The Reproducing Singularity Particle Shape Functions for Problems Containing Singularities

Transcription

1 The Reproducing Singularity Particle Shape Functions for Problems Containing Singularities by Hae-Soo Oh Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC Jae Woo Jeong Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC June G. Kim Department of Mathematics, Kangwon National University, Chunchon, 2-7, Korea March 3, 27 Abstract In this paper, we construct particle shape functions that reproduce singular functions as well as polynomial functions. We also construct piecewise polynomial convolution partition of unity functions by taing the convolution of the scaled conical window function with the characteristic functions of quadrangular patches (we provide the computer code for this construction). We demonstrate that the reproducing singular particle shape functions yield highly accurate numerical solutions for the singularity problems with crac singularity or a jump boundary data singularity. Keywords: reproducing polynomial particle (RPP) shape functions; reproducing singularity particle (RSP) shape functions; patch-wise uniformly spaced particles; interpolation error estimate; the convolution partition of unity function. Corresponding author. Tel.: ; fax: ; hso@uncc.edu supported in part by funds provided by the University of North Carolina at Charlotte supported in part by the Research Grant of the Kangwon National University Visiting Professor of the University of North Carolina at Charlotte

2 Introduction The finite element method(fem) has been widely used to solve many important science and engineering problems. However, the conventional FEM has several obstacles, such as mesh refinement and constructing smooth global basis functions. Recently several generalized finite element methods (GEFM) that circumvent the obstacles of the conventional FEM were introduced. Among many GFEMs that use meshes minimally or do not use meshes at all ([],[2],[3]), those methods related to this paper are Element Free Galerin Method (EFGM) ([],[7],[8], [],[3],[4],[5]), h-p Cloud Method([6]), Partition of Unity Finite Element Method (PUFEM)([9],[26],[27]), and Reproducing Kernel Element Method (RKEM) ([],[6],[7]). This paper is a continuation of the previous paper ([24]) that is closely related to those element free methods: RKPM and RKEM. The Reproducing Kernel Particle Method (RKPM) ([7],[8],[],[3],[4],[5]) is a meshfree method that yields highly accurate approximation to smooth functions by using the reproducing ernel particle shape functions that can exactly interpolate the polynomials of a fixed degree. The RKP(reproducing ernel particle) shape functions can be constructed to be smooth up to any desired order by selecting smooth window functions. However, the RKP shape functions constructed by using specific window functions with compact supports are generally fractional functions with complicated denominators that are solutions of the system of algebraic equations. Thus, these RKP shape functions have the following problems: () They do not satisfy the Kronecer delta property; hence, it has difficulties in dealing with Dirichlet boundary conditions. (2) Accuracy is compromised in numerical integrations for these complex fractional shape functions. In order to alleviate these obstacles, in ([23]), we constructed piecewise polynomial C r - Reproducing Polynomial Particle(RPP) shape functions associated with uniformly (or nonuniformly) distributed particles, that satisfy the Kronecer delta property, for any integer r, and any desired reproducing order. Furthermore, in ([24]), by transforming these piecewise polynomial RPP shape functions via bilinear mappings, we construct piecewise polynomial particle shape functions, associated with patch-wise uniformly (or non-uniformly) distributed particles in a polygonal domain, that have the property of polynomial reproducing of a reduced order. However, elliptic boundary value problems on non-convex domains (especially, craced domains) contain singularities. Moreover, it is well nown that the polynomial shape functions poorly approximate the singular functions. Thus, in this paper, to deal with singularity problems in the framewor of meshfree particle methods, using RPP shape functions of order 2N and special mappings, we construct special particle shape functions that reproduce three inds of singularities: ()r.5+l cos(.5 + l), r.5+l sin(.5 + l), l =,, N ; (2)r.5, r.5, r.5 cos(.5θ); (3)r.5 sin(.5θ), 2

3 and the complete polynomials x y 2, + 2 N. The main ideas of the construction for Reproducing Singularity Particle(RSP) shape functions are as follows: () Let ˆQ s be a rectangular reference patch constructed by tensor product of Reproducing Polynomial Particle(RPP) shape functions on [, ] as shown in ([23]). (2) Suppose a polygonal domain Ω contains a point singularity at the origin (, ) of the rectangular coordinate system. We properly select a rectangular subdomain Q s containing the singularity on which the influence of the singularity is dominant. Next, Ω R := Ω \ Q s, where the singularity effects are tolerable, is divided into quadrangular patches Q j, j = 2,, N Q, as shown in ([23]). (3) Let ˆφ ij, ij Λ, be RPP shape functions on ˆQ s. Then, we construct special mappings T s : ˆQ s Q s such that ˆφ ij Ts, ij Λ, reproduce the singular functions arising in the crac singularity. (4) Through the multiplication of the convolution partition of unity to ˆφ ij Ts, we obtain RSP shape functions on the singular zone Q s with compact supports. The resulting closed form particle shape functions satisfy the Kronecer delta property except at few particles around the singular zone Q s. However, it can be made so that the Kronecer delta property is satisfied at all particle along the boundary of the domain. The paper is organized in the following manner. Section 2 introduces the notations and definitions used in this paper. Section 3 reviews the construction of the (flat-top) convolution partition of unity functions that was given in ([24]). We also present those theorems proved in the previous paper([24]) for the purpose of using them for the construction of reproducing singularity particle shape functions. Section 4 proves three main theorems and tree different mappings that construct the reproducing singular particle shape functions as well as reproducing polynomial particle shape functions. In section 5, by combing the (flat-top) piecewise polynomial partition of unity functions, and the singular particle shape functions, that were constructed in section 3 and section 4, respectively, we construct reproducing singular particle shape functions with compact supports. In section 6, by comparing the interpolation errors in the L 2 -norm and the H -semi norm, we demonstrate the effectiveness of the proposed RSP(Reproducing Singular Particle) shape functions in dealing with the crac singularity as well as the jump boundary data singularity. The concluding remars are stated in section 7. In appendix, we present the diagram of intersection of patches and the support of a scaled window function that are essential in the construction of the convolution partition of unity functions. 2 RKP shape functions and RPP shape functions Throughout this paper, α, β Z d are multi-indices and x = ( x, 2 x,.., d x), x j = ( x j, 2 x j,.., d x j ) denote points in R d. However, if there is no confusion, we also use the conventional notation 3

4 for the points in R d or Z d as x = (x, x 2,, x d ) and α = (α, α 2,, α d ). We also use the following notations: (x x j ) α := ( x x j ) α...( d x d x j ) α d, α x u := α u x α, α := α xα d + α α d. d Let Ω be a domain in R d. For any non-negative integer m, C m (Ω) denotes the space of all functions φ such that φ together with all their derivatives D α φ of orders α m, are continuous on Ω. The support of φ is defined by supp φ = {x Ω : φ(x) }. In the following, a function φ C m (Ω) is said to be a C m - function. We also use the usual Sobolev space denoted by H (Ω). For u H (Ω), the norm and the semi-norm, respectively, are u,ω = α Ω α u 2 dx /2, and u,ω = α = Ω α u 2 dx A weight function(or window function) is a non-negative continuous function with compact support and is denoted by w(x). Two typical window functions are as follows: For x R, (a) Conical: { ( x w(x) = 2 ) l, x, (), x >, which is a C l -function. (b) Gaussian: w(x) = { (e / x 2 ) if x <, if x, which is an infinitely smooth function. In R d, the weight function w(x) can be constructed from a one-dimensional weight function either as w(x) = w( x ) or as w(x) = d i= w(x i), where x = (x,, x d ) and x 2 = x x 2 d. In this paper, we use the latter for a higher dimensional window function. Let Λ be a finite index set and Ω denotes a bounded domain in R d. Let {x j : j Λ} be a set of a finite number of uniformly or non-uniformly spaced points in R d, that are called particles. Definition 2.. Let be a non-negative integer. Then the functions φ j (x) corresponding to the particles x j, j Λ are called the RPP(reproducing polynomial particle) shape functions with the reproducing property of order (or simply, of reproducing order ) if and only if it satisfies the following condition: (x j ) α φ j (x) = x α, for x Ω R d and for α. (3) j Λ /2. (2) 4

