The Closed Form Reproducing Polynomial Particle Shape Functions for Meshfree Particle Methods

The Closed Form Reproducing Polynomial Particle Shape Functions for Meshfree Particle Methods by Hae-Soo Oh Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223 June G Kim Department of Mathematics, Kangwon National University, Chunchon, 200-70, Korea Jaewoo Jeong Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223 April 6, 2007 Abstract It has been known that Reproducing Kernel Particle (RKP) shape functions with Kronecker delta property are not available in simple forms Thus, in this paper, we construct highly regular piecewise polynomial Reproducing Polynomial Particle (RPP) shape functions that satisfy the Kronecker Delta Property Like RKP shape functions, the RPP shape functions reproduce the complete polynomials of degree k for an integer k 0 Moreover, the particles associated with these RPP shape functions are either uniformly distributed in (, ) or non-uniformly distributed in [0, ) (or in a compact set [a, b]) Keywords: The Closed form reproducing polynomial particle(rpp) shape functions; Reproducing kernel particle(rkp) shape functions; Reproducing kernel particle method(rkpm); Translation invariant particles; Non-uniformly distributed particles Corresponding author Tel: +-704-687-4930; fax: +-704-687-645; E-mail: hso@unccedu 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

Introduction The Finite Element Method (FEM) has been a powerful tool in solving many difficult science and engineering problems, especially when the solution domains have complex geometry ([5], [25]) However, there are several difficulties arising in implementing this method The prominent difficulties include the mesh refinements and constructing higher order interpolation fields In order to relax the constraints of the conventional FEM, several generalized finite element methods (GFEM), that use the meshes minimally or do not use the meshes at all, were recently introduced Among many names of GFEM ([],[2],[3],[4]), those methods related to this paper are Element Free Galerkin Method (EFGM) ([],[9],[0],[], [2],[5],[6],[7]), h-p Cloud Method([7],[8]), Partition of Unity Finite Element Method (PUFEM)([8],[23],[24]), and Reproducing Kernel Element Method (RKEM) ([2],[3],[4]) It was shown ([2],[9]) that RKP shape functions with the reproducing property of high order ensure a good approximability in Reproducing Kernel Particle Method (RKPM) ([9],[2],[5], [6],[7]) However, these RKP shape functions have not been given in a simple form, but usually known as the product of window functions and the solutions of the functional linear systems ([2],[3]) Moreover, in application of PU shape functions, the Shepard-type PU functions have been employed In other words, neither the RKP shape functions nor PU shape functions are in a simple form Thus, integrations related to RKP shape functions are not very accurate and require lengthy function evaluations Furthermore, RKP shape functions do not satisfy the Kronecker delta property in general Thus, they have difficulties in dealing with Direchlet boundary conditions In this paper, we construct highly regular piecewise polynomial reproducing polynomial particle (RPP) shape functions that reproduce the complete polynomials of any given order and also satisfy the Kronecker delta property The RKP shape functions are constructed with respect to specific window functions, whereas what we call the RPP(Reproducing Polynomial Particle) shape functions in this paper have no relation with any specific window functions, but are concerned with the sizes of supports However, both are constructed to have the polynomial reproducing property Actually, we may say that the RPP shape functions are special cases of RKP shape functions The difference between RPP and RKP will be elaborated in section 3 This paper is organized as follows In section two, the terminologies and notations used in this paper are introduced A simple example for motivation of our constructions is provided In section three, for r = 0,, 2, 3, we constructed the translation invariant piecewise polynomial C r -RPP shape functions corresponding to uniformly distributed particles in (, ) We also prove the uniqueness theorem for the piecewise polynomial C 0 -RPP shape functions of maximal reproducing order that satisfy the Kronecker delta property Effectiveness of RPP shape functions are tested with the fourth order as well as the second order one dimensional elliptic equation In section 4, for r = 0,, 2, 3, we construct the piecewise polynomial C r -RPP shape functions corresponding to non-uniformly distributed particles in [0, ) (or in a compact set [a, b]) In appendix, we place the proof of Lemma 3 that is used for our construction of RPP shape 2

functions Moreover, for reference purpose, RPP shape functions of high reproducing order with various smoothness are listed for the particles are in (, ) as well as in [0, ) Finally, for the two dimesional extension of RPP shape functions, we refer to ([9],[20]) 2 Definitions and Notations There are two slightly different definitions for RKP shape functions: one in [2] and another in [9] In this paper, for the meaning of the RKP shape functions, we follow the definition in [2] Throughout this paper, α, β Z n are multi indices and x = ( x, 2 x,, n x), x j = ( x j, 2 x j,, n x j ) denote points in R n However, if there are no confusions, we also use the conventional notation for the points in R n or Z n as x = (x, x 2,, x n ) and α = (α, α 2,, α n ) By α β, we mean α β,, α n β n We also use the following notations: (x x j ) α := ( x x j ) α ( n x n x j ) αn, α := α + α 2 + + α n Let Ω be a domain in R n For any nonnegative 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) 0} In the following, a function φ C m (Ω) is said to be a C m - function A weight function(or window function) is a non negative continuous function with compact support and is denoted by w(x) In R n, the weight function w(x) can be constructed from a one-dimensional weight function either as w(x) = w( x ) or as w(x) = n i= w(x i), where x 2 = x 2 + + x2 n Adopting those terminologies and notations of ([2]), we have the following: For j = (j, j 2,, j n ) Z n, and the mesh size 0 < h, let x h j = (j h,, j n h) = hj Then the points x h j are called uniformly distributed particles Let φ be a continuous function with compact support that contains the origin 0 Then the particle shape functions associated to the uniformly distributed particles is defined by φ h j (x) = φ( x jh h ) = φ( x j h h,, x n j n h ), h for j Z n and 0 < h Then these particle shape functions are translation invariant in the sense that x h i+j = x h i + x h j, φ h j (x ih) = φ h i+j(x) In this paper, we assume that the basic shape functions are translation invariant on the uniformly distributed particles, unless stated otherwise The particle shape functions associated with non-uniformly distributed particles are constructed in section 4 3

Without loss of generality, in the theoretical developments and constructions of particle shape functions, we assume that h = and hence φ h j (x) = φ(x j) Let Λ be an index set and Ω denotes a bounded domain in R n Let {x j : j Λ} be a set of uniformly (or non-uniformly) distributed points in R n, that are called particles Definition 2 Let k be a nonnegative 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 k (or simply, of reproducing order k ) if and only if it satisfies the following condition: for x Ω R n, (x j ) α φ j (x) = x α, and for 0 α k, α = α + + α n () j Λ The RKP(Reproducing Kernel Particle) shape function, associated with the particle x j, is constructed by φ j (x) = w(x x j ) (x x j ) α b α (x) (2) 0 α k where b α (x) are chosen so that () is satisfied and w(x) is a window function This gives rise to a linear system in b α (x), namely 0 α k m α+β (x)b α (x) = δ β 0 for 0 β k, (3) where δ 0 β is the Kronecker delta, and m α (x) = j Z w(x x j )(x x j ) α (4) For one dimensional case, this system can be written as M(x) [b 0 (x), b (x),, b k (x)] T = [, 0,, 0] T, where M(x) = w(x x j ) j Z (x x j ) (x x j ) 2 (x x j ) k [, (x x j ),, (x x j ) k ] The coefficient matrix M(x) of the linear system (3) is called the moment matrix 4

