cd~JTk t!J!N:

(2.2)

where T m is any triangle of A and (c~,fl, 2) is the barycentric coordinates of x with respect to T m = v~%~]v~ ] (in counterclockwise order). Associated with any triangulation A, define a domain point set

A C2 Finite Elementand Interpolation

71

M

Oa := ~ {P~ = (iv~1 + jv~ 1 + kv~l)/d: i + j + k = d}

(2.3)

1=1

The ring of order m around the vertex v is

Rm(v) = {Pa-m,j,k ~ T: v = vx(T)} and the disk of order m around v is

O,.(v) = ~ Rj(v) j=o ~[l! in (2.2) a B6zier ordinates associated with pt~ which satisfy the C r We call cij~ smoothness conditions [5]. Associated with each domain point P s O~, we define a linear functional on S~(A) by

2ps = the B6zier ordinate of P. Suppose that F ~ Oa contains K points and that A t ( = {2,: P ~ F}) has the property that it is a determining set for S~(A) in the sense that

s~S~(d)

and

2s=0

forall2~Arimpliess=-O

Then we have Lemma 2.2. dim S~(A) < K.

3. The Dimension Formula of S~(A,) In what follows we will prove that the dimension of S~(A,) is Schumaker's lower bound (cf. [10]). For each vertex v of A,, let e(v) the number of distinct slopes assumed by those edges at v. In particular, when v is a vertex in A, but not in ~o, we have e(v) = 2.

Definition 3.1. For an interior vertex vj of A,, if deg(vj) = 6 (in A,) but e(vj) = 3, we call it a sinoular vertex of A , (or Clough-Tocher vertex [1]). If deg(vl) = 6 but e(vi) > 3, we call it a six-edge vertex of A,. Denote by V~ the number of singular vertices in A ,. It is easy to see that if v is adjacent to some six-edge vertex of 3 , then deg(v) > 6 since co is regular. We will use this fact in the following. Lemma 3.2 (cf. [10]). dimS~(A,) _> 6V + 6E + N + V~ = lb. We shall establish an upper bound to dim S~ (A,) which agrees with the lower bound lb in Lemma 3.2 and in the process obtain an explicit local basis for S~(A,) with Lemma 2.2. Now we introduce some results from [11] without proof. Lemma 3.3. Let A~ be a trianoulation with six interior edges endin9 at only one interior vertex v which just have three different slopes, and vl, wl, ..., v3, w3 in counterclockwise order be the boundary vertices of A ~ Define a subset of domain

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J. Gao

points by F = (D3(v) ~ T Ill) w {v z, w 2, v3}

where T Ell = vv 1 w 1. Then F is a minimal determining set for S~(A~ L e m m a 3.4. Let d 6 be a triangulation with only one interior vertex at v and six boundary vertices at vl, wa, v2, wz, v3, and w3 in counterclockwise order. Suppose that v, v 1 and w z are not collinear and the angle v 1 vw 2 > ~ in counterclockwise direction and denote by F the following subset o f domain points (D3(v) ~ T It]) u {v2, w2} ,

where T It] = vvtw l. Then I" is a minimal determining set for $32(A6).

Let A2. (n >_ 4) be a triangulation with only one interior vertex v and 2n b o u n d a r y vertices, say vl, wl, v2, wz . . . . . w. in counterclockwise order such that wl is on the edge v~vi+l. D e n o t e by 0~ the angle vivvi+~ in counterclockwise direction. Then there exists some 1 < i o < n such that 0~o + 0~o+1 _< 7:. Suppose that i o = 1 without loss of generality. Let vz = a i r + filv3 + Ylw2, v2 = azv +/~2Vl + 72wl, and w2 = ao v +/3o w~ + ?o v2, then we can conclude that/3i < 0, ?i > 0 (i = 1, 2, 3) and 7oY1~2 + 71/3o - 7z > 0

(3.1)

L e m m a 3.5. Let A 2, (n > 4) be a triangulation with the above conditions. Denote by I ~ the following set. {D3(v) c~ z]VWnV1 } k..) {V2 . . . . . Vn, W3,... , Wn_1 } Then F is a minimal determining set for $2(A2,) Proof. First we see # F = dimSZ(A:,) = 6 + 2n. It remains to check that F is a determining set. Suppose that 2ps = 0 for all of P ~ F, then all B6zier ordinates but A, al, az, B, bl and b: (as shown in Fig. 1) are zero by smoothness conditions ts~

V2

V,

w.

