the closed form reproducing polynomial particle shape functions...
TRANSCRIPT
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, 200-701, Korea
Jaewoo JeongDepartment 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 Kro-necker delta property are not available in simple forms. Thus, in this paper, we constructhighly regular piecewise polynomial Reproducing Polynomial Particle (RPP) shape func-tions that satisfy the Kronecker Delta Property. Like RKP shape functions, the RPP shapefunctions 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; Re-producing kernel particle(RKP) shape functions; Reproducing kernel particle method(RKPM);Translation invariant particles; Non-uniformly distributed particles.
∗Corresponding author. Tel.: +1-704-687-4930; fax: +1-704-687-6415; E-mail: [email protected]†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
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1 Introduction
The Finite Element Method (FEM) has been a powerful tool in solving many difficult scienceand engineering problems, especially when the solution domains have complex geometry ([5],[25]). However, there are several difficulties arising in implementing this method. The prominentdifficulties include the mesh refinements and constructing higher order interpolation fields.
In order to relax the constraints of the conventional FEM, several generalized finite ele-ment methods (GFEM), that use the meshes minimally or do not use the meshes at all, wererecently introduced. Among many names of GFEM ([1],[2],[3],[4]), those methods related tothis paper are Element Free Galerkin Method (EFGM) ([1],[9],[10],[11], [12],[15],[16],[17]), h-pCloud Method([7],[8]), Partition of Unity Finite Element Method (PUFEM)([18],[23],[24]), andReproducing Kernel Element Method (RKEM) ([12],[13],[14]).
It was shown ([2],[9]) that RKP shape functions with the reproducing property of high orderensure a good approximability in Reproducing Kernel Particle Method (RKPM) ([9],[12],[15],[16],[17]). However, these RKP shape functions have not been given in a simple form, but usuallyknown as the product of window functions and the solutions of the functional linear systems([12],[13]). Moreover, in application of PU shape functions, the Shepard-type PU functions havebeen employed. In other words, neither the RKP shape functions nor PU shape functions arein a simple form. Thus, integrations related to RKP shape functions are not very accurateand require lengthy function evaluations. Furthermore, RKP shape functions do not satisfythe Kronecker delta property in general. Thus, they have difficulties in dealing with Direchletboundary conditions.
In this paper, we construct highly regular piecewise polynomial reproducing polynomialparticle (RPP) shape functions that reproduce the complete polynomials of any given order andalso satisfy the Kronecker delta property.
The RKP shape functions are constructed with respect to specific window functions, whereaswhat we call the RPP(Reproducing Polynomial Particle) shape functions in this paper haveno 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 maysay that the RPP shape functions are special cases of RKP shape functions. The differencebetween RPP and RKP will be elaborated in section 3.
This paper is organized as follows. In section two, the terminologies and notations used inthis paper are introduced. A simple example for motivation of our constructions is provided.
In section three, for r = 0, 1, 2, 3, we constructed the translation invariant piecewise polyno-mial Cr-RPP shape functions corresponding to uniformly distributed particles in (−∞,∞).We also prove the uniqueness theorem for the piecewise polynomial C0-RPP shape functionsof maximal reproducing order that satisfy the Kronecker delta property. Effectiveness of RPPshape functions are tested with the fourth order as well as the second order one dimensionalelliptic equation.
In section 4, for r = 0, 1, 2, 3, we construct the piecewise polynomial Cr-RPP shape functionscorresponding to non-uniformly distributed particles in [0,∞) (or in a compact set [a, b]).
In appendix, we place the proof of Lemma 3.1 that is used for our construction of RPP shape
2
functions. Moreover, for reference purpose, RPP shape functions of high reproducing order withvarious smoothness are listed for the particles are in (−∞,∞) as well as in [0,∞). Finally, forthe two dimesional extension of RPP shape functions, we refer to ([19],[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 ∈ Zn are multi indices and x = (1x,2 x, .., nx), xj = (1xj ,
2xj , ..,nxj)
denote points in Rn. However, if there are no confusions, we also use the conventional notation
for the points in Rn or Z
n as x = (x1, x2, · · · , xn) and α = (α1, α2, · · · , αn). By α ≤ β, we meanα1 ≤ β1, ..., αn ≤ βn. We also use the following notations: (x − xj)
α := (1x − 1xj)α1 ...(nx −
nxj)αn , |α| := α1 + α2 + · · · + αn. Let Ω be a domain in R
n. For any nonnegative integer m,Cm(Ω) denotes the space of all functions φ such that φ together with all their derivatives Dαφof orders |α| ≤ m, are continuous on Ω. The support of φ is defined by
supp φ = x ∈ Ω : φ(x) 6= 0.
In the following, a function φ ∈ Cm(Ω) is said to be a Cm- function.A weight function(or window function) is a non negative continuous function with compact
support and is denoted by w(x). In Rn, 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=1w(xi), where‖x‖2 = x21 + · · ·+ x2n.
Adopting those terminologies and notations of ([2]), we have the following: For j = (j1, j2, · · · , jn) ∈Zn, and the mesh size 0 < h ≤ 1, let
xhj = (j1h, · · · , jnh) = hj.
Then the points xhj are called uniformly distributed particles. Let φ be a continuous functionwith compact support that contains the origin 0. Then the particle shape functions associatedto the uniformly distributed particles is defined by
φhj (x) = φ(
x− jhh
) = φ(x1 − j1h
h, · · · , xn − jnh
h),
for j ∈ Zn and 0 < h ≤ 1. Then these particle shape functions are translation invariant in
the sense that
xhi+j = xhi + xhj , φhj (x− ih) = φh
i+j(x).
In this paper, we assume that the basic shape functions are translation invariant on the uni-formly distributed particles, unless stated otherwise. The particle shape functions associatedwith non-uniformly distributed particles are constructed in section 4.
3
Without loss of generality, in the theoretical developments and constructions of particleshape functions, we assume that h = 1 and hence φh
j (x) = φ(x− j).Let Λ be an index set and Ω denotes a bounded domain in R
n. Let xj : j ∈ Λ be a set ofuniformly (or non-uniformly) distributed points in R
n, that are called particles.
Definition 2.1. Let k be a nonnegative integer. Then the functions φj(x) corresponding to theparticles xj , j ∈ Λ are called the RPP(reproducing polynomial particle) shape functions with thereproducing property of order k (or simply, “of reproducing order k”) if and only if it satisfiesthe following condition: for x ∈ Ω ⊂ R
n,
∑
j∈Λ
(xj)αφj(x) = xα, and for 0 ≤ |α| ≤ k, |α| = α1 + · · ·+ αn. (1)
The RKP(Reproducing Kernel Particle) shape function, associated with the particle xj , isconstructed by
φj(x) = w(x− xj)∑
0≤|α|≤k
(x− xj)αbα(x) (2)
where bα(x) are chosen so that (1) is satisfied and w(x) is a window function. This gives rise toa linear system in bα(x), namely
∑
0≤|α|≤k
mα+β(x)bα(x) = δ0|β| for 0 ≤ |β| ≤ k, (3)
where δ0|β| is the Kronecker delta, and
mα(x) =∑
j∈Z
w(x− xj)(x− xj)α. (4)
For one dimensional case, this system can be written as
M(x) · [b0(x), b1(x), · · · , bk(x)]T = [1, 0, · · · , 0]T ,
where
M(x) =∑
j∈Z
w(x− xj)
1(x− xj)1(x− xj)2
...(x− xj)k
[1, (x− xj)1, · · · , (x− xj)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 1 by using (2)and (3). For example, let w(x) be a smooth window function whose support is [−1, 1]. Then forx ∈ [0, 1], the moment matrix is
M(x) =
[w(x) + w(x− 1) x · w(x) + (x− 1) · w(x− 1)x · w(x) + (x− 1) · w(x− 1) x2 · w(x) + (x− 1)2 · w(x− 1)
]
and detM(x) = w(x)w(x− 1) is non zero except at x = 0, 1.Thus, for x ∈ (0, 1), the solutions of the linear system (3) are
b0(x) = [x2w(x) + (x− 1)2w(x− 1)]/w(x)w(x− 1),
b1(x) = −[xw(x) + (x− 1)w(x− 1)]/w(x)w(x− 1).
Hence, by the relation (2), φ(x) = w(x)[b0(x) + xb1(x)] = 1 − x, φ(x − 1) = w(x − 1)[b0(x) +(x − 1)b1(x)] = x. Now we extend φ(x) onto [0, 1] by defining φ(0) = 1, and φ(1) = 0. Thenthe translation invariant RKP shape function of reproducing order 1, associated to any weightfunction w(x) with support [−1, 1], is the hat function.