Let us construct a one dimensional RKP shape function of reproducing order by using (2) and (3) For example, let w(x) be a smooth window function whose support is [, ] Then for x [0, ], the moment matrix is [ ] w(x) + w(x ) x w(x) + (x ) w(x ) M(x) = x w(x) + (x ) w(x ) x 2 w(x) + (x ) 2 w(x ) and det M(x) = w(x)w(x ) is non zero except at x = 0, Thus, for x (0, ), the solutions of the linear system (3) are b 0 (x) = [x 2 w(x) + (x ) 2 w(x )]/w(x)w(x ), b (x) = [xw(x) + (x )w(x )]/w(x)w(x ) Hence, by the relation (2), φ(x) = w(x)[b 0 (x) + xb (x)] = x, φ(x ) = w(x )[b 0 (x) + (x )b (x)] = x Now we extend φ(x) onto [0, ] by defining φ(0) =, and φ() = 0 Then the translation invariant RKP shape function of reproducing order, associated to any weight function w(x) with support [, ], is the hat function In this example, for the construction of the RKP shape function, we only used the support size of w(x), but did not use the nature of w(x) Hence the resulting RKP function φ(x) is independent of the smoothness of the weight function w(x) This is because the interior of supp w(x) contains only one particle and hence M(x) is singular at x = 0, It is stated in ([2] and [9]) that if w(x) C q, then φ(x) C q, provided that for all x R n, M(x) is nonsingular In fact, if M (α) (x) is the matrix obtained by replacing the α-th column of M(x) with (, 0, 0,, 0) T, then we have b α (x) = [det M (α) (x)/ det M(x)] C q whenever w(x) C q Hence the relation (2) implies φ(x) C q Now, we are motivated to find another method constructing RPP shape functions without getting weight functions involved For this purpose, we consider the following characterization By applying a similar argument to ([2],[9]), one can show that () is equivalent to (x x j ) β φ j (x) = δ β 0, for 0 β k and x Rn (5) j Z This characterization of the RPP shape functions have no direct relation with the weight functions Thus, by using this equivalent definition, various closed form RPP shape functions are constructed in the following sections 3 Piecewise polynomial RPP shape functions corresponding to uniformly distributed particles in (, ) In this section, by using (5), we construct translation invariant piecewise polynomial particle shape functions that have the reproducing property of any desired reproducing order and also satisfy the Kronecker delta property The following lemma is essential for the construction of RPP shape functions The technical proof of this lemma is in appendix 5

Lemma 3 Suppose K is a positive integer and f(x) is a function defined on such that, if Ω = [ K, K + ) (, 0) (0, ) (K 2, K ) (K, K], f (j,j+) (x), j = K, K 2,, 2,, 0,, 2,, K, are shifted on (0, ), then they are the solutions of the following system of linear functional equations: V (x) [f(x + (K )),, f(x + ), f(x), f(x ),, f(x K)] T = e i, (6) where x (0, ), e i = [0, 0,, 0,, 0,, 0] T, the i-th base vector, and V (x) is the following Vandermonde matrix: (x + (K )) x + x (x (K )) (x K) (x + (K )) 2K (x + ) 2K x 2K (x (K )) 2K (x K) 2K Then () f(x) can be extended to a continuous piecewise polynomial function F (x) defined on Ω, if the right hand side vector is e i for an odd number i In particular, if the right hand side vector is e, F (x) satisfies the Kronecker delta property Moreover, in this case, f(x) is also extendable to a continuous function F (x) that satisfies the Kronecker delta property even when f(x) has a non symmetric support (that is, Ω = [ K, K + ] \ Z or Ω = [ K, K] \ Z) (2) No continuous extensions are possible, if the right side vector is e i for an even number i 3 Existence and uniqueness of a closed form C 0 - RPP shape function of maximum reproducing order Theorem 3 There exits a unique translation invariant C 0 -RPP shape function φ(x), which satisfies the following conditions: (a) The support of φ(x) is [ K, K], where K is a positive integer, (b) The reproducing order of φ(x) is 2K, (c) This satisfies the Kronecker delta property: φ j (i) = δ j i The unique RPP basic shape function is a piecewise polynomial of degree 2K Moreover, this RPP shape function is independent of choice of weight functions Since M(x) is singular in general for some x [ K, K], it is not possible to construct a RPP shape function of reproducing order 2K through the standard construction (2) and(3) 6

Proof For x (0, ), a translation invariant RPP shape function of order 2K satisfies the following system of equations: K k= K+ (x k) α φ(x k) = δ α 0, α = 0,, 2,, 2K (7) This linear system can be written in matrix form as follow V (x) [φ(x + (K )), φ(x + (K 2)),, φ(x),, φ(x (K )), φ(x K)] T = e (8) Lemma 3 shows that the shape function φ(x) determined as the solutions of this system can be continuously extended to [ K, K] so that the extended function satisfies the Kronecker delta property Moreover, such a unique extension makes φ(x) to be also satisfied the definition () at all integer points Since the system (8) has a unique solution and the extension of the solution to [ K, K] is also unique, this is the only RPP basic shape function of order 2K whose support is [ K, K] On each interval bounded by two consecutive integers, φ(x) is a constant times the determinant of (2K ) (2K ) minor matrix of V (x), and hence φ(x) is a piecewise polynomial of degree 2K Related to this existence and uniqueness for C 0 -RPP shape functions, we have observed the followings Theorem 44 of [6] is virtually the same as our uniqueness theorem However, our theorem did not use a sequence of double knots; hence two proofs are different Moreover, the unique solution functions of the system (7) are the Lagrange interpolation polynomials corresponding to the particles K +,, 0,, K Thus, our C 0 -RPP shape functions are just C 0 -spline functions 2 It follows from lemma 3 that Theorem 3 holds for non-symmetric support [ K, K + ] or [ K, K] 3 If the support of φ(x) is exactly [, ], then by Theorem 3, the unique RPP shape function of reproducing order is the hat function because the one dimensional RKP shape function of reproducing order constructed by using any window function with support [, ] in section 2 is the hat function In other words, the C 0 -RPP shape function is the same as the RKP shape function constructed by using a specific window function whenever their supports are [ K, K] and their reproducing order are 2K, for some positive integer K However, suppose φ(x) is a RKP shape function, with reproducing property of order, constructed by using a window function w(x) whose support is larger than [, ], for example, [ 2, 2] Then, we cannot construct this RKP shape function by applying the arguments in Theorem 3 7

4 It is stated in [2] that the regularity of a RKP shape function φ(x) is the same as that of the associated weight function w(x) This statement is not applicable to the foregoing theorem because the support is not large enough to make for the moment matrix M(x) to be non singular for all x R 32 Constructions of closed form C m -RPP shape functions (m 0) that satisfy the Kronecker delta property In parts [A] and [B] of this subsection, by solving the linear system (8), various one dimensional C m RPP shape functions of given reproducing orders are constructed In section 33, the effectiveness of these RPP shape functions are tested by applying them to the second order and the fourth order differential equations, respectively In Theorem 3, for a given symmetric support [ K, K], we considered RPP shape functions with the reproducing property of odd order 2K However, by applying similar arguments for non symmetric support [ K, K + ] (or [ (K + ), K]), one can construct RPP shape functions satisfying the reproducing property of even order 2K Lemma 3 makes it possible to construct the unique closed form C 0 pp(piecewise polynomial) RPP shape function of reproducing order 2K that satisfy the Kronecker delta property with respect to the given support [ K, K] However, by sacrificing one order from the maximal reproducing order 2K (or extending the support to [ (K + ), (K + )] if we want to keep the reproducing order), for a given integer m > 0, we are able to construct a C m -pp RPP shape function on [ K, K] (or [ K, K + ]) that satisfies the Kronecker delta property For this purpose, we start with a piecewise smooth function f(x) that satisfies K k= K+ (x k) α f(x k) = δ α 0, α = 0,, 2,, 2K 2, for x (0, ) (9) Then this system is under-determined In order to make for this system to be an exactly determined system, we add an additional equation to this system as follows: for x (0, ), { K k= K+ (x k)α f(x k) = δ0 α, α = 0,, 2,, 2K 2, K k= K+ (x (0) k)2k f(x k) = G(x) Then the coefficient matrix of this extended system becomes a 2K 2K Vandermonde matrix The right hand side becomes the 2K dimensional column vector [, 0, 0,, 0, G(x)] T If G(0) = 0 and G() = 0, then it follows from Lemma 3 that the solution vector (f(x + K ), f(x + (K 2)),, f(x + ), f(x), f(x ),, f(x K)) T, x (0, ) for f(x) can be extended to be a continuous function F (x) on [ K, K] that satisfies the Kronecker delta property Thus, we impose the following conditions on G(x): G(0) = 0 and G() = 0 () 8

