NURBS based Finite Element Approach for One Dimensional Problems

NURBS based Finite Element Approach for One Dimensional Problems

A. Suresh A

Assistant Professor, Department of Mechanical Engineering,

JNT University College of Engineering Jagityal, Nachupally P.O. Kodimyal Mandal, Karimnagar Dist., Telangana., India, – 505 501

Ch. Yella Reddy B

M. Tech,

Department of Mechanical Engineering, JNT University College of Engineering Jagityal,

Nachupally P.O. Kodimyal Mandal, Karimnagar Dist., Telangana., India, – 505 501

Abstract -In this work, an attempt is made to use the Non Uniform Rational B-Spline(NURBS) basis functions as the shape functions in the finite element method. These basis functions are employed in Collocation Method approximation for the spatial descretization. It uses recursive formula of NURBS basis functions for solving second order differential equations with Neumann's boundary conditions. A test case is considered to study the efficiency of this method. When the degree of the basis increased the stability and efficiency is improved. The result obtained by present method is compared and found to be in good agreement with analytical solution and finite element method.

Keywords: NURBS,B-Splines, Isogeometric Method, Collocation Method, Concrete Pier.

1. INTRODUCTION:

   Mathematical models in the form of differential and partial differential equations are used to represent various engineering problems in the fields, such as Structural mechanics, Solid mechanics, Fluid flow, Heat transfer, Vibration analyses, Contact mechanics etc. The solutions to these mathematical models can be Exact, Analytical or

   In the present work, an attempt is made to use an approximating function for the field variable based on the NURBS basis function to solve the boundary value problem. A non-uniform knot vector for a particular weight vector is used to obtain the second and third degree NURBS basis functions. For the spatial descretization, Collocation Method approximation is employed.

   In this paper the recursive formulation of B-spline and NURBS basis functions [Hughes, T.J.R et al.2005, David

   F. Rogers et al, 2002] are discussed initially then the NURBS collocation method is discussed and formulated. The effectiveness and accuracy of this method is tested using the governing equation of one dimensional structural problem. The structural problem considered is, a loaded typical concrete pier of a bridge with varying cross section, to study the variation of displacement along the pier.

   Considering second order linear differential equations with variable coefficients

   Approximate depending on the nature of these equations. When the Exact solution is not possible, numerical methods are needed to obtain approximate solutions. Many

   d 2U dx2

   • k P(x) dU k Q(x)U F (x)

   1 dx 2

   (1)

   numerical techniques are evolved and has been used increasingly in last few years. Those numerical techniques include Finite Difference Method [R.K.Panday et al, 2004],

   With the boundary conditions U(a)=d1, U(b)=d2. Where

   a,b,d1,d2, k1and k2 are variables, P(x), Q(x) and F(x) are functions of x. Let the approximation solution be

   B-spline Collocation Method [Moshen.A et al, 2008],

   U h (x)

   n1

   C R

   (x)

   Predictor and Corrector Method [Abdalkaleg Hamad et al,

   2014], Finite Element Method [Ch.Sridhar Reddy et al,

   i 2

   i i, p

   (2)

   2014], and many more. In these methods, the approximating function provides higher order of continuity and is capable of providing accurate solutions with continuous gradients throughout the domain.

   The basis functions for B-spline and NURBS are derived using knot space and for a particular degree. A recursive formulation was given Carl.De boor [C.de Boor et al, 1982] for deriving these basis functions. If we use this formulation the evaluation of basis function can be generalized up to any degree. This basis function can be used in collocation method .

   Where Ci are constants to be determined and Ri,p(x) are NURBS basis functions.

   () is the approximate global solution to the exact solution U(x) of the considered second order singular differential equation(1).

2. A BRIEF INTRODUCTION TO B-PLINES/NURBS:

   B-Splines:

   A spline is the mathematical representation of real world geometries. Schoenberg[David F. Rogers et al,2002] was given first reference to the word B-spline and described it as a smooth piecewise polynomial curve. From mathematical point of view, a curve generated by using the

   vertices of a defining polygon and the curve is dependent on some interpolation scheme between the curve and polygon. This scheme is provided by the choice of B-spline basis functions. B-spline basis is generally has non global

   Projecting back into three-dimensional space by dividing through by the homogeneous coordinate yields the rational B-spline curve.

   behaviour due to the property that each vertex of B-spline

   n1

   B h N

   (x)

   Bi is associated with a unique basis function. Thus, each

   i i

   i, p

   n1

   vertex affects the shape of a curve only over a range of

   p(x) i1 Bi Ri, p (x)

   parameter values where its associated basis function and

   n1

   h N

   (x)

   i1

   (7)

   hence the degree of the resulting curve to be changed

   i i1

   i, p

   without changing the number of defining polygon vertices.