In this example, for the construction of the RKP shape function, we only used the supportsize of w(x), but did not use the nature of w(x). Hence the resulting RKP function φ(x) isindependent of the smoothness of the weight function w(x). This is because the interior of suppw(x) contains only one particle and hence M(x) is singular at x = 0, 1.
It is stated in ([2] and [9]) that if w(x) ∈ Cq, then φ(x) ∈ Cq, provided that for all x ∈ Rn,
M(x) is nonsingular. In fact, if M(α)(x) is the matrix obtained by replacing the α-th column
of M(x) with (1, 0, 0, · · · , 0)T , then we have bα(x) = [detM(α)(x)/ detM(x)] ∈ Cq wheneverw(x) ∈ Cq. Hence the relation (2) implies φ(x) ∈ Cq.
Now, we are motivated to find another method constructing RPP shape functions withoutgetting weight functions involved. For this purpose, we consider the following characterization.
By applying a similar argument to ([2],[9]), one can show that (1) is equivalent to∑
j∈Z
(x− xj)βφj(x) = δ0|β|, for 0 ≤ |β| ≤ k and x ∈ Rn. (5)
This characterization of the RPP shape functions have no direct relation with the weightfunctions. Thus, by using this equivalent definition, various closed form RPP shape functionsare constructed in the following sections.
3 Piecewise polynomial RPP shape functions corresponding touniformly distributed particles in (−∞,∞)
In this section, by using (5), we construct translation invariant piecewise polynomial particleshape functions that have the reproducing property of any desired reproducing order and alsosatisfy the Kronecker delta property.
The following lemma is essential for the construction of RPP shape functions. The technicalproof of this lemma is in appendix.
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Lemma 3.1. Suppose K is a positive integer and f(x) is a function defined on
Ω = [−K,−K + 1) ∪ · · · ∪ (−1, 0) ∪ (0, 1) ∪ · · · ∪ (K − 2,K − 1) ∪ (K − 1,K],
such that, iff |(j,j+1)(x), j = K − 1,K − 2, · · · , 2, 1, 0,−1,−2, · · · ,−K,
are shifted on (0, 1), then they are the solutions of the following system of linear functionalequations:
V (x) · [f(x+ (K − 1)), · · · , f(x+ 1), f(x), f(x− 1), · · · , f(x−K)]T = ei, (6)
where x ∈ (0, 1), ei = [0, 0, · · · , 0, 1, 0, · · · , 0]T , the i-th base vector , and V (x) is the followingVandermonde matrix:
1 · · · 1 1 · · · 1 1(x+ (K − 1)) · · · x+ 1 x · · · (x− (K − 1)) (x−K)
......
......
......
...(x+ (K − 1))2K−1 · · · (x+ 1)2K−1 x2K−1 · · · (x− (K − 1))2K−1 (x−K)2K−1
.
Then
(1) f(x) can be extended to a continuous piecewise polynomial function F (x) defined on Ω, ifthe right hand side vector is ei for an odd number i.
In particular, if the right hand side vector is e1, F (x) satisfies the Kronecker delta property.Moreover, in this case, f(x) is also extendable to a continuous function F (x) that satisfiesthe Kronecker delta property even when f(x) has a non symmetric support (that is, Ω =[−K,K + 1] \ Z or Ω = [−K − 1,K] \ Z).
(2) No continuous extensions are possible, if the right side vector is ei for an even number i.
3.1 Existence and uniqueness of a closed form C0- RPP shape function ofmaximum reproducing order.
Theorem 3.1. There exits a unique translation invariant C0-RPP shape function φ(x), whichsatisfies 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 − 1,
(c) This satisfies the Kronecker delta property: φj(i) = δji .
The unique RPP basic shape function is a piecewise polynomial of degree 2K − 1.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 functionof reproducing order 2K − 1 through the standard construction (2) and(3).
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Proof. For x ∈ (0, 1), a translation invariant RPP shape function of order 2K − 1 satisfies thefollowing system of equations:
K∑
k=−K+1
(x− k)αφ(x− k) = δα0 , α = 0, 1, 2, · · · , 2K − 1. (7)
This linear system can be written in matrix form as follow.
V (x) · [φ(x+ (K − 1)), φ(x+ (K − 2)), · · · , φ(x), · · · , φ(x− (K − 1)), φ(x−K)]T = e1. (8)
Lemma 3.1 shows that the shape function φ(x) determined as the solutions of this system canbe continuously extended to [−K,K] so that the extended function satisfies the Kronecker deltaproperty. Moreover, such a unique extension makes φ(x) to be also satisfied the definition (1) atall integer points. Since the system (8) has a unique solution and the extension of the solutionto [−K,K] is also unique, this is the only RPP basic shape function of order 2K − 1 whosesupport is [−K,K].
On each interval bounded by two consecutive integers, φ(x) is a constant times the determi-nant of (2K − 1)× (2K − 1) minor matrix of V (x), and hence φ(x) is a piecewise polynomial ofdegree 2K − 1.
Related to this existence and uniqueness for C0-RPP shape functions, we have observed thefollowings.
1. Theorem 4.4.1 of [6] is virtually the same as our uniqueness theorem. However, our theoremdid not use a sequence of double knots; hence two proofs are different. Moreover, theunique solution functions of the system (7) are the Lagrange interpolation polynomialscorresponding to the particles −K + 1, · · · , 0, · · · ,K. Thus, our C0-RPP shape functionsare just C0-spline functions.
2. It follows from lemma 3.1 that Theorem 3.1 holds for non-symmetric support [−K,K +1]or [−K − 1,K].
3. If the support of φ(x) is exactly [−1, 1], then by Theorem 3.1, the unique RPP shapefunction of reproducing order 1 is the hat function because the one dimensional RKPshape function of reproducing order 1 constructed by using any window function withsupport [−1, 1] in section 2 is the hat function. In other words, the C0-RPP shape functionis the same as the RKP shape function constructed by using a specific window functionwhenever their supports are [−K,K] and their reproducing order are 2K − 1, for somepositive integer K.
However, suppose φ(x) is a RKP shape function, with reproducing property of order 1,constructed by using a window function w(x) whose support is larger than [−1, 1], forexample, [−1.2, 1.2]. Then, we cannot construct this RKP shape function by applying thearguments in Theorem 3.1.
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4. It is stated in [2] that the regularity of a RKP shape function φ(x) is the same as thatof the associated weight function w(x). This statement is not applicable to the foregoingtheorem because the support is not large enough to make for the moment matrix M(x) tobe non singular for all x ∈ R.
3.2 Constructions of closed form Cm-RPP shape functions (m ≥ 0) that satisfythe Kronecker delta property
In parts [A] and [B] of this subsection, by solving the linear system (8), various one dimensionalCm RPP shape functions of given reproducing orders are constructed.
In section 3.3, the effectiveness of these RPP shape functions are tested by applying themto the second order and the fourth order differential equations, respectively.
In Theorem 3.1, for a given symmetric support [−K,K], we considered RPP shape functionswith the reproducing property of odd order 2K−1. However, by applying similar arguments fornon symmetric support [−K,K+1] (or [−(K+1),K]), one can construct RPP shape functionssatisfying the reproducing property of even order 2K.
Lemma 3.1 makes it possible to construct the unique closed form C0 p.p.(piecewise polyno-mial) RPP shape function of reproducing order 2K−1 that satisfy the Kronecker delta propertywith respect to the given support [−K,K]. However, by sacrificing one order from the maximalreproducing order 2K − 1 (or extending the support to [−(K + 1), (K + 1)] if we want to keepthe reproducing order), for a given integer m > 0, we are able to construct a Cm-p.p. RPPshape function on [−K,K] (or [−K,K + 1]) that satisfies the Kronecker delta property.
For this purpose, we start with a piecewise smooth function f(x) that satisfies
K∑
k=−K+1
(x− k)αf(x− k) = δα0 , α = 0, 1, 2, · · · , 2K − 2, for x ∈ (0, 1). (9)
Then this system is under-determined. In order to make for this system to be an exactlydetermined system, we add an additional equation to this system as follows: for x ∈ (0, 1),
∑Kk=−K+1(x− k)αf(x− k) = δα0 , α = 0, 1, 2, · · · , 2K − 2,∑Kk=−K+1(x− k)2K−1f(x− k) = G(x).
(10)
Then the coefficient matrix of this extended system becomes a 2K × 2K Vandermonde matrix.The right hand side becomes the 2K dimensional column vector
[1, 0, 0, · · · , 0, G(x)]T .If G(0) = 0 and G(1) = 0, then it follows from Lemma 3.1 that the solution vector
(f(x+K − 1), f(x+ (K − 2)), · · · , f(x+ 1), f(x), f(x− 1), · · · , f(x−K))T , x ∈ (0, 1)
for f(x) can be extended to be a continuous function F (x) on [−K,K] that satisfies the Kroneckerdelta property. Thus, we impose the following conditions on G(x):
G(0) = 0 and G(1) = 0. (11)
8
Next, we are going to impose some additional condition on G(x) so that f(x) can be extendedto a C2-shape function.