5 By applying a similar argument to ([2],[7]), one can easily prove the following: The condition (3) for the RPP shape functions is equivalent to (x x j ) β φ j (x) = δ β, for β and x Rd. (4) j Λ For the construction of RPP shape functions, we do not use any specific window functions ([23]) in one-dimensional case. For higher dimensional RPP shape functions, we use unbounded RPP shape functions multiplied by the convolution partition of unity functions. On the other hand, the construction of RKP shape function is directly related to the window functions as follows: The RKP (Reproducing Kernel Particle) shape function, associated with the particle x j, j Λ, is constructed by φ j (x) = w(x x j ) (x x j ) α b α (x) (5) α where b α (x) are chosen so that (3) is satisfied and w(x) is a window function. This gives rise to a linear system in b α (x), namely m α+β (x)b α (x) = δ β for β, (6) α where δ β is the Kronecer delta, and m α (x) = j Z w(x x j )(x x j ) α. For one-dimensional case, this system can be written as M(x) [b (x), b (x),, b (x)] T = [,,, ] T, where M(x) = w(x x j ) j Λ (x x j ) (x x j ) 2. (x x j ) [, (x x j ),, (x x j ) ]. The coefficient matrix M(x) of the linear system (6) is called the moment matrix. 5

6 OO Q 3????????????? Ωδ. Ä ÄÄ Ä Ä Ω ÄÄ ÄÄ ÄÄ Ä Ä Q3 2 Q 4 Q4 Q2 Q 2 Q Q //. Figure : Diagram of Ω, Ωδ, for δ =. and the patches Q, Q, =, 2, 3, 4. Z Z χ *β E χ Y E 2* β Y.5 Z.6 Z Y Y Z Z χe * β 3 χ Y E 4* β Y.8.6 Z.4 Z.5.2 Y Y Figure 2: Graphs of the convolution PU functions ψδ (x, y), =, 2, 3, 4, on the Ωδ. In the labels of four figures, χe β stands for χq wδ, for =,, 4. 6

7 3 The convolution (flat-top) partition of unity shape function and the construction non-uniformly distributed particles on quadrangular patches In this section, we briefly review the construction of the flat-top partition of unity shape functions and the construction of patch-wise non-uniformly distributed particles. The detailed descriptions and proofs can be found in ([24]). 3. The construction of the convolution partition of unity shape function For brevity, we denote the coordinates of points of R 2 by x = (x, y) or ξ = (ξ, η). Definition 3.. The convolution of functions f(x) and g(x), which is denoted by (f g)(x), is defined by (f g)(x) = f(y)g(x y)dy R 2 The characteristic function of E R 2, denoted by χ E, is defined by { x E χ E (x) = otherwise The (flat-top) convolution partition of unity(pu) functions are constructed as follows:. Suppose Ω is a polygonal domain and Ω δ = {x : dist(x, Ω) δ}, x = (x, y) which is called the δ-framed Ω (the outer rectangle in Fig. bounded by the dotted line). Now Ω δ is partitioned into large bounded quadrangles (may not be rectangles),, =,, n such that Q Q Ω = Q, Ω δ = n = Q Ω, int(q ) int(q l ) =, for l. Here, for =,, n, the subset int(q ) denote the interior of Q. Then we have n χ int(q ) =, a.e. on Ωδ. (7) = Actually, Q is enlarged from Q by δ only along the boundary part Q Ω (Q = Q if Q Ω =. It is elaborated in section 6.2) 7

8 2. Consider the scaled conical window function, defined by { w δ A( ( ξ (ξ, η) = δ )2 ) l ( ( η δ )2 ) l if ξ δ and η δ, otherwise, (8) where A = δ δ ( ( ξ δ δ )2 ) l dξ ( ( η δ δ )2 ) l dη, and l is an integer with 3 l <. In this paper, this scaled conical window function is said to be the δ-window function. 3. Since χ int(q ) wδ = χ Q w δ, we denote the convolution of the characteristic function of int(q ) and wδ simply by ψ δ := χ Q wδ. (9) Then, by taing convolution of both sides of (7) and w δ and using (9), we have n ψ δ =, for all (x, y) Ω, = supp(ψ δ ) = {x R2 : dist(x, Q ) δ}. Therefore, {ψ δ, =,, n} is the partition of unity subordinate to the covering {supp(ψδ ) : =,, n} of Ω. Now one can see that for each and any positive real number δ, the convolution shape function ψ δ is a wide flat-top piecewise polynomial function. Indeed, let B δ = [ δ, δ] [ δ, δ](to be called a δ-box), P (x,y) = [B δ + (x, y)] Q, where B δ + (x, y) is the δ-box whose center is (x, y). Then we have ψ R δ (x, y) = χ Q (ξ, η)w δ (x ξ, y η)dξdη 2 = w δ (x ξ, y η)dξdη. () P (x,y) Let us note that the integral domain P (x,y) is one of a triangle, a quadrangle, a pentagon, and a hexagon, as shown in appendix A. In other words, ψ δ is an integral of the polynomial wδ over a polygon P (x,y) that is bounded by linear functions. Hence it is a piecewise polynomial whose support is the δ-framed Q (that 8

9 is, the set of all points that are within δ-distance from Q ). Specifically, this flat-top bubble function is as follows:, if [B δ + (x, y)] Q ; ψ δ (x, y) =, if [B δ + (x, y)] R 2 \Q ; r(x, y) >, if [B δ + (x, y)] Q is a proper subset of B δ + (x, y). The piecewise polynomial r(x, y) can be obtained in a closed form function; however, it is complicated except when Q is a rectangle. Thus, r(x, y) can be determined numerically by using the Gaussian quadrature that can yield the exact integral. Fortran code for r(x, y) can be found in our previous paper ([24]). In ([22]), we prove the decomposition of suppψ δ into subsets on which the convolution PU function is a polynomial. It is worth noting that the convolution partition of unity shape function ψ δ (x, y) is as smooth as the window function w δ (ξ, η), since α x [χ Q w δ ] = χ Q α ξ [wδ (ξ, η)], where α = (α, α 2 ) denotes a multi index and x = x x2. The graphs of the convolution partition of unity functions ψ δ (x, y), =, 2, 3, 4 for the quadrangles Q, =, 2, 3, 4 in Fig. are shown in Fig. 2. These figures and the graphs of derivatives of ψ δ can also be found in ([24]). Let us note that in Fig. 2, we have ( 4 ) (x, y) =, for all (x, y) Ω, but not for (x, y) Ω δ \ Ω. = ψ δ 3.2 The construction of reproducing polynomial particle(rpp) shape functions associated with patch-wise non-uniformly distributed particles In order to show that bilinear mappings preserve the property of reproducing polynomials, we adopt the following notations.. Suppose ˆb is a fixed positive real number and the piecewise polynomial reference particle shape functions have the polynomial reproducing property of order 2N: 2N j= () ξ α j ˆϕ j (ξ) = ξ α, for α 2N for all ξ [, ˆb]. (2) Throughout this paper, we use the following notations: h = ˆb/2N, ξ j = h j, j =,,, 2N, ˆϕ j (ξ) = the particle shape functions constructed in appendix B(where ξ j are not uniformly spaced) or, the Lagrange interpolating polynomials corresponding to the nodes ξ j. 9