   Any curve can be represented as a parametric curve, i.e., the coordinate x is represented as a function of a parameter

   t. A parametric B-spline function can be defined by

   Where the Bi s are the 3D defining polygon vertices for the rational B-spline curve and the rational B-spline basis functions given by

   n1

   p(x) B N

   (t)

   R (x)

   hi Ni, p (x)

   i i1

   1. p

      (3)

      i, p

      n1

      hi Ni, p (x)

      (8)

      Where, xmin x xmax, 2 < + 1

      Where the Bi are the position vectors of the n+1 defining polygon

      vertices, p is the order and the Ni,p(x) are the normalized B-spline basis functions. B-spline curve defined as polynomial spline function of order p and degree p-1.

      1. If p=1

         i1

         Here, s are the homogeneous coordinates (occasionally called weights) provide additional blending capability. It is clear that when all =1, ,()

         =,(),thus non-rational B-spline basis functions and curves are included as a special case of rational B-spline basis functions

         Ni, p

         (x) 1

         if xi x xi1

         and curves.

         NURBS derivatives:

         0

      2. If p>1

   otherwise

   (4)

   Since the aim of the collocation Method is to compute approximation solution for differential Equations, so the derivatives of the NURBS basis functions needs to be

   (x xi )Ni, p1(x) (xi p x)Ni1, p1(x)

   N (x)

   calculated.NURBS basis functions are dened by equation (8).

   i, p

   xi p1 xi

   xi p xi1

   (5)

   Equations (9) and (10) are the first and second derivatives of NURBS curve of order 3.

   The values of xi are elements of knot vector satisfying the relation xixi+1. The parameter x varies from xmin to xmax along the curve

   The first derivative of NURBS curve is

   p(x).

   n1

   p' (x) B R'

   (x)

   The sum of the B-spline basis functions is 1 for any parameter value x. Positivity property guarantees that the curve segment lies completely within the convex hll of Pi. The

   partition of unity property ensures that the relationship between the curve and its defining control points is invariant under affine

   Where

   i1

   h N'

   i

   (x)

   i. p

   h N

   n1

   (x) h N '

   (x)

   transformations. Local support property indicates that each segment of B-spline curve is influenced by only p control points.

   R'i, p

   (x) =

   i

   n1

   i, p

   i i, p

   n1

   i

   i1

   i, p

   2

   (9)

   NURBS basis functions:

   Non-Uniform Rational B-Splines (NURBS) were introduced by

   K. Versprille [K. J. Versprille et al, 1975] as significant

   hi Ni, p (x)

   i1

   hi Ni, p (x)

   i1

   improvement that can accurately handle both analytic and modeled curves. NURBS are used in most computer graphics applications, significantly in CAE and renowed industry standards such as IGES (Initial Graphics Exchange Specification), STEP

   The second derivative of NURBS curve is

   n1

   p'' (x) B R''i. p (t)

   (Standard for the Exchange of Product model data).

   A Rational B-spline curve is the projection of a non- rational B-spline curve defined in four-dimensional (4D) homogeneous coordinate space back into three-dimensional (3D) physical space. Specifically,

   Where

   h N''

   i

   i1

   (x)

   n1

   i, p

   i, p

   Ri, p (x)hi N " (x)

   n1

   i, p i i, p

   i, p i i, p

   R' (x) h N ' (x)

   n1 h

   R"i, p (x) =

   i i, p

   i1

   2 *

   i1

   p(x) B

   i1

   i Ni. p

   (x)

   (6)

   n1

   hi Ni, p (x)

   i1

   n1

   hi Ni, p (x)

   i1

   n1

   hi Ni, p (x)

   i1

   Where the s are the 4D homogeneous defining polygon

   vertices for the non-rational 4D B-spline curve. Ni,p(x) is the non- rational B-spline basis function, given in equations (4)& (5).

   (10)

   From the above equations, the basis functions are defined as recursively in terms of previous degree basis function i.e. the pth order basis functions is the combination of ratios of knots and (p-

   1) degree basis function. Again (p-1)th degree basis function is

   defined as the combination ratios of knots and (p-2) degree basis function. In a similar way every NURBS basis function of degree up to (p-(p-2)) is expressed as the combination of the ratios of knots and its previous basis functions.