By differentiating the expanded system (10) twice, we have
V (x) · [f ′′(x+K − 1), · · · , f ′′(x+ 1), f ′′(x), f ′′(x− 1), · · · , f ′′(x−K)]T
= [0, 0, 2, 0, · · · , 0, 0, G′′(x)]T . (12)
Here V (x) is a Vandermonde matrix. Since the right side of the last system has a non-zero entryon the odd number row, Lemma 3.1 implies that the solution vector of this system can be madeto be a continuous function whenever G′′(0) = 0 and G′′(1) = 0. Thus, we impose the followingcondition:
G′′(0) = 0 and G′′(1) = 0. (13)
For the construction of Cr-RPP shape functions (r > 0), we have to start with (10) whenthe freedom parameter function G(x) has the form
G(x) = x(x− 1)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) ∈ Cr),m = the order of the reproducing property.
The graphs of these C0-RPP shape functions are depicted in Fig. 1, from which one canobserve that the RPP functions satisfy the Kronecker delta property.
In the following, we constructed the Cr-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 toany 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].
[A.I] The support is [−1, 1]. The unique RPP shape function of reproducing order 1 whosesupport is [−1, 1] is the following hat function.
φ([−1,1];0;1)(x) =
x+ 1 x ∈ [−1, 0]−x+ 1 x ∈ [0, 1]
0 x ∈ R \ [−1, 1]
[A.II] The support is [−2, 2]. We construct the RPP shape functions of various regularityorder that have the reproducing property of order k ≥ 2.
9
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1 1.5 2-0.2
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-0.2
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3-0.2
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3-0.2
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3 4
Figure 1: The continuous piecewise polynomial RPP shape functions of order k = 1, 2, 3, 4, 5, 6(Left to Right and Top to Bottom).
Consider the following system
V4(x) · [φ(x− 2), φ(x− 1), φ(x), φ(x+ 1)]T = [1, 0, 0, G(x)]T , (14)
where V4(x) is a 4× 4 Vandermonde matrix defined by
V4(x) =
1 1 1 1(x− 2) (x− 1) x (x+ 1)(x− 2)2 (x− 1)2 x2 (x+ 1)2
(x− 2)3 (x− 1)3 x3 (x+ 1)3
. (15)
We already know that G(x) must have a factor x(x − 1). So it suffices to consider thefollowing case:
G(x) = x(x− 1)p(x)
II.1 If p(x) = 0, the system (14) yields the unique C0-RPP shape function φ([−2,2];0;3)(x) ofreproducing order 3.
φ([−2,2];0;3)(x) =
f1,(2;0;3)(x) :=16(x+ 1)(x+ 2)(x+ 3) x ∈ [−2,−1]
f2,(2;0;3)(x) := −12(x− 1)(x+ 1)(x+ 2) x ∈ [−1, 0]f3,(2;0;3)(x) :=
12(x− 2)(x− 1)(x+ 1) x ∈ [0, 1]
f4,(2;0;3)(x) := −16(x− 3)(x− 2)(x− 1) x ∈ [1, 2]
0 x 6∈ [−2, 2].
(16)
10
II.2 If p(x) = 1− 2x, the system (14) yields a C1-RPP shape function of reproducing order 2.
φ([−2,2];1;2)(x) =
g1,(2;1;2)(x) :=12(x+ 1)(x+ 2)2 x ∈ [−2,−1]
g2,(2;1;2)(x) := −12(x+ 1)(3x2 + 2x− 2) x ∈ [−1, 0]g3,(2;1;2)(x) :=
12(x− 1)(3x2 − 2x− 2) x ∈ [0, 1]
g4,(2;1;2)(x) := −12(x− 2)2(x− 1) x ∈ [1, 2]
0 x 6∈ [−2, 2]
(17)
II.3 If p(x) = 1+x−9x2+6x3, the system (14) yields a C2-RPP shape function of reproducingorder 2.
φ([−2,2];2;2)(x) =
−12(x+ 2)3(x+ 1)(2x+ 1) x ∈ [−2,−1],12(x+ 1)(6x4 + 9x3 − 2x+ 2) x ∈ [−1, 0],−12(x− 1)(6x4 − 9x3 + 2x+ 2) x ∈ [0, 1],12(x− 2)3(x− 1)(2x− 1) x ∈ [1, 2],0 x ∈ R \ [−2, 2]
II.4 If p(x) = 1 + x− 30x3 + 45x4 − 18x5, the system (14) yields a C3-RPP shape function ofreproducing order 2.
φ([−2,2];3;2)(x) =
12(x+ 2)4(x+ 1)(6x2 + 9x+ 4) x ∈ [−2,−1],−12(x+ 1)(18x6 + 45x5 + 30x4 + 2x− 2) x ∈ [−1, 0],12(x− 1)(18x6 − 45x5 + 30x4 − 2x− 2) x ∈ [0, 1],−12(x− 2)4(x− 1)(6x2 − 9x+ 4) x ∈ [1, 2],0 x ∈ R \ [−2, 2]
II.5 If p(x) = 1+ x− 105x4 +252x5 − 210x6 +60x7, the system (14) yields a C4-RPP shapefunction of reproducing order 2.
φ([−2,2];4;2)(x) =
−12(x+ 2)5(x+ 1)(20x3 + 50x2 + 44x+ 13) x ∈ [−2,−1],12(x+ 1)(60x8 + 210x7 + 252x6 + 105x5 − 2x+ 2) x ∈ [−1, 0],−12(x− 1)(60x8 − 210x7 + 252x6 − 105x5 + 2x+ 2) x ∈ [0, 1],12(x− 2)5(x− 1)(20x3 − 50x2 + 44x− 13) x ∈ [1, 2],0 x ∈ R \ [−2, 2]
[A.III] The support is [−3, 3].We construct the RPP shape functions of various regularityorder that have the reproducing property of order k ≥ 4. In this case, we have the followingsystem
V6(x) · [φ(x− 3), φ(x− 2), φ(x− 1), φ(x), φ(x+ 1), φ(x+ 2)]T
= [1, 0, 0, 0, 0, G(x)]T , (18)
where V6(x) is the 6× 6 Vandermonde matrix that is similar to (15) and G(x) = x(x− 1)p(x).
11
III.1 If p(x) = 0, then the system (18) yields the unique C0-RPP shape function φ([−3,3];0;5) ofreproducing order 5.
φ([−3,3];0;5)(x) =
f1,(3;0;5) =1120(x+ 1)(x+ 2)(x+ 3)(x+ 4)(x+ 5) x ∈ [−3,−2]
f2,(3;0;5) = − 124(x− 1)(x+ 1)(x+ 2)(x+ 3)(x+ 4) x ∈ [−2,−1]
f3,(3;0;5) =112(x− 2)(x− 1)(x+ 1)(x+ 2)(x+ 3) x ∈ [−1, 0]
f4,(3;0;5) = − 112(x− 3)(x− 2)(x− 1)(x+ 1)(x+ 2) x ∈ [0, 1]
f5,(3;0;5) =124(x− 4)(x− 3)(x− 2)(x− 1)(x+ 1) x ∈ [1, 2]
f6,(3;0;5) = − 1120(x− 5)(x− 4)(x− 3)(x− 2)(x− 1) x ∈ [2, 3]
0 x ∈ R \ [−3, 3]
(19)
III.2 If p(x) = 4− 8x, the system (18) yields a C1-RPP shape function of reproducing order 4.
φ([−3,3];1;4)(x) =
g1,(3;1;4) =1120x(2 + x)(3 + x)2(7 + x) x ∈ [−3,−2],
g2,(3;1;4) = − 124(1 + x)(2 + x)(−24− 3x+ 6x2 + x3) x ∈ [−2,−1],
g3,(3;1;4) =112(1 + x)(12− 12x− 15x2 + 2x3 + x4) x ∈ [−1, 0],
g4,(3;1;4) = − 112(−1 + x)(12 + 12x− 15x2 − 2x3 + x4) x ∈ [0, 1],
g5,(3;1;4) =124(−2 + x)(−1 + x)(24− 3x− 6x2 + x3) x ∈ [1, 2],
g6,(3;1;4) = − 1120(−7 + x)(−3 + x)2(−2 + x)x x ∈ [2, 3],
0 x ∈ R \ [−3, 3]
(20)
III.3 If p(x) = 4 + 4x − 36x2 + 24x3, the system (18) yields a C2-RPP shape function ofreproducing order 4.
