Multivariate Analysis-Suitable T-Splines of Arbitrary Degree

Proof

For any mesh entity 𝙴 in the old mesh T ( n ) with E ℓ ⊂ Q ℓ ̄ for ℓ ≠ j and E j = Q j , the subdivision of 𝚀 removes 𝙴 and inserts three children

with mid ⁡ Q j = 1 2 ⁢ ( inf Q j + sup Q j ) . From the premise of the claim, we know that there is no T-junction 𝚃 with pdir ⁡ ( T ) = j in the 𝑗-orthogonal faces of 𝚀. We define below parents ⁡ ( A ̂ ) for any new anchor A ̂ .

Case 1: p j is odd, i.e. the anchors’ 𝑗-th components are singletons. For any mesh entity

there is also the entity E ( inf Q j ) = P j , inf Q j ⁢ ( E ( sup Q j ) ) and vice versa. This is shown as follows.

Assume for contradiction that there is an entity E ( 1 ) ⊂ ∂ Q ∩ S j ⁢ ( sup Q j ) without counterpart in ∂ Q ∩ S j ⁢ ( inf Q j ) . For arbitrary x ( 1 ) ∈ E ( 1 ) , its counterpart

lies in some 𝑗-orthogonal entity E ( 2 ) ⊂ ∂ Q ∩ S j ⁢ ( inf Q j ) . Since E ( 2 ) ≠ P j , inf Q j ⁢ ( E ( 1 ) ) , there is k ≠ j with E k ( 1 ) ≠ E k ( 2 ) .

If E k ( 1 ) and E k ( 2 ) are both singletons, their inequality implies that they are disjoint in contradiction to

Hence E k ( 1 ) and E k ( 2 ) are not both singletons. If E k ( 1 ) is a singleton, then E k ( 2 ) is an open interval, and we get x ( 1 ) ∈ Sk k ∌ x ( 2 ) . If similarly E k ( 2 ) is a singleton, then E k ( 1 ) is an open interval, and x ( 1 ) ∉ Sk k ∋ x ( 2 ) . If both E k ( 1 ) and E k ( 2 ) are open intervals, then E k ( 1 ) ≠ E k ( 2 ) yields E k ( 1 ) ⊈ E k ( 2 ) or E k ( 2 ) ⊈ E k ( 1 ) , and we assume without loss of generality the first, i.e. that E k ( 1 ) ∖ E k ( 2 ) ≠ ∅ . Since E k ( 1 ) and E k ( 2 ) are open intervals, there is y k ∈ E k ( 1 ) ∩ ∂ E k ( 2 ) , and the point

lies in a 𝑘-orthogonal entity E ( 3 ) ⊂ ∂ E ( 2 ) , hence y ( 2 ) ∈ Sk k , while

satisfies y ( 1 ) ∉ Sk k .

In all cases, Lemma 2.3 yields a T-junction 𝚃 with T ̄ ∩ Q ̄ ≠ ∅ , odir ⁡ ( T ) = k , pdir ⁡ ( T ) = j , ascell ( T ) j ∩ Q j ̄ ≠ ∅ since x ( 1 ) and x ( 2 ) (or y ( 1 ) and y ( 2 ) , respectively) differ only in the 𝑗-th direction. Let T k = { t } ; then Q ̄ ∩ S ℓ ⁢ ( t ) ⊂ GTJ ⁢ ( T ) . By Assumption 4.5, 𝚀 has active neighbor cells in at least three directions ℓ 1 ≠ ℓ 2 ≠ ℓ 3 ≠ ℓ 1 . One of these directions is 𝑗 since 𝚃 is not in the 𝑗-th frame region, as T ̄ ∩ Q ̄ ≠ ∅ and in particular ascell ( T ) j ∩ Q ̄ j ≠ ∅ . At least one of the two remaining directions is not 𝑘; without loss of generality, ℓ 1 ≠ k . The bisection of 𝚀 creates or eliminates a 𝑗-orthogonal T-junction T ′ ⊂ ∂ Q with pdir ⁡ ( T ′ ) = ℓ 1 and ∅ ≠ T ′ ∩ Q ̄ ∩ S k ⁢ ( t ) ⊂ GTJ ⁢ ( T ′ ) ∩ GTJ ⁢ ( T ) , and T ( n ) ∉ WGAS or T ( n + 1 ) ∉ WGAS in contradiction to above. This shows the claim that each entity E ( sup Q j ) ⊂ ∂ Q ∩ S j ⁢ ( sup Q j ) has a counterpart E ( inf Q j ) ⊂ ∂ Q ∩ S j ⁢ ( inf Q j ) .

