PSI - Issue 15

Sharath Chavalla et al. / Procedia Structural Integrity 15 (2019) 8–15 Chavalla et al. / Structural Integrity Procedia 00 (2019) 000–000

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3

2. NURBS and IGA The CAD geometry in FEM is represented as a piece-wise polynomial approximation. For complex engineering designs, generating a FE model from CAD geometry is a cumbersome task than to perform analysis of the models. This was one of the primary motivating factors which lead to the development of the isogeometric concept. In the following section the building blocks of NURBS and isogeometric model is presented. A knot vector in one dimension is a non-decreasing set of coordinates in the parameter space, written as Ξ = {ξ 1 , ξ 2 , …, ξ n + p + 1 }, where ξ i ∈ � is the i th knot, i is the knot index, i = 1, 2, … , n + p + 1, with the polynomial order p , and n is the number of basis functions required to construct the B-spline curve. The parameter space is partitioned by the knots. The B-spline basis functions are defined recursively starting with piecewise constants (for p = 0):

)   

1 0

,

i 1       i

if

(

i,0 N

(1)

otherwise

For higher polynomial orders, p = 1, 2, 3, …, the B – spline basis functions are defined by the Cox-de Boor recursion formula as follows.

ξ

ξ



ξ ξ 

1

i p

 

(

ξ)

(ξ)

(ξ)

N

N

N





i

, 1  (2) B-spline curves in � d are constructed by a linear combination of B-spline basis functions similar to the classical FEA. The vector-valued coefficients of the basis functions are referred as control points. Given n basis functions N i,p with, i = 1, 2, …, n , and corresponding control points B i ∈ � d , i = 1, 2, …, n , a piecewise-polynomial B-spline curve is defined as 1, 1 p 1 , 1 ξ ξ i p   ξ ξ i p i i i i i p p      

n



C

B

(

(

N





(3)

i,p

i

1

i



The piecewise linear interpolation of the control points establishes a control polygon. Subsequently, the tensor product B-spline solids are defined analogously to the B-spline surfaces. For a control lattice defined by { B i,j,k }, i = 1, 2, …, n ; j = 1, 2, …, m ; k = 1, 2, …, l, polynomial orders, p , q and r , and knot vectors Ξ ={ξ 1 , ξ 2 , …, ξ n + p + 1 }, Ҥ = {η 1 , η 2 , …, η m + q + 1 }, and Ẑ = {ς 1 , ς 2 , …, ς m + q + 1 }, a B-spline solid is defined by

1 1 1 j k       n m l i

S

B

(

, , )    

( ) N M L

( ) ( k,r    ) j,q

(4)

i,p

i, j,k

The properties of a solid in IGA are trivariate generalizations of B-spline surfaces. This section briefly outlined the mathematical basis for NURBS and IGA, for a more detailed description it is advised to refer the pioneering work of Hughes et al. (2005). The Bezier extraction methods enables the NURBS models to be simulated using a finite element analysis framework Borden et al. (2011). In this study we used FEAP Taylor (2003) which has an additional suite called nurbfeap enabling the user to perform isogeometric analysis.