Mapping Techniques for Isogeometric Analysis of Elliptic Boundary Value Problems Containing Singularities

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1 Mapping Techniques for Isogeometric Analysis of Elliptic Boundary Value Problems Containing Singularities Jae Woo Jeong Department of Mathematics, Miami University, Hamilton, OH 450 Hae-Soo Oh Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC Sunbu Kang Korea Air Force Academy, Cheongwon, Chungbuk, South Korea and University of North Carolina at Charlotte, Charlotte, NC Hyunju Kim Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC Comput. Methods Appl. Mech. Engrg. (202), Abstract The Method of Auxiliary Mapping (MAM), introduced by Babuška and Oh [2], is an effective method for dealing with singularities in elasticity [6]. MAM was extended to boundary element method (BEM) in the framework of mesh free particle methods [7], reproducing polynomial particle methods [20], and also to infinite domain problems [8]. Similarly, we consider NURBS geometrical mappings that are able to generate crack singularities for isogeometric analysis of elliptic boundary value problems. However, the mapping techniques proposed in this paper are different from MAM. In order to generate singular shape functions, MAM uses conformal mappings that locally change the physical domain, whereas the NURBS mappings used for design of engineering system are not allowed to alter the physical domain for isogeometric analysis. Moreover, unlike MAM, the proposed method makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned. We prove error estimates in Sobolev norms and demonstrate that the proposed mapping technique is highly effective for isogeometric analysis of elliptic boundary value problems with singularities. Mesh refinements to deal with singularities are compared with the mapping technique in the isogeometic analysis framework. Corresponding author. Tel.: ; Fax: ; Supported in part by NSF grant DMS

2 Keywords: Isogeometric analysis; B-spline basis functions; NURBS geometrical mapping; Knot vectors; Control points; Mapping techniques. Introduction NURBS (non-uniform rational B-spline) basis functions that are non interpolant are tools for engineering designs, whereas piecewise polynomial basis functions that are interpolant are shape functions for finite element analysis. Thus, there are barriers between computer aided design (CAD) and finite element analysis (h-fem [5, 9] and p-fem [26]) that make it difficult to synthesize them together. Most recently, introducing NURBS basis functions to finite element analysis, Hughes et al. [0] combined two tasks (engineering design and analysis) into one process. They coined this concept isogeometric analysis. Since then quite a few papers ([3, 7] and references within) have been published to show the theories as well as the computations where isogeometric finite element analysis is a powerful new approach. In the literature, X-FEM in which linear finite element basis functions are enriched with singular functions were successful in dealing with singularity problems. Notably, X-FEM is built into isogeometric analysis of elastic domain with cracks [8]. In order to handle the singularities arising in the partial differential equations, Babuška and Oh [2] introduced the mapping techniques called the Method of Auxiliary Mapping (MAM) into conventional p-fem. It was proved that MAM is successful in dealing with singularities in elasticity [3, 4, 6], oscillating singularities [22], and delamination crack singularities [5]. Recently, Oh et al. extended MAM to the meshfree particle methods [9, 20], the boundary element method in the framework of reproducing polynomial particle methods [7], and also to infinite domain problems [8, 23]. However, MAM has no relevance in the h-fem that uses linear basis functions. In a similar spirit to MAM, we introduce mapping techniques into isogeometric analysis to deal with point singularities (cracks and jump boundary data) that arise in elliptic boundary value problems. The proposed mapping technique implemented in isogeometric analysis is different from MAM in the following aspects: MAM uses conformal mappings that locally change the physical domain (possibly altering design). In contrast, the proposed mapping technique uses NURBS geometrical mappings with proper selection of control points and hence does not alter the system design. Unlike MAM, this mapping technique makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned. The proposed mapping method in the framework of isogeometric analysis yields the same accuracy as MAM in p-fem at a much lower degree of freedom. This is actually a salient feature of isogeometric analysis []. In this paper, we use NURBS basis functions only for the constructions of geometrical mappings that precisely map the parameter space onto a physical domain, however we employ B-spline 2

3 basis functions (continuous piecewise polynomials) for analysis. In other words, by the partition of unity property, B-spline basis functions are NURBS basis functions with unit weights, and hence the finite element analysis of this paper is an isogeometric analysis. However, we do not use full-blown NURBS basis functions (piecewise rational functions whose support consist of a union of several knot spans). Instead we use B-spline basis functions that are interpolant at each knot. Therefore the proofs of error estimates of this paper are easier than those presented in [3] which is a highly technical paper dealing with error estimates of the approximations by rational basis functions whose supports are different from that of FEM. Another popular approach to deal with singularity in the conventional FEM is geometric mesh refinements ([6], [26]). However, it is not simple to implement mesh refinement methods in the isogeometric analysis as shown in [4]. We compare mesh refinement methods with our mapping technique in Example 5.4. It is important to note that the proposed mapping technique is not properly working with neither the B-spline functions by the k-refinement nor the NURBS functions. Moreover, the mapping technique has no effects unless the polynomial degrees of B-spline basis functions are 2 in each variable. This paper is organized as follows: Definitions and terminologies used in this paper are introduced in section 2. In section 3, we construct NURBS geometrical mappings that map the parameter space onto a cracked physical domain and investigate the properties of the geometrical mappings. In section 4, we prove error estimates of the proposed mapping technique. Numerical tests that support the theoretical results are presented in section 5. We apply the proposed methods to the crack singularity, to the jump data singularity (the Motz problem), and to general monotone singularities of type r /q ψ(θ) for q = 2, 4, 5, 8, 0. Finally, the concluding remarks are stated in section 6, in which we describe how to apply the mapping technique for an enrichment of NURBS functions to be used for isogeometric analysis. 2 Preliminaries 2. B-splines, control points, knot vectors, weights, and NURBS In this section, we briefly review definitions and terminologies that are used throughout this paper. We follow those in the books [7, 24], and we thus refer to these texts for details. A knot vector Ξ = {ξ, ξ 2,, ξ m } is a nondecreasing sequence of real numbers in the parameter space [0, ], and the components ξ i are called knots. An open knot vector of order p + is a knot vector that satisfies ξ = = ξ p+ < ξ p+2 ξ m p < ξ m p = = ξ m, in which the first and the last p + knots are repeated and the interior knots can be repeated at most p times. The B-spline functions N i,k (ξ) of order k = p + corresponding to the knot vector Ξ = {ξ, ξ 2,, ξ m } are piecewise polynomials of degree p which are constructed recursively by 3

4 Figure : B-spline functions N i,3 (ξ), i =, 2,, 7 of order k = 3 corresponding to the knot vector Ξ = {0, 0, 0, 0.2, 0.5, 0.8, 0.8,,, }. the formula (Cox-de Boor): { if ξ i ξ < ξ i+, N i, (ξ) = for im, 0 otherwise, N i,t (ξ) = ξ ξ i N i,t (ξ) + ξ i+t ξ N i+,t (ξ), for i m, 2 t k. ξ i+t ξ i ξ i+t ξ i+ (There is a terminology conflict between the design and analysis community. Designers will say a quadratic polynomial has degree 2 and order 3 [, 24]). For example, the piecewise quadratic polynomial B-spline functions N i,3 (ξ) corresponding to the knot vector Ξ = {0, 0, 0, 0.2, 0.5, 0.8, 0.8,,, } are depicted in Fig.. The B-spline functions are useful in design as well as finite element analysis because they have the following properties: variation diminishing, convex hull, non-negativity, piecewise polynomial, compact support, and partition of unity. A B-spline curve is defined as follows: C(ξ) = m k i= N i,k (ξ)b i, where B i are control points that make B-spline functions draw a desired curve as shown in Fig. 2(a). Let Ξ η = {η,, η n } be an open knot vector and let p η and k = p η +, respectively, be the polynomial degree and order of B-spline functions M j,k (η). Then a B-spline surface is defined by S(ξ, η) = m k i= n k j= N i,k (ξ)m j,k (η)b i,j, where B i,j are control points that make a bidirectional control net as shown in Fig. 2(b). 4