For each such pair E ( inf Q j ) , E ( sup Q j ) , the new mesh contains the entity

This particularly holds for the anchors, i.e. A p ( n ) contains pairs ( A ( inf Q j ) , A ( sup Q j ) ) that lie in the boundary of 𝚀, and for each such pair, A p ( n + 1 ) contains an anchor A ̂ = A ( mid ⁡ Q j ) . Consider a new anchor A ̂ ∈ A p ( n + 1 ) ∖ A p ( n ) . We call A ( inf Q j ) , A ( sup Q j ) the parent anchors of A ̂ and write parents ⁡ ( A ̂ ) = { A ( inf Q j ) , A ( sup Q j ) } .

Case 2: p j is even, that is, the anchors’ 𝑗-th components are open intervals. The subdivision of 𝚀 removes any A = A 1 × ⋯ × A d with A j = Q j and inserts A ̂ ( inf Q j ) , A ̂ ( sup Q j ) with

We call 𝐀 the parent anchor of A ̂ ( inf Q j ) and A ̂ ( sup Q j ) and write parents ⁡ ( A ̂ ( inf Q j ) ) = { A } and parents ⁡ ( A ̂ ( sup Q j ) ) = { A } .

In both cases, any new anchor A ̂ ∈ A p ( n + 1 ) ∖ A p ( n ) is in direction 𝑗 aligned with its parent A ∈ parents ⁡ ( A ̂ ) and hence shares the same global index vector I j ⁢ ( A ̂ ) = I j ⁢ ( A ) in the new mesh T ( n + 1 ) . In what follows, we use the local index vector v j ⁢ ( A ) of the old anchor with respect to the old and new mesh, and the local index vector v j ⁢ ( A ̂ ) of the new anchor with respect to the new mesh. In other directions k ≠ j , the skeleton Sk k is unchanged as well as the global and local index vectors I k ⁢ ( A ) , v k ⁢ ( A ) , and therefore these refer to the new mesh, and to the old mesh where applicable. For the existence of T-junctions below, we refer to the new mesh if not stated otherwise.

The subdivision of 𝚀 inserts mid ⁡ Q j to these global index vectors such that we have by construction v j ⁢ ( A ̂ ) ⊂ v j ⁢ ( A ) ∪ { mid ⁡ Q j } and, since mid ⁡ Q j ∈ conv ⁡ v j ⁢ ( A ) , we have conv ⁡ v j ⁢ ( A ̂ ) ⊂ conv ⁡ v j ⁢ ( A ) . If v ℓ ⁢ ( A ̂ ) = v ℓ ⁢ ( A ) for all ℓ ≠ j , this yields supp Ω ⁡ B A ̂ ⊂ supp Ω , T ( n ) ⁡ B A .

Assume for contradiction that there are A ̂ ∈ A p ( n + 1 ) ∖ A p ( n ) , A ∈ parents ⁡ ( A ̂ ) and k ≠ j with v k ⁢ ( A ̂ ) ≠ v k ⁢ ( A ) . Since A ̂ k = A k , the middle entries of v k ⁢ ( A ̂ ) and v k ⁢ ( A ) coincide by construction. Consequently, there is some m < inf Q k or m > sup Q k with

Without loss of generality, we assume the latter cases, i.e. m > sup Q k and v k ⁢ ( A ) ∋ m ∈ conv ⁡ v k ⁢ ( A ̂ ) ∖ v k ⁢ ( A ̂ ) . Lemma 4.7 yields a T-junction 𝚃 with odir ⁡ ( T ) = k , T k = { m } , Q ̃ = ascell ⁡ ( T ) such that

Since A ̂ and 𝐀 differ only in direction 𝑗, we have pdir ⁡ ( T ) = j , and with MBox ( A ̂ , A ) j ⊂ Q j ̄ , we get Q ̃ j ∩ Q j ̄ ≠ ∅ . Similarly, from MBox ⁡ ( A ̂ , A ) ⊂ Q ̄ , we get T j ∩ Q j ̄ , and since T j is a singleton, this is T j ⊂ Q j ̄ .