φ([−3,3];2;4)(x) =
124(2 + x)(3 + x)3(8 + 5x) x ∈ [−3,−2],− 124(1 + x)(2 + x)(48 + 153x+ 114x2 + 25x3) x ∈ [−2,−1],
112(1 + x)(12− 12x− 3x2 + 38x3 + 25x4) x ∈ [−1, 0],− 112(−1 + x)(12 + 12x− 3x2 − 38x3 + 25x4) x ∈ [0, 1],
124(−2 + x)(−1 + x)(−48 + 153x− 114x2 + 25x3) x ∈ [1, 2],− 124(−3 + x)3(−2 + x)(−8 + 5x) x ∈ [2, 3],
0 x ∈ R \ [−3, 3]
[A.IV] The support is [−4, 4]. Similarly, we consider the following system of functions:
V8(x) · [φ(x− 4), φ(x− 3), φ(x− 2), φ(x− 1), φ(x), φ(x+ 1), φ(x+ 2), φ(x+ 3)]T
= [1, 0, 0, 0, 0, 0, 0, G(x)]T , (21)
where V8(x) is the 8× 8 Vandermonde matrix and G(x) = x(x− 1)p(x).
IV.1 If p(x) = 0, the system (21) yields the unique C0-RPP shape function φ([−4,4];0;7) ofreproducing order 7.
12
IV.2 If p(x) = −36 + 72x, the system (21) yields a C1-RPP shape function φ([−4,4];1;6) ofreproducing order 6.
IV.3 If p(x) = −36+−36x+ i324x2− 216x3, the system (21) yields a C2-RPP shape functionsφ([−4,4];2;6) of reproducing order 6.
The piecewise polynomial RPP shape functions φ([−4,4];0;7), φ([−4,4];1;6), φ([−4,4];2;6) are in ap-pendix.
B: The RPP shape functions with non symmetric support [−K,K+1], [−K−1,K].Because of the symmetry, it suffices to construct RPP shape functions with respect to the
support [−K,K +1]. All RPP shape functions constructed in this section satisfy the Kroneckerdelta property.
[B.V] The support is [−1, 2]. When the support is [−1, 2], we also construct RPP shapefunctions that have the reproducing property of order k ≥ 1. In this case, we have the followingequation
1 1 1(x− 1) x (x+ 1)(x− 1)2 x2 (x+ 1)2
·
φ(x− 1)φ(x)
φ(x+ 1)
=
10
G(x)
(22)
We already know that G(x) must have a factor x(x − 1). So it suffices to consider thefollowing case: G(x) = x(x− 1)p(x).
V.1 If p(x) = 0, the system (22) yields the unique C0-RPP shape function of the reproducingproperty of order 2.
φ([−1,2];0;2)(x) =
12(1 + x)(2 + x) x ∈ [−1, 0],−(−1 + x)(1 + x) x ∈ [0, 1],12(−2 + x)(−1 + x) x ∈ [1, 2],0 x ∈ R \ [−1, 2]
V.2 If p(x) = −1+ 2x, the system (22) yields a C1-RPP shape function of reproducing order 1.
φ([−1,2];1;1)(x) =
−(−1 + x)(1 + x)2 x ∈ [−1, 0],(−1 + x)(−1− 2x+ 2x2) x ∈ [0, 1],−(−2 + x)2(−1 + x) x ∈ [1, 2],0 x ∈ R \ [−1, 2]
V.3 p(x) = −1−2x+10x2−6x3, the system (22) yields a C2-RPP shape function of reproducingorder 1.
φ([−1,2];2;1)(x) =
12(1 + x)(2 + 8x3 + 6x4) x ∈ [−1, 0],−(−1 + x)(1 + 2x+ 2x2 − 10x3 + 6x4) x ∈ [0, 1],12(−2 + x)(−1 + x)(−16 + 40x− 28x2 + 6x3) x ∈ [1, 2],0 x ∈ R \ [−1, 2]
13
[B.VI] The support is [−2, 3]. In this case, we construct the RPP shape functions ofreproducing order k ≥ 3. We consider the following equation
V5(x) · [φ(x− 2), φ(x− 1), φ(x), φ(x+ 1), φ(x+ 2)]T = [1, 0, 0, 0, G(x)]T , (23)
where V5(x) is a 5× 5 Vandermonde matrix and G(x) = x(x− 1)p(x).
VI.1 If p(x) = 0, the system (23) yields the unique C0-RPP shape function of reproducing order4.
φ([−2,3];0;4)(x) =
124(1 + x)(2 + x)(3 + x)(4 + x) x ∈ [−2,−1],−16(−1 + x)(1 + x)(2 + x)(3 + x) x ∈ [−1, 0],14(−2 + x)(−1 + x)(1 + x)(2 + x) x ∈ [0, 1],−16(−3 + x)(−2 + x)(−1 + x)(1 + x) x ∈ [1, 2],124(−4 + x)(−3 + x)(−2 + x)(−1 + x) x ∈ [2, 3],0 x ∈ R \ [−2, 3]
VI.2 If p(x) = 2− 4x, the system (23) yields a C1-RPP shape function of reproducing order 3.
φ([−2,3];1;3)(x) =
124(1 + x)(2 + x)(18 + 11x+ x2) x ∈ [−2,−1],−16(1 + x)(−6 + 3x+ 8x2 + x3) x ∈ [−1, 0],14(−1 + x)(−4− 6x+ 5x2 + x3) x ∈ [0, 1],−16(−2 + x)(−1 + x)(−9 + 2x+ x2) x ∈ [1, 2],124(−3 + x)(−2 + x)(−6− x+ x2) x ∈ [2, 3],0 x ∈ R \ [−2, 3]
VI.3 If p(x) = 2 + 3x − 19x2 + 12x3, the system (23) yields a C2-RPP shape function ofreproducing order 3.
φ([−2,3];2;3)(x) =
− 124(1 + x)(2 + x)(16 + 64x+ 52x2 + 12x3) x ∈ [−2,−1],
16(1 + x)(6− 3x− 3x2 + 16x3 + 12x4) x ∈ [−1, 0],−14(−1 + x)(4 + 6x+ 2x2 − 20x3 + 12x4) x ∈ [0, 1],16(−2 + x)(−1 + x)(−29 + 79x− 56x2 + 12x3) x ∈ [1, 2],− 124(−3 + x)(−2 + x)(−180 + 228x− 92x2 + 12x3) x ∈ [2, 3],
0 x ∈ R \ [−2, 3]
[B.VII] The support is [−3, 4]. Similarly, we consider the following system of equations
V7(x) · [φ(x− 3), φ(x− 2), φ(x− 1), φ(x), φ(x+ 1), φ(x+ 2), φ(x+ 3)]T
= [1, 0, 0, 0, 0, 0, G(x)]T , (24)
where V7(x) is a 7× 7 Vandermonde matrix and G(x) = x(x− 1)p(x).
14
X
Y
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
RKP([-2..2];2;2)
X
Y-3 -2 -1 0 1 2 3
-4
-3
-2
-1
0
1
2
3
4DX RKP([-3..3];2;4)DDX RKP([-3..3];2;4)
X
Y
-3 -2 -1 0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
RKP([-3..3];2;4)
X
Y-3 -2 -1 0 1 2 3
-4
-3
-2
-1
0
1
2
3
4DX RKP([-3..3];2;4)DDX RKP([-3..3];2;4)
X
Y
-4 -2 0 2 40
0.2
0.4
0.6
0.8
1
1.2
RKP([-4..4];2;6)
X
Y-4 -2 0 2 4
-4
-2
0
2
DX RKP([-4..4];2;6)DDX RKP([-4..4];2;6)
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.1 If p(x) = 0, the system (24) yields the unique C0-RPP shape function φ([−3,4];0;6)(x) ofreproducing order 6.
VII.2 If p(x) = −12 + 24x, the system (24) yields a C1-RPP function φ([−3,4];1;5)(x) of repro-ducing order 5.
VII.3 If p(x) = −12−16x+112x2−72x3, the system (24) yields a C2-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];1;5)(x) and φ([−3,4];2;5)(x) are in appendix.The C2-RPP shape functions, φ([−2,2];2;2)(x), φ([−3,3];2;4)(x), φ([−4,4];2;6)(x), of reproducing or-
der 2, 4, 6 respectively, and their first and second derivatives are plotted in Fig. 2. We test theseC2-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 fol-lowing remarks.
1. Thus far, the system (9) is extended so that the right hand side can be [1, 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, 1, · · · , H(x)]T . Then, by Lemma 3.1, the corresponding RPP shape functions do notsatisfy 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 Cr-piecewise polynomial RPP shape functions
15
φ([−K,K];r;2K−2) of reproducing order 2K − 2 could not be found in the literatures on thespline functions. Even though the B-spline functions are similar to our RPP particleshape functions, their polynomial reproducing orders are different and the involved linearsystems are different. Actually, the linear systems for the constructions of RPP particleshape functions are smaller than those for the constructions of the B-spline functions ingeneral ([21]).