10 Taing tensor product of one dimensional shape functions, the reference patch is ˆQ = [, ˆb] [, ˆb](see, Fig. 3) and the reference particle shape functions have the extended-polynomial reproducing property as follows: (j,j 2 ) Λ ξ j η 2 j ˆφExt 2 j (ξ) Then, we observe the following. ˆφ Ext j 2 (η) = ξ η 2, for, 2 2N, for all (ξ, η) R 2. (3) (a) It is important to note that the polynomial reproducing property holds not only for (ξ, η) ˆQ, but also for (ξ, η) ˆQ (the complement of ˆQ). (b) If ˆϕ Ext j, j =,,, 2N are the piecewise polynomials shown in appendix, those piecewise polynomials which are not zero on [, h](or [h, ˆb]) are extended to (, h)( or [h, )). Since these piecewise polynomial particle shape functions are global polynomials on (, h]( or [h, )) and satisfy the polynomial reproducing property of reproducing order 2N for all ξ [, h], the extended shape functions also satisfy the polynomial reproducing property of reproducing order 2N for all ξ (, ]( [ˆb, ) ). Abusing notations, the shape functions, that are extended to the outside of [, ˆb], are also denoted by ˆϕ j. 2. Let Q be a quadrangular patch whose four vertices are (x i, y i ), i =, 2, 3, 4. Then, a bijective mapping T : ˆQ Q (see, Fig. 3) is defined by (x, y) = T (ξ, η), where x = x ˆb2 (ˆb ξ)(ˆb η) + x 2 ˆb2 (ξ)(ˆb η) + x 3 ˆb2 (ξ)(η) + x 4 ˆb2 (ˆb ξ)(η), y = y ˆb2 (ˆb ξ)(ˆb η) + y 2 ˆb2 (ξ)(ˆb η) + y 3 ˆb2 (ξ)(η) + y 4 ˆb2 (ˆb ξ)(η). Let where φ ij(x, y) = ˆφ ij (T (x, y)), ˆφ ij (ξ, η) = ˆϕ Ext i (ξ) ˆϕ Ext (η). Then the transformed particle shape functions have the polynomial reproducing property with reduced reproducing order N(one half of the original reproducing order), as stated in the following lemma, which was proved in ([24]). Let Λ 2N be the index set {ij : i, j 2N + }. j

11 ˆQ Q T T 2 Q 2 Figure 3: Mapping from the reference patch to physical patches. Lemma 3.. Suppose the reproducing property (3) holds for ( + 2 ) 2N. Then the transformed particle shape functions have the following polynomial reproducing property x β i y β 2 j φ ij(x, y) = x β y β 2, for (β + β 2 ) N, (x, y) R 2. ij Λ 2N 3.3 RPP Particle shape functions with compact support The supports of the extended piecewise polynomial RPP shape functions, φ ij, constructed in the subsection 3.2, are unbounded. Thus, we need to mae these particle shape functions with small compact support by capping φ ij with the convolution partition of unity functions ψδ constructed in section 3.. Let us define the particle shape functions by φ ij (x, y) := [φ ij ψ δ ](x, y). Then the reduced particle shape functions, φ ij (x, y) become a piecewise polynomial RPP shape functions with polynomial reproducing property of order N with supp(φ ij ) supp(ψ δ ), for some. It was proved in [24] that the capped particle shape functions also have the reproducing polynomial property of order N. That is, x i y 2 j φ ij(x, y) = x y 2, for ( + 2 ) N, (x, y) Ω. ij Λ 2N

12 4 Reproducing Singularity Particle(RSP) Shape functions that reproduce polynomials and Singular functions Suppose a polygonal domain Ω is non convex at a point at which the internal angle is π/α, where α is a real number with < α <. Then an elliptic equation on this non convex domain contains a singularity of type r α f(θ). The number α is called the intensity of a singularity. In this section, we construct the particle shape functions that generate the singular functions arising in the crac singularity or in the jump boundary data singularity. 4. Particle shape functions that generate the singular functions r sin(θ/2), r cos(θ/2) In this section, we construct the (2N + ) (2N + ) particle shape functions that generate the singular functions: r α+ cos((α + )θ), r α+ sin((α + )θ), for =,,, N. (4) as well as the complete polynomials of order N (if α = /2): x y 2, for + 2 N. (5) For this end, we consider a conformal mapping from the z(:= x + iy) plane onto the w(:= ξ + iη) plane. Throughout this paper, we denote the polar coordinates of the points in the z-plane and the w-plane by (r, θ) and (ˆr, ˆθ), respectively. And the corresponding rectangular coordinates, respectively, are denoted by (x, y) and (ξ, η). Definition 4.. The conformal mapping T α from the w-plane to the z-plane is defined by so that z = T α (w) = w /α, T α (ξ, η) = (ˆr /α cos((/α)ˆθ), ˆr /α sin((/α)ˆθ)). (6) We assume that α = n/m is an irreducible rational number with n < m. Even for the cases when n >, we use α = /m for the singular patch mapping T α. Then the inverse mapping is T α (x, y) = (r α cos(αθ), r α sin(αθ)), (7) 2

13 Let β = /α, then their Jacobians are [ J(T α ) = βˆr β cos(β )ˆθ sin(β )ˆθ sin(β )ˆθ cos(β )ˆθ [ J(T α ) = (/βˆr β cos(β )ˆθ sin(β )ˆθ ) sin(β )ˆθ cos(β )ˆθ ], (8) ], (9) J(T α ) = β 2ˆr 2(β ). (2) Next, we construct the particles and the polynomial reproducing particle shape functions on a reference patch, ˆQ = [, â] [, ˆb]. Let ξ i = ( â 2N )(i ), i =, 2,, 2N + η j = ( ˆb 2N )(j ), j =, 2,, 2N +. Let f i (ξ) and g j (η) be the Lagrange interpolating polynomials corresponding to the nodes ξ i, and η j, respectively. The Lagrange interpolating polynomials is simple, but it is not the best choice because the supports are whole interval. Thus, for smaller supports of shape functions, it is recommended to use those particle shape functions shown in appendix B for f i and g j. Let us consider the tensor product of f i (ξ) and g j (η) defined by ˆφ ij (ξ, η) = f i (ξ) g j (η) for the reproducing polynomial shape functions corresponding to the particles (ξ i, η j ) with reproducing polynomial property of order 2N. To generate the singular particle shape functions that can deal with the crac singularity,. we choose the conformal patch mapping with α = /2. Then, z = T α (w) = w 2, w = T α (z) = z. 2. we choose a singular zone Q s. For example, Q s = [.5,.5] [,.5] is the rectangle ABCD around the crac tip (, ) as shown in Fig. 5. It is worth noting that our construction for RSP shape functions is not restricted to the crac singularity. For example, the patch mapping z = T α (w) = w 3, α = /3, can generate RSP shape functions to deal with the re-entrant corner singularity, r 2/3 cos(2θ/3). Remar 4.. There are no particular rules to choose the size of the singular zone Q s. One can choose the singular zone so that the pollution effect of the crac singularity can be tolerable on the outside of this zone. 3

