Interpolating Isogeometric Boundary Node Method and Isogeometric Boundary
Element Method Based on Parameter Space

In this paper, general interpolating isogeometric boundary node method (IIBNM) and isogeometric boundary element method (IBEM) based on parameter space are proposed for 2D elasticity problems. In both methods, the integral cells and elements are defined in parameter space, which can reproduce the geometry exactly at all the stages. In IIBNM, the improved interpolating moving leastsquare method (IIMLS) is applied for field approximation and the shape functions have the delta function property. The Lagrangian basis functions are used for field approximation in IBEM. Thus, the boundary conditions can be imposed directly in both methods. The shape functions are defined in 1D parameter space and no curve length needs to be computed. Besides, most methods for the treatment of the singular integrals in the boundary element method can be applied in IIBNM and IBEM directly. Numerical examples have demonstrated the accuracy of the proposed methods.


Introduction
Isogeometric analysis (IGA) [1,2] has been applied in many areas and it has at least two advantages compared with the traditional methods: (i) the geometry of the model keeps unchanged in the whole procedure; (ii) the refinement can be performed easily in parameter space. Actually, IGA tries to do the analysis directly on the CAD model, in which the solid can be characterized by the boundary with parametric functions [3]. Thus, combining the IGA with the boundary element method (BEM) [4][5][6][7][8][9] is a natural idea and relative investigations for IGA BEM can be found in [10][11][12][13]. In most types of IGA BEM, B-spline and NURBS basis functions are applied. However, some regular geometry can be construed by simple analytical parametric functions, such as circle and sphere. Besides, the parametric function for a curve or surface is not uniform and different CAD packages may use different functions. Is there any possibility to construct an isogeometric method that is independent of parametric functions? One of the most possible method is the boundary face method (BFM) [14]. In BFM [15,16], only the forms of parametric functions are needed to be known. Actually, the translations between parameter coordinates and physical coordinates can be done easily by functions in some CAD packages [17][18][19][20].
In this paper, the idea is applied in an interpolating boundary node method (IBNM), or can also be called an interpolating boundary element-free method (IBEFM) [21][22][23]. In the original boundary node method (BNM) [24] or Galerkin BNM [25,26] the moving least-square (MLS) approximation [27][28][29][30] is used for field approximation, which leads to difficulty for applying boundary conditions. The interpolating moving least-square (IMLS) method [27] with singular weight functions can be implemented to obtain shape function with delta function property. However, it may lead to difficulties while implementation because of the singular weight functions. An improved interpolating moving least-square (IIMLS) approximation was further presented [21,31] with no singular weight functions and has been applied in BEFM [32]. Wang et al. [33] proposed regularized improved interpolating moving least-square method and adaptive orthogonal improved interpolating moving least-square method [34] to avoid the singular moment matrix while computing the shape function.
In this paper, an interpolating isogeometric boundary node method (IIBNM) is proposed. The method can be considered as a general isogeometric method based on parameter space. The parametric functions are only used to characterize the boundary and the IIMLS method is implemented to interpolate the fields, including the displacement and traction fields. An isogeometric boundary element method (IBEM) can also be obtained if the Lagrangian basis functions are implemented to interpolate the fields. In both methods, there is no discretization error and the integrals are performed on isogeometric cells or elements. Besides, the boundary conditions can be applied easily in both methods, since the shape functions obtained by the IIMLS method and Lagrangian basis functions have the interpolating property. Compared with the conventional BEM and BNM, the proposed methods can avoid the discretization error of geometry and can be integrated with Computer-Aided Design (CAD) easily. Compared with BNM and the isogeometric BEMs [10,35,36], the shape functions for field approximation have interpolating property, and it leads to a simpler way to impose the boundary conditions.