3. Those closed form RPP shape functions are much easier in implementing RKPM ([12]).However, we do not claim that our closed form (piecewise polynomial) RPP shape functionshave better approximability than those non-closed form RKP shape functions constructedby the standard method (2) and specific window functions. For example, polynomialshape functions are not effective in approximating singular functions. In ([2], [4]), a toolfor checking the approximability of RKP shape functions is given. That is, the largesteigenvalue of the matrix related to gradients of the amount failed to have the reproducingproperty of order k + 1 is one indicator (Theorem 4.4 of [2]).
3.3 Numerical Examples for translation invariant RPP shape functions
In order to demonstrate the effectiveness of the RPP shape functions constructed in the previoussection, we apply some of those RPP basic shape functions to the numerical solutions of a secondorder and a fourth order differential equations.
(A) [Second order equation:] The unique C0-RPP shape functions of reproducing orderk = 1, 3, 5 are applied to a second order elliptic equation. Consider the following model problem.
− d2
dx2u(x) = f(x) in (−1, 1) (25)
u(±1) = 0, (26)
where f(x) = −ex((1−x2)6−24x(1−x2)5−12(1−x2)5+120x2(1−x2)4). Then, u(x) = ex(1−x2)6is 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 − 1,K = 1, 2, 3, · · · . Then theuniformly distributed particles x1, · · · , xN+2K−2 are
[−1− h(K − 1)], · · · , [−1− h], Â [−1], [−1 + h], · · · , [1− h], [1] Á, [1 + h], · · · , [1 + h(K − 1)].
Let us note that 2(K − 1) particles are located in the outside of the domain [−1, 1] and two endpoints are particles.
Let ψj(x) : [−K,K] → R be a shift function defined by ψj(x) = hx + xj , where xj is thejth particle. Then the RPP shape functions used for the solution of the variational equation ofthe model problem are φ([−K,K];0;2K−1)(ψj(x)) : j = 1, · · · , (N + 2K − 2). For various sizes ofh and various reproducing orders, relative errors in energy norm(%) of the resulting numericalsolutions are computed in Table 1.
16
Table 1: Relative Error in Energy Norm (%) obtained by applying the C0-RPP shape functionsφ([−1,1];0;1)(x), φ([−2,2];0;3)(x), φ([−3,3];0;5)(x) of reproducing order k = 1, 3, 5, respectively to the
model problem −d2udx2 = f. The true solution of the model problem is u(x) = ex(1− x2)6.
h k = 1 k = 3 k = 5
0.2e-0 25.1e-0 5.4e-0 1.8e-0
0.1e-0 12.7e-0 8.1e-1 8.9e-2
0.5e-1 6.4e-0 1.1e-1 3.6e-3
2.5e-2 3.2e-0 1.3e-2 1.2e-4
1.25e-2 1.6e-0 1.7e-3 5.4e-5
Remark 3.1. It was proved in ([2], [9]) that the interpolation error of a smooth function inenergy norm associated with RKP shape functions of reproducing order k is O(hk). Thus, weplot the relative error in energy norm versus the mesh sizes in log-log scales in Fig. 2. Theslopes of those lines in this figure are close to 1, 3, 5, respectively.
(B) [Fourth order equation:] We apply the closed form C2-RPP basic shape functions ofreproducing 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.
d4
dx4u(x) = f(x) in (−1, 1) (27)
u(±1) =d2u
dx2(±1) = 0, (28)
where
f(x) = ex[(1− x2)6 − 48(1− x2)5x+ 720(1− x2)4x2 − 72(1− x2)5 − 3840(1− x2)3x3
+ 1440(1− x2)4x+ 5760(1− x2)2x4 − 5760(1− x2)3x2 + 360(1− x2)4].
Then the true solution is u(x) = ex(1− x2)6. Percentage relative errors in energy norm for eachcase are depicted in Fig. 4. Numeric data of these results are also shown in Table 2.
4 Piecewise polynomial RPP shape functions corresponding tonon-uniformly distributed particles in [0,∞)
Suppose the domain is [0,∞), a1, a2, · · · , an (an 6= an−1) are positive real numbers, and theparticles are distributed as follows: x0, x1, x2, · · · , xn, xn+1, · · · where
|xj − xj−1| = aj , (j = 1, 2, · · · , n− 1),
|xj − xj−1| = an, (j ≥ n), and x0 = 0.
17
h(Mesh Size)
RE
LA
TIV
EE
RR
OR
(%)
INE
NE
RG
YN
OR
M
10-3 10-2 10-1 10010-5
10-4
10-3
10-2
10-1
100
101
102
([-1,1];C0; Order=1)([-2,2];C0; Order=3)([-3,3];C0; Order=5)
h(MESH SIZE)
RE
LA
TIV
EE
RR
OR
(%)
INE
NE
RG
YN
OR
M
0.05 0.1 0.15 0.20.2510-5
10-4
10-3
10-2
10-1
100
101
102([-2,2]; C2; Order=2)([-3,3]; C2; Order=4)([-4,4]; C2; Order=6)
D u = fxxxx
Figure 3: (Left) The relative error in Energy Norm (%) obtained by applying the C0-RPP shapefunctions φ([−1,1];0;1)(x), φ([−2,2];0;3)(x), φ([−3,3];0;5)(x) of reproducing order k = 1, 3, 5, respec-
tively to the model problem − d2udx2 = f. (Right) The relative error (%) in Energy Norm obtained
by applying the C2-RPP shape functions of reproducing order 2, 4, 6, respectively, to the modelfourth order problem − d4u
dx4 = f. In both figures, the relative errors are plotted with respect tothe mesh size h in log-log scale.
Table 2: The relative error in Energy Norm (%) of the fourth order equation, − d4udx4 = f, obtained
by using C2-RPP shape functions of reproducing order 2, 4, 6, respectively. The true solution isu(x) = ex(1− x2)6.
h φ([−2,2];2;2)(x) φ([−2,2];2;4)(x) φ([−2,2];2;6)(x)
0.2e-0 52.79e-0 16.57e-0 5.94e-0
0.1e-0 33.86e-0 3.22e-0 4.61e-1
0.5e-1 18.60e-0 4.52e-1 1.94e-2
2.5e-2 9.56e-0 5.95e-2 9.26e-4
1.25e-2 4.82e-0 7.95e-3 5.75e-4
18
The actual coordinates of particles are x0 = 0, x1 = a1, x2 = a1 + a2, · · · , xn =∑n
j=1 aj ,xn+1 = an +
∑nj=1 aj , xn+2 = 2an +
∑nj=1 aj , and so on.
//•x0
//•x1
//•x2
//•x3 · · ·
//•xn−1
//•xn · · ·
For each particle xj , we will construct a C0-piecewise polynomial RPP shape function φ(xj)(x)of reproducing order 2K − 1, as well as a Cr- piecewise polynomial RPP shape function ofreproducing order 2K − 2, for r ≥ 1.
In the previous section, it was shown that the unique translation invariant C0-piecewisepolynomial basic RPP shape functions with reproducing order 2K − 1 must have support aslarge as [−K,K], where K is a positive integer. Thus, in order to construct piecewise polynomialRPP shape functions associated with non-uniformly distributed particles in [0,∞), we first givethe conditions on an individual shape function and the set of particles that should be in thesupport of this shape function.
In the previous section, the translation invariant C0-RPP shape functions correspondingto uniformly distributed particles in (∞,∞), have supports which consist of 2K consecutiveintervals. However, when the particles are restricted to be in the interior of the domain or onthe boundary of the domain, the supports of RPP shape functions corresponding to particles nearboundary have various lengths. For example, the support of the shape function correspondingto the boundary particle consists of K subintervals. In this section, we will use the followingnotations for RPP shape functions.
φ(xj)(x) := the shape function associated with the particle xj (the shape function in thespace coordinate system).
φj(x) := the shape function centered at 0, obtained by translating φ(xj)(x) (the shapefunction in the reference coordinate system).
Of course, these RPP shape functions corresponding to non-uniformly distributed particles arenot translation invariant.