14 For, i =, 2,, 2N +, and j =, 2,, 2N +, we denote the particles and the corresponding singular shape functions as follows:. Let T α (ξ i, η j ) = (x ij, y ij ), T α (r ij, θ ij ) = (ξ i, η j ), (ˆr ij, ˆθ ij ) = the polar coordinates of (ξ i, η j ), (r ij, θ ij ) = the polar coordinates of T α (ξ i, η j ). Then { Tα ξi = r : α ij cos(αθ ij ) η j = r α ij sin(αθ ij ) ; T α : { x ij = ˆr /α ij cos(ˆθ ij /α) y ij = ˆr /α ij sin(ˆθ ij /α). (2) 2. The singular shape functions corresponding to the particles (x ij, y ij ) are where Henceforth, we simply write ˆφ Ext ij Lemma 4.. () For ij, l Λ 2N, we have φ Ext ij(x, y) = ( ˆφ ij ˆφ ij (ξ, η) = f Ext i Tα )(x, y), (ξ) gj Ext (η). by ˆφ ij. Let Λ 2N = {l :, l 2N + }. Then we have ˆφ ij (ξ, η l ) = δ i δ jl. (2) For any point (ξ, η) in the w-plane (and hence in Tα (Q s )), the reference global polynomial shape functions, ˆφ ij (ξ, η), (i, j) Λ 2N, have the reproducing polynomial property of order 2N: ˆφ ij (ξ, η) = ξ η 2, for + 2 2N. (22) ξ i η 2 j ij Λ 2N Let us note that the mapped particles in the singular zone are not uniformly distributed (see, Fig. 4). Moreover, the corresponding singular shape functions φ ij (x, y) do not have compact supports. Theorem 4.. Suppose the reproducing polynomial particle (RPP) shape functions (Lagrange interpolants or the piecewise polynomials in appendix B) ˆφ ij (ξ, η), for ij Λ 2N 4

15 corresponding to the particles (ξ i, η j ), ij Λ 2N satisfy the following relation: ξ i η 2 j ij Λ 2N ˆφ ij (ξ, η) = ξ η 2, for + 2 2N. (23) Let Then, we have φ ij(x, y) = ˆφ ij (Tα (r, θ)) = ˆφ ij (r α cos(αθ), r α sin(αθ)). () φ ij satisfy the Kronecer delta property. (2) For α = /2, the particle shape functions φ ij, ij Λ 2N, reproduce the complete polynomial of order N : x y 2, + 2 N. (24) (3) For any α, the particle shape functions φ ij, ij Λ 2N, reproduce the following singular functions: r (2+)α cos((2 + )αθ), r (2+)α sin((2 + )αθ), for =,, (N ). (25) Henceforth, because of the property (3) of Theorem 4., the shape function φ ij = ˆφ ij Tα are said to be the Reproducing Singularity Particle (RSP) shape functions. However, the supports of φ ij are unbounded, and hence they are not yet the required RSP shape functions. As stated in section 3.3, after multiplying the convolution PU function ψs, δ the adjusted shape functions φ ij := ψs δ φ ij with compact supports will be actually called the RSP shape functions. Proof. By the conformal mapping (6), the Eqt. (23) is transformed to [ r α ij cos(αθ ij ) ] [ r α ij sin(αθ ij ) ] 2 ˆφij (r α cos(αθ), r α sin(αθ)) (i,j) Λ 2N = [r α cos(αθ)] [r α sin(αθ)] 2, for + 2 2N. (26) In what follows, we denote the right-hand side of Eqt. (26) by P (, 2 ; r, αθ) = [r α cos(αθ)] [r α sin(αθ)] 2 = r α( + 2 ) [cos(αθ)] [sin(αθ)] 2. [A] Suppose + 2 is an even integer and α = /2, we have the following cases: (A-) If + 2 =, there is only one case (, 2 ) = (, ). P (, 2 ; r, θ/2) =. 5

16 (A-2) If + 2 = 2, there are three case (, 2 ) = (2, ), (, ), (, 2). P (2, ; r, θ/2) = r cos 2 ( θ 2 ) = r + cos θ 2 = (x + r), 2 P (, ; r, θ/2) = r cos θ 2 sin θ 2 = 2 r sin θ = 2 y, P (, 2; r, θ/2) = r sin 2 θ 2 = r cos θ 2 = (r x) 2 (A-4) If + 2 = 4, there are five cases (, 2 ) = (4, ), (3, ), (2, 2), (, 3), (, 4). P (3, ; r, θ/2) = (r /2 cos(θ/2)) 2 (r /2 cos(θ/2))(r /2 sin(θ/2)) = 2 (x + r) 2 y = (xy + yr), 4 P (, 3; r, θ/2) = P (, ; r, θ/2)p (, 2; r, θ/2) = ( xy + yr), 4 P (2, 2; r, θ/2) = P (, ; r, θ/2)p (, ; r, θ/2) = 4 y2, P (4, ; r, θ/2) = r 2 cos 4 θ 2 = r2 4 ( + 2 cos θ + cos2 θ) = 4 (r2 + 2rx + x 2 ), P (, 4; r, θ/2) = r 2 sin 4 θ 2 = 4 (r2 2rx + x 2 ) From these relations, we can generate polynomials of degree 2 as follows: = P (, ; r, θ/2), x = P (2, ; r, θ/2) P (, 2; r, θ/2), y = 2P (, ; r, θ/2), y 2 = 4P (2, 2; r, θ/2) xy = 2P (3, ; r, θ/2) 2P (, 3; r, θ/2), x 2 = [P (2, ; r, θ/2) P (, 2; r, θ/2)] 2 = P (4, ; r, θ/2) 2P (2, 2; r, θ/2) + P (, 4; r, θ/2). (27) 6

17 (A-6) If + 2 = 6, there are the following seven relations: P (4, 2; r, θ/2) = P (2, ; r, θ/2)p (2, 2; r, θ/2) = 8 (xy2 + ry 2 ), P (2, 4; r, θ/2) = P (, 2; r, θ/2)p (2, 2; r, θ/2) = 8 (ry2 xy 2 ), P (5, ; r, θ/2) = P (, ; r, θ/2)p (4, ; r, θ/2) = 8 (2x2 y + y 3 + 2rxy), P (, 5; r, θ/2) = P (, ; r, θ/2)p (, 4; r, θ/2) = 8 (2x2 y + y 3 + 2rxy), P (3, 3; r, θ/2) = P (, ; r, θ/2)p (2, 2; r, θ/2) = 8 (y3 ), P (6, ; r, θ/2) = P (2, ; r, θ/2) 3 = 8 (x + r)3, P (, 6; r, θ/2) = P (2, ; r, θ/2)p (2, 2; r, θ/2) = 8 (r x)3, Using all of above cases, we obtain all of the monomials of degree 3: xy 2 = 4(P (4, 2; r, θ/2) P (2, 4; r, θ/2)), y 3 = 8P (3, 3; r, θ/2), x 3 = P (6, ; r, θ/2) P (, 6; r, θ/2) 3(P (4, 2; r, θ/2) P (2, 4; r, θ/2)), x 2 y = 2(P (5, ; r, θ/2) P (, 5; r, θ/2)) 4P (3, 3; r, θ/2) Inductively, we can combine all cases up to + 2 = 2N, to show that the RSP shape functions ˆφ ij ( α T α ), ij Λ 2N, reproduce the complete polynomials of order N: x y 2, for + 2 N. (28) [B] Next, let us consider the cases when + 2 is an odd integer. Suppose + 2 = 2, =, 2,, N. Using the indetity cos(2 )Θ + i sin(2 )Θ = (e iθ ) 2 = (cos Θ + i sin Θ) 2, we obtain cos(2 )Θ = sin(2 )Θ = ( ) 2 ( ) l cos 2 2l Θ sin 2l Θ, 2l l= ( ) 2 ( ) l cos 2 2l Θ sin 2l Θ. 2l l= 7