The Boundary Integral Equation with Isogeometric Cells
In this section, the boundary integral equation (BIE) for 2D elasticity problems is reviewed and discretized by isogeometric cells. The basic equations can be written as with where r ij are the stresses, e ij are the strains and u i are the displacements. C ijkl are the components of the elastic tensor and d ij are the Kronecker-delta functions. l is the shear modulus, k ¼ 2vl=ð1 À 2vÞ is the Lamé constant, and v is the Poisson ratio.
One can have the boundary integral equation (BIE) as [37] 0 ¼ Z À Uðx; yÞtðyÞdÀðyÞÀ Z À Tðx; yÞ½uðyÞ À uðxÞdÀðyÞ where À is the boundary of a bounded domain in R 2 . t represents the traction vector on À. For plane strain state, the components of Uðx; yÞ and Tðx; yÞ are U ki ðx; yÞ ¼ À1 8pð1 À vÞl fð3 À 4vÞ lnðrÞd ki À r ;k r ;i g T ki ðx; yÞ ¼ À1 4pð1 À vÞr f½ð1 À 2vÞd ki þ 2r ;k r ;i @r @n À ð1 À 2vÞðr ;k n i À r ;i n k Þg where r ¼ x À y k k, and nðyÞ is the unit outward normal. Points x and y are the source point and field point in R 2 , respectively.
In conventional BEM or BNM, the boundary À in Eq. (5) is approximated by the discrete cells. In BNM, the cells are used for the convenience of computing the integrals on the boundary. In BEM, the cells are also used for approximating the fields and they are called elements. In general, the straight line or the parabola can be used as each cell to approximate the boundary. However, for complex curves, these kinds of cells cannot reproduce the geometries exactly and the discretization error occurs.
In most of the CAD/CAE packages, the curves are represented by parametric functions and only the local coordinates in one-dimensional parameter space are needed to obtain the global coordinates in twodimensional space. In general, a curve can be represented by parametric functions by where xfx 1 ; x 2 g are the global coordinates and n is the parameter coordinate. Usually one can use xðnÞ to clarify that point x is represented by the parameter n in parametric form.
The normal n of a point x on the curve can be obtained by Now suppose the boundary À can be represented in parametric form and one can partition it as where À j ½n j ; n jþ1 is the jth cell with range ½n j ; n jþ1 , n j is the jth parameter and N is the number of cells. À j is also represented in parametric form by Eq. (8), thus, the boundary À can be reconstituted exactly no matter how many cells are used. This is why the cells are called as isogeometric cells.
Then Eq. (5) can be written as Tðxð&Þ; yðnÞÞ½uðyðnÞÞ À uðxð&ÞÞJ dn (11) where & and n are parameter coordinates for points x and y, respectively. J can be evaluated by Eq. (11) has partitioned the boundary with isogeometric cells for the purpose of integration. The rest of the problem is how to approximate the field and two methods will be introduced in the next two sections.

The Interpolating Isogeometric Boundary Node Method
In this section, the IIMLS method is implemented to obtain the shape function for field approximation and the interpolating isogeometric boundary node method (IIBNM) is derived.
3.1 The Improved Interpolating Moving Least-Square Method [21,31] Suppose point xfx 1 ; …; x d g is a point in a domain in space R d ðd ¼ 1; 2; 3Þ and the local support domain RðxÞ is centered at x. For function uðxÞ, one can define another formula as where qðx; y i Þ is defined as The function qðx; y i Þ has the following properties: and The local function ofũðx; yÞ can be evaluated bỹ where a j ðxÞ are coefficients needed to be determined. g j ðx; yÞ is defined as In Eq. (19), p j ðxÞ are basis functions. To obtain the unknown a j ðxÞ, one can define the weighted discrete L 2 norm as where wðx À y i Þ are weight functions.

The Interpolating Isogeometric Boundary Node Method
With the IIMLS approximation, a meshless interpolating isogeometric boundary node method (IIMLS) can be obtained. The displacements uðyÞ and tractions tðyÞ on the boundary point y can be interpolated by where n is the parameter coordinate for point y, u i l and t i l are the nodal displacement and traction, and È i ðnÞ is the shape function obtained by IIMLS method as ΦðnÞ ¼ĝðn; nÞA À1 ðnÞBðnÞ þ qðnÞ By substituting Eqs. (34) and (35) into Eq. (11), one can obtain For boundary node xðn p Þ, Eq. (37) can be rewritten as: Finally, Eq. (38) can be rewritten as In Eqs. (40) and (41), K i is the set of cells that have contributions to È i ðnÞ and K p is the set of cells contain node xðn p Þ (see Fig. 1).

