1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME ENGINEERING AND TECHNOLOGY (IJARET)ISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online) IJARETVolume 3, Issue 2, July-December (2012), pp. 322-336© IAEME: www.iaeme.com/ijaret.asp ©IAEMEJournal Impact Factor (2012): 2.7078 (Calculated by GISI)www.jifactor.com AN APPLIED TWO-DIMENSIONAL B-SPLINE MODEL FOR INTERPOLATION OF DATA Mehdi Zamani Civil Engineering Department, Faculty of Technology and Engineering, Yasouj University, Yasouj, IRAN, mahdi@mail.yu.ac.ir ABSTRACT The literature about the interpolation of data is less than the ones for the approximation of data especially for three dimension data. The three methods of interpolation, two-dimensional Lagrange, two-dimensional cubic spline and two-dimensional explicit cubic spline are investigated. In the present study a new model with two-dimensional B-spline approach has been developed. The presented model has the advantage of simplicity and applicability with respect to the volume of operation calculations and using non-uniform and non-symmetric data. Three problems were selected for the testing and verifying the model with the square arrangement of the data and its non-uniform distribution. The results indicate that this model is simple, efficient and applicable for the two-dimensional interpolation of data. The model can be generalized for non-regular and non-uniform distribution of data, which can result in a bounded sparse matrix for the governing linear system of equations. Keywords: approximation, B-spline, cubic spline, interpolation, Lagrange methodI. INTRODUCTION The interpolation models are applied considerably in all branches of engineering activities. Examples are the evaluation of surface area and volume of non-uniform and irregular objects, determination of volumes of cutting, filling and embankment in earth work, and engineering surveying. Volume of surface topography and morphology forms such as hills and valleys. Here interpolation means to find a model function such as p (x) which satisfies n data points fi therefore; p ( x ) = f for i = 1,2,3, L, n . In approximation approach, the i i model curve does not necessarily cross the data points. The classic methods of interpolation are explained in details by (Fomel, 1997b), (Collins, 2003), (Dubin, 2003), (Kiusalaas, 2005), (Nocedal and Wright, 2006) and (Zamani, 2009a). The methods for two-dimensional 322
2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEMEinterpolation in engineering literature are not frequent. The most important models in thisarea are Lagrange method, B-spline method, two-dimensional cubic spline method, andexplicit two-dimensional cubic spline method. The introduction of the paper should explainthe nature of the problem, previous work, purpose, and the contribution of the paper. Thecontents of each section may be provided to understand easily about the paper.1.1 Lagrange Method The Lagrange method for this case is the generation of one-dimensional Lagrangeformulation for two-dimensional interpolation problems (Dierckx, 1993) and (Burden andFairs, 2010). For a problem with a set of 9 triple data A ε {(xi, yi, zi) i=1,2,…,9)} Figure 1.The two-dimensional Lagrange equation is as follows, ( x − x 2 )( x − x3 )( y − y 4 )( y − y 7 ) ( x − x1 )( x − x3 )( y − y 4 )( y − y 7 ) z ( x, y ) = z1 + z2 + ( x1 − x 2 )( x1 − x3 )( y1 − y 4 )( y1 − y 7 ) ( x2 − x1 )( x 2 − x3 )( y 2 − y 4 )( y 2 − y 7 ) (1) ( x − x1 )( x − x2 )( y − y 4 )( y − y 7 ) z3 + ( ) z 4 + L + ( ) z9 ( x3 − x1 )( x3 − x2 )( y3 − y 4 )( y 3 − y 7 ) y x Fig: 1 Simple data distributionThe two-dimensional Lagrange model is limited in terms of applicability. The degree andorder of the Lagrange polynomial increases as the number