Next, we are going to impose some additional condition on G(x) so that f(x) can be extended to a C 2 -shape function By differentiating the expanded system (0) twice, we have V (x) [f (x + K ),, f (x + ), f (x), f (x ),, f (x K)] T = [0, 0, 2, 0,, 0, 0, G (x)] T (2) Here V (x) is a Vandermonde matrix Since the right side of the last system has a non-zero entry on the odd number row, Lemma 3 implies that the solution vector of this system can be made to be a continuous function whenever G (0) = 0 and G () = 0 Thus, we impose the following condition: G (0) = 0 and G () = 0 (3) For the construction of C r -RPP shape functions (r > 0), we have to start with (0) when the freedom parameter function G(x) has the form G(x) = x(x )p(x), where p(x) is a polynomial In what follows, the indices of RPP shape function φ ([a,b];r;m) (x) indicate the following: [a, b] = the support of φ(x), r = the order of the regularity(that is, φ(x) C r ), m = the order of the reproducing property The graphs of these C 0 -RPP shape functions are depicted in Fig, from which one can observe that the RPP functions satisfy the Kronecker delta property In the following, we constructed the C r -RPP shape function up to the regularity order r = 2 However, by using a similar argument, one can construct RPP functions that are regular up to any desired regularity order All RPP shape functions constructed in this section satisfy the Kronecker delta property A: The RPP shape functions with symmetric support [ K, K] [AI] The support is [, ] The unique RPP shape function of reproducing order whose support is [, ] is the following hat function φ ([,];0;) (x) = x + x [, 0] x + x [0, ] 0 x R \ [, ] [AII] The support is [ 2, 2] We construct the RPP shape functions of various regularity order that have the reproducing property of order k 2 9

08 08 08 06 04 06 04 02 06 04 02 02 0 0 0 - -05 0 05-02 - -05 0 05 5 2-02 -2-5 - -05 0 05 5 2 08 08 08 06 06 06 04 04 04 02 02 02 0 0 0-02 -2-0 2 3-02 -3-2 - 0 2 3-02 -3-2 - 0 2 3 4 Figure : The continuous piecewise polynomial RPP shape functions of order k =, 2, 3, 4, 5, 6 (Left to Right and Top to Bottom) Consider the following system V 4 (x) [φ(x 2), φ(x ), φ(x), φ(x + )] T = [, 0, 0, G(x)] T, (4) where V 4 (x) is a 4 4 Vandermonde matrix defined by V 4 (x) = (x 2) (x ) x (x + ) (x 2) 2 (x ) 2 x 2 (x + ) 2 (x 2) 3 (x ) 3 x 3 (x + ) 3 (5) We already know that G(x) must have a factor x(x ) following case: G(x) = x(x )p(x) So it suffices to consider the II If p(x) = 0, the system (4) yields the unique C 0 -RPP shape function φ ([ 2,2];0;3) (x) of reproducing order 3 f,(2;0;3) (x) := 6 (x + )(x + 2)(x + 3) x [ 2, ] f 2,(2;0;3) (x) := 2 (x )(x + )(x + 2) x [, 0] φ ([ 2,2];0;3) (x) = f 3,(2;0;3) (x) := 2 (x 2)(x )(x + ) x [0, ] (6) f 4,(2;0;3) (x) := 6 (x 3)(x 2)(x ) x [, 2] 0 x [ 2, 2] 0

II2 If p(x) = 2x, the system (4) yields a C -RPP shape function of reproducing order 2 g,(2;;2) (x) := 2 (x + )(x + 2)2 x [ 2, ] g 2,(2;;2) (x) := 2 (x + )(3x2 + 2x 2) x [, 0] φ ([ 2,2];;2) (x) = g 3,(2;;2) (x) := 2 (x )(3x2 2x 2) x [0, ] g 4,(2;;2) (x) := 2 (x 2)2 (x ) x [, 2] 0 x [ 2, 2] II3 If p(x) = + x 9x 2 + 6x 3, the system (4) yields a C 2 -RPP shape function of reproducing order 2 2 (x + 2)3 (x + )(2x + ) x [ 2, ], 2 (x + )(6x4 + 9x 3 2x + 2) x [, 0], φ ([ 2,2];2;2) (x) = 2 (x )(6x4 9x 3 + 2x + 2) x [0, ], 2 (x 2)3 (x )(2x ) x [, 2], 0 x R \ [ 2, 2] II4 If p(x) = + x 30x 3 + 45x 4 8x 5, the system (4) yields a C 3 -RPP shape function of reproducing order 2 2 (x + 2)4 (x + )(6x 2 + 9x + 4) x [ 2, ], 2 (x + )(8x6 + 45x 5 + 30x 4 + 2x 2) x [, 0], φ ([ 2,2];3;2) (x) = 2 (x )(8x6 45x 5 + 30x 4 2x 2) x [0, ], 2 (x 2)4 (x )(6x 2 9x + 4) x [, 2], 0 x R \ [ 2, 2] II5 If p(x) = + x 05x 4 + 252x 5 20x 6 + 60x 7, the system (4) yields a C 4 -RPP shape function of reproducing order 2 2 (x + 2)5 (x + )(20x 3 + 50x 2 + 44x + 3) x [ 2, ], 2 (x + )(60x8 + 20x 7 + 252x 6 + 05x 5 2x + 2) x [, 0], φ ([ 2,2];4;2) (x) = 2 (x )(60x8 20x 7 + 252x 6 05x 5 + 2x + 2) x [0, ], 2 (x 2)5 (x )(20x 3 50x 2 + 44x 3) x [, 2], 0 x R \ [ 2, 2] [AIII] The support is [ 3, 3] We construct the RPP shape functions of various regularity order that have the reproducing property of order k 4 In this case, we have the following system V 6 (x) [φ(x 3), φ(x 2), φ(x ), φ(x), φ(x + ), φ(x + 2)] T (7) = [, 0, 0, 0, 0, G(x)] T, (8) where V 6 (x) is the 6 6 Vandermonde matrix that is similar to (5) and G(x) = x(x )p(x)