5 (a) B-spline Curve (b) B-spline Surface Figure 2: (a) B-spline curve and control points. (b) B-spline surface and control net. Let {w i : i =,, m k} be the set of weights. Then the corresponding NURBS basis functions are defined by R i,k (ξ) = N m k i,k(ξ)w i, W (ξ) = N s,k (ξ)w s > 0. W (ξ) The NURBS basis functions are now piecewise rational functions. A NURBS curve corresponding to the control points {B i : i =,, m k}, NURBS basis functions {R i,k (ξ) : i =,, m k}, and the weights {w i : i =,, m k} is C(ξ) = m k i= s= R i,k (ξ)b i. Let {w i,j : i =,, m k, j =,, n k } be the set of weights. Then NURBS basis functions corresponding to the open knot vectors Ξ ξ and Ξ η and the weights {w i,j } are defined by where W (ξ, η) = R i,j (ξ, η) = N i,k(ξ)m j,k (η)w i,j, W (ξ, η) m k s= n k t= N s,k (ξ)m t,k (η)w s,t > 0. Let {B i,j : i =,, m k, j =,, n k } be a set of control points in R d, d 2. Then a NURBS surface corresponding to the control points {B i,j }, NURBS basis functions {R i,j (ξ, η)}, and the weights {w i,j } is S(ξ, η) = m k i= n k j= R i,j (ξ, η)b i,j. 5

6 2.2 Weak solution in Sobolev space Let Ω be a connected open subset of R d. We define the vector space C m (Ω) to consist of all those functions φ which, together with all their partial derivatives α φ(= α α d d φ) of orders α = α + +α d m, are continuous on Ω. A function φ C m (Ω) is said to be a C m -function. If Ψ is a function defined on Ω, we define the support of Ψ as supp Ψ = {x Ω Ψ(x) 0}. For an integer k 0, we also use the usual Sobolev space denoted by H k (Ω). For u H k (Ω), the norm and the semi-norm, respectively, are u k,ω = /2 α u 2 dx, u k,,ω = max α k {ess.sup α u(x) : x Ω} ; α k u k,ω = α =k Ω Ω α u 2 dx /2, u k,,ω = max α =k {ess.sup α u(x) : x Ω}. Suppose we are concerned with an elliptic boundary value problem on a domain Ω with Dirichlet boundary condition g(x, y) along the boundary Ω. Let W = {w H (Ω) : w Ω = g} and V = {w H (Ω) : w Ω = 0}. The variational formulation of the Dirichlet boundary value problem can be written as: Find u W such that B(u, v) = L(v), for all v V, () where B is a continuous bilinear form that is V-elliptic ([5]) and L is a linear functional. The solution to () is called a weak solution which is equivalent to the strong (classical) solution corresponding elliptic PDE whenever u is smooth enough. The energy norm of the trial function u is defined by [ ] /2 u eng = B(u, u). 2 Let W h W, V h V be finite dimensional subspaces. Since the NURBS basis functions do not satisfy the Kronecker delta property, in this paper we approximate the nonhomogenuous Dirichlet boundary condition by the least squares method as follows: g h W h such that g g h 2 dγ = minimum. Ω We can write the Galerkin form (a discrete variational equation) of () as follows: Given g h, find u h = w h + g h, where w h V h, such that B(u h, v h ) = L(v h ), for all v h V h, which can be rewritten as: Find the trial function w h V h such that B(w h, v h ) = L(v h ) B(g h, v h ), for all test functions v h V h. (2) 6

7 3 NURBS geometrical mappings that generate singular basis functions The geometrical mappings we are concerned with are the NURBS curves or the NURBS surfaces defined in the previous section. In order to clearly address how the proposed NURBS geometrical mapping handles the singularity problems in higher dimensional cases, we first show how a B-spline geometrical mapping F handles effectively one-dimensional singularities. Let Ξ η = {0,, 0,,, } be an open knot vector of order k = p η +. Then the B-spline functions M j,k (η) corresponding to Ξ η are ( ) pη M j,k (η) = η j ( η) pη j+ for j =,, k. j Here, M j,k, j =,, k, are also called the Bernstein polynomials of degree p η. Let B j = (0, 0), for j =,, k, and B k = (0, c) be control points, for a constant c. Then the B-spline geometrical mapping F(η) = M j,k (η)b j = (0, cη pη ) k j= maps the parameter space [0, ] onto the physical space {0} [0, c] R 2 and its inverse is η = F (0, y) = (/c) /pη y /pη. Thus, the approximation space V h = span{m i,k F i =,, k }, where k is an integer greater than or equal to k and M i,k are the Bernstein polynomials (B-spline functions) of degree k, contains the following singular as well as smooth functions: y l/pη, l = 0,,, k. In other words, the geometrical mapping F is able to generate the singularity of type r λ, where 0 < λ = /p η <. For example, if p = 2, then the Bernstein polynomials of degree 2 are and M,3 = ( η) 2, M 2,3 = 2η( η), M 3,3 = η 2. B = (0, 0), B 2 = (0, 0), B 3 = (0, /2) are control points. Then the geometrical mapping obtained by these control points and its inverse, respectively, are F(η) = (0, 2 ) η2 and F (0, y) = 2 y. 7

8 Suppose Sη h = span{m j,5 j =,, 5} where M j,5 are the Bernstein polynomials corresponding the the open knot vector Ξ = {0, 0, 0, 0, 0,,,,, } of order 5, then Sη h contains, η,, η 4. Hence the approximation space Vy h = span{m j,5 F : j =,, 5} for isogeometric analysis contains, y, y, y 3/2, y 2. However, the approximation space V h y = span{r j,5 (NURBS) F : j =,, 5} may not be able to generate these singular functions. Thus, the mapping technique is not applicable to the NURBS based isogeometric analysis. We are now going to extend this argument to NURBS geometrical mappings from the parameter space Ω = [0, ] [0, ] onto a cracked unit disk Ω R 2 in the next two subsections. 3. NURBS geometrical mapping to deal with crack singularity of type r /2 ψ(θ) Suppose the physical domain Ω is a unit disk with a crack along the positive x-axis as shown in Fig. 3(b). We now consider a NURBS isogeometric mapping from the parameter space Ω = [0, ] [0, ] to the physical domain Ω. Consider the knot vectors: Ξ ξ = {0, 0, 0, 4, 4, 2, 2, 34, 34 },,,, Ξ η = {0, 0, 0,,, }. Let N i,3, i =,, 9 be the B-splines corresponding to the knot vector Ξ ξ and let M j,3, j =,, 3 be the B-splines corresponding to the knot vector Ξ η. Then these B-spline functions are { ( 4ξ) 2 if ξ [0, N,3 (ξ) = 4 ] 0 if ξ / [0, 4 ] N 2,3 (ξ) = N 3,3 (ξ) = N 5,3 (ξ) = N 7,3 (ξ) = N 9,3 (ξ) = { 8ξ( 4ξ) if ξ [0, 4 ] 0 if ξ / [0, 4 ] (4ξ) 2 if ξ [0, 4 ] { (2 4ξ) 2 if ξ [ 4, 2(4ξ )(2 4ξ) if ξ [ 2 ] N 4,3 (ξ) = 4, 2 ] 0 if ξ / [0, 2 ] 0 if ξ / [ 4, 2 ] (4ξ ) 2 if ξ [ 4, 2 ] { (3 4ξ) 2 if ξ [ 2, 3 2(4ξ 2)(3 4ξ) if ξ [ 4 ] N 6,3 (ξ) = 2, 3 4 ] 0 if ξ / [ 4, 3 4 ] 0 if ξ / [ 2, 3 4 ] (4ξ 2) 2 if ξ [ 2, 3 4 ] (4 4ξ) 2 if ξ [ 3 4, ] N 8,3 (ξ) = 0 if ξ / [ 2 {, ] (4ξ 3) 2 if ξ [ 3 4, ] 0 if ξ / [ 3 4, ] { 8(4ξ 3)( ξ) if ξ [ 3 4, ] 0 if ξ / [ 3 4, ] M,3 (η) = ( η) 2, M 2,3 (η) = 2η( η), M 3,3 (η) = η 2, for η [0, ]. (4) Consider the control points B i,j and the weights w i,j for i 9, j 3, that are listed in Table. With the B-spline functions shown in (3) and (4), the 27 control points and (3) 8