of data point increases. Therefore,it results in the oscillation and sinus behavior of the governing curve. Hence, the Lagrangecurves give high inclination and deviations from the real curves for the points between thedata or nodes. Therefore, the model is not applicable and efficient for huge engineering data.1.2 B-spline Approximation Method B-spline methods have been extended since 1970s. The designated B stands for Basis,so the full name of this approach is the basic spline. B-spline methods are mostly used forcurves and surfaces in computer graphics. The 2nd and 3rd degree B-splines which are usedextensively for approximation of data are less applicable for the interpolation of data. Withsome modifications and corrections the B-spline models specially the second and thirddegrees can be applied for interpolation of huge data, particularly for one-dimensionalinterpolation problems. The 2nd and 3rd degrees B-spline polynomials can be used forapproximation of n control points ( pi , i = 0,1,2, L , n − 1) (Saxena and Sahay, 2005) and (Salomon,2006) as, 323
3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME  1 − 2 1  pi −1    t 1  − 2 2 0  pi  1 Pi (t ) = t 2  i = 1,2, L , n − 2 (2) 2 1  1 0  pi +1    −1 3 − 3 1  pi −1   1 3 −6 3 0  pi    Pi (t ) = t 3 t2 t 1   = t  [B]{p} i = 1,2,L, n − 3 (3) 6 − 3 0 3 0  pi +1    1 4 1 0  p i + 2   The above B-spline curves will not cross through the control points but pass near them. The initialand terminal points (q1 and q2) of the cubic B-spline curve which are joint points can be obtainedfrom Eq. (4). 1 q1 = [ pi −1 + 4 pi + pi +1 ] 6 1 (4) q2 = [ pi + 4 pi +1 + pi + 2 ] 6The two-dimensional cubic B-spline can be obtained from the product of Eq. (3) in two dimensionsas, [ pi (t , u ) = t  [B]{pi } u  [B]{p j } ] T (5) [ ] p i (t , u ) = t  [B ] z i , j [B ] {u} = T − 1 3 − 3 1  z i −1, j −1 z i , j −1 z i +1, j −1 z i + 2, j −1   − 1 3 − 3 1  u 3    z i + 2, j  1  3 − 6 0   1 3 −6 3 0  z i −1, j z i, j z i +1, j   4 u 2  t 3 t2 t 1  6 − 3 0 3  0  z i −1, j +1 z i , j +1 z i +1, j +1 z i + 2, j +1  6  − 3 3 3   1  u  (6)      1 4 1 0  z i −1, j + 2  zi, j +2 z i +1, j + 2 z i + 2, j + 2   1  0 0 0  1   Where t , u ∈ [0,1] . Using the linear transformation from t and u to x and y; respectively for theuniform B-spline surface patch in Fig. (2), the graph equation pi ( x, y) results as follows, pi ( x, y ) = x [ A][B ][Z ][B ] [ A] {y} T T (7)where x  = x 2 x 1 ,  y  = y 2 y 1 and  a2 0 0  a 2 2 a b b 2      1 x1 A =  2 ab a 0 , AT =  0 a b  , a = , b=− , x2 − x1 x2 − x1  b2 b 1 0 0 1 (8)     1 y1 a = , b = − y 2 − y1 y 2 − y1 324
4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME y x Fig: 2 Data distribution for B-spline surface patchA network of data with (m+1)×(n+1) control points P00 through Pmn has (m+1) rows and(n+1) columns with uniform x and y increments (∆x, ∆y ) . From the above network about (m-2)×(n-2) local cubic B-spline graphs or patches can be obtained which have continuity at leastc1 on each element side. The final graph which consists of local graphs passes (m-1)×(n-1)internal points qij . If the data points transfer to those internal joint points qij the final graphwould be a two-dimensional cubic B-spline which is obtained by the interpolation method.Hence, the final graph crosses all the data points. For the details of this method, the situationof internal joint points, and their coordinates refer to (Salomon, 2006). The qij can bedetermined with respect to z value as Eq. (8). 