4.1 C0-piecewise polynomial RPP shape functions for non-uniformly distributedparticles in [0,∞)
For a clear description of constructing C0-RPP shape functions associated with the particlesthat are non-uniformly distributed in [0,∞), we start with a specific example in which K = 2.In this example, we also assume that a1 =
14 , a2 =
14 , a3 =
12 and a4 = 1 are the step sizes for
non-uniformly distributed particles. In this case, the coordinates of the related non-uniformlydistributed particles on [0,∞) are
x0 = 0, x1 =1
4, x2 =
1
2, x3 = 1, x4 = 2, x5 = 3, · · ·
19
X
Y
0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
node = 0node = 1/4node = 1/2
φ
φ
φ
0
1/4
1/2
(x)
(x-1/4)
(x-1/2)
C -RKP shape function for K=20
X
Y
0 1 2 3 4 5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
node=1node=2node=3
C -RKP shape functions for K = 20
φ
φ
φ
(x )
(x )
(x )
1
2
3
X
Y
0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
(node=0)(node=1/4)(node=1/2)
1C -RKP shape functions for K=2
X
Y
0 1 2 3 4 5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
(node=1)(node=2)(node=3)
1C -RKP shape functions for K=2
Figure 4: (Up Left:) The graph of the C0 RPP shape function of order 3,φ(x0)(x), φ(x1=1/4)(x)φ(x2=1/2)(x); (Up Right:) The graph of the C0 RPP shape function of or-der 3, φ(x3=1)(x), φ(x4=2)(x), φ(x5=3)(x); (Down Left:) The graph of the C1 RPP shape functionof order 2, φ(x0)(x), φ(x1=1/4)(x)φ(x2=1/2)(x); (Down Right:) The graph of the C1 RPP shapefunction of order 2, φ(x3=1)(x), φ(x4=2)(x), φ(x5=3)(x). Let us note that these shape functionssatisfy the Kronecker delta property, however parts of graphs are higher than 1. Actually, themaximum vale of these shape functions do not occur at nodal points.
20
Let us note that the C0-RPP shape functions φ(xj) corresponding to the above particles havethe reproducing order 2K − 1 = 3 if and only if, for all x ∈ [0,∞),
∞∑
j=0
(x− xj)αφ(xj)(x) = δα0 , for α = 0, 1, 2, 3. (29)
In order for the above system of four equations to be solvable, we consider this system sep-arately on each of subintervals [0, 1/2], [1/2, 1], [1, 2], [2, 3], [3, 4], · · · , with four shape functionscorresponding to particles in the the following index sets:
Λ0 = x0, x1, x2, x3, Λ1 = x1, x2, x3, x4, Λ2 = x2, x3, x4, x5,Λ3 = x3, x4, x5, x6, Λ4 = x4, x5, x6, x7, · · · · · ·
(I-C0:) Let us start with the index set Λ4 = x4 = 2, x5 = 3, x6 = 4, x7 = 5. Sincethe particle in this set is uniformly distributed, and the radius of the support is K = 2, suppφ(xj)(x) = [xj − 2, xj + 2], j = 4, 5, 6, 7. Hence, the intersection of supports of φ(xj), xj ∈ Λ4 is[3, 4]. If x ∈ (3, 4), then the system (29) becomes the following system for 4 unknowns:
∑
xj∈Λ4
(x− xj)αφ(xj) |[3,4] (x) = δα0 , for α = 0, 1, 2, 3.
Solving this equation, we have
φ(x4) |[3,4] (x) = −1
6(x− 5)(x− 4)(x− 3),
φ(x5) |[3,4] (x) =1
2(x− 5)(x− 4)(x− 2),
φ(x6) |[3,4] (x) = −1
2(x− 5)(x− 3)(x− 2),
φ(x7) |[3,4] (x) =1
6(x− 4)(x− 3)(x− 2).
(30)
Using the unique C0-RPP shape function of reproducing order 3 corresponding to uniformlydistributed particles that is defined by (16), one can easily verify that
φ(x4) |[3,4] (x) = f4,(2;0;3)(x− 2), φ(x5) |[3,4] (x) = f3,(2;0;3)(x− 3),
φ(x6) |[3,4] (x) = f2,(2;0;3)(x− 4), φ(x7) |[3,4] (x) = f1,(2;0;3)(x− 5).
By a similar manner, the solution functions of (29) for x ∈ [k, k + 1], k ≥ 3, with respectto an appropriate index set Λk+1 of four particles are translations of φ([−2,2];0;3)(x) defined by(16). Thus, we assign the translated RPP shape function φ([−2,2];0;3)(x− xj) to each particle xjif j ≥ 6.
21
(II-C0:) Next, if x ∈ (2, 3), then we use the set Λ3 = x3 = 1, x4 = 2, x5 = 3, x6 = 4 so thatthe system (29) becomes the following system for four unknowns:
∑
xj∈Λ3
(x− xj)αφ(xj) |[2,3] (x) = δα0 , α = 0, 1, 2, 3.
The solutions of this system are
φ(x3) |[2,3] (x) = −1
6(x− 4)(x− 3)(x− 2),
φ(x4) |[2,3] (x) =1
2(x− 4)(x− 3)(x− 1),
φ(x5) |[2,3] (x) = −1
2(x− 4)(x− 2)(x− 1),
φ(x6) |[2,3] (x) =1
6(x− 3)(x− 2)(x− 1).
(31)
(III-C0:) If x ∈ (1, 2), then we use the index set Λ2 =x2 =
12 , x3 = 1, x4 = 2, x5 = 3
.
Then the system (29) becomes the following system for four unknowns:∑
xj∈Λ2
(x− xj)αφ(xj) |(1,2) (x) = δα0 , α = 0, 1, 2, 3.
The solutions of this system are
φ(x2) |[1,2] (x) = −8
15(x− 3)(x− 2)(x− 1),
φ(x3) |[1,2] (x) =1
2(x− 3)(x− 2)(2x− 1),
φ(x4) |[1,2] (x) = −1
3(x− 3)(x− 1)(2x− 1),
φ(x5) |[1,2] (x) =1
10(x− 2)(x− 1)(2x− 1).
(32)
(IV-C0:) If x ∈ (1/2, 1), then we use the index set Λ1 =x1 =
14 , x2 =
12 , x3 = 1, x4 = 2
. Then
the system (29) becomes the following system for four unknowns:∑
xj∈Λ1
(x− xj)αφ(xj) |(1/2,1) (x) = δα0 , α = 0, 1, 2, 3.
The solutions of this system are
φ(x1) |[ 12,1] (x) = −
32
21(x− 2)(x− 1)(2x− 1)
φ(x2) |[ 12,1] (x) =
4
3(x− 2)(x− 1)(4x− 1)
φ(x3) |[ 12,1] (x) = −
1
3(x− 2)(2x− 1)(4x− 1)
φ(x4) |[ 12,1] (x) =
1
21(x− 1)(2x− 1)(4x− 1).
(33)
22
(V-C0:) Thus far, for x ∈ (1/2,∞), we have solved
∞∑
j=0
(x− xj)αφ(xj)(x) = δα0 , α = 0, 1, 2, 3.
In order to get the RPP shape functions that satisfy the above system for all x ∈ [0,∞), wefinally use the index set Λ0 = 0, 14 , 12 , 1 as follows: for x ∈ [0, 1/2), solve the following system
∑
xj∈Λ0
(x− xj)αφ(xj) |[0, 12] (x) = δα0 , α = 0, 1, 2, 3.
Then the solutions of this system for the four unknown functions, are
φ(x0) |[0, 12] (x) = −(x− 1)(2x− 1)(4x− 1)
φ(x1) |[0, 12] (x) =
32
3(x− 1)x(2x− 1)
φ(x2) |[0, 12] (x) = −4(x− 1)x(4x− 1)
φ(x3) |[0, 12] (x) =
1
3x(2x− 1)(4x− 1).
(34)
Now, assembling the solutions listed in Eqns. (30), (31), (32), (33), and (34), we define theRPP shape functions of reproducing order 3 corresponding to the particles, x0, x1, x2, x3, x5, · · · ,that have the following supports, respectively,
supp φ(x0) = [0, 1/2],
supp φ(x1) = [0, 1/2] ∪ [1/2, 1],
supp φ(x2) = [0, 1/2] ∪ [1/2, 1] ∪ [1, 2],
supp φ(x3) = [0, 1/2] ∪ [1/2, 1] ∪ [1, 2] ∪ [2, 3],
supp φ(x4) = [1/2, 1] ∪ [1, 2] ∪ [2, 3] ∪ [3, 4],
supp φ(x5) = [1, 2] ∪ [2, 3] ∪ [3, 4] ∪ [4, 5],
supp φ(xj) = [xj − 2, xj + 2], if j ≥ 6.
(35)
Since all partially generated RPP shape functions satisfy the Kronecker delta property(i.e.φ(xj)(xi) = δij for all i, j ≥ 0), the assembled shape functions are continuous piecewise poly-nomials that satisfy the Kronecker delta property.
