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 October 23, 26 Abstract In this paper, we construct particle shape functions that reproduce the singular functions as well as polynomial functions. We also construct the piecewise polynomial wide-flat-top 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 accurate solutions to 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], [],[4],[5],[6]), h-p Cloud Method([5]), Partition of Unity Finite Element Method (PUFEM)([2],[27],[28]), and Reproducing Kernel Element Method (RKEM) ([],[7],[8]). This paper is a continuation of our previous paper ([25]) that is closely related to those element free methods: RKPM and RKEM. The Reproducing Kernel Particle Method (RKPM) ([7],[8],[],[4],[5],[6]) is a mesh free 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 our previous papers([],[24]), we constructed piecewise polynomial C r -Reproducing Polynomial Particle(RPP) shape functions associated with uniformly (or non-uniformly) distributed particles, that satisfy the Kronecer delta property, for any integer r, and any desired reproducing order. Furthermore, in ([25]), 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, we construct special mappings that transform the RPP shape functions with polynomial reproducing property of order 2n into another RPP shape functions with polynomial reproducing property of order n as well as n numbers of singular shape functions that resemble the nown singularities. The main ideas of the construction for Reproducing Singularity Particle(RSP) shape func- 2

3 tions are as follows: () Let ˆQ be a rectangular reference patch constructed by tensor product of Reproducing Polynomial Particle(RPP) shape functions on [, ] as shown in ([],[24]). (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 wea, is divided into quadrangular patches Q j, j = 2,, n, as shown in ([25]). (3) Let T s be a conformal mapping on the complex plane defined by Ts (z) = z α, where α (, ). Then the transformation is a singular function from a modified curved reference rectangle ˆQ onto Q S. That is, T s (ξ, η) = (ˆr /α cos ˆθ/α, ˆr /α sin ˆθ/α), where (ˆr, ˆθ) is the polar coordinates of (ξ, η). Now, the transformed RPP shape function by the singular mapping T s becomes RSP shape functions on Q S. For all other rectangular patches Q j, the mappings T j are bilinear. (4) Capping the transformed particle shape functions via multiplication of the flat-top partition of unity, we obtain RSP shape functions on the singular zone Q S with compact supports as well as RPP shape functions with compact supports on all other patches Q j, j =, 2,, n. 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 the boundary of the domain. We organize this paper in the following manner. In section 2, we explain notations and definitions used in this paper. In section 3, we review the construction of the flat-top partition of unity functions that was given in ([25]). And we also present those theorems shown in the previous paper for the purpose of using the theorems for the construction of reproducing singularity particle shape functions in the following section. In section 4, we prove theorems about constructions of reproduce singular shape functions as well as polynomial shape functions through properly chosen conformal mappings. In section 5, by combing the flat-top piecewise polynomial partition of unity functions, and the singular shape function, 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 energy 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 data singularity. Some concluding remars are stated in section 7. Finally, in appendix, we gave 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. 3

4 2 RKP shape functions and RPP shape functions Throughout this paper, α, β 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 for the points in R d or d as We also use the following notations: x = (x, x 2,, x d ) and α = (α, α 2,, α d ). (x x j ) α := ( x x j ) α...( d x d x j ) α d, α := α + α α d, α x u := α u x α xα 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. 4 /2. (2)

5 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 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) (4) 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 α m α+β (x)b α (x) = δ β for β, (5) where δ β is the Kronecer delta, and m α (x) = j 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 (5) is called the moment matrix. 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. (6) j Λ 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. χ *β E χ E 2* β χe * β 3 χ E 4* β 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 ([25]). 3. The construction of flat-top 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 partition of unity functions are constructed as follows:. Suppose Ω is a polygonal domain and Ω δ = {(x, y) : dist((x, y), Ω) δ} 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) = The relation between the patch Q and the enlarged patch Q is elaborated in section

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). By the symbolic calculations of definite integrals, the polynomial r(x, y) can be obtained in a closed form function; however, it is complicated. 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 ([25]). 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 ([25]). Let us note that ( 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 2m: n ξj α ˆφ j (ξ) = ξ α, for α 2m for all ξ [, ˆb]. j= Throughout this paper, we use the following notations: h = ˆb/n, ξ j = h j, j =,,, n, ˆφ j (ξ) = the particle shape functions constructed in appendix B 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: ξ α j η α 2 ˆφExt Ext j (ξ) ˆφ j 2 (η) = ξ α η α 2, for α, α 2 2m, for all (ξ, η) R 2. (2) (j,j 2 ) Λ j 2 Then, we observe the following. (a) If ˆφ Ext j, j =,,, n are the Lagrange interpolating polynomials, these product of global polynomials satisfy polynomial reproducing property for all (ξ, η) R 2. (b) If ˆφ Ext j, j =,,, n 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 2m for all ξ [, h], the extended shape functions also satisfy the polynomial reproducing property of reproducing order 2m 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 Let (x i, y i ), i =, 2, 3, 4. Then, a bijective mapping T : ˆQ Q (see, Fig. 3) is defined by where (x, y) = T (ξ, η) = (T x (ξ, η), T y (ξ, η)), 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). φ (j,j 2 ) (x, y) = ˆφ (j,j 2 )(T (x, y))(see, Fig. 6), where ˆφ (j,j 2 )(ξ, η) = ˆφ j (ξ) ˆφ j2 (η). Then the transformed particle shape functions have the polynomial reproducing property with reduced reproducing order m, as stated in the following lemma, which was proved in ([25]).