   NURBS derivative basis of second degree over uniform knot vector and equal weights for all control points (i.e., hi =1) is

   unknowns Cis in equation (2). Extra knot values are taken into consideration both sides of the domain of problem when evaluating the second degree NURBS basis functions at the nodes. These extra knots are taken to satisfy the partition of unity property and to get accurate NURBS basis functions.

   First derivative of approximation function (2) is

   shown graphically below in figures 2 and 3.

   10

   dU h (x) dx

   n1

   i

   i

   C R'i, p (x)

   i2

   (11)

   Second derivative of approximation function (2) is

   5

   d 2U h

   0

   n1

   ''

   dx2

   -5

   Ci R i, p (x)

   i2

   (12)

   Substituting, the approximate solution (2) in (1) we have,

   -10

   -15

   -0.2 0 0.2 0.4 0.6 0.8 1 1.2

   d 2U h

   dx2

   k1P(x)

   dU h

   dx

   k2Q(x)U

   F(x)

   h

   h

   (13)

   Figure 1: First derivative of second degree NURBS basis function with Knot vector X= {-0.2,-0.1, 0, 0.1, 0.2, 0.3,

   0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2}

   Substituting the Approximation function and its derivatives (2),(11) and (12) in the equation (13), we have

   n1

   200 C R"

   n1

   (x) k P(x) C R'

   n1

   (x) k Q(x) C R

   1. F(x)

      i

      150 i2

      i, p

      1. i i2

         i, p

      2. i i2

   i, p

   100

   50

   0

   -50

   -100

   -150

   -200

   -250

   -0.2 0 0.2 0.4 0.6 0.8 1 1.2

   Figure 2: Second derivative of second degree NURBS basis function with Knot vector X= {-0.2,-0.1, 0, 0.1, 0.2, 0.3, 0.4, 0.5, 6, 0.7, 0.8, 0.9, 1.0,

   1.1, 1.2}

3. COLLOCATION METHOD:

   Collocation method is used widely in approximation methods

   1. Expanding the equation (14)

      1

      1

      [C2 R"2, p (x) C1R"1, p (x) C1R" , p (x) ……. Cn1R"n1, p (x)]

      1

      1

      k1P(x)[C2 R'2, p (x) C1R'1, p (x) C1R' , p (x) ……. Cn1R'n1, p (x)]

      k2Q(x)[C2 R 2, p (x) C1R 1, p (x) C1R 1, p (x) ……. Cn1R n1, p (x)] F (x)

      (15)

      Now let the coefficients of C-2, C-1, C1, Cn-1 are assumed as R- 2(x), R-1(x), R1(x).Rn-1(x), now we have the equation 15, as

      [R2 (x)]C2 [R1(x)]C1 [R1(x)]C1 ……. [Rn1(x)]Cn1 F(x)

      (16)

      In matrix form, we have

      C2

      C1

      particularly solving differential equations. The collocation method

      C1

      together with NURBS (Non-Uniform Rational Basis Spline) approximations represents an economical alternative since it only requires the evaluation of the unknown parameters at the grid

      [R2 (x)

      R1 (x)

      R1 (x). . . .

      .Rn1 (x)] .

      .

      .

      F (x)

      points or nodes or collocation points. In normal collocation method we use polynomials whereas in NURBS collocation method we use NURBS basis functions. The selection of nodes or collocation points is arbitrary. The basis functions vanishes at the boundary values. The success of this Collocation method is dependent on the choice of basis. The main aim is to analyse the efciency of the NURBS based collocation method for such problems with sufcient accuracy.

      Cn1

      (17)

      Equation (16) is evaluated at xis, i=1,2,3,.n-1 gives the system of (n-1)×(n+1) equations in which (n+1) arbitrary constants are involved.

      The Matrix (17) can be written as

      Formulation of NURBS Collocation Method:

      R2 (1)

      R1 (1)

      R1 (1)…..

      Rn1 (1)

      C2

      F (1)

      As mentioned earlier NURBS functions are used as basis in

      R (2)

      R (2)

      R (2)…..

      R (2)

      F (2)

      collocation method whereas the base functions which are used in

      2

      R2 (3)

      1

      R1 (3)

      1

      R1 (3)…..

      n 1

      Rn1 (3)

      C1

      C1

      F (3)

      the normal collocation method are polynomials, vanishes at the

      . . .