18 Thus, for =, 2,, N, we have the following relations: r α(2 ) cos α(2 )θ = = r α(2 ) sin(2 )θ = = ( ) 2 ( ) l r α(2 ) cos 2 2l αθ sin 2l αθ, 2l l= ( ) 2 ( ) l P (2 2l, 2l; r, αθ). 2l l= l= l= Specifically, when + 2 =, we have When + 2 = 3, we have r 3α cos(3αθ) = When + 2 = 5, we have ( ) 2 ( ) l r α(2 ) cos 2 2l θ sin 2l θ 2l ( ) 2 ( ) l r α(2 ) P (2 2l, 2l ; r, αθ). 2l r α cos(αθ) = P (, ; r, αθ), r α sin(αθ) = P (, ; r, αθ). ( ) 3 ( ) l P (3 2l, 2l; r, αθ) 2l l= = P (3, ; r, αθ) 3P (, 2; r, αθ), r 3α sin(3αθ) = 2 ( ) 3 ( ) l P (3 2l, 2l ; r, αθ) 2l r 5α cos(5αθ) = l= = 3P (2, ; r, αθ) P (, 3; r, αθ). 2 ( ) 5 ( ) l P (5 2l, 2l; r, αθ) 2l l= = P (5, ; r, αθ) P (3, 2; r, αθ) + 5P (, 4; r, αθ), r 5α sin(5αθ) = 3 ( ) 5 ( ) l+ P (6 2l, 2l ; r, αθ) 2l l= = 5P (4, ; r, αθ) P (2, 3; r, αθ) + P (, 5; r, αθ). Taing α = /2 in Theorem 4., we have the following 8

19 Corollary 4.. Suppose the reproducing singularity particle (RSP) shape functions on the z- plane are constructed by the conformal mapping, Tα (z) = z /2, as follows: Then, we have φ ij(x, y) = ( ˆφ ij Tα )(x, y) = ˆφ ij (r /2 cos(θ/2), r /2 sin(θ/2)). () The shape functions, φ ij (x, y), ij Λ 2N, have the Kronecer delta property: φ ij(x l, y l ) = δ l ij. (2) The shape functions, φ ij (x, y), ij Λ 2N, have the polynomial reproducing property of order N: x y 2, for + 2 N. (29) (3) The shape functions, φ ij (x, y), ij Λ 2N, reproduce the singular functions: r /2+l cos(/2 + l)θ, r /2+l sin(/2 + l)θ, l =,, (N ). (3) Remar 4.2. The transformed relation (26) reproduces polynomials when the reproducing orders are even numbers (that is, ( + 2 ) is, 2, 4,, 2N), whereas the same relation reproduces singular shape functions when the reproducing orders are add numbers (that is, ( + 2 ) is, 3, 5,, 2N ). Corollary 4.2. Suppose the intensity of singularity is a rational number α = n/m with < n < m. The RSP shape functions constructed via the singular patch mapping defined by β T α (ξ, η) = (ˆr /β cos((/β)ˆθ), ˆr /β sin((/β)ˆθ)), (3) where β = /m, generate the singular functions: r n/m+l cos(n/m + l)θ, r n/m+l sin(n/m + l)θ, l =,, 2,,. (32) Proof. Without loss of generality, we assume that α = 2/3(the re-entrant corner singularity). In order to generate the singular functions: r 2/3+l cos(2/3 + l)θ, r 2/3+l sin(2/3 + l)θ, l =,, 2,, (33) we can consider the following patch mapping: where β = /3. Observing that β T α (ξ, η) = (ˆr /β cos((/β)ˆθ), ˆr /β sin((/β)ˆθ)), (34) 2β = 2/3, 5β = (2/3 + ), 8β = (2/3 + 2),, Theorem 4. implies that the RSP shape functions generate the required singular functions 9

20 Corollary 4.3. Suppose u(x, y) is a linear combination of the singular functions, listed in (3), and the complete polynomials, listed in (29). Then the RSP shape functions φ ij, ij Λ 2N exactly interpolate u(x, y). That is, for all (x, y) R 2, u(x, y) = 2N+ i= 2N+ j= u(x ij, y ij )φ ij(x, y). Proof. It is sufficient to prove this claim when u(x, y) has two terms. For example, suppose From the relation (27), we have ij Λ 2N C u(x, y) = C r /2 sin(θ/2) + C 2 xy. u(x, y) = C P (, ; r, θ/2) + C 2 [2P (3, ; r, θ/2) 2P (, 3; r, θ/2)] = [ ] { r /2 [ ] ij sin(θ ij /2) ˆφij (T α (x, y)) + C 2 2 r /2 3 [ ij cos(θ ij /2) r /2 ij sin(θ ij /2) [ ] 2 r /2 [ ] } ij cos(θ ij /2) r /2 3 ij sin(θ ij /2) ˆφ ij (T α (x, y)) = [ r /2 ij sin(θ ij /2)] ˆφij (T α (x, y)) + C 2 [(x ij y ij + y ij r ij ) + ( x ij y ij + y ij r ij )] ˆφ ij (T α (x, y)) ij Λ 2N C = ij Λ 2N [ ] } {C r /2 ij cos(θ ij /2) + C 2 [x ij y ij ] ˆφij (T α (x, y)) = u(x ij, y ij )φ(x, y). ij Λ 2N ] In the following three subsections, without loss of generality, we use 25 particles (that is, N = 2) in the reference patch ˆQ. 4.2 Particle shape functions that generate the singular functions r, r cos(3θ/2) Consider the following domains D cos = {(r, θ) : < r /2, 2π/3 θ π}; D 2 = {(r, θ) : < r /2, θ π/3}; D 3 = (, log(/2)] [, π 2 ]; ˆQcos = (, / 2] [, ]; ˆQ = [, ] [, ]. Now we define the following bijective mappings: f 2 (r, θ) = (r, θ 2 3 π) : D cos D 2 f 23 (r, θ) = (log(r), 3 2 θ) : D 2 D 3 f 34 (ˆx, ŷ) = (eˆx/2, cos(ŷ)) : D 3 ˆQ (35) cos ϕ(ξ, η) = (2 2ξ, 2η ) : ˆQ cos ˆQ 2