11 ˆQ Q T T 2 Q 2 Figure 3: Mapping from the reference patch to physical patches. Lemma 3.. Suppose the reproducing property (2) holds for (α + α 2 ) 2m. Then the transformed particle shape functions have the following reproducing order x β j y β 2 j 2 φ (j,j 2 ) (x, y) = xβ y β 2, for (β + β 2 ) m, (x, y) R 2. (j,j 2 ) Λ 3.3 RPP Particle shape functions with compact support The supports of the extended piecewise polynomial RPP shape functions, φ (j,j 2 ) constructed in the subsection 3.2, are unbounded. Thus, we need to mae these particle shape functions with small compact support by capping φ (j,j 2 ) with the flat-top convolution partition of unity functions constructed in section 3.. Let us define the particle shape functions by φ j j 2 (x, y) := [φ (j,j 2 ) ψδ ](x, y). Then the reduced particle shape functions, φ j j 2 (x, y) become a piecewise polynomial RPP shape functions with polynomial reproducing property of order m with supp(φ j j 2 ) supp(ψ δ )), for some. It was proved in [25] that the capped particle shape functions also have the reproducing polynomial property of order m. That is, x β j y β 2 j 2 φ (j,j 2 )(x, y) = x β y β 2, for (β + β 2 ) m, (x, y) Ω. (j,j 2 ) Λ

12 4 Reproducing Singularity Particle(RSP) Shape functions that reproduce polynomials and Singular functions Without loss of generality, in this section, we are mainly concerned with construction of particle shape functions that reproduce the following four singular functions: r /2+ cos((/2 + )θ), r /2+ sin((/2 + )θ), for =,. (3) as well as the complete polynomials of order 2: x β y β 2, for β + β 2 2. (4) 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.. In this paper, the conformal mapping T s from the w-plane to the z-plane is defined by so that Ts : z = T s (w) = w β, β = /α ξ = r α cos(αθ), η = r α sin(αθ), ; T s : { x = ˆr β cos(β ˆθ), y = ˆr β sin(β ˆθ). Here α denotes a real number with α (, ), called the intensity of the singularity. Then, in the computations of stiffness matrices, the following formulas are used. ] J(T s ) = [ x ξ x η y ξ y η [ ] = βˆr β cos(β )ˆθ sin(β )ˆθ sin(β )ˆθ cos(β )ˆθ [ ] J(T s ) = (/βˆr β cos(β )ˆθ sin(β )ˆθ ) sin(β )ˆθ cos(β )ˆθ (5) (6) J(T s ) = β 2ˆr 2(β ). (7) Next, we construct the particles and the polynomial reproducing particle shape functions on a reference patch, ˆQ = [, â] [, ˆb]. Let ξ i = ( â 4 )(i ), i =, 2,, 5 η j = ( ˆb 4 )(j ), j =, 2,, 5 2

13 Let f i (ξ) and g j (η) be the Lagrange interpolating polynomials corresponding to the nodes ξ i, i =, 2,, 5 and η j, j =, 2,, 5, respectively. However, the interpolating polynomials 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 4. To generate the singular particle shape functions that can deal with the crac singularity,. we consider the conformal mapping with the mapping size β = 2(that is, the intensity of the singularity is α = /2). Then, z = T s (w) = w 2, w = T s (z) = z. 2. we choose a size of singular zone Ω S. For example, Ω S is the quadrangle ABCD around the crac tip (, ) as shown in Fig. 4. 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 s (w) = w 3/2 can generate RSP shape functions to deal with the re-entrant corner singularity. Remar 4.. There are no particular rules to choose the size of the singular zone. One can choose the singular zone so that the pollution effect of the crac singularity can be tolerable on the outside of this zone. Instead of covering a neighborhood of the crac tip with two quadrangles (Fig. 4), one can choose a singular zone Ω S, as one piece as shown in Fig. 5. For example, as shown in Fig. 5, Ts (Ω S ) contains the reference rectangle ˆQ = [,.4] [.4,.4] and hence ξ i = (.)(i ), i =, 2,, 5 η j =.4 + (.2)(j ), j =, 2,, 5. For, i =, 2,, 5, and j =, 2,, 5, we denote the particles and the corresponding singular shape functions as follows:. Let T s (ξ i, η j ) = (x ij, y ij ), (ˆr ij, ˆθ ij ) = the polar coordinates of (ξ i, η j ), (r ij, θ ij ) = the polar coordinates of T s (ξ i, η j ), Ts (r ij, θ ij ) = (ξ i, η j ). 3