Having the existence of 𝚃, there is also a T-junction T ( 0 ) with the same properties as 𝚃, which is closest to 𝚀 in direction 𝑘. We therefore consider the minimal m 0 > sup Q k such that there is a T-junction T ( 0 ) ∈ T k ⁢ ( T ̂ ) with

Case 1: v j ⁢ ( T ( 0 ) ) ∩ Q j ̄ ⊆ v j ⁢ ( A ̂ ) ∩ Q j ̄ . Since v j ⁢ ( A ̂ ) ∩ Q j ̄ = { inf Q j , mid ⁡ Q j , sup Q j } , this leads to mid ⁡ Q j ∈ conv ⁡ v j ⁢ ( T ( 0 ) ) by construction of local knot vectors.

Since 𝚃 with T k = { m } from above is a 𝑘-orthogonal T-junction, it is not in the 𝑘-th frame region, and sup Q k < m < N k − ⌊ p k + 1 2 ⌋ , i.e. 𝚀 does not touch the 𝑘-th frame region in positive direction. Hence, for any x ( 0 ) ∈ Q ̄ ∩ S k ⁢ ( sup Q k ) ∩ S j ⁢ ( mid ⁡ Q j ) , the subdivision of 𝚀 creates or eliminates a T-junction T ( 1 ) with

We choose x ( 0 ) such that x k ( 0 ) = sup Q k , x j ( 0 ) = mid ⁡ Q j and x ℓ ( 0 ) ∈ T ℓ ( 0 ) ̄ ∩ A ̂ ℓ for all ℓ ∉ { k , j } . This yields

Case 1.1: v k ⁢ ( T ( 1 ) ) ∩ ( sup Q k , m 0 ) ⊆ v k ⁢ ( A ̂ ) ∩ ( sup Q k , m 0 ) . By construction, we know that # ⁢ v k ⁢ ( A ̂ ) = p k + 2 , and from sup A ̂ k ≤ sup Q k , we get that

From (A.1) and m 0 ∈ ( sup Q k , m ] , we know that m 0 ∈ conv ⁡ v k ⁢ ( A ̂ ) , and hence

Moreover, from (A.3), we have T k ( 1 ) = { sup Q k } , and hence, by construction,

Together with (A.5), there exists z ∈ v k ⁢ ( T ( 1 ) ) with z ≥ m 0 , and hence m 0 ∈ conv ⁡ v k ⁢ ( T ( 1 ) ) . Together with (A.2), this is m 0 ∈ conv ⁡ v k ⁢ ( T ( 0 ) ) ∩ v k ⁢ ( T ( 1 ) ) ≠ ∅ , and together with (A.4), T ( n ) or T ( n + 1 ) is not WGAS in contradiction to the assumption.

Case 1.2: There exists some m 2 ∈ v k ⁢ ( T ( 1 ) ) ∩ ( sup Q k , m 0 ) ∖ v k ⁢ ( A ̂ ) . Lemma 4.6 yields that, for any x ( 1 ) ∈ P k , m 2 ⁢ ( A ̂ ) , y ( 1 ) ∈ P k , m 2 ⁢ ( T ( 1 ) ) ̄ , it holds that x ( 1 ) ∉ Sk k ∋ y ( 1 ) . Choose x ( 1 ) , y ( 1 ) such that x ℓ ( 1 ) = y ℓ ( 1 ) for all ℓ ≠ j . This is possible because x k ( 1 ) = y k ( 1 ) holds trivially and

from (A.4) and above. Lemma 2.3 yields another T-junction T ( 2 ) and Q ( 2 ) = ascell ⁡ ( T ( 2 ) ) with

From T ( 2 ) ̄ ∩ conv ⁡ { x ( 1 ) , y ( 1 ) } ≠ ∅ and T j ( 2 ) being a singleton, we get

in contradiction to the minimality of m 0 . This ends Case 1.

Case 2: There is m 1 ∈ v j ⁢ ( T ( 0 ) ) ∩ Q j ̄ with m 1 ∉ v j ⁢ ( A ̂ ) ∩ Q j ̄ . Lemma 4.6 yields that x ( 0 ) ∉ Sk j ∋ y ( 0 ) holds for all x ( 0 ) ∈ P j , m 1 ⁢ ( A ̂ ) , y ( 0 ) ∈ P j , m 1 ⁢ ( T ( 0 ) ) ̄ . We choose x ( 0 ) , y ( 0 ) such that

Lemma 2.3 yields T ( 2 ) ∈ T j with

From (A.1) and sup Q k < m 0 ≤ m , we get m 0 ∈ conv ⁡ v k ⁢ ( A ̂ ) .