9 Table : Control points B i,j and weights w i,j. i j B i,j w i,j i j B i,j w i,j i j B i,j w i,j (0, 0) 2 (0, 0) 3 (, 0) 2 (0, 0) (0, 0) (, ) 2 3 (0, 0) 3 2 (0, 0) 3 3 (0, ) 4 (0, 0) (0, 0) (, ) 2 5 (0, 0) 5 2 (0, 0) 5 3 (, 0) 6 (0, 0) (0, 0) (, ) 2 7 (0, 0) 7 2 (0, 0) 7 3 (0, ) 8 (0, 0) (0, 0) (, ) 2 9 (0, 0) 9 2 (0, 0) 9 3 (, 0) weights, we now construct a NURBS geometrical mapping from the parameter space Ω onto Ω as follows: 9 3 F(ξ, η) = R i,j (ξ, η)b i,j. i= j= Here R i,j (ξ, η), i 9, j 3, are NURBS basis functions defined by where R i,j (ξ, η) = N i,3(ξ)m j,3 (η)w i,j, W (ξ, η) W (ξ, η) = 9 s= t= 3 N s,3 (ξ)m t,3 (η)w s,t. Noting that from Table, w s,j = if s =, 3, 5, 7, 9 and w s,j = / 2 if s = 2, 4, 6, 8, and using the partition of unity property: 3 t= M t,3(η) =, we have [ 9 3 ] W (ξ, η) = N s,3 (ξ) M t,3 (η)w s,t s= t= = [N 2,3 (ξ) + N 4,3 (ξ) + N 6,3 (ξ) + N 8,3 (ξ)] / 2 w(ξ), + [N,3 (ξ) + N 3,3 (ξ) + N 5,3 (ξ) + N 7,3 (ξ) + N 9,3 ( ξ)] which becomes a function of ξ only. Since B i, = B i,2 = (0, 0) for all i by the choice of the control points B i,j and the weights w i,j in Table, we have F(ξ, η) = η 2 9 i= N i,3 (ξ)w i,3 B i,3 := (x(ξ, η), y(ξ, η)), w(ξ) 9 (5)

10 where the coordinate functions are as follows: x(ξ, η) = η2 [N,3 + N 2,3 / 2 N 4,3 / 2 N 5,3 N 6,3 / 2 + N 8,3 / ] 2 + N 9,3 (ξ) w(ξ) ( ) X(ξ) = η 2, w(ξ) y(ξ, η) = η2 w(ξ) = η 2 ( Y (ξ) w(ξ) [ N 2,3 / 2 N 3,3 N 4,3 / 2 + N 6,3 / 2 + N 7,3 + N 8,3 / ] 2 (ξ) ). Moreover, the weight function w(ξ) is bounded away from zero: (6) W (ξ, η) w(ξ). Lemma 3.. Suppose u(r, θ) = rψ(θ) for a smooth function ψ. Then û(ξ, η) := u F(ξ, η) = ηψ(ξ), where Ψ(ξ) C 0 [0, ] and Ψ(ξ) C (0, ) unless ξ = /4, /2, 3/4. Proof. From Fig. 3 and the weights w i,j in Table, we have We thus have which implies = x(ξ, η) 2 + y(ξ, η) 2 = η 4. ( ) X(ξ) 2 + w(ξ) From (7), the pull back of r onto Ω becomes Since ( ) Y (ξ) 2, (7) w(ξ) w(ξ) 2 = X(ξ) 2 + Y (ξ) 2. [ (X(ξ) ) 2 r F(ξ, η) = η + w(ξ) ( ) ] Y (ξ) 2 /2 = η. w(ξ) ( ) ( ) 2π + tan y(ξ,η) x(ξ,η) = 2π + tan Y (ξ) X(ξ) if ξ [0, 4 ), 3 2 π ( ) ( ) if ξ = 4, θ(ξ, η) = π + tan y(ξ,η) x(ξ,η) = π + tan Y (ξ) X(ξ) if ξ ( 4, 3 4 ), 2 π ( ) ( ) if ξ = 3 4, tan y(ξ,η) x(ξ,η) = tan Y (ξ) X(ξ) if ξ ( 3 4, ]. and X(ξ) and Y (ξ) are smooth, and X(ξ) 0 unless ξ = /4, /2, 3/4, the pull back of ψ, Ψ(ξ) ψ(θ(ξ, η)), is smooth unless ξ = /4, /2, 3/4. Since F and ψ are continuous, so does Ψ. 0

11 (a) Parameter space with coarse mesh (b) Physical domain with control points Figure 3: The parameter space and the physical domain for the NURBS mapping F. Lemma 3. shows that the pull back of a singular function r /2 ψ(θ) through the NURBS mapping F becomes a piecewise smooth function on the parameter space Ω. For the error analysis in the forthcoming section, we estimate an upper bound of the determinant of the Jacobian of F: det(j(f)) = x y ξ η x y η ξ = 2η3 w(ξ) 2 X (ξ)y (ξ) X(ξ)Y (ξ), 3 X (ξ)y (ξ) X(ξ)Y [ (ξ) = (X (ξ)y (ξ) X(ξ)Y (ξ))χ [i/4,(i+)/4) (ξ) ], i=0 where X(ξ) and Y (ξ) are defined by (6). Here χ [i/4,(i+)/4) (ξ) = for ξ [i/4, (i + )/4) and χ [i/4,(i+)/4) (ξ) = 0 otherwise, i = 0,, 3. From those quadratic B-spline functions N i,3 (ξ) defined by (3), one can see that X (ξ)y (ξ) X(ξ)Y (ξ) is a bounded cubic polynomial on each subinterval [i/4, (i + )/4). We actually get 2( + 2) (X (ξ)y (ξ) X(ξ)Y (ξ))χ [0,/4) (ξ) 4 2, 2( + 2) (X (ξ)y (ξ) X(ξ)Y (ξ))χ [/4,/2) (ξ) 4 2, 2( + 2) (X (ξ)y (ξ) X(ξ)Y (ξ))χ [/2,3/4) (ξ) 4 2, 2( + 2) (X (ξ)y (ξ) X(ξ)Y (ξ))χ [3/4,) (ξ) 4 2. (8) We thus have det(j(f)) 64 2(3 2 2) 6. (9) Remark 3.. () Unlike the usual NURBS geometrical mapping in the isogeometric analysis literature, the NURBS mapping constructed above is not one-to-one along η = 0 where the determinant of Jacobian matrix J(F) is zero. (2) In order to make det(j(f)) approach zero slower than η does as η goes to 0, it is possible to use ηe ωη in place of η for the construction of the geometrical mapping F with damping factor ω. However, we tested this mapping to see that the alternative non-nurbs mapping does not yield better results in dealing with the singularity of type r λ ψ(θ), where 0 < λ /2.