14 Then { Ts ξi = r : ij cos(θ ij /2) η j = r ij sin(θ ij /2) ; T s : { x ij = ˆr ij 2 cos(2ˆθ ij ) y ij = ˆr ij 2 sin(2ˆθ ij ). (8) 2. The singular shape functions, defined on the singular zone, corresponding to the particles (x ij, y ij ) are φ ij(x, y) = ( ˆφ ij Ts )(x, y), where ˆφ ij (ξ, η) = f i (ξ) g j (η). Let Λ 4 = {(, l);, l 5}. Then we have Lemma 4.. () For (i, j), (, l) Λ 4, we have ˆφ ij (ξ, η l ) = δ i δ jl. (2) For any point (ξ, η) in the w-plane (and hence in Ts (Q S )), the reference global polynomial shape functions, ˆφ ij (ξ, η), (i, j) Λ 4, have the reproducing polynomial property of order 4: ˆφ ij (ξ, η) = ξ α η α 2, for α + α 2 4. (9) ξ α i η α 2 j (i,j) Λ 4 In Fig. 4, the relation between the non rectangular reference patch ˆQ S and a neighborhood of the crac tip Q S are as follows: ˆQ S = T s (Q S = a neighborhood of the crac tip) [,.6] [,.6]. Let us note that the mapped particles in the singular zone are not uniformly distributed. Moreover, the corresponding singular shape functions φ ij (x, y) do not have compact supports. Theorem 4.. Suppose the reproducing singularity particle (RSP) shape functions on the z- plane are constructed by the conformal mapping, Ts (z) = z /2, as follows: Then, we have φ ij(x, y) = ( ˆφ ij Ts )(x, y) = ˆφ ij (r /2 cos(θ/2), r /2 sin(θ/2)). () The shape functions, φ ij (x, y), (i, j) Λ 4, have the Kronecer delta property: φ ij(x l, y l ) = δ l ij. 4

15 (2) The shape functions, φ ij (x, y), (i, j) Λ 4, have the polynomial reproducing property of order 2: For (x, y) R 2, x β ij yβ 2 ij φ ij(x, y) = x β y β 2, for (β + β 2 ) 2. (i,j) Λ 4 (3) The shape functions, φ ij (x, y), (i, j) Λ 4, reproduce not only the singular functions: r /2 cos(θ/2), r /2 sin(θ/2), r 3/2 cos(3θ/2), r 3/2 sin(3θ/2) (2) but also the complete polynomials of degree 2:, x, y, x 2, xy, y 2. (2) Hence after, because of the property (3) of Theorem 4., the shape function φ ij = ˆφ ij Ts 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 exactly the RSP shape functions that we require. As stated in section 3.3, after multiplying the flat-top partition of unity 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 T s and (8), the Eqt. (9) is transformed to [ ] r /2 α [ ij cos(θ ij /2) r /2 α2 ij sin(θ ij /2)] ˆφij (r /2 cos(θ/2), r /2 sin(θ/2)) (i,j) Λ 4 = [r /2 cos(θ/2)] α [r /2 sin(θ/2)] α 2, for α + α 2 4. (22) Let us denote the right hand side of the eqt. (22) by [ ] P (α, α 2 ; r, θ) = r /2 α [ cos(θ/2) r sin(θ/2)] /2 α2. Then we have the following relations: ( P (2, ; r, θ) = r 2 P (, 2; r, θ) = r P (, ; r, θ) = 2 y, ) (cos θ + ) ) ( ( cos θ) 2 = (x + r), 2 = (r x), 2 P (3, ; r, θ) = (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) = P (, ; r, θ)p (, 2; r, θ) = ( xy + yr), 4 P (2, 2; r, θ) = P (, ; r, θ)p (, ; r, θ) = 4 y2. 5