III If p(x) = 0, then the system (8) yields the unique C 0 -RPP shape function φ ([ 3,3];0;5) of reproducing order 5 f,(3;0;5) = 20 (x + )(x + 2)(x + 3)(x + 4)(x + 5) x [ 3, 2] f 2,(3;0;5) = 24 (x )(x + )(x + 2)(x + 3)(x + 4) x [ 2, ] f 3,(3;0;5) = 2 (x 2)(x )(x + )(x + 2)(x + 3) x [, 0] φ ([ 3,3];0;5) (x) = f 4,(3;0;5) = 2 (x 3)(x 2)(x )(x + )(x + 2) x [0, ] f 5,(3;0;5) = 24 (x 4)(x 3)(x 2)(x )(x + ) x [, 2] f 6,(3;0;5) = 20 (x 5)(x 4)(x 3)(x 2)(x ) x [2, 3] 0 x R \ [ 3, 3] (9) III2 If p(x) = 4 8x, the system (8) yields a C -RPP shape function of reproducing order 4 φ ([ 3,3];;4) (x) = g,(3;;4) = 20 x(2 + x)(3 + x)2 (7 + x) x [ 3, 2], g 2,(3;;4) = 24 ( + x)(2 + x)( 24 3x + 6x2 + x 3 ) x [ 2, ], g 3,(3;;4) = 2 ( + x)(2 2x 5x2 + 2x 3 + x 4 ) x [, 0], g 4,(3;;4) = 2 ( + x)(2 + 2x 5x2 2x 3 + x 4 ) x [0, ], g 5,(3;;4) = 24 ( 2 + x)( + x)(24 3x 6x2 + x 3 ) x [, 2], g 6,(3;;4) = 20 ( 7 + x)( 3 + x)2 ( 2 + x)x x [2, 3], 0 x R \ [ 3, 3] (20) III3 If p(x) = 4 + 4x 36x 2 + 24x 3, the system (8) yields a C 2 -RPP shape function of reproducing order 4 φ ([ 3,3];2;4) (x) = 24 (2 + x)(3 + x)3 (8 + 5x) x [ 3, 2], 24 ( + x)(2 + x)(48 + 53x + 4x2 + 25x 3 ) x [ 2, ], 2 ( + x)(2 2x 3x2 + 38x 3 + 25x 4 ) x [, 0], 2 ( + x)(2 + 2x 3x2 38x 3 + 25x 4 ) x [0, ], 24 ( 2 + x)( + x)( 48 + 53x 4x2 + 25x 3 ) x [, 2], 24 ( 3 + x)3 ( 2 + x)( 8 + 5x) x [2, 3], 0 x R \ [ 3, 3] [AIV] The support is [ 4, 4] Similarly, we consider the following system of functions: V 8 (x) [φ(x 4), φ(x 3), φ(x 2), φ(x ), φ(x), φ(x + ), φ(x + 2), φ(x + 3)] T = [, 0, 0, 0, 0, 0, 0, G(x)] T, (2) where V 8 (x) is the 8 8 Vandermonde matrix and G(x) = x(x )p(x) IV If p(x) = 0, the system (2) yields the unique C 0 -RPP shape function φ ([ 4,4];0;7) reproducing order 7 of 2

IV2 If p(x) = 36 + 72x, the system (2) yields a C -RPP shape function φ ([ 4,4];;6) of reproducing order 6 IV3 If p(x) = 36 + 36x + i324x 2 26x 3, the system (2) yields a C 2 -RPP shape functions φ ([ 4,4];2;6) of reproducing order 6 The piecewise polynomial RPP shape functions φ ([ 4,4];0;7), φ ([ 4,4];;6), φ ([ 4,4];2;6) are in appendix B: The RPP shape functions with non symmetric support [ K, K +], [ K, K] Because of the symmetry, it suffices to construct RPP shape functions with respect to the support [ K, K + ] All RPP shape functions constructed in this section satisfy the Kronecker delta property [BV] The support is [, 2] When the support is [, 2], we also construct RPP shape functions that have the reproducing property of order k In this case, we have the following equation φ(x ) (x ) x (x + ) φ(x) = 0 (22) (x ) 2 x 2 (x + ) 2 φ(x + ) G(x) We already know that G(x) must have a factor x(x ) following case: G(x) = x(x )p(x) So it suffices to consider the V If p(x) = 0, the system (22) yields the unique C 0 -RPP shape function of the reproducing property of order 2 2 ( + x)(2 + x) x [, 0], ( + x)( + x) x [0, ], φ ([,2];0;2) (x) = 2 ( 2 + x)( + x) x [, 2], 0 x R \ [, 2] V2 If p(x) = + 2x, the system (22) yields a C -RPP shape function of reproducing order ( + x)( + x) 2 x [, 0], ( + x)( 2x + 2x φ ([,2];;) (x) = 2 ) x [0, ], ( 2 + x) 2 ( + x) x [, 2], 0 x R \ [, 2] V3 p(x) = 2x+0x 2 6x 3, the system (22) yields a C 2 -RPP shape function of reproducing order 2 ( + x)(2 + 8x3 + 6x 4 ) x [, 0], ( + x)( + 2x + 2x φ ([,2];2;) (x) = 2 0x 3 + 6x 4 ) x [0, ], 2 ( 2 + x)( + x)( 6 + 40x 28x2 + 6x 3 ) x [, 2], 0 x R \ [, 2] 3

[BVI] The support is [ 2, 3] In this case, we construct the RPP shape functions of reproducing order k 3 We consider the following equation V 5 (x) [φ(x 2), φ(x ), φ(x), φ(x + ), φ(x + 2)] T = [, 0, 0, 0, G(x)] T, (23) where V 5 (x) is a 5 5 Vandermonde matrix and G(x) = x(x )p(x) VI If p(x) = 0, the system (23) yields the unique C 0 -RPP shape function of reproducing order 4 φ ([ 2,3];0;4) (x) = 24 6 4 6 24 ( + x)(2 + x)(3 + x)(4 + x) x [ 2, ], ( + x)( + x)(2 + x)(3 + x) x [, 0], ( 2 + x)( + x)( + x)(2 + x) x [0, ], ( 3 + x)( 2 + x)( + x)( + x) x [, 2], ( 4 + x)( 3 + x)( 2 + x)( + x) x [2, 3], 0 x R \ [ 2, 3] VI2 If p(x) = 2 4x, the system (23) yields a C -RPP shape function of reproducing order 3 φ ([ 2,3];;3) (x) = 24 ( + x)(2 + x)(8 + x + x2 ) x [ 2, ], 6 ( + x)( 6 + 3x + 8x2 + x 3 ) x [, 0], 4 ( + x)( 4 6x + 5x2 + x 3 ) x [0, ], 6 ( 2 + x)( + x)( 9 + 2x + x2 ) x [, 2], 24 ( 3 + x)( 2 + x)( 6 x + x2 ) x [2, 3], 0 x R \ [ 2, 3] VI3 If p(x) = 2 + 3x 9x 2 + 2x 3, the system (23) yields a C 2 -RPP shape function of reproducing order 3 φ ([ 2,3];2;3) (x) = 24 ( + x)(2 + x)(6 + 64x + 52x2 + 2x 3 ) x [ 2, ], 6 ( + x)(6 3x 3x2 + 6x 3 + 2x 4 ) x [, 0], 4 ( + x)(4 + 6x + 2x2 20x 3 + 2x 4 ) x [0, ], 6 ( 2 + x)( + x)( 29 + 79x 56x2 + 2x 3 ) x [, 2], 24 ( 3 + x)( 2 + x)( 80 + 228x 92x2 + 2x 3 ) x [2, 3], 0 x R \ [ 2, 3] [BVII] The support is [ 3, 4] Similarly, we consider the following system of equations V 7 (x) [φ(x 3), φ(x 2), φ(x ), φ(x), φ(x + ), φ(x + 2), φ(x + 3)] T = [, 0, 0, 0, 0, 0, G(x)] T, (24) where V 7 (x) is a 7 7 Vandermonde matrix and G(x) = x(x )p(x) 4