      . .

      .

      nodes. Let [a, b] be the domain of the governing differential equation and is partitioned as X={a=x1, x2, .. xn-1, xn,=b}with non-uniform values between [a b] with n sub domains. for a

      .

      .

      .

      .

      .

      .

      . . .

      . . .

      . . .

      s R (n 1)

      R (n 1)

      R (n 1)…..

      R (n 1)Cn1

      F (n 1)

      particular homogeneous coordinate(weights) i.e., hi' . The xis are known as nodes, these nodes are treated as knots in collocation NURBS method, where NURBS basis functions are defined and those nodes are used to make the residue equal zero to determine

      2 1 1

      n 1

      (18)

      Applying boundary conditions to approximate the solution, we Governing equation is

      have

      d 2u 1 du 25

      n1

      C R

      1. n1

         C R

      2. d

   dx2 (1 x) dx

   E for 0 x 2 (22) With

   i i2

   i, p d1

   i

   i2

   i, p 2

   boundary coditions u(2)=0, (1 x)E du

   5 and having a

   (19)

   A square matrix of size (n+1)x(n+1) is obtained from equations

   dx

   dx

   4

   4

   exact solution

   x0

   (18), (19)

   56.25 6.25(1 x)2 7.5ln1 x

   R2 (a)

   R1 (a)

   R1 (a). . . .

   Rn1 (a) C2

   d1

   3

   E

   E

   R (1)

   R (1)

   R (1).

   . . .

   R (1)

   C

   F (1)

   u(x)

   (23)

   2

   R (2)

   1

   R (2)

   1

   (2).

   . . .

   n 1 1

   (2) C

   2

   1 R1

   Rn1 1

   F (2)

   R2 (3) R1 (3) R1 (3). . . . Rn1 (3) . F (3)

   Comparing the given differential equation with equation (1), we

   . .

   .

   .

   .

   .

   . .

   .. . . .

   .. . . .

   .. . . .

   . . .

   .

   .

   .

   .

   .

   . . .

   have

   F(x)=25/E, a=0, b=2 and d1=0, d2=-20/(E(1+x)), P(x)=1/(1+x).

   R (n 1) R (n 1)

   R (n 1). . . . R

   (n 1) F (n 1)

   2

   1 1

   n1

   .

   R2 (b)

   R1 (b)

   R1 (b). . . .

   Rn1 (b)

   Cn1

   d2

   (20)

   Taking the approximation function from the equation (2), it can be written as

   It is in the form of [][] = []

   n1

   U h (x) C R

   (x)

   i

   i2

   i, p

   The matrix [R] is diagonally dominated square matrix of size (n+1) because of local support of basis functions. So that the system of equations are easily solved for arbitrary constants Cis.

   We have

   [] = [][]1 (21)

   Taking number of intermittent segments (or sub domains) as 11 (i.e. n=11), order of NURBS curve as 3 (i.e. p=3). X={a=X0=0, X1, X2, . Xn-1, Xn=b} with non-uniform values between [a b],for a homogeneous coordinates(weights) p=1.12, hi=1,for i1 and knot vector having 15 elements or knot values. Now the above equation can be modified as

   The approximate solution becomes as known solution is

   10

   U h (x) C R

   (x)

   (24)

   obtained. Now the final approximation solution obtained by substituting these constants in equation(2). This

   i

   i2

   i,3

   approximate solution is used to evaluate the field variable at each node(Collocation point) in the considered domain.

   Substituting the approximation function in governing equation,

   we have

   The exact solution is also evaluated at these points and the

   10

   C R''

   (x) 1 10 C R'

   (x) 25

   result values are compared with each other to find out the accuracy of the NURBS Collocation Method.

   i i2

   i,3

   i i2

   i,3 E

   (25)

   1 x

   1 x

4. Test Problem:

   A numerical example is considered to study the efficiency and convergence of the Collocation Method.

   Consider a loaded typical concrete pier of a bridge with varying cross section, to study the variation of displacement along the pier. the geometry and loads of a pier are shown in figure 3[J.N. Reddy et al, 2005]. The load 20(kN/m2) represents the weight of the bridge and an assumed distribution of traffic on the bridge. The concrete weighs approximately 25(kN/m3) and its modulus is E=28*106(kN/m2),the displacement along the pier at different points is analyzed.