16 From these relations, we can generate polynomials of degree 2 as follows: = P (, ; r, θ), x = P (2, ; r, θ) P (, 2; r, θ), y = 2P (, ; r, θ), y 2 = 4P (2, 2; r, θ) xy = 2P (3, ; r, θ) 2P (, 3; r, θ), x 2 = [P (2, ; r, θ) P (, 2; r, θ)] 2 = P (4, ; r, θ) 2P (2, 2; r, θ) + P (, 4; r, θ). (23) Using the relations (23), we obtained the following polynomial reproducing property: (i,j) Λ 4 x β ij yβ 2 ij ˆφ ij (r /2 cos(θ/2), r /2 sin(θ/2)) = x β y β 2, for β + β 2 2. (24) Next, in order to generate the singular functions, we now apply the following triple angle formulas: Since we have [r /2 sin(θ/2)] 3 = 4 sin 3θ = 3 sin θ 4 sin 3 θ, (25) cos 3θ = 4 cos 3 θ 3 cos θ. (26) [ ] r 3/2 (3 sin(θ/2) sin(3θ/2) = ( 3 4 )r3/2 (sin(θ/2) ( 4 )r3/2 sin(3θ/2), and r 3/2 (sin(θ/2) = r[r /2 sin(θ/2)] we obtain the singular function Since we have = [r /2 cos(θ/2)] 2 [r /2 sin(θ/2)] + [r /2 cos(θ/2)] [r /2 sin(θ/2)] 3, r 3/2 sin(3θ/2) = 3P (2, ; r, θ) P (, 3; r, θ). [r /2 cos(θ/2)] 3 = ( 4 )r3/2 [cos(3θ/2) + 3 cos(θ/2)] = ( 4 )r3/2 cos(3θ/2) + ( 3 4 )r3/2 cos(θ/2), and r 3/2 cos(θ/2) = r [r /2 cos(θ/2)] = [r /2 cos(θ/2)] 2 [r /2 cos(θ/2)] + [r /2 sin(θ/2)] 2 [r /2 cos(θ/2)], 6

17 we obtain the singular function r 3/2 cos(3θ/2) = P (3, ; r, θ) 3P (, 2; r, θ). Since r /2 cos(θ/2) = P (, ; r, θ), r /2 sin(θ/2) = P (, ; r, θ), we obtain all of the singular shape functions in (2). Remar 4.2. The transformed relation (22) reproduces polynomials when the reproducing orders are even numbers (that is, (α +α 2 ) is, 2, or, 4), whereas the same relation reproduces singular shape functions when the reproducing orders are add numbers (that is, (α + α 2 ) is, or 3). Corollary 4.. Suppose u(x, y) is a linear combination of the singular functions, listed in (2), and the complete polynomials, listed in (2). Then the RSP shape functions φ ij, (i, j) Λ 4 exactly interpolate u(x, y). That is, for all (x, y) R 2, u(x, y) = 5 5 u(x ij, y ij )φ ij(x, y). i= j= Proof. It is sufficient to prove this claim when u(x, y) has two terms. For example, suppose From the relation (23), we have (i,j) Λ 4 C u(x, y) = C r /2 sin(θ/2) + C 2 xy. u(x, y) = C P (, ; r, θ) + C 2 [2P (3, ; r, θ) 2P (, 3; r, θ)] [ ] { = r /2 [ ] ij sin(θ ij /2) ˆφij (T s (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 s (x, y)) [ r /2 ij sin(θ ij /2)] ˆφij (T s (x, y)) + C 2 [(x ij y ij + y ij r ij ) + ( x ij y ij + y ij r ij )] ˆφ ij (T s (x, y)) (i,j) Λ 4 C (i,j) Λ 4 [ ] } {C r /2 ij cos(θ ij /2) + C 2 [x ij y ij ] ˆφij (T s (x, y)) = u(x ij, y ij )φ(x, y). (i,j) Λ 4 7

18 By applying similar arguments to the proof of above theorem and using the following multiple angle formula: sin(nθ) = 2 sin[(n )θ] cos θ sin[(n 2)θ], cos(nθ) = 2 cos[(n )θ] cos θ cos[(n 2)θ], one can easily show that Theorem 4. can be generalized as follows: Theorem 4.2. Suppose the reproducing polynomial particle (RPP) shape functions (Lagrange interpolants or the piecewise polynomials with compact supports in appendix B) associated with the nodes (ξ i, η j ), (i, j) Λ 2n = {(i, j) : i, j 2n} are denoted by and if Then, we have ˆφ ij (ξ, η), for (i, j) Λ 2n φ ij(x, y) = ˆφ ij (r /2 cos(θ/2), r /2 sin(θ/2)). () φ ij, (i, j) Λ 2n, have the reproducing polynomial property of order n. That is, (i,j) Λ 2n x α ij yα 2 ij ˆφ ij (r /2 cos(θ/2), r /2 sin(θ/2)) = x α y α 2, α n, which is equivalent to (x x ij ) α (y y ij ) α2 ˆφij (r /2 cos(θ/2), r /2 sin(θ/2)) = δ α, α n. (27) (i,j) Λ 2n (2) φ ij satisfy the Kronecer delta property. (3) φ ij, (i, j) Λ 2n, reproduce the following singular functions: r /2+ cos((/2 + )θ), r /2+ sin((/2 + )θ), for =,, (n ). (28) Thus, the arguments in this section can be extended to (2n + ) (2n + ) singular particle shape functions so that they can reproduce the singular functions r /2+l cos(/2 + l)θ, r /2+l sin(/2 + l)θ, l =,, (n ) as well as the complete polynomials of order n x β y β 2, for β + β 2 n. 8