Adopting those polynomials fi,(2;0;3) defined by (16), the assembled C0-RPP shape functionsof reproducing order 3 are as follows:
1. φ(x0)(x) = φ0(x) and φ0(x) is as follows:
φ0(x) =
−(x− 1)(2x− 1)(4x− 1) x ∈ [0, 12 ]
0 x 6∈ [0, 12 ]
23
2. φ(x1)(x) = φ1(x− 1/4) and φ1(x) is as follows:
φ1(x) =
13(4x− 3)(4x− 1)(4x+ 1) x ∈ [− 1
4 ,14 ]
− 121(4x− 7)(4x− 3)(4x− 1) x ∈ [ 14 ,
34 ]
0 x 6∈ [−14 , 34 ]
3. φ(x2)(x) = φ2(x− 1/2) and φ2(x) is as follows:
φ2(x) =
−(2x− 1)(2x+ 1)(4x+ 1) x ∈ [− 12 , 0]
13(2x− 3)(2x− 1)(4x+ 1) x ∈ [0, 12 ]
− 115(2x− 5)(2x− 3)(2x− 1) x ∈ [ 12 ,
32 ]
0 x 6∈ [−12 , 32 ]
4. φ(x3)(x) = φ3(x− 1) and φ3(x) is as follows:
φ3(x) =
13(x+ 1)(2x+ 1)(4x+ 3) x ∈ [−1,− 1
2 ]
−13(x− 1)(2x+ 1)(4x+ 3) x ∈ [− 12 , 0]
12(x− 2)(x− 1)(2x+ 1) x ∈ [0, 1]
f4,(2;0;3)(x) x ∈ [1, 2]
0 x 6∈ [−1, 2]
5. φ(x4)(x) = φ4(x− 2) and φ4(x) is as follows:
φ4(x) =
121(x+ 1)(2x+ 3)(4x+ 7) x ∈ [− 3
2 ,−1]−13(x− 1)(x+ 1)(2x+ 3) x ∈ [−1, 0]f3,(2;0;3)(x) x ∈ [0, 1]
f4,(2;0;3)(x) x ∈ [1, 2]
0 x 6∈ [−32 , 2]
6. φ(x5)(x) = φ5(x− 3) and φ5(x) is as follows:
φ5(x) =
110(x+ 1)(x+ 2)(2x+ 5) x ∈ [−2,−1]f2,(2;0;3)(x) x ∈ [−1, 0]f3,(2;0;3)(x) x ∈ [0, 1]
f4,(2;0;3)(x) x ∈ [1, 2]
0 x 6∈ [−2, 2]
7. The shape function corresponding to the particle xj (j ≥ 6) is
φ(xj)(x) = φ([−2,2];0;3)(x− xj)
24
The graph of these shape functions φ(xj)(x), j = 0, 1, 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 C0-RPP shape functions φ(xj) corresponding to non-uniformly distributed particles in
[0,∞) have the polynomial reproducing order 2K − 1 if and only if, for all x ∈ [0,∞),
∞∑
j=0
(x− xj)αφ(xj)(x) = δα0 for α = 0, 1, 2, 3, · · · , 2K − 1. (36)
In order for the above system of 2K equations to be solvable, we consider this system onthe subintervals, as in the above example, with 2K shape functions corresponding to particlesin the the following index sets
Λ0 = x0, x1, · · · , x2K−1, Λ1 = x1, x2, · · · , x2K, Λ2 = x2, x3, · · · , x2K+1,· · · · · · Λn = xn, xn+1, · · · , xn+2K−1, · · · · · ·
(nG) : Suppose the particles in Λn are uniformly distributed and are not affected by thenon-uniformly distributed particles. The intersection of supports of the RPP shape functionsfor the particles in Λn is [xn+K−1, xn+K ]. For x ∈ [xn+K−1, xn+K ], we solve the system of 2Kequations for 2K unknowns:
∑
xj∈Λn
(x− xj)αφ(xj) |[xn+K−1,xn+K ] (x) = δα0 , α = 0, 1, 2, 3, · · · , 2K − 1.
((n−1)G) : For x ∈ [xn+K−2, xn+K−1], we solve the system of 2K equations for 2K unknowns:
∑
xj∈Λn−1
(x− xj)αφ(xj) |[xn+K−2,xn+K−1] (x) = δα0 , α = 0, 1, 2, 3, · · · , 2K − 1.
. . . . . . . . .(1G) : For x ∈ [xK , xK+1], we solve the system of 2K equations for 2K unknowns:
∑
xj∈Λ1
(x− xj)αφ(xj) |[xK ,xK+1] (x) = δα0 , α = 0, 1, 2, 3, · · · , 2K − 1.
(0G) : Finally, for x ∈ [0, xK ], we solve the system of 2K equations for 2K unknowns:
∑
xj∈Λ0
(x− xj)αφ(xj) |[0,xK ] (x) = δα0 , α = 0, 1, 2, 3, · · · , 2K − 1.
By a similar method to the above example, assembling those partially defined RPP shapefunctions in the steps (nG), ((n− 1)G), · · · , (1G), and (0G) together, we have the required C0-RPP shape functions corresponding to non-uniformly distributed particles xj (j ≥ 0) in [0,∞).
25
4.2 C1-piecewise polynomial RPP shape functions for non-uniformly dis-tributed particles in [0,∞)
In the construction of C0-RPP shape functions in the previous section, we solved the system of2K equations for 2K unknowns on each interval [0, xK ], [xK , xK+1], · · · , [xn+K−1, xn+K ], · · · .Without any restrictions, the solution functions are assembled to be C0-RPP shape functions ofreproducing order 2K − 1.
However, as we discussed in section 3, in order for the assembled shape functions to beCr, r > 0 functions, we have to impose some conditions to the system of 2K equations for 2Kunknowns on each of the above intervals. More specifically, we need to impose some conditionsso that the solution functions have the same derivative at the nodal points xj , j > 0.
In order to impose such conditions for smoothness at particles, it is necessary to sacrificethe reproducing order by 1, by imposing a polynomial G(x) on the right hand side of the lastequation as follows:
∑∞j=0(x− xj)αφ(xj)(x) = δα0 , α = 0, 1, 2, · · · , 2K − 2,∑∞j=0(x− xj)2K−1φ(xj)(x) = G(x)
(37)
Then, obviously, the shape functions that are solutions of the system (37) of 2K equationshave the reproducing property of order 2K − 2. Here G(x) will be properly selected on eachinterval so that the solution of this system can be Cr, r > 0, functions.
In this section, we sketch how to construct C1-RPP shape functions with polynomial repro-ducing order 2K − 2. In order to make the system (37) being coupled with 2K unknowns oneach interval, we use the same subsets of particles, Λ0,Λ1, · · · ,Λn, · · · , as before.
For specific construction of C1-RPP shape functions, we describe it for the RPP shapefunctions with reproducing order 2K − 2,K = 2, corresponding to the following non-uniformlydistributed particles:
x0 = 0, x1 =1
4, x2 =
1
2, x3 = 1, x4 = 2, x5 = 3, · · ·
Following the same argument as the previous subsection, we consider the system (37) sepa-rately on each of subintervals
[0, 1/2], [1/2, 1], [1, 2], [2, 3], [3, 4], · · · ,
with four shape functions corresponding to particles in the the following index sets:
Λ0 = x0, x1, x2, x3, Λ1 = x1, x2, x3, x4, Λ2 = x2, x3, x4, x5,Λ3 = x3, x4, x5, x6, Λ4 = x4, x5, x6, x7, · · · .
If we consider only the particles corresponding to the index set Λk, the system (37) can berewritten as follows:
∑xj∈Λk
(x− xj)αφ(xj)(x) = δα0 , α = 0, 1, 2,∑xj∈Λk
(x− xj)3φ(xj)(x) = Gk(x)(38)
26
(I-C1:) By the same reason as the previous section, if we start with the index set Λ4 = x4 =2, x5 = 3, x6 = 4, x7 = 5, then we only need to solve the system (38) for x ∈ [3, 4]. By Lemma3.1, in order for this system being uniquely solvable, 3 and 4 should be zeros of G4(x).
Solving the constrained system (38) for G4(x) = (x − 4)(x − 3)(7 − 2x),Λ4, and x ∈ [3, 4],we have the following solutions:
φ(x4) |[3,4] (x) = −1
2(x− 4)2(x− 3),
φ(x5) |[3,4] (x) =1
2(x− 4)(3x2 − 20x+ 31),
φ(x6) |[3,4] (x) = −1
2(x− 3)(3x2 − 22x+ 38),
φ(x7) |[3,4] (x) =1
2(x− 4)(x− 3)2.
(39)
Let us note that in this section, the choice of the constraint functions Gk(x) are generally notunique.
Using the C1-RPP shape function, gj,(2;1;2), j = 1 · · · 4, defined by (17), one can see that thosefunctions in (39) can be written as follows:
φ(x4) |[3,4] (x) = g4,(2;1;2)(x− 2), φ(x5) |[3,4] (x) = g3,(2;1;2)(x− 3),
φ(x6) |[3,4] (x) = g2,(2;1;2)(x− 4), φ(x7) |[3,4] (x) = g1,(2;1;2)(x− 5).