   Figure 3: A typical concrete pier

   Knot vector is xi={0, 0.0093, 0.1564, 0.2133, 0.8854, 1.5498,

   1.6346, 1.7374, 1.9238, 1.9923, 2}.

   By solving the set of equations we get the constants Ci where i=0,1,2,….,9 and by substituting these constant values in approximation solution equation then we get the final solution for the given problem.

   Now the final approximation solution is evaluated at each node (Collocation point)i.e. xi={0, 0.0093,0.1564,0.2133,

   ,1.9923} and the values field variable u(x) at each node are calculated. The exact solution also evaluated at these points and result values of field variable u(x) are compared with each other to find out the accuracy of the NURBS Collocation Method and shown in table 1, and by increasing the degree of NURBS basis from second to third degree (i.e., order from p=3 to 4) the accuracy of method increases, result obtained is compared with exact and second degree solution. The values are tabulated as below in table 2.

   Table1:Comparison of field variable (u) with exact solutions for knot vector xi ={0,0.0093,0.1564,0.2133,…..

   ..,109923,2} and weights hi={1.12,1,1,1….1,1}

   Node (Knot values)	Exact Solution (*10-5)	NURBS Collocation Method Solution (unequal weights) (*10-5)
   0	0.2080	0.2016
   0.0093	0.2073	0.2009
   0.1564	0.1966	0.1903
   0.2133	0.1923	0.1860
   0.8854	0.1340	0.1292
   1.5498	0.0601	0.0583
   1.6346	0.0494	0.0480
   1.7374	0.0361	0.0350
   1.9238	0.0108	0.0105
   1.9923	0.0011	0.0011
   2.0	0.0	0.0

   Table 2: Comparison of field varible u(x) with exact and NURBS Collocation Method for second

   Node (Knot Values)	Exact Sol (*10-5)	NURBSCM Sol. with unequal weights, for p=3(second degree) (*10-5)	NURBSCM sol. with unequal weights, for p=4(third degree) (*10-5)
   0	0.2080	0.2016	0.2073
   0.0093	0.2073	0.2009	0.2067
   0.1564	0.1966	0.1903	0.1959
   0.2133	0.1923	0.1860	0.1916
   0.8854	0.1340	0.1292	0.1334
   1.5498	0.0601	0.0583	0.0599
   1.6346	0.0494	0.0480	0.0492
   1.7374	0.0361	0.0350	0.0359
   1.9238	0.0108	0.0105	0.0107
   1.9923	0.0011	0.0011	0.0011
   2.0	0	0	0

   td>

   1.7374

   Node (Knot Values)	Exact Sol (*10-5)	NURBSCM Sol. with unequal weights, for p=3(second degree) (*10-5)	NURBSCM sol. with unequal weights, for p=4(third degree) (*10-5)
   0	0.2080	0.2016	0.2073
   0.0093	0.2073	0.2009	0.2067
   0.1564	0.1966	0.1903	0.1959
   0.2133	0.1923	0.1860	0.1916
   0.8854	0.1340	0.1292	0.1334
   1.5498	0.0601	0.0583	0.0599
   1.6346	0.0494	0.0480	0.0492
   0.0361	0.0350	0.0359
   1.9238	0.0108	0.0105	0.0107
   1.9923	0.0011	0.0011	0.0011
   2.0	0	0	0

   ((p-1)=2)and third((p-1)=3) degree basis.

   From the tables, it can be stated that the values of the field variable obtained by NURBS Collocation Method using unequal weights are nearer to the Exact solution values and by increasing the degree of the NURBS basis the accuracy of collocation method increases. Thus maximum modulus error is constantly decreasing as the degree of the basis functions increased.

   The following figures compares the approximate solution obtained by NURBS Collocation Method with exact solution calculated at same collocation points.

   Figure4: Comparison of field variable (u) with exact solutions for non- uniform knot spacing

   Figure5: Comparison of field varible u(x) with exact and NURBS Collocation Method for second and third degree basis for unequal weights

5. CONCLUSIONS:

In this work, an attempt is made to use the NURBS basis functions as the shape functions in the finite element method.NURBS basis functions are defined recursively and incorporated in the collocation method. The accuracy and efficiency of the present method is illustrated by a structural test problem. The NURBS Collocation Method solution is compared with exact solution and found to be in best fit approximation.

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3. C.de Boor and K.H"ollig, (1982), B-spline from parallelepipeds, J.

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