19 5 The construction of RSP and RPP global shape functions with compact supports In section 3.2, through the bilinear patch mapping T : ˆQ Q we constructed reproducing polynomial particle (RPP) shape functions with reproducing order m satisfying x β ij yβ 2 ˆφ ij (i,j) T (x, y) = xβ y β 2, β + β 2 m, for all (x, y) R 2. (i,j) Λ 2m In section 4, through the singular patch mapping T s : ˆQ S Q S, we constructed reproducing singularity particle shape functions satisfying x β ij yβ 2 ˆφ ij (i,j) Ts (x, y) = x β y β 2, β + β 2 m, for all (x, y) R 2, (i,j) Λ 2m and generating the singularity functions: r /2+ cos((/2 + )θ), r /2+ sin((/2 + )θ), for =,, (n ). The highly smooth piecewise polynomial particle shape functions φ ij = ˆφ ij T correspond to the mapped particle T (ξ i, η j ), whereas the singular particle shape functions φ l = ˆφ l Ts correspond to the mapped particle T s (ξ, η l ). Let us note that the supports of these particle shape functions are unbounded; the mapped particles are overlapping along the common edges of patches. [A] We reduce these particle shape functions to the functions with compact supports by multiplying the flat-top convolution partition of unity functions ψ δ = wδ χ Q constructed in section 3.. For (x, y) Ω and for (i, j) Λ 2m, 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 ) = {(x, y) : dist((x, y), Q δ} is compact for all. Here Q is enlarged from the patch Q by δ (see, section 6.2 for details). In Fig. 6, the unbounded particle shape function [ ˆφ (2,2) Ts (x, y)] and the corresponding capped particle shape function Φ s 22 (x, y) on [.5,.5] [, ] are shown when m = 8 and Q s is the quadrangle Q in Fig. 4. Because of the flat-top shape function ψs(x, δ y), the capped RSP shape function is zero outside of Q δ, as one can see from the right side figure of Fig. 6. [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 n, T s (ξ i, η j ),, l n( s stands for the index of the patch containing the singularity). 9

20 To each of these local particle numbers, we assign one global particle number to the mapped particles by assigning the same global number to those local particles on which several (two, three or four) mapped particles, for example, T (ξ, η ), T 2 (ξ 2, η 2 ), T 3 (ξ 3, η 3 ), T 4 (ξ 4, η 4 ), are the same particle on the common edge of two patches or the common vertex of several patches. That is, 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 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 in Ω. Let Λ Ω denotes the set of indices of the globally numbered particles. following theorem. Then we have the Theorem 5... Φ I, I Λ Ω are the reproducing polynomial particle shape functions of order m for all (x, y) Ω. 2. Φ I, I Λ Ω generate the singular functions r /2+l sin(/2 + l)θ, r /2+l cos(/2 + l)θ, l =,,, m. 3. Φ I C l (Ω) for all I Λ Ω, except at the singularity points, if the window function w δ C l (R 2 ). 4. All Φ I has a compact support and satisfy the Kronecer delta property at all boundary particles as well as most inside particles. Moreover, if we chose δ for the scaled window function w δ properly, all Φ I satisfy the Kronecer delta property at every globally numbered particles. 5. Because of the wide-flat-top partition of unity function in the construction of Φ I, I Λ Ω, they are strongly linear independent. Proof. Suppose T (ξ i, η j ) = T 2 (ξ s, η t ) = T (ξ l, η m ) = (x I, y I ), then ˆφ T (x I, y I ) =, for =, 2, 3. Hence, [ 3 ] Φ I (x I, y I ) = ˆφ T l ψ δ l (x I, y I ) = l= [ 3 ] ψ δ l (x I, y I ) =. l= 2

21 If (x, y) is inside a patch Q and dist((x, y), Q ) > δ, then ψ δ (x, y) =. Hence, Φ I, I Λ Ω satisfy the reproducing polynomial (as well as singular) shape function property at (x, y). On the other hand, if dist((x, y), Q ) δ and dist((x, y), Q 2 ) δ, then x β I yβ 2 I Φ I(x, y) = I Λ Ω x β ij yβ 2 ij φ ij(x, y) ψ δ (x, y) + (i,j) Λ 2m = x β y β 2 [ ] ψ δ (x, y) + ψ δ 2 (x, y) = x β y β 2. (,l) Λ 2m x β l yβ 2 l φ l(x, y) ψ δ 2 (x, y) Suppose δ is small enough so that, if dist((x I, y I ), Q ) δ, then the particle (x I, y I ) is on the edges of quadrangular patch Q. Hence, all of RPP (or RSP) shape functions Φ I, I Λ Ω satisfy the Kronecer delta property. In section 6.2, a specific construction of the RSP shape functions Φ I and global numbering rules are more clearly explained in conjunction with Fig. 4 6 Interpolation error associated with reproducing singularity particle shape functions 6. Methods of calculating interpolation error in energy norm The formulas of this section, that are useful for calculation of the patch-wise stiffness matrices as well as computation of interpolation error, can be found in ([25]). The interpolation of u(x, y) associated with these piecewise polynomial RSP shape functions (or RPP shape functions) is defined by [ ] [ Np Np I h u = u( i q st )[ i φ st (x, y)] = i= st i= st u( i Ext q st )( ˆφ st Ti ) ψi δ where N p stands for the number of quadrangular patches. Note that for the particles along the common side of two adjacent patches, corresponds two different RPP shape functions. However, their sum is defined to be the shape function corresponding to this common particle. Thus, the Kronecer delta property is satisfied at these common particles. [A:] The squared L 2 (Ω)-norm of the interpolation error is computed by the following mater ], 2