2 RKP([-22];2;2) 4 3 DX RKP([-33];2;4) DDX RKP([-33];2;4) 2 RKP([-33];2;4) Y 08 06 04 02 0-2 - 0 2 X 0-3 -2-0 2 3 X - Y 2-2 -3-4 Y 08 06 04 02 0-3 -2-0 2 3 X -02 4 3 2 DX RKP([-33];2;4) DDX RKP([-33];2;4) 2 08 RKP([-44];2;6) 2 DX RKP([-44];2;6) DDX RKP([-44];2;6) 0-3 -2-0 2 3 X - Y Y 06 04 0-4 -2 0 2 4 X Y -2 02-2 -3-4 0-4 -2 0 2 4 X -4 Figure 2: φ ([ 2,2];2;2) (x), φ ([ 3,3];2;4) (x), φ ([ 4,4];2;6) (x), and their first and second derivatives VII If p(x) = 0, the system (24) yields the unique C 0 -RPP shape function φ ([ 3,4];0;6) (x) of reproducing order 6 VII2 If p(x) = 2 + 24x, the system (24) yields a C -RPP function φ ([ 3,4];;5) (x) of reproducing order 5 VII3 If p(x) = 2 6x+2x 2 72x 3, the system (24) yields a C 2 -RPP function φ ([ 3,4];2;5) (x) of reproducing order 5 The RPP shape functions φ ([ 3,4];0;6) (x), φ ([ 3,4];;5) (x) and φ ([ 3,4];2;5) (x) are in appendix The C 2 -RPP shape functions, φ ([ 2,2];2;2) (x), φ ([ 3,3];2;4) (x), φ ([ 4,4];2;6) (x), of reproducing order 2, 4, 6 respectively, and their first and second derivatives are plotted in Fig 2 We test these C 2 -RPP shape functions with respect to a fourth order equation in the following subsection We now close our constructions of translation invariant RPP shape functions with the following remarks Thus far, the system (9) is extended so that the right hand side can be [, 0,, G(x)] T However, in order to construct smooth RPP shape functions with high reproducing order, the same system can be extended in such a way that the extended right hand side can be [0, 0,,, H(x)] T Then, by Lemma 3, the corresponding RPP shape functions do not satisfy the Kronecker delta property 2 Our RPP shape functions have some similarity to the the B-spline functions However, if r be a positive integer, then most of our C r -piecewise polynomial RPP shape functions 5

φ ([ K,K];r;2K 2) of reproducing order 2K 2 could not be found in the literatures on the spline functions Even though the B-spline functions are similar to our RPP particle shape functions, their polynomial reproducing orders are different and the involved linear systems are different Actually, the linear systems for the constructions of RPP particle shape functions are smaller than those for the constructions of the B-spline functions in general ([2]) 3 Those closed form RPP shape functions are much easier in implementing RKPM ([2]) However, we do not claim that our closed form (piecewise polynomial) RPP shape functions have better approximability than those non-closed form RKP shape functions constructed by the standard method (2) and specific window functions For example, polynomial shape functions are not effective in approximating singular functions In ([2], [4]), a tool for checking the approximability of RKP shape functions is given That is, the largest eigenvalue of the matrix related to gradients of the amount failed to have the reproducing property of order k + is one indicator (Theorem 44 of [2]) 33 Numerical Examples for translation invariant RPP shape functions In order to demonstrate the effectiveness of the RPP shape functions constructed in the previous section, we apply some of those RPP basic shape functions to the numerical solutions of a second order and a fourth order differential equations (A) [Second order equation:] The unique C 0 -RPP shape functions of reproducing order k =, 3, 5 are applied to a second order elliptic equation Consider the following model problem d2 u(x) dx2 = f(x) in (, ) (25) u(±) = 0, (26) where f(x) = e x (( x 2 ) 6 24x( x 2 ) 5 2( x 2 ) 5 +20x 2 ( x 2 ) 4 ) Then, u(x) = e x ( x 2 ) 6 is the true solution of this problem Suppose h is the given mesh size such that N = 2/h Z, the degree of freedom, and φ(x) is a basic RPP shape function of reproducing order k := 2K, K =, 2, 3, Then the uniformly distributed particles x,, x N+2K 2 are [ h(k )],, [ h], [ ], [ + h],, [ h], [], [ + h],, [ + h(k )] Let us note that 2(K ) particles are located in the outside of the domain [, ] and two end points are particles Let ψ j (x) : [ K, K] R be a shift function defined by ψ j (x) = hx + x j, where x j is the jth particle Then the RPP shape functions used for the solution of the variational equation of the model problem are {φ ([ K,K];0;2K ) (ψ j (x)) : j =,, (N + 2K 2)} For various sizes of h and various reproducing orders, relative errors in energy norm(%) of the resulting numerical solutions are computed in Table 6

Table : Relative Error in Energy Norm (%) obtained by applying the C 0 -RPP shape functions φ ([,];0;) (x), φ ([ 2,2];0;3) (x), φ ([ 3,3];0;5) (x) of reproducing order k =, 3, 5, respectively to the model problem d2 u dx 2 = f The true solution of the model problem is u(x) = e x ( x 2 ) 6 h k = k = 3 k = 5 02e-0 25e-0 54e-0 8e-0 0e-0 27e-0 8e- 89e-2 05e- 64e-0 e- 36e-3 25e-2 32e-0 3e-2 2e-4 25e-2 6e-0 7e-3 54e-5 Remark 3 It was proved in ([2], [9]) that the interpolation error of a smooth function in energy norm associated with RKP shape functions of reproducing order k is O(h k ) Thus, we plot the relative error in energy norm versus the mesh sizes in log-log scales in Fig 2 The slopes of those lines in this figure are close to, 3, 5, respectively (B) [Fourth order equation:] We apply the closed form C 2 -RPP basic shape functions of reproducing order k = 2, 4, 6, φ ([ 2,2];2;2) (x), φ ([ 3,3];2;4) (x), φ ([ 4,4];2;6) (x), respectively, to the following fourth order differential equation where d 4 u(x) = f(x) in (, ) (27) dx4 u(±) = d2 u (±) = 0, (28) dx2 f(x) = e x [ ( x 2 ) 6 48( x 2 ) 5 x + 720( x 2 ) 4 x 2 72( x 2 ) 5 3840( x 2 ) 3 x 3 + 440( x 2 ) 4 x + 5760( x 2 ) 2 x 4 5760( x 2 ) 3 x 2 + 360( x 2 ) 4] Then the true solution is u(x) = e x ( x 2 ) 6 Percentage relative errors in energy norm for each case are depicted in Fig 4 Numeric data of these results are also shown in Table 2 4 Piecewise polynomial RPP shape functions corresponding to non-uniformly distributed particles in [0, ) Suppose the domain is [0, ), a, a 2,, a n (a n a n ) are positive real numbers, and the particles are distributed as follows: x 0, x, x 2,, x n, x n+, where x j x j = a j, (j =, 2,, n ), x j x j = a n, (j n), and x 0 = 0 7