21 Define a singular mapping from D cos onto ˆQ cos by Tcos(r, θ) = ( r, cos( 3 (θ 2π/3))), (36) 2 which is the composition of f 2, f 23, and f 34. Then the patch mapping from a reference patch ˆQ cos onto D cos is T cos (ξ, η) = (ξ 2, 2 3 π cos (η)). (37) Theorem 4.2. Let ˆϕ i (ξ), j 5, be the reference particle shape function defined by (2), ˆφ ij (ξ, η) = ˆϕ i (ξ) ˆϕ j (η) and suppose 5 ξ i η 2 j i,j= Then the particle shapes ˆφ ij (ξ, η) = ξ η 2, for, 2 4, (ξ, η) ˆQ cos. (38) [φ cos] ij ( ˆφ ij ) T cos, i =,, 5, j =,, 5, satisfy the Kronecer delta property and generate the singular functions Proof. It follows from (36) and (38) that 5 T cos (ξ i,j= i η 2 j r /2, r /2 cos(3θ/2), r 3/2. )( ˆφ ij T cos(r, θ) = (r /2 ) (cos 3θ/2) 2, for, 2 4. By selecting proper combinations of and 2, we obtain the the required singular functions. Remar 4.3. The tensor product shape functions, ˆφij = ˆϕ i ˆϕ j, i, j 5, has stronger reproducing polynomial property with serendipity reproducing order than the order restricted by Corollary 4.4. The particles corresponding to the RSP shape functions [φ cos] ij are all in the domain D cos, and hence these RSP shape functions exactly interpolate a linear combination of those singular functions, a r /2 + a 2 r /2 cos((3/2)θ) + a 3 r 3/2, on the domain D cos. 4.3 Particle shape functions that generate the singular functions r, r sin(3θ/2) Let D sin = {(r, θ) : < r /2, π/3 θ π}; D 2 = {(r, θ) : < r /2, 2π/3 θ }; D 3 = (, log(/2)] [, π]; ˆQsin = (, / 2] [, ]. 2

22 Then we have the following bijective mappings: g2(r, θ) = (r, θ π/3) : D sin D 2 g 23 (r, θ) = (log(r), 3 2 θ) : D 2 D 3 g 34 (ˆx, ŷ) = (eˆx/2, cos(ŷ)) : D 3 ˆQ (39) sin ϕ (ξ, η) = (2 2ξ, η) : ˆQ sin ˆQ Define a singular patch mapping from D sin onto ˆQ sin by T sin (r, θ) = ( r, cos((3/2)(θ π/3)), (4) which is the composition of g 2, g 23, and g 34. Then the patch mapping from a reference patch ˆQ sin onto D sin is T sin (ξ, η) = (ξ 2, π/3 + (2/3) cos (η)). (4) By the similar methods to the proof of Theorem 4.3, we have the following: Theorem 4.3. Suppose ˆφ ij, i =,, 5, j =,, 5, respectively, are the reproducing polynomial particle shape functions associated the particles (ξ i, η j ) ˆQ sin such that Then the particle shapes 5 ξ i η 2 j i,j= ˆφ ij (ξ, η) = ξ η 2, for, 2 4. (42) [φ sin] ij ( ˆφ ij ) Tsin, i =,, 5, j =,, 5, satisfy the Kronecer delta property and generate the singular functions r /2, r /2 sin(3θ/2), r 3/2. Corollary 4.5. The particles corresponding to the RSP shape functions [φ sin ] ij are all in the domain D sin, and hence these RSP shape functions exactly interpolate a linear combination of those singular functions, b r /2 + b 2 r /2 sin((3/2)θ) + b 3 r 3/2, on the domain D sin. 4.4 Interpolation error of the Crac singularity associated with RSP shape functions Let us consider the following three linear combinations of those singular functions that are generated in the previous three subsections: U α (r, θ) = r.5 cos(.5θ) + r.5 cos(.5θ) + r.5 sin(.5θ) + r.5 sin(.5θ), U cos (r, θ) = r.5 + r.5 cos(.5θ), (43) U sin (r, θ) = r.5 + r.5 sin(.5θ). 22

23 Then, It follows from Corollary 4.3 through Corollary 4.5 that the RSP shape functions { ˆφ ij (T /2 ) : i, j 5}, { ˆφ ij (Tcos) : i, j 5} and { ˆφ ij (Tsin ) : i, j 5}, exactly interpolate U α (r, θ), U cos (r, θ), and U sin (r, θ), respectively. Let D α = {(r, θ) : r <.5, θ π} and ˆQ α = [, / 2] [, / 2]. Then the conformal : D α ˆQ α with α = /2 is defined by mapping T α Tα (r, θ) = (r.5 cos(.5θ), r.5 cos(.5θ)). For the purpose of computing the derivatives of the interpolation errors, we compute the Jacobian of the mappings,tα, Tcos, Tsin, respectively, as follows: [ ] [ ] J(Tα ξ/ x η/ x ) = = αr α cos(α )θ sin(α )θ, ξ/ y η/ y sin(α )θ cos(α )θ [ J(Tcos) (/2)r =.5 cos θ (3/2) sin( 3 ] 2θ)(sin θ)/r (/2)r.5 sin θ ( 3/2) sin( 3, 2θ)(cos θ)/r [ J(Tsin (/2)r ) =.5 cos θ ( 3/2) cos( 3 ] 2θ)(sin θ)/r (/2)r.5 sin θ (3/2) cos( 3. 2θ)(cos θ)/r Now, the gradient of ˆφ ij (Tcos) is as follows: [ ˆφij (Tcos(r, ] θ))/ x ˆφ ij (Tcos(r, θ))/ y = J(T cos) J(ϕ) [ αi (s)α j (t)/ s α i (s)α j (t)/ t ], (44) where ϕ : ˆQcos ˆQ is defined by (35), and α i (t) is the Lagrange interpolation polynomial associated with the node i in [, ]. The gradients of ˆφij (Tα ) and ˆφ ij (Tsin ) can be computed in a similar manner. Let us note that, as r, the test singular functions (43) approach zero, whereas their derivatives become infinitely large. However, numerical tests show that for each case, the absolute maximum interpolation errors are virtually zero as shown in Table. From Table, we observe the followings:. we compute the maximum interpolation errors by evaluating the interpolation errors at the following 6 points: (r, θ), θ = (π/6), =,,, 6, for each of the ten layers: r =.5,.2,.,.E-,,.E Rerr stands for the sum of the maximum of D x (error) and the maximum of D y (error). 3. Since U α / x at only one point (r, π/2) is very small, the relative error is about four digits larger than all the other numbers even though the absolute error is virtually zero. Thus, in estimating the maximum relative D x (error) of the first case, the relative error at the point (r, π/2) was excluded. 23