22 patch approach. I h u 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 [B:] Let x := ( x, y )T, and ξ := ( ξ, η )T. 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 (I h u u) 2 = x (I hu u) + y (I hu u) dxdy, Ω where each term of this integral can be computed on the reference patch as follows: [ Ω x (I hu 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 ), and [ Ω y (I hu u)] 2 dxdy = Np [ u( i q st ) ([(J 2, J 22) ξ ˆφst ] (ψ δi T i ) + ˆφ st ( y ) ψδi ) T i i= ˆQ st + ( u( q st ) Λ i st y [ ˆφ st T ] T i ψ δ T i + [ ˆφ st T T i ] ) y ψδ T i u y T i 2 J(T i ). 22

23 Here, 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 )]. (29) 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 (Ts (z) = z /2 ). 6.2 Numerical Examples In order to demonstrate the effectiveness of reproducing singularity particle (RSP) shape functions in dealing with the two dimensional singularity problems, we compute the interpolation errors of the functions containing two dimensional singularities. For this end, we explain the procedures of constructing RSP shape functions as well as RPP(reproducing polynomial particle) shape functions in conjunction with Fig. 4. We assume that our test problems have either a jump boundary data singularity at (, ) (Fig. 4) or a crac along the negative x-axis (Fig. 4 and Fig. 5). () Mappings for patchwise non uniformly spaced particles. T s : (the w-plane) (the x-plane)(conformal mapping) T 2 : [, ] [, ] Q 2 (bilinear mapping) T 3 : [, ] [, ] Q 3 (bilinear mapping) T 4 : [, ] [, ] Q 4 (bilinear mapping) (2) With δ =., the convolution partition of unity shape functions are constructed by ψ δ = wδ χ Q, =, 2, 3, 4, where Q are quadrangles whose vertices are as follows : Q Q 2 Q 3 Q 4 : (.37, δ), (.37, δ), (.3,.6), (.3,.6), : (.37, δ), ( + δ, δ), ( + δ, + δ), (.3,.6), : (.3,.6), (.3,.6), ( + δ, + δ), ( δ, + δ), : ( δ, δ), (.37, δ), (.3,.6), ( δ, +δ, ), and the patch corresponding to the singular zone in Fig. 4 is 23

24 .8.6 D* Reference Patch and Reference Particles for Singular one C *.4 B*.2 A* Mapped Particles & Quadrangular Patches.4.2 G.8 Q3 F C.6 B Q4.4 Q Q 2.2 D A H E Figure 4: The singular reference Domain ˆQ s (Top) and a neighborhood patch of the crac tip Q s = ABCD (Bottom). The conformal mapping T s (ˆr, ˆθ) = (ˆr 2 cos 2ˆθ, ˆr 2 sin 2ˆθ) maps the curved pentagon A B C D O to the quadrangle ABCD. This mapping sends some parts of the reference particles to outside of the neighborhood quadrangle Q s of the crac singularity. 24

25 . Singular Refenence Domain. Mapped Particles.8 Reference Patch for Singular Functions B.6 A.4 C.2 O D E -.4 G -.6 F -.8 Mapped Particles.5 B.4 C.3.2. A crac D G F O E Figure 5: The Singular Reference Domain ˆΩ S (Left) and a neighborhood of the crac tip Ω S (Right). The Conformal mapping T S (ˆr, ˆθ) = (ˆr 2 cos 2ˆθ, ˆr 2 sin 2ˆθ) maps the curved quadrangles ODCBA and ODEF G to the rectangles O D C B A and O D E F G, respectively. Q : (.36, ), (.36, ), (.3,.6), (.3,.6). Let us note that the enlarged quadrangle Q contains Q so that the convolution PU shape function ψ δ becomes along Q Ω. We also note that Q Ω = Q for =, 2, 3, 4. (3) Global numbering of the mapped particles. (a) To the mapped particles along the common edges CG and BF (Fig. 4) and two mapped particles at (.36, ) and (.36, ) on the boundary, there correspond two local particles. To each of these overlapped particles, we assign one global particle number. (b) Since all other mapped particles correspond to different points, we assign different global particle numbers to those non overlapped particles (Fig. 4). (4) Global reproducing polynomial particle shape functions Φ I φ ij (x, y) = ˆφ ij (r /2 cos θ 2, r/2 sin θ 2 ) ψδ (x, y), Φ J φ ij (x, y) = [ ˆφ ij T (x, y)] ψδ (x, y) for 2, Φ K 2 φ ij (x, y) + 3 φ st (x, y) if two patch-wise particles are T 2 (ξ i, η j ) = T 3 (ξ s, η t ) on BF, Φ K2 3 φ ij (x, y) + 4 φ st (x, y) if two patch-wise particles are T 3 (ξ i, η j ) = T 4 (ξ s, η t ) on CG, Φ K3 φ ij (x, y) + 4 φ ij (x, y), at T s (,.6) = (.36, ), 25