12 (3) Since N i,3 (ξ), i 9, are interpolant when ξ = 0, /4, /2, 3/4, we have F(0, η) = η 2 B,3 = (η 2, 0), F(/4, η) = η 2 B 3,3 = (0, η 2 ), F(/2, η) = η 2 B 5,3 = ( η 2, 0), F(3/4, η) = η 2 B 7,3 = (0, η 2 ). 3.2 NURBS mapping to deal with stronger singularity of type r λ ψ(θ) with 0 < λ < /2 Suppose λ = /q for a positive integer q that is greater than 2. Then we consider the two open knot vectors: Ξ ξ = {0, 0, 0, 4, 4, 2, 2, 34, 34 },,,, Ξ η = {η,, η q+, η q+2,, η 2(q+) }, where η j = 0 if j q + ; η j = if q + < j. We select 9(q + ) control points so that B i,j = (0, 0), if j q +, and the corresponding weights w i,j in a similar way to those in Table. Then the corresponding NURBS geometrical mapping F : Ω Ω is of the form F(ξ, η) = (x(ξ, η), y(ξ, η)), where ( ) ( ) X(ξ) Y (ξ) x(ξ, η) = η q, y(ξ, η) = η q, (0) w(ξ) w(ξ) where X(ξ), Y (ξ), w(ξ) are the same as those in (6). It has the following property: r λ ψ(θ) F(ξ, η) = ηψ(ξ), where Ψ(ξ) is piecewise smooth by a similar argument to Lemma 3.. That is, the pull back of a singular function onto the parameter space through the NURBS mapping F becomes a continuous piecewise smooth function. Lemma 3.2. Suppose ψ(θ) is a smooth function such that Ψ(ξ) = (ψ F)(ξ, η) is smooth except ξ = 0, /4, /2, 3/4,. Then, for each non-negative integer p, there is a constant C(p) such that d p dξ p Ψ(ξ) C(p) ψ p,,[0,2π], j =,, 4, () ((j )/4,j/4) where C() 7.8 and C(2) 3.25, and the constant C(p) is increasing as p is getting larger. Proof. By the definition of the geometrical map F and Eq. (7), we have θ(ξ) = C j + tan Y (ξ) X(ξ), where C j is a constant that depends on the region [(j )/4, j/4], j =, 2, 3, 4. Since Ψ(ξ) = ψ(c j + tan Y (ξ) X(ξ) ), by the Faà di Bruno s Formula [2] (a formula for derivatives of a composite function), we have d p dξ p Ψ(ξ) = p! s!s 2! s p! ψ(s) (θ) ( θ (ξ) 2! ) s ( θ ) ( ) (ξ) s2 θ (p) sp (ξ) 2! p!

13 where the sum runs all possible solutions of the system s +s 2 + +s p = s, s +2s 2 + +ps p = p with s i 0. Therefore, for each j =,, 4, we get d p dξ p Ψ 0,,([j ]/4,j/4) p C(s,, s p ) ψ s, C(p) ψ p,. s=0 For example, if ψ(θ) = sin(θ/2), by (7), X 2 +Y 2 (3+2 2)/8 and by (8), Y X X Y 4 2, and hence Y X X Y X 2 +Y Error estimates The NURBS geometrical mapping F : Ω Ω constructed with coarse mesh on Ω (Fig. 3(a)) does not change as the mesh on Ω is further refined. Let S h S(Ξ ξ, Ξ η, p ξ, p η ) = span{n i,pξ +(ξ)m j,pη+(η) i m, j n}, Sξ h S(Ξ ξ, p ξ ) = span{n i,pξ +(ξ) i m}, V h = span{[n i,pξ +(ξ)m j,pη+(η)] F i m, j n}, Vξ h = span{[n i,p ξ +(ξ)η l ] F i m}, for a fixed l, Q h = mesh on (0, ) 2, K = F(Q), Q Q h. (2) Here p ξ and p η, respectively, are the polynomial degrees of B-spline basis functions in the ξ- and the η-directions. Let us note that these approximation spaces are for the isogeometric analysis of elliptic problems, but not for the construction of F. In this section, we assume the following:. The degree p without suffix means the degree of polynomial in the ξ-variable. M j,n (η) are Bernstein polynomials for all j whose supports are [0, ] in the construction of the geometrical mapping F. 2. u is the weak solution of () and u h is the Galerkin approximate solution of (2). Each knot value of the open knot Ξ ξ has multiplicity p. That is, each member in S h ξ is a C0 -function. Because of these facts, the error estimates of our isogeometric analysis now becomes similar to those of FEM. Lemma 4.. Suppose v C[0, h ] is (p + 2)-times differentiable on (0, h ). If are p + distinct numbers, then there exists 0 = x 0 < x < x 2 < < x p = h Π p v S p (x) span{x i ( x) p i : i = 0,, 2,, p} 3

14 such that Moreover, if h <, Π p v(0) = v(0) and Π p v(h ) = v(h ). (3) v(x) Π p v(x) h p+ v p+, /(p + )!; d dx [v(x) Π pv(x)] hp ( v p+, + v p+2, ) /p!. (4) Proof. If a projection operator from C[0, ] onto S p (x) is defined by the p-th Lagrange polynomial that interpolates v C[0, ] at x i, i = 0,, 2,, p: Π p v = p v(x l )L p,l (x), l=0 where L p,l (x) = p (x x i ) i=0,i l (x l x i ), then, for each x (0, h ), there exists c(x) (0, h ) such that v(x) Π p v(x) = v(p+) (c(x)) β(x), with β(x) = (x x 0 ) (x x p ). (5) (p + )! Since β(x) h p+, we have v (p+) v(x) Π p v(x) 0, h p+ (p + )! 0, Since f(x) := [v(x) Π p v(x)] (p + )! and β(x) are differentiable for all x (0, h ), we have which implies f(x + h) f(x) h. [ ] = β(x + h) v (p+) (c(x + h)) v (p+) (c(x)) /h + v (p+) (c(x)) [β(x + h) β(x)] /h, (6) d dx [v(x) Π pv(x)] h p [ v p+, ]/(p!), if lim h 0 β(x + h) = 0. (7) [ ] If lim β(x + h) 0, in the relation (6), then lim v (p+) (c(x + h)) v (p+) (c(x)) /h exists, h 0 h 0 and hence d [ ] dx [v(x) Π pv(x)] h p+ v (p+2) 0, + (p + )h p v(p+) 0, /(p + )! h p [ v (p+2) 0, + v (p+) 0, ] /(p!) (since h < ). 4