RELATIVE ERROR(%) IN ENERGY NORM 0 2 0 0 0 0-0 -2 0-3 0-4 ([-,];C0; Order=) ([-2,2];C0; Order=3) ([-3,3];C0; Order=5) 0-5 0-3 0-2 0-0 0 h(mesh Size) RELATIVE ERROR(%) IN ENERGY NORM 0 2 ([-2,2]; C2; Order=2) ([-3,3]; C2; Order=4) ([-4,4]; C2; Order=6) 0 0 0 0-0 -2 0-3 0-4 0-5 Dxxxx u = f 005 0 05 02025 h(mesh SIZE) Figure 3: (Left) The relative error in Energy Norm (%) obtained by applying the C 0 -RPP shape functions φ ([,];0;) (x), φ ([ 2,2];0;3) (x), φ ([ 3,3];0;5) (x) of reproducing order k =, 3, 5, respectively to the model problem d2 u = f (Right) The relative error (%) in Energy Norm obtained dx 2 by applying the C 2 -RPP shape functions of reproducing order 2, 4, 6, respectively, to the model fourth order problem d4 u = f In both figures, the relative errors are plotted with respect to dx 4 the mesh size h in log-log scale Table 2: The relative error in Energy Norm (%) of the fourth order equation, d4 u = f, obtained dx 4 by using C 2 -RPP shape functions of reproducing order 2, 4, 6, respectively The true solution is u(x) = e x ( x 2 ) 6 h φ ([ 2,2];2;2) (x) φ ([ 2,2];2;4) (x) φ ([ 2,2];2;6) (x) 02e-0 5279e-0 657e-0 594e-0 0e-0 3386e-0 322e-0 46e- 05e- 860e-0 452e- 94e-2 25e-2 956e-0 595e-2 926e-4 25e-2 482e-0 795e-3 575e-4 8

The actual coordinates of particles are x 0 = 0, x = a, x 2 = a + a 2,, x n = n j= a j, x n+ = a n + n j= a j, x n+2 = 2a n + n j= a j, and so on x 0 x x 2 x 3 x n x n For each particle x j, we will construct a C 0 -piecewise polynomial RPP shape function φ (xj )(x) of reproducing order 2K, as well as a C r - piecewise polynomial RPP shape function of reproducing order 2K 2, for r In the previous section, it was shown that the unique translation invariant C 0 -piecewise polynomial basic RPP shape functions with reproducing order 2K must have support as large as [ K, K], where K is a positive integer Thus, in order to construct piecewise polynomial RPP shape functions associated with non-uniformly distributed particles in [0, ), we first give the conditions on an individual shape function and the set of particles that should be in the support of this shape function In the previous section, the translation invariant C 0 -RPP shape functions corresponding to uniformly distributed particles in (, ), have supports which consist of 2K consecutive intervals However, when the particles are restricted to be in the interior of the domain or on the boundary of the domain, the supports of RPP shape functions corresponding to particles near boundary have various lengths For example, the support of the shape function corresponding to the boundary particle consists of K subintervals In this section, we will use the following notations for RPP shape functions φ (xj )(x) := the shape function associated with the particle x j (the shape function in the space coordinate system) φ j (x) := the shape function centered at 0, obtained by translating φ (xj )(x) (the shape function in the reference coordinate system) Of course, these RPP shape functions corresponding to non-uniformly distributed particles are not translation invariant 4 C 0 -piecewise polynomial RPP shape functions for non-uniformly distributed particles in [0, ) For a clear description of constructing C 0 -RPP shape functions associated with the particles that are non-uniformly distributed in [0, ), we start with a specific example in which K = 2 In this example, we also assume that a = 4, a 2 = 4, a 3 = 2 and a 4 = are the step sizes for non-uniformly distributed particles In this case, the coordinates of the related non-uniformly distributed particles on [0, ) are x 0 = 0, x = 4, x 2 = 2, x 3 =, x 4 = 2, x 5 = 3, 9

2 0 C -RKP shape function for K=2 2 0 C -RKP shape functions for K = 2 08 06 node = 0 node = /4 node = /2 08 06 φ (x ) node= node=2 node=3 Y 04 02 φ(x) 0 φ (x-/4) /4 Y 04 02 φ (x ) 2 φ (x ) 3 0-02 0 2 3 X φ (x-/2) /2 0 0 2 3 4 5 X -02 2 C -RKP shape functions for K=2 2 C -RKP shape functions for K=2 08 08 (node=) (node=2) (node=3) Y 06 04 (node=0) (node=/4) (node=/2) Y 06 04 02 02 0-02 0 2 3 X 0 0 2 3 4 5 X -02 Figure 4: (Up Left:) The graph of the C 0 RPP shape function of order 3, φ (x0 )(x), φ (x =/4)(x)φ (x2 =/2)(x); (Up Right:) The graph of the C 0 RPP shape function of order 3, φ (x3 =)(x), φ (x4 =2)(x), φ (x5 =3)(x); (Down Left:) The graph of the C RPP shape function of order 2, φ (x0 )(x), φ (x =/4)(x)φ (x2 =/2)(x); (Down Right:) The graph of the C RPP shape function of order 2, φ (x3 =)(x), φ (x4 =2)(x), φ (x5 =3)(x) Let us note that these shape functions satisfy the Kronecker delta property, however parts of graphs are higher than Actually, the maximum vale of these shape functions do not occur at nodal points 20

Let us note that the C 0 -RPP shape functions φ (xj ) corresponding to the above particles have the reproducing order 2K = 3 if and only if, for all x [0, ), (x x j ) α φ (xj )(x) = δ0 α, for α = 0,, 2, 3 (29) j=0 In order for the above system of four equations to be solvable, we consider this system separately on each of subintervals [0, /2], [/2, ], [, 2], [2, 3], [3, 4],, with four shape functions corresponding to particles in the the following index sets: Λ 0 = {x 0, x, x 2, x 3 }, Λ = {x, x 2, x 3, x 4 }, Λ 2 = {x 2, x 3, x 4, x 5 }, Λ 3 = {x 3, x 4, x 5, x 6 }, Λ 4 = {x 4, x 5, x 6, x 7 }, (I-C 0 :) Let us start with the index set Λ 4 = {x 4 = 2, x 5 = 3, x 6 = 4, x 7 = 5} Since the particle in this set is uniformly distributed, and the radius of the support is K = 2, supp φ (xj )(x) = [x j 2, x j + 2], j = 4, 5, 6, 7 Hence, the intersection of supports of φ (xj ), x j Λ 4 is [3, 4] If x (3, 4), then the system (29) becomes the following system for 4 unknowns: x j Λ 4 (x x j ) α φ (xj ) [3,4] (x) = δ α 0, for α = 0,, 2, 3 Solving this equation, we have φ (x4 ) [3,4] (x) = (x 5)(x 4)(x 3), 6 φ (x5 ) [3,4] (x) = (x 5)(x 4)(x 2), 2 φ (x6 ) [3,4] (x) = (x 5)(x 3)(x 2), 2 (30) φ (x7 ) [3,4] (x) = (x 4)(x 3)(x 2) 6 Using the unique C 0 -RPP shape function of reproducing order 3 corresponding to uniformly distributed particles that is defined by (6), one can easily verify that φ (x4 ) [3,4] (x) = f 4,(2;0;3) (x 2), φ (x5 ) [3,4] (x) = f 3,(2;0;3) (x 3), φ (x6 ) [3,4] (x) = f 2,(2;0;3) (x 4), φ (x7 ) [3,4] (x) = f,(2;0;3) (x 5) By a similar manner, the solution functions of (29) for x [k, k + ], k 3, with respect to an appropriate index set Λ k+ of four particles are translations of φ ([ 2,2];0;3) (x) defined by (6) Thus, we assign the translated RPP shape function φ ([ 2,2];0;3) (x x j ) to each particle x j if j 6 2