26 Φ K4 φ ij (x, y) + 2 φ ij (x, y) at T s (.6, ) = (,.36). Then the (piecewise-polynomial) reproducing singularity particle(rsp) and the (piecewisepolynomial) reproducing polynomial particle(rpp) shape functions Φ J, J Λ Ω have compact supports. Here, Λ Ω is the index set of globally numbered particles. These shape functions satisfy the Kronecer delta property at each particles on the boundary. (5) The interpolation of u(x, y) associated with the global particle shape functions, Φ I, I Λ Ω, is defined by I h u(x, y) = I Λ Ω u(x I, y I )Φ I (x, y) where u(x I, y I ) = u(t (ξ i, η j )), (i, j) is a local index corresponding to the global index I. The interpolation errors are estimated in the following two norms: the L 2 -norm and the H -semi norm, respectively, defined by { } /2 { } /2 error H = (I h u u) 2 dxdy, error H = (I h u u) 2 dxdy. Q Q [A] In the computational perspectives, we observe the followings:. For the particle shape functions corresponding to the particles in the patch Q (Fig. 4), we use the conformal mapping T s (z = w 2 ). Thus, the components Jij of the inverse matrix and the determinant J(T s ) in the formula in [B] of section 5. are those in (6) and (7). 2. If the RSP reproducing order is 2 (hence, the reference particle shape shape functions are of degree 4), then one can chose the quadrangular patch Q so that it can contain all particles corresponding to those in the reference patch (as shown in Fig. 5). However, if the RSP reproducing order is 4 (hence, the reference particle shape shape functions are of degree 8), one should not mae Q containing all mapped particles because extended reference RPP shape functions φ Ext (i,j)(ξ, η) tae very large values at those points outside the reference patch. Thus, in Fig. 4, we chose the singular zone Q so that almost all parts of Ts (Q )(the curved pentagon in Top of Fig. 4) are contained in the reference patch ˆQ = [,.6] [,.6]. The numerical results will be optimal when the curved pentagon Ts (Q ) is best fitted to the singular reference patch ˆQ = [,.6] [,.6]. 3. At those points in T s ( ˆQ ) \ Q, the RSP shape functions Φ I, I Λ Ω, do not satisfy the Kronecer delta property. In particular, Φ K (T s (.6,.6)) =, for all particle shape functions Φ K that correspond to the mapped particles that fall inside Q. However, that does not mean the amplitude of the global RSP shape function Φ J corresponding to the particle T s (.6,.6) is zero in the interpolation approximations as well as the finite element approximations. 26

27 It is important to note that Q contains all those particles fall onto the boundaries and hence the Kronecer delta property is satisfied at the all of particles on the boundaries. Hence, our RSP shape functions Φ I, I Λ Ω, can handle the Dirichlet boundary conditions in the standard way. [B] The first test is the interpolation error of a singular function arising from the jump boundary data. Let us consider the benchmar problem, nown as the Motz Problem ([9], p335): u = in Ω(shown in Fig. 4), { 5 along the vertical line x = u = along the negative x-axis, u = along the non negative x-axis. n Then u has a jump boundary data at (, ). It is nown that around the singularity point (, ), the solution of the Motz problem is dominated by b l r l+/2 cos(l + /2)θ. l= Example. Let us compute the interpolation error of a singular function with jumpboundary data singularity u (r, θ) =.5r /2 cos(θ/2) +.4r /2 sin(θ/2) +.2r 3/2 cos(3θ/2) +.r 3/2 sin(3θ/2), (3) associated with reproducing singularity particle shape functions Φ I, I Λ Ω. From table, we have the followings:. the columns RPP and RSP, respectively, stand for the interpolation errors associated with reproducing polynomial particle shape functions and with reproducing singularity particle shape functions. 2. m = 2 and m = 4, respectively, stand for the cases when the reference patch ˆQ have 4 4 particles and 8 8 particles. 3. The results by using RSP are far better than those by RPP because the RSP shape functions generate the test functions. 4. From Corollary 4., the interpolation error of u associated with the reproducing singular particle shape functions is zero on a neighborhood of the singularity. Actually, since we used.e-2 as the computer zero for these computations, we cannot get any better results than those in RSP column of Table. In Figs. 7 and 8, we show that 27