15 Let T k : [0, ] I k = [ξ k, ξ k+ ] be the linear transformation. operator Π [k] p : H (I k ) S p [k] (ξ) Sξ h I k is defined by Then a local projection Π [k] p w = Π p (w T k ) T k for w H (I k ). Let us note that S p [k] (ξ) T k is the space of Bernstein polynomials of degree p because of the choice of the open knot vector Ξ ξ with multiplicity p at each knot. Theorem 4.. Let 0 < λ /2. Suppose u(r, θ) = N k=0 c kr (λ+k) ψ k (θ) with smooth functions ψ k (θ) solves the Poisson equation. Assume that each node in the open knot vector Ξ ξ for the approximation space S h ξ has the multiplicty p ξ. (i) If F is the NURBS geometrical mapping from the parameter space Ω onto the physical domain Ω, defined by (0) with q = /λ, then u F = N c l η +l/λ Ψ l (ξ) = l=0 N c l η +ql Ψ l (ξ), l=0 for piecewise smooth functions Ψ l (ξ). (ii) If u h V h is an isogeometric finite element solution of u, then we have [ N ] u u h,ω C h p c l ( Ψ l p+, + Ψ l p+2, ) /(p!). (8) l=0 (iii) Assume u H0 (Ω). If h 0 (0, ) is determined so that the projection P h of S onto S h and the complement operator I P h are continuous with respect to a weighted Sobolev norm (Theorem 22.6 of [5]), h min{h 0, /e}, and p = p ξ 2, p η max{2, qn + }, then we have [ N ] u u h,ω C h p+ c l ( Ψ l p+, + Ψ l p+2, ) /(p!), l=0 [ N ] u u h 0,Ω C 0 h p+ c l ( Ψ l p+, + Ψ l p+2, ) /(p!). l=0 Ψ l p+i, := k Ψ l p+i,,ik, where Ψ l is smooth on I k, for each k and k I k = [0, ]. Here p = p ξ is the polynomial degree of N i,p+ (ξ) for an approximation of the angular direction (the polynomial degree p η 2 for an approximation of the radial direction is held fixed), and h = max{ ξ i+ ξ i } is the maximum length of knot spans of the open knot vector Ξ ξ and the constants C, C 0, C are independant of h and p. Proof. (i) An error estimate on each element K = F(Q), Q Q h can be summed together to have a global error estimate on Ω. Thus we have an error estimate on an element K in the following. Without loss of generality, we may assume that λ = /2 and u(r, θ) is a sum of two terms: u(r, θ) = r /2 ψ (θ) + r 3/2 ψ 2 (θ). (9) 5

16 Using the arguments similar to the proof of Lemma 3. we have u(r, θ) F = ηψ (ξ) + η 3 Ψ 2 (ξ), (20) where the continuous functions Ψ (ξ), Ψ 2 (ξ) are smooth unless ξ = /4, 2/4, 3/4. (ii) In what follows, we let p η = p 3 and p ξ = p 2.. Let T = {Q k = [ξ k, ξ k+ ] [0, ] k =,, N ξ } be a partition of ˆΩ, such that {i/4} [0, ], i = 0,, 2, 3, 4, are edges of some elements in T. (η η i ) 2. L p,s (η) = p i=0,i s (η s η i ), where η i = i/ p, i = 0,,, p. For each k =,, N ξ, we let L [k] p,l (ξ) = p (ξ ξi k) i=0,i l (ξl k ξk), where ξk i = ξ k + i(ξ k+ ξ k )/p, i = 0,,, p. i 3. The p-th and p-th Lagrange interpolating polynomials, respectively, are Π η (Φ(η)) = p s=0 Φ(η s )L p,s (η) on [0, ] and p Π ξ (Ψ(ξ)) Ik := Π [k] ξ (Ψ) = Ψ(ξ s [k] )L k p,s(ξ), on I k = [ξ k, ξ k+ ]. s=0 Π ξ is the interpolant defined on C[0, ] such that Π ξ Ik = Π [k] ξ for k =,, N ξ. Suppose Φ(η) is a polynomial of order p and Ψ(ξ) is a piecewise smooth function. Π η Φ = Φ and since Π ξη = Π ξ Π η, we have Then ΦΨ Π ξη ΦΨ = ΦΨ Π η ΦΠ ξ Ψ = Φ[Ψ Π ξ Ψ]. (2) Using this equation, Lemma 3.2, and Lemma 4., we obtain the following inequalities that will be used throughout the proof: [Ψ Π [k] ξ Ψ]2 dξ h 2(p+) { Ψ p+, /(p + )!} 2 [meas(i k )]; I k Q k [ΦΨ Π ξη ΦΨ] 2 dξdη = 0 Φ 2 dη [Ψ Π [k] ξ Ψ]2 dξ I k h 2(p+) { Ψ p+, /(p + )!} 2 Φ 2 0,[0,] [meas(i k)]; ΦΨ Π ξη ΦΨ 0, Φ 0, Ψ Π ξ Ψ 0, C h p+ { Ψ p+, /(p + )!} ; d ΦΨ Π ξη ΦΨ, Φ, Ψ Π ξ Ψ 0, + Φ 0, dξ [Ψ Π ξψ] 0, C 2 h p Φ, ( Ψ p+, + Ψ p+2, ) /p!. Let us note that k meas(i k) =. Hence, the terms, meas(i k ), will be cancelled out after summing the error bounds over Q k. (22) 6

17 Let û(ξ, η) = u(r, θ) F(ξ, η), Π ξη û Qk = ηπ [k] ξ Ψ (ξ) + η 3 Π [k] ξ Ψ 2(ξ), Π xy u K = Π ξη û Q F (x, y). Then, since the interpolating operator Π ξη is linear, (23) (û Π ξη û) Qk = η(ψ Π [k] ξ Ψ ) + η 3 (Ψ 2 Π [k] ξ Ψ 2). (24) First, we estimate an upper bound of u Π xy u 0,K. Using (9), (22) and (24) yields u Π xy u 2 0,K = {(û Π ξη û) Qk } 2 det(j(f)) dξdη Q k [ 6 η(ψ Π [k] ξ Ψ ) + η 3 (Ψ 2 Π [k] 2 ξ 2)] Ψ dξdη Q k 32 { [ ] Ψ (ξ) Π [k] 2 [ ] } 3 ξ Ψ (ξ) + Ψ 2 (ξ) Π [k] 2 ξ Ψ 2(ξ) dξ I k C 02 h 2(p+) ( Ψ 2 p+, + Ψ 2 2 p+, ) /(p + )![meas(ik )]. (25) Second, we estimate an upper bound of u Π xy u,k. Since the geometrical mapping F is unchanged at all levels of refinement (the h-p-k refinement, see page 45 of [7]), we do not lose a generality even though we work with a fixed geometrical mapping. Using the geometrical mapping defined by (6) and the property (7) of F, we have [ ] q q 2 = det(j(f)) [J(F) ] T [J(F) ] q 2 q 22 { (X ) 4 2 ( ) } Y 2 {( ) ( ) Y Y ( ) ( ) X X } +, 2 + η w w w w w w = = w 2 2 X Y XY {( ) ( ) Y Y 2 + w w ( X w 2w 2 η X Y XY, 0 0, { [(Y ηw 2 ) ] 2 2 X Y XY + w ) ( ) X } { [(Y, η w w [( ) X ] 2 } w. ) ] 2 + [( ) X ] 2 } w From (3) and (5), X (ξ)w(ξ) X(ξ)w (ξ) and Y (ξ)w(ξ) Y (ξ)w (ξ) are piecewise cubic polynomials and (2 + 2)/4 w(ξ). Moreover, the common factor of denominators is bounded away from zero: from (8), we have 4.8 X (ξ)y (ξ) X(ξ)Y (ξ). Thus we have max{ ηq, q 22 } C. 7