Case 2.1: v k ⁢ ( T ( 2 ) ) ∩ ( sup Q k , m 0 ) ⊆ v k ⁢ ( A ̂ ) ∩ ( sup Q k , m 0 ) . This leads to m 0 ∈ conv ⁡ v k ⁢ ( T ( 2 ) ) and hence

which means that T ( n + 1 ) is not WGAS in contradiction to the assumption.

Case 2.2: There exists m 2 ∈ v k ⁢ ( T ( 2 ) ) ∩ ( sup Q k , m 0 ) ∖ v k ⁢ ( A ̂ ) . For any x ( 1 ) ∈ P k , m 2 ⁢ ( A ̂ ) , y ( 1 ) ∈ P k , m 2 ⁢ ( T ( 2 ) ) ̄ , Lemma 4.6 yields x ( 1 ) ∉ Sk k ∋ y ( 1 ) . Choose y ( 1 ) such that

Lemma 2.3 yields another T-junction T ( 3 ) and Q ( 3 ) = ascell ⁡ ( T ( 3 ) ) with

in contradiction to the minimality of m 0 . This ends Case 2.2 and concludes the proof. ∎

Proof

We set

Then we have by construction for p ℓ ≥ 1 that

The combination of (5.2) and (A.6) yields

We distinguish eight cases illustrated in Table 3.

Case 1: x ℓ ∈ A ℓ , i ≠ ℓ = pdir ⁡ ( T ) and p ℓ is odd. Since p ℓ is odd, A ℓ is a singleton, i.e. A ℓ = { x ℓ } , in contradiction to the existence of y ∈ A ℓ with y ≠ x ℓ from (5.4) above.

Case 2: x ℓ ∈ A ℓ , i ≠ ℓ = pdir ⁡ ( T ) and p ℓ is even. Then A ℓ is an open interval and T ℓ = { t } is a singleton. From (A.7), we obtain

From the definition of local index vectors, we also know t ∈ v ℓ ⁢ ( T ) , which yields v ℓ ⁢ ( A ) ⋈̸ v ℓ ⁢ ( T ) in contradiction to (5.5).

Case 3: x ℓ ∈ A ℓ , i ≠ ℓ ≠ pdir ⁡ ( T ) and p ℓ is odd. Then T ℓ is an open interval, and partsupp ⁡ ( A , x , ℓ ) = A ℓ = { x ℓ } . Hence x ℓ ∈ v ℓ ⁢ ( A ) . Inequality (A.7) yields x ℓ ∈ T ℓ ̄ ⊆ conv ⁡ v ℓ ⁢ ( T ) .

Case 4: x ℓ ∈ A ℓ , i ≠ ℓ ≠ pdir ⁡ ( T ) and p ℓ is even. Then T ℓ and A ℓ are open intervals, and (A.7) yields that T ℓ ̄ ∩ A ℓ ≠ ∅ . Together with (5.5), we have x ℓ ∈ A ℓ = T ℓ ⊂ conv ⁡ v ℓ ⁢ ( T ) .

Case 5: x ℓ ∉ A ℓ , i ≠ ℓ ≠ pdir ⁡ ( T ) and p ℓ is even. Assume without loss of generality that x ℓ > y for all y ∈ A ℓ . In this case, partsupp ⁡ ( A , x , ℓ ) = A ℓ ∪ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] with A ℓ being an open interval and x ℓ ∈ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] . Also T ℓ is an open interval, and (A.7) and (5.5) yield that either T ℓ = A ℓ or inf T ℓ ∈ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] , that is, T ℓ ̄ ∩ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] ≠ ∅ . The knot vector v ℓ ⁢ ( T ) contains p ℓ 2 + 1 entries that are not smaller than sup T ℓ and p ℓ 2 + 1 entries that are not greater than inf T ℓ . The interval [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] contains p ℓ 2 + 1 entries of v ℓ ⁢ ( A ) . Together with (5.5), all entries of v ℓ ⁢ ( A ) ∩ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] match with entries of v ℓ ⁢ ( T ) , and we get x ℓ ∈ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] ⊆ conv ⁡ v ℓ ⁢ ( T ) .

Case 6: x ℓ ∉ A ℓ , i ≠ ℓ ≠ pdir ⁡ ( T ) and p ℓ is odd. Then A ℓ is a singleton A ℓ = { s } . Assume without loss of generality that x ℓ > s ; then partsupp ⁡ ( A , x , ℓ ) = A ℓ ∪ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] = [ s , max ⁡ v ℓ ⁢ ( A ) ] which contains ⌈ p ℓ 2 ⌉ + 1 entries of v ℓ ⁢ ( A ) . As in Case 5 above, we have T ℓ ̄ ∩ [ s , max ⁡ v ℓ ⁢ ( A ) ] ≠ ∅ , and v ℓ ⁢ ( T ) containing ⌈ p ℓ 2 ⌉ + 1 entries that are at least sup T ℓ and ⌈ p ℓ 2 ⌉ + 1 entries that are at most inf T ℓ . Together with (5.5), we get x ℓ ∈ [ s , max ⁡ v ℓ ⁢ ( A ) ] ⊆ conv ⁡ v ℓ ⁢ ( T ) .