28 Table : (Jump Boundary Data Singularity) The L 2 -Norm and the H -Semi Norm of the Interpolation error (I h u u), on the patches Q, Q 2, Q 3, Q 4, respectively. Here, the test function is u(x, y) of (3) that contains the jump boundary data singularity. Jump Boundary Data Singularity Norm Error in L 2 -norm Error in H -semi norm Basis RPP RSP RPP RSP Order m = 2 m = 4 m = 2 m = 4 m = 2 m = 4 m = 2 m = 4 Q 9.34E E-3 4.8E-6 9.2E-8 2.5E- 3.93E- 4.23E E-6 Q E E-3.9E E-7 3.E E- 3.2E-4.86E-5 Q 3.57E E E-5.8E E E-2.44E E-4 Q 4 7.9E E-3.2E E E E- 3.38E-4.72E-4 a. In Fig.7, the true solution u (r, θ) with jump boundary data singularity agrees with the interpolation associated with RSP shape function up to five digits at all most all plotting points. Thus, two figures are virtually the same. However, the graph of the interpolation of u asociated with RPP shape functions (Lower center graph of Fig. 7) is quite different from the graph of the true u (Upper left figure). b. In Fig. 8, the interpolations of the true solution u (r, θ) associated with capped RSP shape functions is compared with the interpolations of u (r, θ) associated with capped RPP shape functions over the singular zone Q. From this figure, one can see that approximability of RSP shape functions is far better than that of RPP shape functions in dealing with singularities. [B2] The second test is the interpolation error of a singular function that arise from the singularity of a single notched crac. 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([26], 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. Example 2. Thus, by taing the Poisson ratio ν =.3 and the oung s modulus E =, we assume that a test function is the first term of the θ-component of the displacement vector 28

29 Table 2: (Crac Singularity) The L 2 -Norm and the H -Semi Norm of the Interpolation error (I h u u), on the patches Q, Q 2, Q 3, Q 4, respectively. Here, the test function is u 2 (r, θ) of (3) that contains the crac singularity. Crac Singularity Norm Error in L 2 -norm Error in H -semi norm Basis RPP RSP RPP RSP Order m = 2 m = 4 m = 2 m = 4 m = 2 m = 4 m = 2 m = 4 Q 4.22E-2 3.4E E E E-.56E-.32E- 4.39E-2 Q E E-2 5.2E E-6 9.4E-2.22E-.92E E-3 Q E-3.57E E E-5 5.E E- 4.76E E-3 Q E-3 3.8E E-4.88E-5.5E-.45E-.57E-2 9.3E-3 as follows: u 2 (r, θ) =.6r /2 { 4.6 sin(θ/2) + sin(3θ/2)}.r /2 {4.6 cos(θ/2) 3 cos(3θ/2)}. (3) From Table 2, we observe the followings.. The columns RPP, RSP, m = 2, and m = 4 stand for the same things as Table. 2. The results by using RSP shape functions are better than those by RPP shape functions. However, the improvement is not as good as Table because the RSP shape functions are not able to generate the term r /2 cos(3θ/2) of (3). 3. Reducing the size of supports by using those particle shape function in appendix B for the reference shape functions, one can obtain much improved results. In Fig. 9, we compared the true solution u 2 (r, θ) (Left hand side figure) with the interpolation of the true solution associated with the RSP shape functions (Right hand side figure). Graphically, two figures are the same, because two figures are the same at all plotting points up to four digits (three digits at few points near the singularity). Since errors of Finite element solutions are bounded by interpolation errors, from the results in Tables and 2, one can conclude that the solutions by meshless particle methods associated with reproducing singular particle(rsp) shape functions should be at least as accurate as the finite element solutions obtained by using Method of Auxiliary Mapping (MAM)([4],[9],[2]) in the framewor of the conventional p-fem. 7 Concluding Remars The point singularity of elliptic boundary value problems are either the monotone singularity of type x α, x α (log x) l 29

30 or the oscillating singularity of type x α cos ε log x, where < α < is called an intensity of singularity and a small real number ε is called the oscillating factor. In this paper, we constructed the reproducing singularity particle shape functions that can reproduce singular functions which resemble the monotone singularities. More wor should be done for constructions of particle shape functions that can handle the oscillating singularities, arising at delamination cracs of the composite materials. The RSP shape functions capture the jump boundary data singularity ( r (.5+) cos((.5 + )θ), =,, 2, ), however, could not exactly capture the singular functions ( r (.5+) cos((.5+ )θ), =,, 2 ). In other words, our RSP shape functions cannot exactly interpolate the crac singularities, even though they are able to interpolate exactly the jump boundary data singularity. The proof of error estimate for an interpolation associated with RPP shape function was given in ([25]). An error estimate of an interpolation associated with the RSP shape functions will be reported in a forthcoming paper. There are three different types of three-dimensional domain singularities: vertex singularity, edge singularity and vertex-edge combined singularity. We expect that it is not very difficult to extend the two-dimensional construction of reproducing singularity particle shape functions to the three-dimensional cases. However, it may not be easy to extend the convolution partition of unity shape functions to the three-dimensional cases. A Diagram of intersecting polygons and their vertices 3