1 qij = 36 [ ( zi −1, j −1 + 4 z i , j −1+ zi +1, j −1 ) + 4( zi −1, j + 4 z i , j + zi +1, j ) + ( zi −1, j +1 + 4 z i , j +1+ zi +1, j +1 ) ] (8)The interpolation by two-dimensional cubic B-spline of a network data with mesh m×n anduniform ∆x and ∆y requires 2(m+n+2) more equations in order to satisfy the uniqueness ofthe governing system of equations. However, it relates to the degree of polynomial B-splineconsidered. The above extra equations should be obtained by satisfying the boundaryconditions. The governed linear system of equations is a kind of nonadigonal system. It isclear that this model is not practical for problems having huge data points.1.3 Two-dimensional Cubic Spline MethodThis method is obtained by the product of two cubic spline equations in directions x and y asEq. (9). s ij ( x, y ) = aij + bij ( x − x i ) + cij ( y − y j ) + d ij ( x − x i ) 2 + eij ( x − x i )( y − y j ) + f ij ( y − y j ) 2 + g ij ( x − x i ) 2 ( y − y j ) + hij ( x − x i )( y − y j ) 2 + iij ( x − x i ) 3 ( y − y j ) + jij ( x − x i )( y − y j ) 3 (9) + k ij ( x − x i ) 2 ( y − y j ) 2 + lij ( x − x i ) 3 ( y − y j ) 2 + mij ( x − x i ) 2 ( y − y j ) 3 + nij ( x − x i ) 3 ( y − y j ) 3 + oij ( x − x i ) 3 + pij ( y − y j ) 3Where sij is bicubic spline which defines for each rectangular element and has 16 parameters.Those parameters are obtained by satisfying the following 16 continuity equations on thecorners of each element. 325
5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME sij ( xi , y j ) = f ij ⇒1 sij ( xi ±1 , y j ±1 ) = f i +1, j +1 ⇒ 3 ∂sij ∂ 2 sij ( xi , y j ) ⇒ 3 ( xi , y j ) ⇒ 3 (10) ∂x ∂x 2 2 ∂sij ∂ sij ( xi , y j ) ⇒ 3 ( xi , y j ) ⇒ 3 ∂y ∂y 2The above 16 equations of continuity for each element are written and combined together toform a nonadiagonal linear system of equations. By adding the governing boundaryconditions to the above linear system it is complemented to a linear system of equationswhich has a unique solution. This method of solution gives the c2 continuity along each nodeand at least c1 continuity at the element sides.1.4 Explicit Two-dimensional Cubic Spline MethodAn explicit two-dimensional interpolation model has been developed by (Zamani, 2010) thatis the simplification of the above two-dimensional cubic spline method. He applied uniformnetwork of data set (rectangular distribution). The interpolation equation which isimplemented in his model is as follows, sij ( x, y ) = a 0 + a1 ( x − x i ) + a 2 ( y − y j ) + a3 ( x − x i ) 2 + a 4 ( x − x i )( y − y j ) + a 5 ( y − y j ) 2 + g ij ( x − x i ) 2 ( y − y j ) + a 6 ( x − x i ) 3 + a 7 ( x − x i ) 2 ( y − y j ) + a8 ( x − x i )( y − y j ) 2 (11) + a9 ( y − y j ) 3The coefficients a0 to a9 are obtained by using c1 continuity on each element sides. Theyconsist of 4 equations of continuity for function and 6 equations of continuity for the firstpartial derivatives along the x and y axis at 4 nodes of each element, Eqs. (12) and (13).  sij ( xi , y j ) = f ij  s (x , y ) = f  ij i +1 j i +1, j  (12)  sij ( xi , y j +1 ) = f i , j +1 sij ( xi +1 , y j +1 ) = f i +1, j +1   ∂ sij (i, j )  ∂ sij (i + 1, j )  = f x,i , j  = f x,i +1, j  ∂x  ∂x  ∂ sij (i, j )  ∂ sij (i + 1, j )  = f y ,i , j  = f y ,i +1, j (13)  ∂y  ∂y  ∂ sij (i, j + 1)  ∂ sij (i , j + 1)  = f x,i , j +1  = f y ,i , j +1  ∂x  ∂yWith having the above partial derivatives the problem is unique and it consists of 10equations and 10 unknowns which can be solved explicitly (it’s not necessary to form anylinear system of equations). In most cases the partial derivatives are not available at each 326