Figure 1

A C 2 Finite Element and Interpolation

73

Thus we have the following system a t = yi A,

a2 = y2A,

al = fl20bl + 2floYob2,

b 1 = y2 B ,

b2 = y22B

a2 = gob2

Hence we obtain yxA = fl2yzB + 2floYoy2B

y~A = floy2B,

(3.2)

The determinant of (3.2) is given by D = 717230(2707172 -q- 7130

-

-

72)

Thus we conclude that D < 0 with (3.1). This completes the proof of determining property. L e m m a 3.6. Let A o be the Q T triangulation of a single quadrilateral with four vertices

at vl, v2, v 3 and va, and denote by Vo the cross-point of vlv 3 and v2v 4. Then the following domain-point set is a minimal determining set for S2(Ao).

1) For each triangle T ttl = VoVtVt+t (l = 1, 2, 3,4), choose the points D3(vl) (~ Tt0;

2) Choose the points in R4(vo) but outside of D3(vt) (l = 1, 2, 3, 4); 3) Choose the points in the following set D2(vo) ~ [VoV 1 u VoV2].

We can directly prove lemma 3.5 by complicated computation. Now we are in a position to state the main result. Theorem 3.7. dim S2(A) = 6V + 6E + N + V, and the following union set F o r 1)-5) is a minimal determining set for $62(A,): 1) For each interior vertex v of 09 with deg(v) = 3,/f e(v) = 3 in A,, then use Lemma 3.3 to choose points in D3(v) of A , such that vvl in Lemma 3.3 is corresponding to edge in 09; if e(v) >_ 4 in A,, then use Lemma 3.4 to choose points in D3(v) of A,, such that vvl in Lemma 3.4 corresponding to an edge in c9, and mark the edge L of 09 whose point R3(v) c~ L is not chosen. 2) For each interior vertex v of 09 with deg(v) >_ 4, use Lemma 3.5 to choose points in D3(v) in A , such that vv~s in Lemma 3.5 corresponding to the edges in co, except for R3(v) c~ L' s where all L in 09 are marked. 3) For each boundary vertex v of oJ. Let ei, ..., e2,~+~ be all edges in A , ending at v and T [tl is the triangle with edges et and ez, choose the following set ((D3(v) c~ Ttll) w (R3(v) c~ (e t w . . . w e2m+l))/{R3(v) c~ ei; e~ marked and i --- 1 (mod 2)}.

4) For each edge e = vt vz of ~o, choose the points, in one triangle of A , with e, which is a distance 2 from e but outside of D3(vl) and D3(v2). 5) For each quadrilateral of 09, use the item 3) of Lemma 3.6 to choose five points in

A,.

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J. Gao

Proof. We first check the cardinality of the above union set. The number of items 1) and 2) and 3) is 6V + 3E + E B + Vs. The number of items 4) and 5) is E + 5N. Thus the total number is 6 V + 4E + E B + 5N + V~ = 6V + 6E + N + V~ = Ib, where EB is number of boundary edges of 09. It remains to check that this set is a determining set for SZ6(A,). This is simple. First, it follows from Lemma 3.3-3.5 that 2ps = 0 for any P ~ Do(v), v ~ 09 and then by Lemma 3.6 and items 4) and 5) we can conclude that 2pS = 0 for all domain points. This completes the proof. Theorem 3.8. There exists a local basis of Sz(A.), say B = {Bp; ~QBe = ~QV,for all P, Q ~ F}. In addition, for P in item 1) the supp Bp contained in subpartition of 09 with interior vertices at v and w such that vw is marked, or with only one interior vertex; for P in item 2) or 3) the supp Bp consists of all quadrilaterals of co with the corresponding vertex at v; for P in item 4) the supp Bp is triangles of A , with the corresponding edge e; for P in item 5) the supp Be is the corresponding single quadrilateral ofw.

4. The Interpolation by ~(A,) Let A 0 be the Q T triangulation of only one quadrilateral t2 = v l v2 v3/)4 and denote T t~l = Vo/)~v~§ (i ~ Z4). Then the following Lemma will play an important role in computing methods for the interpolants. Lemma 4.1. There exists a unique s ~ Sg(A.) for any f

I D~_v,D~+l_o,s(vl) = D~_v D~+ _o,f(v), 2

1

__

2

1

2

0 < ~i + o~2 ~ 3,

[Do~-ooS(/)o) = Do~_~of(Vo),

i ~ Z~,

t = 1,2,3,4,

Dv,-voS(Vo)= Dv_oof(Vo), 2

C3(~) such that

9

D;,,s(xvlvi+l) - D~f(~vi/)i+l), S(Vo) = f(vo),

~

2

(4.1)

D~-ooS(Vo) = D,_vof(Vo), 2

Doz_~oS(/)o)= Do~_vof(/)o)

where n~ is the unit normal vector of v~/)~+~ Proof. Let D be the minimal determining set for S~(Ao) defined in Lemma 3,6. With the relations between B-form and Taylor expansion of polynomials (cf. [4]) and C z smoothness conditions, it is clear that all of B6zier ordinates of s associated with P ~ D can be represented by values in (4.1). Hence s is uniquely determined by the interpolation conditions. Let A, be a Q T triangulation of o9. For each interior vertex v of 09, let v~, w~, v2.... , v~, w~ be all of vertices in A, such that the conditions are satisfied in Lemma 3.4 and 3.5, respectively. For each boundary vertex v of A let v~, w~, v2, ..., w~_~, v, be all vertices in A, adjacent to v such that v~ and vn are also boundary vertices. For S~ (A,) we give the following interpolation scheme. Let f ~ C 3(12), then we solves a spline s ~ Sg(A,) such that

A C 2 Finite Element and Interpolation

75

1) F o r each vertex v of co.