(II-C 0 :) Next, if x (2, 3), then we use the set Λ 3 = {x 3 =, x 4 = 2, x 5 = 3, x 6 = 4} so that the system (29) becomes the following system for four unknowns: x j Λ 3 (x x j ) α φ (xj ) [2,3] (x) = δ α 0, α = 0,, 2, 3 The solutions of this system are φ (x3 ) [2,3] (x) = (x 4)(x 3)(x 2), 6 φ (x4 ) [2,3] (x) = (x 4)(x 3)(x ), 2 φ (x5 ) [2,3] (x) = (x 4)(x 2)(x ), 2 (3) φ (x6 ) [2,3] (x) = (x 3)(x 2)(x ) 6 (III-C 0 :) If x (, 2), then we use the index set Λ 2 = { x 2 = 2, x 3 =, x 4 = 2, x 5 = 3 } Then the system (29) becomes the following system for four unknowns: x j Λ 2 (x x j ) α φ (xj ) (,2) (x) = δ α 0, α = 0,, 2, 3 The solutions of this system are φ (x2 ) [,2] (x) = 8 (x 3)(x 2)(x ), 5 φ (x3 ) [,2] (x) = (x 3)(x 2)(2x ), 2 φ (x4 ) [,2] (x) = (x 3)(x )(2x ), 3 φ (x5 ) [,2] (x) = (x 2)(x )(2x ) 0 (IV-C 0 :) If x (/2, ), then we use the index set Λ = { x = 4, x 2 = 2, x 3 =, x 4 = 2 } Then the system (29) becomes the following system for four unknowns: x j Λ (x x j ) α φ (xj ) (/2,) (x) = δ α 0, α = 0,, 2, 3 The solutions of this system are φ (x ) [,] (x) = 32(x 2)(x )(2x ) 2 2 φ (x2 ) [ 2,] (x) = 4 (x 2)(x )(4x ) 3 φ (x3 ) [ 2,] (x) = (x 2)(2x )(4x ) 3 φ (x4 ) [,] (x) = (x )(2x )(4x ) 2 2 22 (32) (33)

(V-C 0 :) Thus far, for x (/2, ), we have solved (x x j ) α φ (xj )(x) = δ0 α, α = 0,, 2, 3 j=0 In order to get the RPP shape functions that satisfy the above system for all x [0, ), we finally use the index set Λ 0 = {0, 4, 2, } as follows: for x [0, /2), solve the following system x j Λ 0 (x x j ) α φ (xj ) [0, 2 ] (x) = δα 0, α = 0,, 2, 3 Then the solutions of this system for the four unknown functions, are φ (x0 ) [0, ] (x) = (x )(2x )(4x ) 2 φ (x ) [0, 2 ] (x) = 32 (x )x(2x ) 3 φ (x2 ) [0, ] (x) = 4(x )x(4x ) 2 φ (x3 ) [0, 2 ] (x) = x(2x )(4x ) 3 (34) Now, assembling the solutions listed in Eqns (30), (3), (32), (33), and (34), we define the RPP shape functions of reproducing order 3 corresponding to the particles, x 0, x, x 2, x 3, x 5,, that have the following supports, respectively, supp φ (x0 ) = [0, /2], supp φ (x ) = [0, /2] [/2, ], supp φ (x2 ) = [0, /2] [/2, ] [, 2], supp φ (x3 ) = [0, /2] [/2, ] [, 2] [2, 3], supp φ (x4 ) = [/2, ] [, 2] [2, 3] [3, 4], supp φ (x5 ) = [, 2] [2, 3] [3, 4] [4, 5], supp φ (xj ) = [x j 2, x j + 2], if j 6 (35) Since all partially generated RPP shape functions satisfy the Kronecker delta property(ie φ (xj )(x i ) = δj i for all i, j 0), the assembled shape functions are continuous piecewise polynomials that satisfy the Kronecker delta property Adopting those polynomials f i,(2;0;3) defined by (6), the assembled C 0 -RPP shape functions of reproducing order 3 are as follows: φ (x0 )(x) = φ 0 (x) and φ 0 (x) is as follows: φ 0 (x) = { (x )(2x )(4x ) x [0, 2 ] 0 x [0, 2 ] 23

2 φ (x )(x) = φ (x /4) and φ (x) is as follows: 3 (4x 3)(4x )(4x + ) x [ 4, 4 ] φ (x) = 2 (4x 7)(4x 3)(4x ) x [ 4, 3 4 ] 0 x [ 4, 3 4 ] 3 φ (x2 )(x) = φ 2 (x /2) and φ 2 (x) is as follows: (2x )(2x + )(4x + ) x [ 2, 0] φ 2 (x) = 3 (2x 3)(2x )(4x + ) x [0, 2 ] 5 (2x 5)(2x 3)(2x ) x [ 2, 3 2 ] 0 x [ 2, 3 2 ] 4 φ (x3 )(x) = φ 3 (x ) and φ 3 (x) is as follows: 3 (x + )(2x + )(4x + 3) x [, 2 ] 3 (x )(2x + )(4x + 3) x [ 2, 0] φ 3 (x) = 2 (x 2)(x )(2x + ) x [0, ] f 4,(2;0;3) (x) x [, 2] 0 x [, 2] 5 φ (x4 )(x) = φ 4 (x 2) and φ 4 (x) is as follows: 2 (x + )(2x + 3)(4x + 7) x [ 3 2, ] 3 (x )(x + )(2x + 3) x [, 0] φ 4 (x) = f 3,(2;0;3) (x) x [0, ] f 4,(2;0;3) (x) x [, 2] 0 x [ 3 2, 2] 6 φ (x5 )(x) = φ 5 (x 3) and φ 5 (x) is as follows: (x + )(x + 2)(2x + 5) x [ 2, ] 0 f 2,(2;0;3) (x) x [, 0] φ 5 (x) = f 3,(2;0;3) (x) x [0, ] f 4,(2;0;3) (x) x [, 2] 0 x [ 2, 2] 7 The shape function corresponding to the particle x j (j 6) is φ (xj )(x) = φ ([ 2,2];0;3) (x x j ) 24

The graph of these shape functions φ (xj )(x), j = 0,, 2, 3, 4, 5, 6, are depicted in Fig 4 The construction of RPP shape functions for the general case are similar to the above example Let us briefly describe the general procedure The C 0 -RPP shape functions φ (xj ) corresponding to non-uniformly distributed particles in [0, ) have the polynomial reproducing order 2K if and only if, for all x [0, ), (x x j ) α φ (xj )(x) = δ0 α for α = 0,, 2, 3,, 2K (36) j=0 In order for the above system of 2K equations to be solvable, we consider this system on the subintervals, as in the above example, with 2K shape functions corresponding to particles in the the following index sets Λ 0 = {x 0, x,, x 2K }, Λ = {x, x 2,, x 2K }, Λ 2 = {x 2, x 3,, x 2K+ }, Λ n = {x n, x n+,, x n+2k }, (ng) : Suppose the particles in Λ n are uniformly distributed and are not affected by the non-uniformly distributed particles The intersection of supports of the RPP shape functions for the particles in Λ n is [x n+k, x n+k ] For x [x n+k, x n+k ], we solve the system of 2K equations for 2K unknowns: x j Λ n (x x j ) α φ (xj ) [xn+k,x n+k ] (x) = δ α 0, α = 0,, 2, 3,, 2K ((n )G) : For x [x n+k 2, x n+k ], we solve the system of 2K equations for 2K unknowns: x j Λ n (x x j ) α φ (xj ) [xn+k 2,x n+k ] (x) = δ α 0, α = 0,, 2, 3,, 2K (G) : For x [x K, x K+ ], we solve the system of 2K equations for 2K unknowns: x j Λ (x x j ) α φ (xj ) [xk,x K+ ] (x) = δ α 0, α = 0,, 2, 3,, 2K (0G) : Finally, for x [0, x K ], we solve the system of 2K equations for 2K unknowns: x j Λ 0 (x x j ) α φ (xj ) [0,xK ] (x) = δ α 0, α = 0,, 2, 3,, 2K By a similar method to the above example, assembling those partially defined RPP shape functions in the steps (ng), ((n )G),, (G), and (0G) together, we have the required C 0 - RPP shape functions corresponding to non-uniformly distributed particles x j (j 0) in [0, ) 25

