### Table 3: POD Basis Functions From U = B-splines and 4 Time Snapshots, Energy= 80% N, Linear B-splines K, POD Basis Functions

1999

Cited by 16

### Table 2: Number of simulations that select the given number of B-splines

2005

"... In PAGE 12: ... The chosen architecture depends on the simulation, but in general, small architectures are prefered, as summarized by the following tables. Table2 gives the number of time each B-splines basis has been chosen and table 3 gives the number of time each number of hidden neurons has been chosen. Number of B-splines 5 7 10 15 20 FMLP 18 17 8 3 4 FpMLP 38 8 4 0 0 Table 2: Number of simulations that select the given number of B-splines... ..."

Cited by 4

### Table 1. Condition number of B as a function of the degree n of the B-spline

"... In PAGE 5: ... That is, given the closed form continuous description D of the FT of a zero-centered B-spline se- quence f kg (see [8], [10]) with uneven degree as1 D = 0 + 2 bn+2c+1 X i=1 i cos(2 if) : (17) We find the condition number easily by maximizing and minimizing (17). Some algebraic transforms which are omitted here for brevity reveal (B) = 0 @1 ? 4 (n?2) DIV 4 X i=0 2i+1 1 A ?1 : (18) Table1 depicts the condition number of B as a function of the B-spline degree n. We observe a moderate increase with n making the problem tractable for all spline orders used in practice.... ..."

### Table 2: Example surfaces created using an interactive editor for hierarchical b-splines.

"... In PAGE 5: ... Most were created using an interactive editor for hierarchical B-splines, one created using SoftImage, and the last from a digitized surface data created using a Cyberware laser scan- ning system. The results are illustrated in Section 8 and sum- marized in Table2 and Table 3. All of the resulting surfaces (with exceptions noted below) approximate the initial input mesh with 0.... ..."

### Table 1: Difference between cost functions and the distance between the symmetric cubic spiral and the B-spline minimizers Bn.

1000

### Table 2a: Same as Table 1a but for B-spline FEM.

"... In PAGE 25: ... In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown. In Table2 a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table 1a shows that they are identical for equal mesh point numbers.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table2 c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used ( Table2 d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table2 e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required ( Table2 f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 28: ...f [10?1; 10?10] are su cient in most applications. In Fig. 8b there is an indication for 6th order to become optimal order at precisions better than 10?10. This is in agreement with the conclusion that has been drawn form the data in Table2... ..."

### Table 2c: Same as Table 1c but for B-spline FEM.

"... In PAGE 25: ... In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown. In Table2 a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table 1a shows that they are identical for equal mesh point numbers.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table2 c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used ( Table2 d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table2 e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required ( Table2 f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 28: ...f [10?1; 10?10] are su cient in most applications. In Fig. 8b there is an indication for 6th order to become optimal order at precisions better than 10?10. This is in agreement with the conclusion that has been drawn form the data in Table2... ..."

### Table 2e: Same as Table 1e but for B-spline FEM.

"... In PAGE 25: ... In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown. In Table2 a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table 1a shows that they are identical for equal mesh point numbers.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table2 c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used ( Table2 d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table2 e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required ( Table2 f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 28: ...f [10?1; 10?10] are su cient in most applications. In Fig. 8b there is an indication for 6th order to become optimal order at precisions better than 10?10. This is in agreement with the conclusion that has been drawn form the data in Table2... ..."

### TABLE I RESULTS OF SHAPE CODING USING B-SPLINES AND A QP OF 20 ( KIDS FRAME 0).

1998

Cited by 15