D~lO~2s(v) = O~lO~2f(v),

0 <_ ~i + ~2 <-- 2.

2) F o r each singular vertex v (cf. Definition 3.1) in A, 3 V) = O~3 _vf(v ) Dw,-vs(

D~:_,D$]_~s(v) = D~_vD~]_vf(v),

i = 1, 2,

~i + a2 = 3,

~l > 1,

~2 > 1.

~l >-- 1,

~2 >- 1.

3) F o r each six-edge vertex v (cf. Definition 3.1) in A,

,_ vs(v) = D,~3 _,f(v) D~:_,D~]_vs(v) = D~_~D~]_,f(v),

i = 1, 3

~1 + ~2 = 3,

4) F o r each other interior vertex v of co

D~_~s(v) = D~_~f(v), D~:_~D~,~_.s(v) = D~:_~D~,~_vf(v),

i = 3, 4 , . . . , n, O~1 + O~2 = 3 ,

O~1 > 1 ,

O~2 ~

1.

5) F o r each boundary vertex v of r

D ~ _ , s ( v ) = D~_~f(v) 3 D~:_~D~_~s(v) = D~_,D$]_~f(v),

i = 1, 2 . . . . . n - 1,

~l + ~2 = 3,

~1 > 1,

~2 > 1.

6) F o r each unmarked edge e of co, s(89 = f( 89 and for each edge e, D2s(89 = DZ,f( 89 where n is normal vector to e and 89 is the middle point of e. 7) F o r each quadrilateral of co with vertices at vl, v2, v3, v4, denote v0 = vl v3 c~ v2 v4,

D~?D;~s(Vo) = D~,lD;=f(vo),

0 <_ ~l + ~2 <- 2.

D,2_vS(Vo) = D~rof(vo)

i = 1, 2.

Theorem 4.2. There exists only one spline s in S~(A .) such that s satisfies the above interpolation conditions 1)-7).

Proof. With interpolation conditions and smoothness conditions it is easy to show all of B$zier ordinate of s on the domain point in F of Theorem 3.7 are determined. Thus the minimal property of Fimplies that s uniquely determined since the number of interpolation conditions is # F. At least, under the aid of L e m m a 4.1 s is easily computed from the interpolation conditions.

Acknowledgement The author thanks Professor W. Kn6del of Universit/it Leipzig for his kind help. Thanks are also due to the Referee for his good suggestions.

References [1] Alfeld, P., Schumaker, L. L.: The dimension of bivariate spline spaces of smoothness r for degree d > 4r + 1. Constructive Approximation 3, 189-197 (1987).

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J. Gao: A C 2 Finite Element and Interpolation

[2] Alfeld, P., Piper, B., Schumaker, L. L.: Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d > 4r + 1. Computer Aided Geometric Design 4, 105-112 (1987). I-3] Alfeld, P., Piper, B., Schumaker, L. L.: On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1. Numer. Math. 57, 651-661 (1990). [4] de Boor, C.: B--form basics. In: Farin, G. (ed.) Geometric modeling. Philadelphia: SIAM P.A. 1987. [5] Farin, G.: Triangular Bernstein-Brzier patches. CAGD 3, 83-127 (1986). [6] Gao, J. B.: A new finite element of C 1 cubic splines, J. Comput. Appl. Math. 40, 305-312 (1992). I-7] Lawson, C.: Software for C 1 surface interpolation. In: Rice, J. R. (ed.) Mathematical software III. New York: Academic Press, pp. 161-194 (1977). 1-8] Powell, M. J. D., Sabin, M. A.: Piecewise quadratic approximation on triangles. ACM. Trans. Math. Software 3, 316-325 (1977). [9] Renka, R. J.: Algorithm 624, triangulations and interpolations at arbitrarily distributed points in the plane. ACM Trans. Math. Software 10, 440-446 (1984). 1-10] Schumaker, L. L.: On the dimension of spaces of piecewise polynomials in two variables. In: Schempp, W., Zeller, K. (eds.) Multivariate approximation theory. Birkhauser, pp. 396-412 (1979). [11] Schumaker, L. L.: Dual bases for spline spaces on cells. CAGD 5, 277-284 (1988). J. B. Gao Department of Mathematics Wuhan University Wuhan 430072 People's Republic of China