42 C -piecewise polynomial RPP shape functions for non-uniformly distributed particles in [0, ) In the construction of C 0 -RPP shape functions in the previous section, we solved the system of 2K equations for 2K unknowns on each interval [0, x K ], [x K, x K+ ],, [x n+k, x n+k ], Without any restrictions, the solution functions are assembled to be C 0 -RPP shape functions of reproducing order 2K However, as we discussed in section 3, in order for the assembled shape functions to be C r, r > 0 functions, we have to impose some conditions to the system of 2K equations for 2K unknowns on each of the above intervals More specifically, we need to impose some conditions so that the solution functions have the same derivative at the nodal points x j, j > 0 In order to impose such conditions for smoothness at particles, it is necessary to sacrifice the reproducing order by, by imposing a polynomial G(x) on the right hand side of the last equation as follows: { j=0 (x x j) α φ (xj )(x) = δ0 α, α = 0,, 2,, 2K 2, j=0 (x x j) 2K φ (xj )(x) = G(x) (37) Then, obviously, the shape functions that are solutions of the system (37) of 2K equations have the reproducing property of order 2K 2 Here G(x) will be properly selected on each interval so that the solution of this system can be C r, r > 0, functions In this section, we sketch how to construct C -RPP shape functions with polynomial reproducing order 2K 2 In order to make the system (37) being coupled with 2K unknowns on each interval, we use the same subsets of particles, Λ 0, Λ,, Λ n,, as before For specific construction of C -RPP shape functions, we describe it for the RPP shape functions with reproducing order 2K 2, K = 2, corresponding to the following non-uniformly distributed particles: x 0 = 0, x = 4, x 2 = 2, x 3 =, x 4 = 2, x 5 = 3, Following the same argument as the previous subsection, we consider the system (37) separately on each of subintervals [0, /2], [/2, ], [, 2], [2, 3], [3, 4],, with four shape functions corresponding to particles in the the following index sets: Λ 0 = {x 0, x, x 2, x 3 }, Λ = {x, x 2, x 3, x 4 }, Λ 2 = {x 2, x 3, x 4, x 5 }, Λ 3 = {x 3, x 4, x 5, x 6 }, Λ 4 = {x 4, x 5, x 6, x 7 }, If we consider only the particles corresponding to the index set Λ k, the system (37) can be rewritten as follows: { x j Λ k (x x j ) α φ (xj )(x) = δ0 α, α = 0,, 2, x j Λ k (x x j ) 3 (38) φ (xj )(x) = G k (x) 26

(I-C :) By the same reason as the previous section, if we start with the index set Λ 4 = {x 4 = 2, x 5 = 3, x 6 = 4, x 7 = 5}, then we only need to solve the system (38) for x [3, 4] By Lemma 3, in order for this system being uniquely solvable, 3 and 4 should be zeros of G 4 (x) Solving the constrained system (38) for G 4 (x) = (x 4)(x 3)(7 2x), Λ 4, and x [3, 4], we have the following solutions: φ (x4 ) [3,4] (x) = 2 (x 4)2 (x 3), φ (x5 ) [3,4] (x) = 2 (x 4)(3x2 20x + 3), φ (x6 ) [3,4] (x) = 2 (x 3)(3x2 22x + 38), (39) φ (x7 ) [3,4] (x) = 2 (x 4)(x 3)2 Let us note that in this section, the choice of the constraint functions G k (x) are generally not unique Using the C -RPP shape function, g j,(2;;2), j = 4, defined by (7), one can see that those functions in (39) can be written as follows: φ (x4 ) [3,4] (x) = g 4,(2;;2) (x 2), φ (x5 ) [3,4] (x) = g 3,(2;;2) (x 3), φ (x6 ) [3,4] (x) = g 2,(2;;2) (x 4), φ (x7 ) [3,4] (x) = g,(2;;2) (x 5) (II-C :) Solving the system (38) for G 3 (x) = (x 3)(x 2)(5 2x), Λ 3, and x [2, 3], we have the following solutions: φ (x3 ) [2,3] (x) = 2 (x 3)2 (x 2) φ (x4 ) [2,3] (x) = 2 (x 3)(3x2 4x + 4) φ (x5 ) [2,3] (x) = 2 (x 2)(3x2 6x + 9) (40) φ (x6 ) [2,3] (x) = (x 3)(x 2)2 2 (III-C :) Solving the system (38) for G 2 (x) = (x 2)(x )(2 3 2 x), Λ 2, and x [, 2], we have the following solutions: φ (x2 ) [,2] (x) = 4 3 (x 2)2 (x ) φ (x3 ) [,2] (x) = 2 (x 2)(5x2 4x + 7) φ (x4 ) [,2] (x) = 3 (x )(5x2 7x + ) (4) φ (x5 ) [,2] (x) = (x 2)(x )2 2 27

(IV-C :) Solving the system (38) for G (x) = (x )(x 2 )( 3 2 5 2 x), Λ, and x [/2, ], we have the following solutions: φ (x ) [,] (x) = 6 2 3 (x )2 (2x ) φ (x2 ) [ 2,] (x) = 4 3 (x )(4x2 20x + 5) φ (x3 ) [ 2,] (x) = 3 (2x )(4x2 25x + 8) φ (x4 ) [,] (x) = (x )(2x )2 2 3 (42) (V-C :) Finally, solving the system (38) for G 0 (x) = (x )(x 2 )(x 4 )x(6x ), Λ 0, and x [0, /2], we have the following solutions: φ (x0 ) [0, 2] (x) = (x )(2x )2 (3x + )(4x ) φ (x ) [0, 2] (x) = 8 3 (x )x(2x )(24x2 0x 3) φ (x2 ) [0, 2] (x) = 2(x )x(4x )(2x2 8x ) φ (x3 ) [0, 2] (x) = 3 x2 (2x )(4x )(6x 7) (43) By assembling those solutions of (39), (40), (4), (42), and (43), we obtain the C -RPP shape functions of reproducing order 2 corresponding to the the following non-uniformly distributed particles x 0 = 0, x = 4, x 2 = 2, x 3 =, x 4 = 2, x 5 = 3, Using those polynomials g j,(2;;2) defined by (7), the assembled C -RPP shape functions of reproducing order 2 that satisfy the Kronecker delta property are as follows: φ (x0 ) = φ 0 (x 0), where φ 0 (x) is as follows: φ 0 (x) = 2 φ (x ) = φ (x /4), where φ (x) is as follows: { (x )(2x ) 2 (3x + )(4x ) x [0, 2 ] 0 x [0, 2 ] 6 (4x 3)(4x )(4x + )(2x2 + x 2) x [ 4, 4 ] φ (x) = 6 (4x 3)2 (4x ) x [ 4, 3 4 ] 0 x [ 4, 3 4 ] 28