6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME node but can be removed or eliminated by using numerical derivatives approaches. The coefficients a0 to a9 of Eq. (11) can be determined from the following equations. a0 = f ij , a1 = f x,ij , a2 = f y ,ij , 3 2 3 1 a3 = − a0 − a1 + 2 f i +1, j − f x,i +1, j (14) hi2 hi hi hi 1 1 1 1 1 1 a4 = − a0 − a1 + ( f i +1, j + f i , j +1 − f i +1, j +1 ) + f y ,i +1, j − a2 + f x ,i , j +1 hi k j kj hi k j hi hi kj 3 2 3 1 a5 = − a − a 2 + 2 f i , j +1 − 2 0 f y ,i , j +1 kj kj kj kj 2 1 2 1 a6 = a + 2 a1 − 3 f i +1, j + 2 f x,i +1, j 3 0 hi hi hi hi 1 1 1 1 a7 = 2 a0 + a1 − 2 ( f i +1, j + f i , j +1 − f i +1, j +1 ) − f x,i , j +1 h kj i hi k j hi k j hi k j 1 1 1 1 a8 = a + 2 0 a2 − 2 ( f i +1, j + f i , j +1 − f i +1, j +1 ) − f y ,i +1, j hi k j hi k j hi k j hi k j 2 1 2 1 a9 = 3 a0 + 2 a 2 − 3 f i , j +1 + 2 f y ,i , j +1 kj kj kj kj Where hi and ki are length and width of element ij ; respectively. For a huge set of data which usually exists in engineering problems the above model is simple and requires less calculation operations with respect to the other methods of two-dimensional interpolation. The only limitation for this model is determining the partial derivatives with respect to x and y axis at each node and they should be approximated by the governing methods.II. MODEL FORMULATION The model is the generation of B-spline method in two dimensional, therefore; it is better to explain a brief review of the one-dimensional B-spline. 2.1 One-dimensional B-spline This model is explained in more detail by (Salamon, 2006) but in this study it is briefly discussed. The B-spline curve consists of a series of B-spline basic functions that are linearly independent in domain [a, b] as, n s ( x ) = ∑ ci β i [ui ( x )] =c 0 β 0 (u ) + c1 β1 (u ) + L + c n β n (u ) (15) i =1 where u ∈ [−2,2] , u i ( x) = ( x − xi ) / h , h = (b − a ) / n and n is the number of elements. Each B- spline basic function consists of four cubic polynomial functions as Eqs. (16). 327
7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME July December  (2 + u) 3 , − 2 ≤ u ≤ −1  2 3 1 + 3(1 + u ) + 3(1 + u ) − 3(1 + u ) , −1 ≤ u ≤ 0  β i (u ) = 1 + 3(1 − u ) + 3(1 − u ) 2 − 3(1 − u )3 , 0 ≤ u ≤1  (16) (2 − u )3 , 1≤ u ≤ 2    0 , otherwiseThe following basic equation is applied for B-spline basic function for increasing its B splineapplicability and continuity order (Zamani, 2009b).  x − xi 2 −b   −b u 2  h  (17) β i (u ) = ae − c = ae −cWhere a = 4.016478 , b = 1 .3740615 and c = 0.016478 . Fig. (3) shows the B-spline basic Bfunction for Eq. (17). Fig: 3 B-spline basic functionIn order to promote the accuracy and efficiency of B-spline method two basic functions B splinec−1 β −1 (u ) and cn+1 β n +1 (u ) are added to Eq. (15) for adding the effects of boundary conditions.The coefficients c−1 and cn +1 can be calculated with respect to the B-spline curve at boundary B splinepoints according to the following equations. c−1β −1 ( x ) + 4c0 β 0 ( x ) + c1β1 ( x ) = f (x ) (18) c −1 β −1 ( x0 ) + 4c0 β 0 ( x0 ) + c1 β 1 ( x0 ) = f ( x0 ) (19) B-spline basic functions β −1 ( x ) , β 0 ( x) and β1 ( x) in Eq.After making derivatives from the B(19) it can be written as, 328
8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME − c−1α + c1α = f ( x0 ) = f 0 (20) b f f he 0 0 c−1 = c1 − = c1 − (21) α 2ab hwhere α = . In the same way the coefficient cn+1 for the last boundary point can be 2abe −hobtained. f n f n h e b cn +1 = cn −1 + = cn −1 + (22) α 2abWith regarding to f 0 and f n the derivatives at the first and the last points are not availableand should be approximated. This model gives a tridiagonal linear system of equations withdimensions (n+1) × (n+1) with 4 on the main diagonal entries and 1 on the sub diagonal andsup diagonal components. The obtained linear system of equations is easy to solve. This canbe done by Thomas algorithm. The above model results in smooth curves for data of havingnonlinear distributions and uniform elements. The model can also be extended for data ofnon-uniform element length.2.2 Two-dimensional B-splineThe two-dimensional B-spline model is the generation of one-dimensional model in twodirections and in cylindrical coordinates is as follows, ρ2 −b β k ( x, y ) = a e l2 h2 −c (23) − b l2 h2 [( x− x ) +( y− y ) ] i 2 j 2 (24) β k ( x, y) = a e −cwhere k is number of node having (xi, yi) coordinates. Eq. (24) for l = 2 / 2 is, − 2b h2 [( x − x ) i 2 +( y − y j ) 2 ] (25) β ij ( x, y ) = a e −cWhere h is element length. The two-dimensional B-spline Eq. (25) for h=1 and ρ = 2 , 2 / 2 and 0 equal 0, 1 and 4; respectively. The effective radius for B-spline function of Eq.