18 Let (u Π xy u) F = (u Πxy u), xy = [ x, y ]T, and ξη = [ ξ, η ]T. Then using the inequalities in (4), we have u Π xy u 2,K = [ xy (u Π xy u)] T [ xy (u Π xy u)] dxdy K [ T [ ] = ξη(u Πxy u)] det(j(f)) [J(F) ] T [J(F) ] ξη(u Πxy u) dξdη Q ( [ ] C 2 [ ξη η(ψ Π [k] ξ Ψ ) + η 3 (Ψ 2 Π [k] T ξ 2))] Ψ q q 2 Q q 2 q 22 ( [ ξη η(ψ Π [k] ξ Ψ ) + η 3 (Ψ 2 Π [k] dξdη ξ 2))] Ψ [ { d C 3 η 2 q Q dξ (Ψ Π [k] ξ Ψ ) + η 2 d } 2 dξ (Ψ 2 Π [k] ξ Ψ 2) { } + q 22 (Ψ Π [k] ξ Ψ ) + 3η 2 (Ψ 2 Π [k] 2 ] ξ Ψ 2) dξdη C 4 B 2 [meas(i k )]h 2p, (26) where B = ( Ψ p+, + Ψ p+2, + Ψ 2 p+, + Ψ 2 p+2, ) /(p!). Adding (25) and (26), we have u Π xy u,k C 5 h p meas(i k )B. (27) Since Π xy û K = ηπ [k] ξ Ψ (ξ) + η 3 Π [k] ξ Ψ 2(ξ) S h Qk and V h = S h F, using the well known argument (Céa s Lemma, [5]) with (27), we have Summing (28) over all K Q F, we have u u h,k C 6 h p meas(i k )B. (28) u u h,ω C 7 h p B, (29) where the constant C 7 is independent of h and p. (iii) For a maximum norm estimate, we use those arguments in Section 22 of Ciarlet[5]: let P h be the projection of H 0 (ˆΩ) onto S h such that ˆB((I P h )û, ˆv h ) = 0, for all ˆv h S h, (30) where I is an identity operator on S. By Theorem 22.6 of [5], there is h 0 (0, ) such that for all h min{h 0, /e}, ln h /2 P h û 0, + h P h û, C 3 ( û 0, + h ln h û, ), ln h /2 Iû 0, + h Iû, ln h 0 /2 ( û 0, + h ln h û, ), (3) for all û H 0 (ˆΩ) W, (ˆΩ). Here W, (ˆΩ) is the set of functions v such that v, <. By a similar manner to the proof of Theorem 22.7 of [5], we apply (3) to û û h = û P h û = (I P h )û = (I P h )(û Π ξη û) 8

19 to obtain the following ln h /2 û û h 0, + h û û h, (C 3 + ln h /2 ) ( û Π ξη û 0, + h ln h û Π ξη û, ). (32) Moreover, by (22.8) of [5], if, for each v S h, v Qk (ξ, η) is a polynomial of order 2, one can drop ln h terms from the above inequality (32). Thus, using û Π ξη û = η(ψ (ξ) Π ξ Ψ (ξ)) + η 3 (Ψ 2 (ξ) Π ξ Ψ 2 (ξ)) and those estimates in (22) yields û û h 0,,Ω C 4 ( û Π ξη û 0, + h û Π ξη û, ) C 5 h p+ ( Ψ p+, + Ψ p+2, + Ψ 2 p+, + Ψ 2 p+2, ) /(p!). (33) The L 2 -norm estimate is followed by the maximum norm estimate: since meas(ˆω) =, we have u u h 0,Ω C 03 û û h 0,,Ω C 0 h p+. (34) Under the same assumptions of Theorem 4. and with the same notations of the proof, using results in Schwab ([25]), we prove the following error estimates: Let ϕ : Ω = [, ] I k = [ξ k, ξ k+ ] [0, ] be the linear mapping defined by ξ = ϕ(t) = (h/2)(t + ) + ξ k, where h = ξ k+ ξ k. Let Ψ(t) = Ψ(ξ) ϕ(t). Let 0 j m be integers. Then V m j ( Ω) is the space of all w L 2 ( Ω) for which m w 2 Vj m( Ω) = ( t 2 ) i w (i) (t) 2 dt <. i=j Ω Suppose Ψ(t) H ( Ω) V m 0 ( Ω) for some m, then by Theorem 3.4 and Corollary 3.5 of Schwab ([25]), there exist Π p Ψ S p ( Ω) (which denotes the space of polynomials of degree p) such that Π p Ψ(±) = Ψ(±) and Ψ Π p C Ψ 0, Ω 2 (m)p m Ψ V m s ( Ω), Ψ [Π p Ψ] 0, Ω C (m)p m Ψ V m s ( Ω), where 0 s min(p, m). We assume m p. Then scaling back to I k yields Π p Ψ(±) = Ψ(±), (35) Ψ [Π p Ψ] 0,Ik C (m)( h 2 )m p m Ψ V m m (I k ), (36) Ψ Π p Ψ 0,Ik C 2 (m)( h 2 )m+ p m Ψ V m m (I k ). (37) Let Π p Ψ ϕ = Π p Ψ Ik = Π [k] p Ψ, then Π p Ψ S h ξ. If u(r, θ) F = ηψ (ξ) + η 3 Ψ 2 (ξ), we have ηπ p Ψ (ξ) + η 3 Π p Ψ 2 (ξ) S h = span{n(ξ)η j N(ξ) Sξ h, j = 0,, 3}. 9

20 Now using (35)-(37), we have (ηψ + η 3 Ψ 2 ) (ηπ p Ψ + η 3 Π p Ψ 2 ) 2 0,Q k = η(ψ Π p Ψ ) + η 3 (Ψ 2 ΠΨ 2 ) 2 0,Q k (Ψ Π p Ψ ) 2 0,I k + (Ψ 2 ΠΨ 2 ) 2 0,I k C (m) 2 ( h 2 )2(m+) p 2m ( Ψ 2 V m m (I k) + Ψ 2 2 V m m (I k) ); (38) and from (26) [ Q k Q k ξη C 3 T [ (u Πxy u)] det(j(f)) [J(F) ] T [J(F) ] [ ξη ( η(ψ Π [k] p Ψ ) + η 3 (Ψ 2 Π [k] ( [ ξη Q k [ η 2 q + q 22 η(ψ Π [k] p Ψ ) + η 3 (Ψ 2 Π [k] ξη [ T q q p Ψ 2 ))] 2 q 2 q 22 p Ψ 2 ))] dξdη { d dξ (Ψ Π [k] p Ψ ) + η 2 d dξ (Ψ 2 Π [k] p Ψ 2 ) { (Ψ Π [k] p Ψ ) + 3η 2 (Ψ 2 Π [k] } 2 ] p Ψ 2 ) dξdη ] C 5 (m)( h [ 2 )2m p 2m Ψ 2 Vm (I k) + Ψ 2 2 Vm (I k) + C 6 (m)( h [ ] 2 )2(m+) p 2m Ψ 2 Vm (I k) + Ψ 2 2 Vm (I. k) } 2 ] (u Πxy u) dξdη ] (39) By applying the inequalities (38) and (39) and Céa s Lemma([5]) on each Q k and then summing over all K k = F(Q k ), we have the following: Theorem 4.2. Under the same assumption of the previous theorem, if m p, we have [ ] N u u h,ω C(m) hm 2 m p m d dξ Ψ l(ξ) V m (I k ). (40) Remark 4.. From Lemma 3.2, for each l, Ψ l (ξ) V m m (I k ) in (40), is sharply increased as m is increased. Thus, in Theorem 4.2, if m( p) is large, then h should be small for an optimal result. However, the error estimate (40) shows that the error goes to zero as p is increased even if the regularity m of Ψ l is fixed to be low. Thus, the estimate (40) has an advantage over the estimate (8). 5 Numerical tests In order to show that the proposed mapping techniques are effective in dealing with singularity problems, our mapping method is applied to the elliptic boundary value problems with singularity of type k l=0 r λ ψ(θ), where 0 < λ <, and ψ is a smooth function. 20