Case 7: x ℓ ∉ A ℓ , i ≠ ℓ = pdir ⁡ ( T ) and p ℓ is even. Then A ℓ is an open interval, and we assume without loss of generality x ℓ > y for all y ∈ A ℓ , obtaining partsupp ⁡ ( A , x , ℓ ) = A ℓ ∪ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] with x ℓ ∈ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] . Since ℓ = pdir ⁡ ( T ) , T ℓ is a singleton T ℓ = { t } = T ℓ ̄ , and (A.7) yields t ∈ partsupp ⁡ ( A , x , ℓ ) . Together with t ∈ v ℓ ⁢ ( T ) and (5.5), we get that t ∈ v ℓ ⁢ ( A ) ∩ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] . The partial index vector v ℓ ⁢ ( A ) ∩ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] contains p ℓ 2 + 1 entries of v ℓ ⁢ ( A ) , while v ℓ ⁢ ( T ) contains p ℓ 2 + 1 entries that are at least 𝑡 and p ℓ 2 + 1 entries that are at most 𝑡. As in previous cases, we obtain with (5.5) that v ℓ ⁢ ( A ) ∩ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] ⊂ v ℓ ⁢ ( T ) and consequently x ℓ ∈ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] ⊂ conv ⁡ v ℓ ⁢ ( T ) .

Case 8: x ℓ ∉ A ℓ , i ≠ ℓ = pdir ⁡ ( T ) and p ℓ is odd. Then A ℓ is a singleton A ℓ = { s } . Assume without loss of generality that x ℓ > s ; then we have partsupp ⁡ ( A , x , ℓ ) = A ℓ ∪ [ sup A ℓ , max ⁡ v ℓ ⁢ ( A ) ] = [ s , max ⁡ v ℓ ⁢ ( A ) ] which contains ⌈ p ℓ 2 ⌉ + 1 entries of v ℓ ⁢ ( A ) . Since ℓ = pdir ⁡ ( T ) , T ℓ is a singleton T ℓ = { t } = T ℓ ̄ , and (A.7) yields

Moreover, t ∈ ∂ Q ℓ = { inf Q ℓ , sup Q ℓ } ⊆ v ℓ ⁢ ( T ) for the associated cell Q = ascell ⁡ ( T ) from definition (4.2). By construction of the knot vector, we have Q ℓ ⊂ conv ⁡ v ℓ ⁢ ( T ) ∖ v ℓ ⁢ ( T ) , and with (5.5), we obtain Q ℓ ∩ v ℓ ⁢ ( A ) = ∅ . Consequently, s , max ⁡ v ℓ ⁢ ( A ) ∉ Q ℓ , and hence we have either Q ℓ ⊂ [ s , max ⁡ v ℓ ⁢ ( A ) ] or Q ℓ ∩ [ s , max ⁡ v ℓ ⁢ ( A ) ] = ∅ . Together with (5.3), we have Q ℓ ⊂ [ s , max ⁡ v ℓ ⁢ ( A ) ] , and since [ s , max ⁡ v ℓ ⁢ ( A ) ] is closed, Q ℓ ̄ ⊂ [ s , max ⁡ v ℓ ⁢ ( A ) ] . The combination with (5.5) yields that { inf Q ℓ , sup Q ℓ } ⊆ v ℓ ⁢ ( A ) ∩ [ s , max ⁡ v ℓ ⁢ ( A ) ] . Since v ℓ ⁢ ( T ) contains ⌈ p ℓ 2 ⌉ + 1 entries that are at least inf Q ℓ and ⌈ p ℓ 2 ⌉ + 1 entries that are at most sup Q ℓ , (5.5) yields v ℓ ⁢ ( A ) ∩ [ s , max ⁡ v ℓ ⁢ ( A ) ] ⊆ v ℓ ⁢ ( T ) and hence x ℓ ∈ [ s , max ⁡ v ℓ ⁢ ( A ) ] ⊂ conv ⁡ v ℓ ⁢ ( T ) .

We have shown the claim in all cases, which concludes the proof. ∎