(25) is 2 . Figs. (4) and (5) show the graph of Eq. (25) and its contour line. Fig: 4 Two-dimensional B-spline basic function graph 329
9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME July December Fig: 5 Two-dimensional B-spline basic function contour dimensional BSuppose there exist n data points with uniform rectangular or square mesh. The element dimensionsare constant all over the mesh with length h = hi = xi +1 − xi and k = k j = y j +1 − y j . The two-dimension B-spline function for this set of data can be written as, spline n s ( x, y ) = ∑ cm β m ( x, y ) = c1β1 + c3 β 3 + c3 β 3 + L cn β n (26) m =1where n is the total number nodes and m is the node number. For simplification of the formulationlet’s consider square elements with h = k = 1 . The B-spline function s ( x, y ) for node m is, s ( xm , y m ) = f m ,m = cm − k β m − k + cm −1 β m −1 + cm β m + cm +1β m +1 + cm + k β m + k (27)The values of B-spline basic functions from Eq. (27) equal,β m − k = β m −1 = β m +1 = β m + k = α = a e −2b (28) βm = 4If Eq. (27) is applied on each node the following linear system of equations Eq. (29) will be 4 α 0 0 0 α O 0   c1   f1  α  4 α O O O α O  c2   f 2      0 α 4 α α  M   M       0 O α 4 α O     (29) 0   =  = [ A] {c} = { f } O α 4 α O       α O α 4 α O  M   M  O     α α 4 α  cn −1   f n −1     O O α O O O α 4   cn   f n     obtained where it is pentadiagonal system with 4 on the main diagonal entries and α on the four othersdiagonal components. Since the values of diagonal entries are constant, therefore; it is only necessaryto save the values of 4, α and their addresses to save matrix A. This saves computer memoryespecially for problems with huge number of data.Three problems are chosen for investigation of checking and efficiency of the above formulation abovewhich are explained here. 330
10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME2.2.1 Problem 1There are 25 data points {xi , yi , z i ; i = 1,2,L,25} in this problem. They form a mesh point with5 rows and 5 columns and square elements as Fig. (6). The element side is h = k = 1 . Thedomain of function f ( x, y ) is x ∈ [0,4] , y ∈ [0,4] . The data are obtained from Eq. (30). z = f ( x, y ) = e [0.71( x −1) ]0.5 2 +1.13 ( y − 2 ) 2 (30)As indicated in Fig. (6) 20 auxiliary points or nodes around the data are considered forincluding the effects of boundary conditions. This action causes an increase in accuracy andefficiency of presented model for solving these kinds of problems. Fig: 6 Data distribution in problem 1Partial derivatives are required with respect to x axis and y axis at boundary nodes for thispurpose. For the most engineering data the above partial derivatives for boundary points arenot available and should be somehow approximated. The B-spline model which is used forthis problem is, 25 20 s ( x, y ) = ∑ ci β i ( x, y ) + ∑ cib β ib ( x, y ) (31) i =1 i =1In Eq. (31) the second set of equation is related to the effects of boundary conditions whichcan be expanded as Eq. (32). 20 ∑c i =1 ib β ib ( x, y ) = c−1 y β −1 y + c− 2 y β − 2 y + L + c−5 y β −5 y + c21 y β 21 y + c22 y β 22 y + L + c25 y β 25 y + (32) c−1 x β −1 x + c− 2 x β − 2 x + L + c− 21 x β − 21 x + c5 x β 5 x + c6 x β 6 x + L + c25 x β 25 xFor calculation s( x, y ) or z value for internal points in each element it is necessary to consideronly a few terms of Eq. (31). Most of its terms are zero for each point inside the elements 331
11. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEMEbecause of the nature of B-spline basic function. Fig. (7) shows the amount of theeffectiveness with regard to the number of B-spline basic functions which contributes todetermining s ( x, y ) . For example in elements with nodes 8, 9, 13, and 14 the number ofrequired B-spline basic functions for calculation s ( x, y ) are from minimum 4 to maximum 7Fig. (7). This problem is solved by the presented model. Figs. (8), (8a), (9) and (9a) showthe comparative behavior between the model and the real values of f ( x, y ) function for thecentral points of elements. As it can be seen from the figures there are a suitable relationshipand agreements between the model output and the real data. Fig: 7 Required number of B-spline basic function 4 3 y 2 1 0 0 1 2 3 4 x 0 0. 1 1. 2 Fig: 8 Graph of real data Fig: 8a Contour lines of real data 332
12. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME July December 4 3 y 2 1 0 0 1 2 3 4 x Fig: 9 Graph of B-spline model spline Fig: 9a Contour lines of B-spline model B spline2.2.2 Problem 2This problem in relation to the previous one has more complexity with respect to thevariations of data. In this example there are 121 data points {xi , yi , zi ; i = 1,2,L,121}. The dataform a uniform mesh having 11 rows and 11 columns with the domain of function f ( x, y ) x ∈ [1,6] , y ∈ [1,6] . The data are generated by the following equation. 2 2 ( y − 4)π (33)f ( x, y ) = [10 e −( x − 4.5) + 6.817 e −1.2( x − 2.5) ] Cos 2.53 8About 44 auxiliary boundary nodes for applying the effects of boundary conditions areconsidered. The partial derivatives with respect to x and y axis at boundary points arecalculated by the forward and backward divided difference formulas. The graph and contourlines for Eq. (33) are in Figs. (10) and (10a). They are shown from the B-spline model in s. B splineFigs. (11) and (11a) as well. Comparison of the figures shows a good agreement and suitablerelationship between the real data and the interpolated data. middle points f(x,y) Fig: 10 Graph of real data Fig: 10a Contour lines of real data 333
13. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME July December Fig: 11 Graph of B-spline model spline Fig: 11a Contour lines of B-spline model B spline2.2.3 Problem 3This example consists of 49 triple data points {xi , y i , z i ; i = 1,2,L,49} . According to Fig. (12),they form a uniform square mesh with 7 rows and 7 columns of data and 36 elements withelement size h = k = 1.0 . The data which are highly nonlinear are obtained from Eq. (34). z = ( x 2 + y 2 )Sin [8 Arctg ( x / y )] (34)About 28 auxiliary points are considered for including the effects of boundary conditions. 43y +44 +45 +46 +47 +48 +49y -43x 43 44 45 46 47 48 49 +49x -36 36 37 38 39 40 41 42 +42 -29 29 30 31 32 33 34 35 +35 -22 22 23 24 25 26 27 28 +28 -15 15 16 17 18 19 20 21 +21 -8 8 9 10 11 12 13 14 +14 -1x 1 2 3 4 5 6 7 +7y -1y -2 1y -3 -4 -5 -6 -7y Fig: 12 Network of data and boundary pointsThe domain of data for this problem is x ∈ [1,7] , y ∈ [1,7] . Similar to the previous problemthe partial derivatives with respect to x axis and y axis at the boundary points are calculatedby the numerical method differentiation. The parameters governing to the boundary pointscan be obtained from the above deri derivatives. The parameters c7 x and c −7 y for example can bedetermined from Eqs. (35) and (36). 334
14. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME 1 c7 x = c6 + f x (7 ) (35) A 1 c− 7 y = c14 − f y (7) (36) Awhere A = 8abe −b . The two dimensional B-spline basic function which is applied for this problemfor h = 1.0 and l = 2 / 2 is, β k ( x, y ) = a e [ −2 b ( x − xi ) 2 + ( y − y j ) 2 ]−c (37)The graphs and contour lines obtained by the model and from the function f ( x, y ) are in Figs. (13),(13a), (14) and (14a). A comparison between these figures presents a good and suitable relationshipfor real data and the two-dimensional B-spline model. f(x,y) f(x,y) 6.5 6 5.5 5 4.5 y 4 3.5 3 2.5 2 1.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Fig: 13 Graph of real data x Fig: 13a. Contour lines of real data B-spline model B-spline model 6.5 6 5.5 5 4.5 y 4 3.5 3 2.5 2 1.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 x Fig: 14 Graph of B-spline model Fig: 14a Contour lines of B-spline model 335
15. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEMEIII. CONCLUSIONS AND DISCUSSION The interpolation methods especially the two-dimensional ones have many applications in all branches of engineering science. Therefore; the development, extension and improvement of these methods are necessary. The presented model is easy to implement and results in a linear pentadiagonal systems of equations which is diagonally dominant and simply solvable. The presented model for interpolation of data with rectangular and square elements is powerful. Also it can be expanded for non-uniform mesh or triangular elements of data. The final linear system of equations for those cases is sparse and bounded while about more than 90% of matrix entries are zero. The graphs obtained by this method have no limitations of continuity properties inside and across the elements because of the continuity conditions of the applied B-spline basic function. These conditions explain the advantage and robustness of the presented model to the available methods of two-dimensional interpolation. REFERENCES Burden, R. and Fairs, J. (2010). Numerical Analysis, 9th Ed. Brooks-Cole Pub. Comp.Boston. Collins, G. W. (2003). II Fundamental Numerical Methods and Data Analysis, (internet pdf file). Dierckx, P. (1993.) Curve and Surface Fitting with splines, Clarendon Press, NY. Dubin, D. (2003). Numerical and Analytical Methods for Scientists and Engineers Using Mathematica, John Wiley & Sons, Inc., Pub., Hokoben, New Jersy. Fomel, S. (1997b). On the General Theory of Data Interpolation, SEP. 94, 165-179. Kiusalaas, J. (2005). Numerical Methods In Engineering with MATLAB, Cambridge Univ. Press,UK. Lobo, N. (1995). Curve fitting using spline sections of different orders, Proceedings of the First International Mathematics Symposium, P. 267-274. Nocedal, J. and Wright, S. J. (2006). Numerical Optimization, 2nd Ed., Springer, Science +Business Media, LLC, USA. Salomon, D. (2006). Curves and Surfaces for computer Graphics, Springer Science +Business Media, Inc. USA. Saxena, A. and Sahay B. (2005). Computer Aided Engineering Design, Springer, Anamaya Publishers, New Delhi, India. Zamani, M. (2009 a). Three simple spline methods for approximation and interpolation of data, Contemporary Engineering Science, Vol.2, No.8, P. 373-381. Zamani, M. (2009 b). An investigation of B-spline and Bezier methods for interpolation of data, Contemporary Engineering Science, Vol.2, No.8, P. 361-371. Zamani, M. (2010). A simple 2-D interpolation model for analysis of nonlinear data, Natural Science, Vol. 2, No. 6, June 2010, P. 641-646. M. Nagaraju Naik and P. Rajesh Kumar, “Spectral Approach To Image Projection With Cubic B-Spline Interpolation” International journal of Electronics and Communication Engineering &Technology (IJECET), Volume3, Issue3, 2012, pp. 153 - 161, Published by IAEME 336
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