21 Table 2: The degree of polynomials and the open knot vectors for the geometrical mapping F. variables polynomial degrees open knot vectors ξ p ξ = 2 Ξ ξ = {0, 0, 0, 4, 4, 2, 2, 3 4, 3 4,,, } η p η = 2 Ξ η = {0, 0, 0,,, } For example, the crack singularity and the jump-boundary data singularity have λ = /2 and the interface problems and the elasticity problems with exotic boundary conditions could have λ close to 0. Throughout this section, we measure the error (u u h ) of the computed solutions obtained by the proposed mapping method in the following norms: u u h,rel (%) = u uh 00, u u h u L 2,rel(%) = u uh L 2 u L 2 [ ] u u u h 2 eng u h 2 2 eng eng,rel (%) = 00. u 2 eng 00, 5. The crack singularity of type r /2 In this subsection,. the parameter space is Ω = [0, ] [0, ], the physical space is Ω = {(r, θ) : r, 0 < θ < 2π} with crack along the positive x-axis, 2. the geometrical mapping F is the NURBS surface corresponding to the knot vectors in Table 2 and the 27 control points and the 27 weights in Table. We emphasize that our geometrical mapping F does not change as the degrees of B-splines are elevated or the knots are inserted. We consider two Poisson s equations on Ω: One with homogeneous Dirichlet BC (Example 5.) and another with non-homogeneous Dirichlet BC (Example 5.2). Example 5.. Let u (r, θ) = [ r( r) sin ( ) θ + sin 2 ( )] 3θ 2 and f = u. Then u is the analytic solution of the Poisson equation with homogeneous boundary condition: u = f in Ω with u = 0 on Γ = Ω, (4) that has the crack singularity at (0, 0). By Lemma 3., the pull back of u (x, y) onto the parameter space is of the form û = (u F)(ξ, η) = ηψ (θ) + η 3 Ψ 2 (θ) 2

22 and hence we need to have order elevations up to degree p η = 3 (or higher) in η-direction by choosing the open knot vector Ξ η = {0, 0, 0, 0,,,, }. Note that the knot vector for the mapping F is unaltered (that is, those in Table 2). To improve the isogeometric analysis of (4) in the angular direction we elevate the degrees of polynomials with the mesh fixed (the p-version): [0, ] = [0, /4] [/4, /2] [/2, 3/4] [3/4, ]. For an isogeometric finite element solution of (4), the relative error (%) in the maximum norm as well as the L 2 -norm with respect to a p-refinement are listed in Table 3. The computed strain energy and the relative error (%) in the energy norm with respect to a p-refinement are listed in Table 4. Moreover, the relative error (%) in the maximum norm and the L 2 -norm listed in Table 3 are depicted in Fig. 4. The relative error (%) in the energy norm of Table 4 are depicted in Fig. 5(a). We use the tolerance and double precision for our calculation. In Table 4, the strain energy of the computed solution reaches the true strain energy when p = 3. The computed energies corresponding to p = 4, 5 and 6 in Table 4 are spurious because these are larger than the the true energy outside the tolerance. However, the computed energies shown in Table 3 corresponding to p = 3, 4, 5, 6 agree with the true energy within the tolerance. In this section, all of the figures are plotted in semi-log scales except four figures that are plotted in log-log scale (Figs. 5(b), 6, 2 and 3). The energy norm is equivalent to the - norm for u. The degrees of freedom (DOF) is the number of B-spline basis functions that are employed for the finite element analysis. In order to numerically justify the error analysis of the previous section, we plot the relative error of the computed solutions obtained by the p-refinements (Figs. 4 and 5(a)) as well as the h-refinements (Figs. 5(b) and 6). By Theorem 4., we have log u u h (p + ) log h + log C, log u u h 0 (p + ) log h + log C 0, log u u h p log h + log C, (42) where we choose h = /4 for the p-version and p(= p ξ ) = 2 or 3 for the h-version, from which we observe the following: The slopes of convergence profiles in the maximum norm, the L 2 -norm, and the -norm for the relative error against the p-degrees (the p-refinement) are about log(/4) = 0.602, as shown in Figs. 4 and 5(a). The slopes of convergence profiles of the relative error against the mesh size h in the angular direction (the h-refinement) are p in the -norm; (p + ) in the L 2 -norm and the maximum norm, as shown in Figs. 5(b) and 6. The corresponding numerical data of the h-refinement are listed in Tables 7, 8, and 9 in appendix. 22

23 Table 3: The relative error (%) in the maximum norm and the L 2 -norm of the computed solutions of the Poisson equation (4) with homogeneous Dirichlet boundary condition. The slopes are computed by using two consecutive nodes on the line in Fig. 4. (p ξ, p η ) dof u u h,rel (%) slope u u h L 2,rel(%) slope (2, 3) E E + 0 (3, 3) E E (4, 3) E E (5, 3) E E (6, 3) E E (7, 3) E E (8, 3) E E (9, 3) 70.59E E (0, 3) E E (, 3) E E (2, 3) E E (3, 3) E E (4, 3) E E (5, 3) 8.263E E (6, 3) E E From Theorem 4., the slope of the convergence profile in the -norm can be estimated as follows: log u u h with (p + ) log u u h with p [(p + ) log h + log C ] [p log h + log C ] = log h = 0.602, (43) when the relative error in -norm is plotted against the p degrees. However, the computed slopes of the convergence profile are slightly better than the estimated slopes in Tables 3 and 4. Together with the h-convergence in Figs. 5(b) and 6 and Tables 7-9, we can see that numerical tests support our theory on the error analysis. The performance of the k-refinement combined with mapping technique is shown in Table 0 in appendix. We do not observe a big advantage of the k-refinement over the p-refinement in the presence of singularity. Let us note that in the support of the B-spline functions used in our p-refinements, û is smooth, however, in the support of the B-spline function by the k-refinements, û is not smooth. In general, if the η direction is also k-refined, the mapping technique are not able to generate η, η 3. Hence, the corresponding approximation space do not contain the singular functions r /2 ψ (θ), r 3/2 ψ 2 (θ) which resemble the true solution u. However, the B-spline functions obtained by the p-refinement generate the complete polynomials. Thus, the corresponding approximation space on Ω contains the singular functions that resemble the true solution. 23