24 4. The errors for the second case is lager than the other two cases. The error would be as good as that of the first case if calculating error were restricted on D cos = {(r, θ) : < r 5, π θ 3π/2}. Let us note that the particles corresponding to the RSP shape functions ˆφ ij (T cos(r, θ)) are in the domain D cos that is one third of the upper half dis with radius r =.5 (see, Fig. 4). Remar 4.4. Under stress boundary condition along the crac front, the solution vector u(r, θ) = (u r, u θ ) of the linear elasticity equations around the crac tip can be expressed as follows([25], p29): u r = 4µ ( r 2π )/2 {[(2 ) cos(θ/2) cos(3θ/2)]k I [(2 ) sin(θ/2) 3 sin(3θ/2)]k II } + o(r /2 ) u θ = 4µ ( r 2π )/2 {[ (2 + ) sin(θ/2) + sin(3θ/2)]k I [(2 + ) cos(θ/2) 3 cos(3θ/2)]k II } + o(r /2 ), where = 3 4ν for plane strain, µ = E/(2( + ν)), and K I and K II are the opening mode and the sliding mode stress intensity factors, respectively. We have seen that those singular functions appeared in the singular displacement vector functions can be exactly interpolated by using one of the RSP shape functions { ˆφ ij (T /2 (r, θ)) : i, j 5}, { ˆφ ij (Tcos(r, θ)) : i, j 5} and { ˆφ ij (Tsin (r, θ)) : i, j 5}. Thus, the meshfree method incorporated with these RSP shape functions around the crac tip could yield highly accurate stress analysis. From the RSP shape function constructed in this section, we observe the followings:. Our RSP shape functions are different from adopting the enriched base function r.5 incorporated with a cut-off function in the classical FEM ([5]). Assuming that the boundary is along the line θ = π, the five particles along the ξ-axis in the reference rectangles ˆQ cos, ˆQ sin (the η-axis in the reference rectangle ˆQ α ) are sent to the boundary part Q s Ω of the patch Q s = [.5,.5] [,.5]. Since the RSP shape functions satisfy the Kronecer delta property at these particles, it is easy to handle the Dirichlet boundary condition. However, it is important to note that, if the reference patches are ˆQ α = [, H ] [, H ], ˆQ cos = [, H 2 ] [, ], ˆQ cos = [, H 3 ] [, ], then all three types RSP shape functions concurrently satisfy the Kronecer delta property at the five particles on the negative x-axis (θ = π) only when H = H 2 = H 3 = H. We used H = / 2 for Fig One can use 6 particles on the reference patches ˆQ instead of 25 particles, however, the five particles on the boundary handles the boundary condition better than the four particles does. 24

25 Table : Max Relative errors along the half circle S r = {(r, θ) : θ π} for the radii r =.5,.2,.,.E-,.E-2,.E-3,.E-4,,.E-7. Rerr indicates the sum of x-derivative and y-derivative of errors in the maximum norm. Rerr stands for the relative errors in the maximum norm. Maximum Relative Errors S. type r /2 cos(θ/2), r /2 sin(θ/2) r /2, r /2 cos(3θ/2) r /2, r /2 sin(3θ/2) Layers(r) Rerr- Rerr- Rerr-2 Rerr-2 Rerr-3 Rerr-3.5E+ 8.3E E E E-3 2.2E E-5.2E+ 5.5E-6.E-3 8.8E-4 3.2E E E-5.E+ 8.69E E E E E E-5.E- 2.75E E E-4 6.7E E-5.7E-4.E E E-4.E-3.6E-2.46E-4.9E-4.E E E-4.2E-3.77E-2.3E-4 3.7E-4.E E E E-4.84E-2.24E-4 3.9E-4.E-5 2.8E-4 2.E-4.E-3.33E E E-4.E E E-4 9.9E-4 2.8E E E-3.E E E-4.55E-3.9E-2.28E E-3 3. The particles corresponding to the RSP shape functions constructed in sections 4.(for the case I singularity), 4.2(for the case II singularity), 4.3(for the case III singularity), respectively, are plotted in Fig. 4. If ˆQ α were [,.5] [,.5], the particles ( diamonds in Fig. 4) would have been inside Q = [.5,.5] [,.5]. In that case, the accuracy along the layer r =.5 drops by one digit, whereas the accuracy along the layer r =.E-7 is slightly improved. Moreover, in this case, the Kronecer delta property is not satisfied for the RSP shape functions for the case II and the case III singularities, unless the reference patches for these cases are also adjusted to ˆQ cos = [,.5] [, ], ˆQ cos = [,.5] [, ]. 5 The construction of RSP and RPP global shape functions with compact supports Without loss of generality, we assume that all reference patches, ˆQ, ˆQ α, ˆQ cos, ˆQ sin, are equipped with the standard RPP shape functions of reproducing order 4. In section 3, using the bilinear patch mapping T : ˆQ Q we constructed the RPP shape functions that reproduce the complete polynomial of m (m = 4 if Q is rectangle, m = 2 if Q is a quadrangle): x β y β 2, β + β 2 m, for all (x, y) R 2. 25

26 Particles for Case I Singularity.8.6 Particles for Case II Singularity.6 Y.4.4 Y Paricles for Case III singularity.4 Y Figure 4: Particles corresponding to the RSP shape functions for the Case I singularity (U α ), the Case II singularity(u cos ), and the Case III singularity(u sin ), respectively. 26

27 In section 4, through the singular patch mapping T α : ˆQ α Q α, α = /2, we constructed the RSP shape functions that reproduce the complete polynomials as well as the singular functions: x β y β 2, β + β 2 2, for all (x, y) R 2, r.5+ cos((.5 + )θ), r.5+ sin((.5 + )θ), for =,. Using the singular patch mappings T cos and T sin, we constructed the RSP shape functions that reproduce the singular functions: r.5, r.5, r.5 cos(.5θ), r.5 sin(.5θ). From now on, the singular patch mappings T α, T cos, T sin are denoted by T s and the bilinear patch mapping is denoted by T. Let us note that the supports of these particle shape functions are unbounded; the mapped particles by the bilinear patch mapping T are overlapping along the common edges of patches that do not contain singularities. [A] We reduce these particle shape functions to the functions with compact supports by multiplying the convolution partition of unity functions ψ δ = wδ χ Q constructed in section 3.. For (x, y) Ω and for ij Λ 4, we define Φ ij(x, y) = [ ˆφ (i,j) T (x, y)] [ψδ (x, y)], Φ s l (x, y) = [ ˆφ (,l) Ts (x, y)] [ψs(x, δ y)]. Then supp(φ ij ) and supp(φs l ) are compact subsets {(x, y) : dist((x, y), Q δ}. Here Q is enlarged by δ only along the boundary part Q Ω (see, section 6.2 for details). [B] (Global numbering of particles). Consider the following mapped particles obtained by the patch mappings T (ξ i, η j ), =, 2, 3,, n Q, i, j 5, T s (ξ i, η j ), i, j 5( s stands for the index of the patch mappings for the singularities). [B: Numbering among RPP shape functions] To each of these local particle numbers T (ξ i, η j ) corresponding to the RPP shape functions, we assign one global particle number. If several mapped particles associated with RPP shape functions share one point in common, we assign the same global number to thses particles and these RPP shape function is the sum of the associated RPP shape functions. Suppose, for example, T (ξ, η ), T 2 (ξ 2, η 2 ), T 3 (ξ 3, η 3 ), T 4 (ξ 4, η 4 ), 27

