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Deriving T
r
ðuÞ (4) with respect to u yields
T
0
r
ðuÞ¼
P
0
0
ðuÞW
0
ðuÞP
0
ðuÞW
0
0
ðuÞ
W
2
0
ðuÞ
,
P
1
ðuÞ
W
1
ðuÞ
: ð6Þ
According to (4), the h-degree spline functions P
0
ðuÞ and W
0
ðuÞ are defined on the knot vector (5). It is easy to show that
P
0
0
ðuÞ and W
0
0
ðuÞ are both defined on
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
h
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
0<i
2
6hþ1
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
0<i
a
1
6hþ1
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
h
and their degree is ðh 1Þ. Based on Lemma 1, P
1
ðuÞ in (6) is defined on knot vector
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2h
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þh
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þh
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2h
and its degree is ð2h 1Þ. On the other hand, W
1
ðuÞ¼W
2
0
ðuÞ is defined on knot vector
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2hþ1
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þh
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þh
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2hþ1
and its degree is 2h. Therefore, the statement holds for n ¼ 1.
Assume that the statement holds for some unspecified value of n 1; n P 2. That is,
T
ðn1Þ
r
ðuÞ¼
P
n1
ðuÞ
W
n1
ðuÞ
; ð7Þ
where P
n1
ðuÞ is a ð2
n1
h n þ 1Þ-degree spline function with knot vector
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n1
hnþ2
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þð2
n1
1Þh
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þð2
n1
1Þh
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n1
hnþ2
;
W
n1
ðuÞ is a 2
n1
h-degree spline function with knot vector
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n1
hþ1
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þð2
n1
1Þh
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þð2
n1
1Þh
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n1
hþ1
:
Deriving T
ðn1Þ
r
ðuÞ (7) with respect to u yields
T
ðnÞ
r
ðuÞ¼
P
0
n1
ðuÞW
n1
ðuÞP
n1
ðuÞW
0
n1
ðuÞ
W
2
n1
ðuÞ
,
P
n
ðuÞ
W
n
ðuÞ
:
Also based on Lemma 1, P
0
n1
ðuÞW
n1
ðuÞ is a ð2
n
h nÞ-degree spline function defined on
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n
hnþ1
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þð2
n
1Þh
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þð2
n
1Þh
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n
hnþ1
; ð8Þ
P
n1
ðuÞW
0
n1
ðuÞ is a ð2
n
h nÞ-degree spline function defined on
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n
hnþ1
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þð2
n
1Þh1
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þð2
n
1Þh1
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n
hnþ1
: ð9Þ
By inserting u
2
; u
3
; ...; u
a
1
into the knot vector (9) using the de Boor algorithm, both knot vectors (8) and (9) are identical.
Therefore, P
n
ðuÞ¼P
0
n1
ðuÞW
n1
ðuÞP
n1
ðuÞW
0
n1
ðuÞ is a spline function of degree ð2
n
h nÞ with the knot vector (8).
On the other hand, W
n
ðuÞ¼W
2
n1
ðuÞ is a 2
n
h-degree spline function defined on knot vector
u
1
; ...; u
1
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n
hþ1
; u
2
; ...; u
2
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
i
2
þð2
n
1Þh
; ...; u
a
1
; ...; u
a
1
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
i
a
1
þð2
n
1Þh
; u
a
; ...; u
a
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
n
hþ1
:
Therefore, it is shown that indeed the statement holds for n. Since both the basic and the inductive steps have been
proved, then the statement holds for all n. And the theorem is proved. h
According to Theorem 1, the NURBS function T
r
ðuÞ (4) and its derivatives of arbitrary order have the same breakpoint
sequence fu
1
; u
2
; ...; u
a
g. Since DT
r
ðuÞ is a combination of T
r
ðuÞ and its derivatives, DT
r
ðuÞ has the same breakpoint sequence
as T
r
ðuÞ and its derivatives of arbitrary order. Therefore, they are defined on the same knot intervals, i.e.,
Corollary 1. The NURBS function T
r
ðuÞ (4), its derivatives of arbitrary order, and DT
r
ðuÞ are defined on the same knot intervals,
½u
1
; u
2
; ½u
2
; u
3
; ...; ½u
a
1
; u
a
:
474 H. Lin et al. / Comput. Methods Appl. Mech. Engrg. 267 (2013) 471–486
这个微分算子必须是个线性算
子,如果是个非线性算子,则会
出现一个关于NURBS函数未知系数
的非线性方程组。