24 Relative error in maximum norm (%) Relative error in L2 norm (%) Polynomial degree (a) Rel error in max-norm: p-version, homogenuous BC Polynomial degree (b) Rel error in L 2 -norm: p-version, homogenous BC Figure 4: Relative error of computed solutions of the Poisson equation (4) with homogeneous Dirichlet boundary condition: (a) The relative error in the maximum norm (%), (b) The relative error in the L 2 norm (%). Relative error in energy norm (%) Relative error in energy norm (%) Quadratic Cubic Polynomial degree (a) Rel error in -norm: p-version, homogenuous BC 0 3 h size 0 (b) Rel error in -norm: h-version, homogenuous BC Figure 5: Relative error in energy norm (%) of the Poisson equation (4) with homogeneous Dirichlet boundary condition: (a) Relative error in -norm versus polynomial degrees p. The slope of the line is about log(/4) (b) Relative error versus the mesh size h, hence the slopes of the lines are 2 and 3, respectively, according as p = 2 and p = 3. 24

25 Table 4: The computed strain energy and the relative error in energy norm (%) with respect to a p-refinement for the Poisson equation (4) with homogeneous Dirichlet boundary condition. The computed energy becomes the true energy when p = 3. The energies corresponding to p = 4, 5, 6 are spurious because these energies are larger than the true energy. (p ξ, p η ) dof u h 2 eng u u h eng,rel(%) slope (2, 3) E E + 0 (3, 3) E E (4, 3) E E (5, 3) E E (6, 3) E E (7, 3) E E (8, 3) E E (9, 3) E E (0, 3) E E (, 3) E E (2, 3) E E (3, 3) E E (4, 3) E E 06 (5, 3) E E 06 (6, 3) E E E

26 Relative error in maximum norm (%) Quadratic Cubic 3 4 Relative error in L2 norm (%) Quadratic Cubic 3 4 h size 0 (a) Rel error in max-norm: h-version, homogenuous BC h size 0 (b) Rel error in L 2 -norm: h-version, homogenuous BC Figure 6: (a) The relative errors in the maximum norm (%) and (b) The relative errors in the L 2 -norm (%) of computed solutions of the Poisson equation (4) with homogeneous Dirichlet boundary condition. The plot of the relative errors versus the mesh sizes h in log-log scale make straight lines whose slopes are 3 and 4, respectively, according as p = 2 and p = 3. Next, we test the proposed mapping method to the Poisson equation with non-homogeneous boundary condition. Example 5.2. Let u 2 (r, θ) = r( + e r ) [ sin ( ) θ + cos 2 ( ) θ + sin 2 ( ) 3θ + cos 2 ( )] 3θ 2 and f = u 2. Then u 2 is the analytic solution of the Poisson equation with non-homogeneous boundary condition: u = f in Ω with u = u 2 on Γ = Ω, (44) that has a crack singularity at (0, 0). This example is different from Example 5. in the following aspects:. The true solution contains the term e r and e r F can not be a polynomial in the parameter space, hence it would be better to have refinement in the radial direction as well as in the angular direction. Thus, the error analysis of Theorem 4. does not fit to this case. 2. The problem has non-homogenous Dirichlet BC along the entire boundary and B-spline basis functions do not have the Kronecker delta property. Hence, we use the least squares method to impose the non-homogenous Dirichlet BC. We expect additional errors in imposing BC. Moreover, the energy norm is not equivalent to the -norm for u 2. 26

27 0 0 Relative error (%) Maximum norm L2 norm Energy norm Polynomial degree Figure 7: The relative error (%) in the maximum norm, the L 2 -norm, and the energy norm of the computed solutions of the Poisson equation (44) with non-homogeneous Dirichlet boundary condition. Therefore, log(/4) is not an expected slope of the convergence profile for Example 5.2. Theorem 4. is concerned with the p-refinement in the ξ-direction only for the error bound because there are no errors in the η-direction for isogeometric analysis. However, for the numerical solutions of Example 5.2, we use the p-refinement in the ξ-direction as well as in the η-direction. Therefore, the error bound in the energy norm for Example 5.2 should be stated in terms of p ξ, p η and h. Nevertheless, we observe that the convergence profile in the maximum norm, the L 2 -norm and the energy norm depicted in Fig. 7 are almost the same as those of Example 5.. Numerical data corresponding to Fig. 7 are in Tables and 2 in appendix. Next we test our mapping methods to other prominent singularity problems. However, the error analysis of Theorem 4. is not applicable to these cases. Further investigation of NURBS mapping for error analysis of the general cases in which the p-refinement in the radial direction as well as in the angular direction is required. 5.2 The Motz problem The Motz problem is a well known benchmarking problem containing a point singularity ([, 4, 7] and references within). It contains a jump boundary data singularity of type O(r /2 ) at the origin (0, 0). 27

28 (,) u n = 0 Γ 3 (,) u n = 0 Γ 4 Γ 2 u = 500 (,0) Γ 5 Γ (0,0) (,0) u u = 0 n = 0 (a) Domain & jump boundary data at (0, 0) (b) control points Figure 8: (a) The domain of the Motz problem and boundary conditions. (b) The control points for NURBS geometrical mapping to deal with the Motz problem. The eight physical elements are formed by solid curves and lines, and 8 control points are repeated at the origin (0, 0). Example 5.3. (Motz problem) Let Ω = [, ] [0, ]. Consider the following Laplace s equation with mixed boundary conditions: u = 0 in Ω, u = 500 on Γ 2, u = 0 on Γ 5, u n = 0 on Γ Γ 3 Γ 4, where Γ = [0, ] {0}, Γ 2 = {} [0, ], Γ 3 = [, ] {}, Γ 4 = { } [0, ], and Γ 5 = [, 0] {0}, as shown in Fig. 8. The true solution of the Motz problem can be expressed asymptotically as follow: u 3 (r, θ) = A k r (/2+k) cos ((/2 + k)θ). (45) k=0 Oh et al.([7]) introduced a benchmarking numerical solution of this problem by accurately estimating the first 50 coefficients of the solution (45). We use this computed solution (the partial sum of the first 50 terms of (45)) as the true solution of the Motz problem for the errors of the computed solutions. The control points and the mesh on the physical domain are illustrated in Fig. 8(b) in which to generate the singularity of the form r 2 +k ψ(θ), 8 control points are located at the origin (0, 0). The relative error (%) in the maximum norm, the L 2 -norm and the energy norm are shown in Table 5 and are depicted in Fig. 9. We observe that the proposed mapping method yields as accurate numerical solutions as those in [4, 7] at lower DOF. The control points and the corresponding weights to construct the geometrical mapping to deal with the Motz problem are listed in Table 3 in appendix. 28

29 Table 5: The relative error(%) in the maximum norm, the L 2 -norm and the energy norm of the computed solutions (with respect to a p-refinement) of the Motz problem are listed. (p ξ, p η ) dof u 3 u h 3,rel(%) u 3 u h 3 L 2,rel(%) u 3 u h 3 eng,rel(%) (2, 2) E E E + 00 (3, 3) E E E 0 (4, 4) E E E 02 (5, 5) 24.6E E E 02 (6, 6) E 04.05E E 03 (7, 7) E E E 04 (8, 8) E E E 04 (9, 9) E E E 05 (0, 0) E E E 05 (, ) E E 08.62E 05 (2, 2) E E E 05 (3, 3) E E E 05 (4, 4) E E E 05 (5, 5) E E E 05 (6, 6) E E E Relative errors (%) Maximum Norm L2 Norm Energy Norm Polynomial degree Figure 9: The relative error (%) in the maximum norm, the L 2 -norm, and the energy norm of computed solutions of the Motz problem are depicted. 29