The Isogeometric Boundary Element Method
In IIBNM, the integral cells and field approximations are independent. In this section, the isogeometric cells are also used to interpolate the fields and the shape functions used in traditional BEM are applied. The method can also be called an isogeometric boundary element method (IBEM). The interpolation is also performed in parameter space and a coordinate transformation is defined in a cell À j ½n j ; n jþ1 as where s 2 ½À1; 1 is the local coordinate (see Fig. 2). If the values at nodes n j and n jþ1 are used to interpolate the field in the cell À j ½n j ; n jþ1 , then uðyÞ and tðyÞ at point y 2 À j can be interpolated by Figure 1: Cells and nodes in IIBNM where u j and t j are the displacements and tractions at point x j , and w i j ðsÞ is the shape function on À j defined as w 0 j ðsÞ ¼ 0:5 À 0:5s (47) By substituting Eqs. (45) and (46) into Eq. (11), one can obtain where q is the new local coordinate of & in the relative cell and J 1 ¼ ðn jþ1 À n j Þ=2. Then for xðnðs p ÞÞ, we can have: The shape function in Eq. (50) also has the interpolating property, thus, one can obtain Finally, Eq. (51) will be In Eqs. (53) and (54), K i are cells cover node xðnðs i ÞÞ (see Fig. 3).

Numerical Examples
In all the examples, E ¼ 2:5 and v ¼ 0:3 are applied for the material properties. To treat the singular integrals appeared in the proposed methods, the rigid body motion method is applied to compute the strongly singular integrals, and the self-adaptive coordinate transformation method [38] is used to evaluate the weakly singular integrals [37].

Dirichlet Problems on a Ring
A ring shown in Fig. 4 is tested in this example. The radius of the inner circle is 1, and the minimum radius and maximum radius of the outer ellipse are 2 and 4, respectively. Displacements are known on all the boundaries and the following solutions are considered: (i) Linear solution: (ii) Quadratic solution: Figure 3: Cells and nodes in IBEM (iii) Cubic solution: The relative errors of the displacements and stresses are shown in Figs. 5-7. One can indicate that both IIBNM and IBEM have good agreement with the exact results. The errors of IIBNM and IBEM are very close to each other for the cases with few nodes. It can also be observed that the results obtained by IIBNM have a little concussion when the boundary nodes are few.

Lame Problem
The Lame problem shown in Fig. 8 is considered. Plane strain state is considered and geometry shown in Fig. 8(b) is analyzed.
The exact solutions for this problem are The results of traction t 2 on line x 2 ¼ 0 are plotted in Fig. 9. All the results shown in Fig. 9 are solved by 69 boundary nodes. Fig. 10 shows the relative errors of traction t 2 on line x 2 ¼ 0 for different methods. It can be observed that the accuracy of IIBNM is the highest.

Plate with an Elliptical Hole
In this example, a geometry under tensile load shown in Fig. 11 is studied and the parameters are L ¼ 20, r 1 ¼ 2, r 2 ¼ 4 and T ¼ 1. Plane stress statement is assumed. 3845 nodes are used in IBEM and IIBNM while 344363 CPS4R elements are used in FEM to solve the problem.
The stresses r 2 on line x 2 ¼ 0 are shown in Fig. 12 and the stresses r 1 on line x 1 ¼ 0 are shown in Fig. 13. One can find out that similar results are obtained by the proposed methods, compared with FEM. The contours of stresses r 1 and r 2 are shown in Figs. 14 and 15. Conventional BEM with 3845 nodes is also used to simulate the problem. It can be observed that the results obtained by FEM, BEM, IBEM, and IIBNM have little difference.

Conclusions
Two general isogeometric methods based on the parameter space are proposed. The integral cells and elements are defined in parameter space, and the geometry can be reproduced at all the stages in both methods. The displacement and traction fields are approximated by the IIMLS method and Lagrangian basis functions in parameter space in IIBNM and IBEM, respectively.