28 are the same point on the common edge of two patches or the common vertex of several patches. Then the global particle number and the global RPP shape functions are determined as follows: Φ I (x, y) = Φ ij (x, y) if one mapped particle corresponds to one point in Ω, Φ I (x, y) = Φ ij (x, y) + Φ i j (x, y), if two mapped particles corresponds to one point in an edge Ω, Φ I (x, y) = Φ i j (x, y) + Φ 2 i 2 j 2 (x, y) + Φ 3 i 3 j 3 (x, y) + Φ 4 i 4 j 4 (x, y), if several mapped particles(e.g. four) correspond to one point at a vertex Ω. [B2: Numbering among RSP shape functions] To each local particle number T s (ξ i, η j ) corresponding to the RSP shape function, we assign a different global particle number even though several local particles fall on the same point in Ω. For example, since T α (r, π/2) = T cos (r, ) = T sin (r, ) = (r 2, π), three different RSP shape functions are corresponded to each of five points, ([/(4 2)] 2, π), =,, 2, 3, 4. Thus, at least three different global particle numbers are assigned to each of these five points. The reason for this assignment is that three different types of RSP shape functions corresponding to the particles T α (ξ i, η j ), T cos (ξ i, η j ), T sin (ξ i, η j ) should be used within the same rectangular patch to deal with the crac singularity. [B3: Numbering among RSP and RPP shape functions whose corresponding particles are the same point] If one local particle corresponds to an RSP shape function as well as an RPP shape function, then this point has two different global particle numbers. Let Λ Ω = Λ R Ω ΛS Ω denotes the set of indices of the globally numbered particles, where ΛR Ω is the index set for the global RPP shape functions and Λ S Ω is the index set for the global RSP shape functions. Then we have the following theorem. Theorem 5... Φ I, I Λ R Ω, are the reproducing polynomial particle shape functions of order at least 2 for all (x, y) Ω that are not in the patches containing singularities. 2. For I Λ R Ω, Φ I has a compact support and satisfy the Kronecer delta property at all boundary particles as well as the inside particles, except for the singular particles that correspond to the RSP shape functions. 3. If all RSP shape functions (of three different types) constructed in section 4 are used, the vector space spanned by Φ I, I Λ S Ω, include the singular functions r.5+l sin(.5 + l)θ, r.5+l cos(.5 + l)θ, l =, ; r.5, r.5, r.5 sin(.5)θ, r.5 sin(.5)θ, multiplied by the convolution PU function, which is around the crac tip. 4. If the window function w δ C l (R 2 ), Φ I C l (Ω) for all I Λ Ω, except at the singularity points. Proof. () If (x, y) is an interior point of Q that is the δ-distance away from Q, then the convolution PU function ψ δ becomes at (x, y). Hence, Φ I, I Λ Ω satisfy the reproducing polynomial (as well as singular) shape function property at (x, y). 28

29 On the other hand, if (x, y) is inside of the 2δ-band along Q (hence < ψ δ (x, y) < ), then this point is also inside the 2δ-band along Q 2, where Q 2 is a patch adjacent to the patch Q. Thus, if dist((x, y), Q ) δ and dist((x, y), Q 2 ) δ, then (2) Suppose x β I yβ 2 I Φ I(x, y) = I Λ Ω ij φ ij(x, y) ψ δ (x, y) + ij Λ 4 x β ij yβ 2 = x β y β 2 x β l yβ 2 l Λ 4 [ ] ψ δ (x, y) + ψ δ 2 (x, y) = x β y β 2. l φ l(x, y) ψ δ 2 (x, y) T (ξ i, η j ) = T 2 (ξ i2, η j 2) = T 3 (ξ i3, η j3 ) = (x I, y I ), then ˆφ il j l T l (x I, y I ) = ˆφ il j l (ξ il, η jl ) =, for l =, 2, 3. Hence, [ 3 ] [ 3 ] Φ I (x I, y I ) = ˆφ il j l T l ψ δ l (x I, y I ) = ψ δ l (x I, y I ) =. (3) and (4) are obvious. l= 6 Interpolation error associated with RSP shape functions For brevity, we consider the case when u(x, y) contains only the type I singularity stated in subsection 4.. The interpolation of u(x, y) associated with the combined RSP shape functions and the RPP shape functions is defined by Iu = N Q [ ] u(x I, y I )Φ I (x, y) = u( q st )[ φ st (x, y)] + u( s q st )[ s φ st (x, y)] I Λ = st st N Q [ ] [ ] = u( q st )( ˆφ st T ) ψδ + u( s q st )( ˆφ st Tα ) ψs δ, = st where s q st = T α (ξ s s, η s t ), α = /2, stand for the particles ( diamonds in Fig. 5) corresponding to the RSP shape functions in the rectangular patch Q s = [.5,.5] [,.5] containing singularity in Fig. 5. A local interpolation of u on the patch Q l is defined by st l= I l u = st u(t l (ξ s, η t ))( ˆφ st T l (x, y)). 29

30 Lemma 6.. Suppose x p L l= [supp ψδ l ]. If the local interpolation error of a function u(x, y) at the point x p is denoted by (I l u u)(x p ), then the global interpolation error of u at x p is the sum of the local errors multiplied by the convolution PU functions as follows: L ψl δ (x p) (I l u u)(x p ). l= Proof. Since L l= ψδ l (x p) =, we have L ψl δ (x p) (I l u u)(x p ) = l= = L L ψl δ (x p) I l u(x p ) [ ψl δ (x p)] u(x p ) l= l= [ L ] ψl δ (x p) I l u(x p ) u(x p ), l= which is the global interpolation error at the point x p. 6. Interpolation error in energy norm The formulas for computation of interpolation error in energy norm can be found in ([24]). We briefly describe parts of the formulas for our numerical example. The squared L 2 (Ω)-norm of the interpolation error is computed by the following mater patch approach. Iu u 2 = = Np i= Np i= Q i ˆQ + Λ i ( st u( i q st )[ ˆφ st Ti ] ψi δ + st Λ i st [ u( i q st )[ ˆφ st ] ψi δ T i st u( q st )[ ˆφ ij T T i ] ψ δ T i) u T i u( q st )[ ˆφ st T 2 J(T i ). ] ψδ u 2 Let x := ( x, y )T, and Jij denotes the (i, j)-component of the inverse of the Jacobian of T i. Then, the squared H -semi norm of the interpolation error is defined by { [ ] 2 [ ] } 2 x (Iu u) 2 = (Iu u) + (Iu u) dxdy, x y Ω 3

31 where each term of this integral can be computed on the reference patch, for example [ Ω x (Iu u)]2 dxdy = Np ( ) u( i q st )[ i= Q i x ˆφ st Ti ] ψi δ + u( q st )[ x ˆφ st T ] ψδ u x st Λi st Np [ ( = u( i q st ) [(J, J2) ξ ˆφst ] (ψi δ T i ) + ˆφ st ( ) x ψδ i ) T i i= ˆQ + Λ i st st ( u( q st ) x [ ˆφ st T ] T i ψ δ T i + [ ˆφ st T T i ] ) x ψδ T i u x T i 2 2 J(T i ). Here Λ i = { : [suppψ δ ] Q i }(the index set of the neighboring patches around Q i ). Moreover, for effective evaluation of these integrals, we observe the following:. x [ ˆφ st T ] T i and y [ ˆφ st T ] T i can be computed by the following chain rules: x [ ˆφ st T ] T i = x [ ˆφ st T ] (T T ) T i ] = [J(T ) ξ ˆφst (T T i ) = [J(T ) (T T i )] [ ξ ˆφst (T T i )]. (45) 2. An explicit form of the inverse function T is not available in general. Thus, (T T i ) is evaluated by Newton s method, that yield the desired inverse coordinates in two or three iterations because T is bilinear mapping. For = s, the inverse of the singular patch mapping T s is available in an explicit form in section Numerical example We explain the procedures of constructing RSP shape functions as well as RPP(reproducing polynomial particle) shape functions in conjunction with Fig. 5. We assume that our test problem contains a jump boundary data singularity at (, ) (see, Fig. 5). () Mappings for patch-wise non uniformly spaced particles. T α : ˆQ s [, / 2] [, / 2] Ω (conformal mapping:t α (w) = w 2, α = /2) T : [, ] [, ] Q, =,, 5 (bilinear mapping), where Q,, Q 5 are rectangles in Fig. 5. 3