(II-C1:) Solving the system (38) for G3(x) = (x − 3)(x − 2)(5 − 2x),Λ3, and x ∈ [2, 3], wehave the following solutions:
φ(x3) |[2,3] (x) = −1
2(x− 3)2(x− 2)
φ(x4) |[2,3] (x) =1
2(x− 3)(3x2 − 14x+ 14)
φ(x5) |[2,3] (x) = −1
2(x− 2)(3x2 − 16x+ 19)
φ(x6) |[2,3] (x) =1
2(x− 3)(x− 2)2
(40)
(III-C1:) Solving the system (38) for G2(x) = (x− 2)(x− 1)(2− 32x),Λ2, and x ∈ [1, 2], we
have the following solutions:
φ(x2) |[1,2] (x) = −4
3(x− 2)2(x− 1)
φ(x3) |[1,2] (x) =1
2(x− 2)(5x2 − 14x+ 7)
φ(x4) |[1,2] (x) = −1
3(x− 1)(5x2 − 17x+ 11)
φ(x5) |[1,2] (x) =1
2(x− 2)(x− 1)2
(41)
27
(IV-C1:) Solving the system (38) for G1(x) = (x − 1)(x − 12)(
32 − 5
2x),Λ1, and x ∈ [1/2, 1],we have the following solutions:
φ(x1) |[ 12 ,1] (x) = −16
3(x− 1)2(2x− 1)
φ(x2) |[ 12 ,1] (x) =4
3(x− 1)(14x2 − 20x+ 5)
φ(x3) |[ 12 ,1] (x) = −1
3(2x− 1)(14x2 − 25x+ 8)
φ(x4) |[ 12 ,1] (x) =1
3(x− 1)(2x− 1)2
(42)
(V-C1:) Finally, solving the system (38) for G0(x) = (x− 1)(x− 12)(x− 1
4)x(6x− 1),Λ0, andx ∈ [0, 1/2], we have the following solutions:
φ(x0) |[0, 12 ] (x) = (x− 1)(2x− 1)2(3x+ 1)(4x− 1)
φ(x1) |[0, 12 ] (x) = −8
3(x− 1)x(2x− 1)(24x2 − 10x− 3)
φ(x2) |[0, 12 ] (x) = 2(x− 1)x(4x− 1)(12x2 − 8x− 1)
φ(x3) |[0, 12 ] (x) = −1
3x2(2x− 1)(4x− 1)(6x− 7)
(43)
By assembling those solutions of (39), (40), (41), (42), and (43), we obtain the C1-RPP shapefunctions of reproducing order 2 corresponding to the the following non-uniformly distributedparticles
x0 = 0, x1 =1
4, x2 =
1
2, x3 = 1, x4 = 2, x5 = 3, · · ·
Using those polynomials gj,(2;1;2) defined by (17), the assembled C1-RPP shape functions ofreproducing order 2 that satisfy the Kronecker delta property are as follows:
1. φ(x0) = φ0(x− 0), where φ0(x) is as follows:
φ0(x) =
(x− 1)(2x− 1)2(3x+ 1)(4x− 1) x ∈ [0, 12 ]
0 x 6∈ [0, 12 ]
2. φ(x1) = φ1(x− 1/4), where φ1(x) is as follows:
φ1(x) =
−16(4x− 3)(4x− 1)(4x+ 1)(12x2 + x− 2) x ∈ [− 14 , 14 ]−16(4x− 3)2(4x− 1) x ∈ [14 ,
34 ]
0 x 6∈ [−14 , 34 ]
28
3. φ(x2) = φ2(x− 1/2), where φ2(x) is as follows:
φ2(x) =
(2x− 1)(2x+ 1)(4x+ 1)(6x2 + 2x− 1) x ∈ [− 12 , 0]13(2x− 1)(28x2 − 12x− 3) x ∈ [0, 12 ]
−16(2x− 3)2(2x− 1) x ∈ [12 ,32 ]
0 x 6∈ [−12 , 32 ]
4. φ(x3) = φ3(x− 1), where φ3(x) is as follows:
φ3(x) =
−13(x+ 1)2(2x+ 1)(4x+ 3)(6x− 1) x ∈ [−1,− 12 ]
−13(2x+ 1)(14x2 + 3x− 3) x ∈ [−( 12), 0]12(x− 1)(5x2 − 4x− 2) x ∈ [0, 1]
g4,(2;1;2)(x) x ∈ [1, 2]
0 x 6∈ [−1, 2]
5. φ(x4) = φ4(x− 2), where φ4(x) is as follows:
φ4(x) =
13(x+ 1)(2x+ 3)2 x ∈ [−32 ,−1]−13(x+ 1)(5x2 + 3x− 3) x ∈ [−1, 0]g3,(2;1;2)(x) x ∈ [0, 1]
g4,(2;1;2)(x) x ∈ [1, 2]
0 x 6∈ [−32 , 2]
6. φ(x5) = φ5(x− 3), where φ5(x) is as follows:
φ5(x) =
12(x+ 1)(x+ 2)2 x ∈ [−2,−1]g2,(2;1;2)(x) x ∈ [−1, 0]g3,(2;1;2)(x) x ∈ [0, 1]
g4,(2;1;2)(x) x ∈ [1, 2]
0 x 6∈ [−2, 2]
7. For the particle xj (j ≥ 6)
φ(xj)(x) = φ([−2,2];1;2)(x− xj)
The graphs of the C1-RPP shape functions of reproducing order 2, associated with those particles0, 1/4, 1/2, 1, 2, 3, are depicted in Fig. 4.
29
In general, to construct Cr, r ≥ 1, RPP shape function of reproducing order 2K− 2, we haveto solve the following system of 2K equations with one constrained equation for index subsetsΛk containing 2K particles:
∑xj∈Λk
(x− xj)αφ(xj)(x) = δα0 α = 0, 1, 2, · · · , 2K − 2,∑xj∈Λk
(x− xj)2K−1φ(xj)(x) = Gi(x)(44)
Here Gk(x) is properly selected so that the resulting solutions have the same derivatives atthe nodes and satisfy the Kronecker delta property.
Since the construction procedures of RPP shape functions associated with the particlesin [0,∞) do not depend on non-uniformity of particles, we put C0-RPP shape functions ofreproducing order 5(K = 3) as well as C1-RPP shape functions of reproducing order 4(K = 3)associated with the uniformly distributed particles in [0,∞) in appendix.
4.3 Cr-piecewise polynomial RPP shape functions for arbitrary distributedparticles that are in a closed interval [a, b]
In the previous subsections, we constructed piecewise polynomial RPP shape functions when theparticles are non-uniformly(or uniformly) distributed in [0,∞). By taking linear transformationof those shape functions, we can also have piecewise polynomial RPP shape functions of thesame degree of reproducing order, associated with the particles distributed in (−∞, b]. Thus,we can construct piecewise polynomial RPP shape functions associated with non-uniformly(oruniformly) distributed particles in a compact set [a, b].
Without loss of generality, we construct C0-RPP shape functions when the particles are in[a, b], a < b. Suppose K = 3 and the particles are uniformly distributed as follows:
x0 = 0, x1 = 1, x2 = 2, · · · , x5 = 5, x6 = 6, x7 = 7, · · · , x10 = 10, x11 = 10, x12 = 12, (45)
on the closed bounded domain [0, 12].Let φ(xj)(x) be the C0-piecewise polynomial RPP shapes functions, that are constructed in
appendix, associated with the particles: x0 = 0, x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6.Let us note that
φ(x6)(x) = φ([−3,3];0;5)(x− x6)which is symmetric about x6.
Now, we assign piecewise polynomial C0-shape functions, ψxj, associated with the uniformly
distributed particles on the bounded domain [0, 12], like those in (45), as follows:
1. ψxj(x) := φ(xj)(x), j = 0, 1, 2, 3, 4, 5,
2. ψx6(x) := φ([−3,3];0;5)(x− x6),
3. ψxj(x) := φ(12−xj)(−(x− 12)), j = 7, 8, 9, 10, 11, 12.
30
Then, these shape functions have the property of reproducing of order 5. Indeed,
12∑
j=0
(x− xj)αψxj(x) =
8∑
j=0
(x− xj)αψxj(x) for x ∈ [0, 6]
12∑
j=4
(x− xj)αψxj(x) for x ∈ [6, 12]
Since φx4|[3,7](x) = φ([−3,3];0;5)|[−1,3](x− x4), we have
ψx7(x) = φ(x7)(−(x− 12)) = φ([−3,3];0;5)((−(x− 12))− x5)
= φ([−3,3];0;5)((−(x− 12))− 5) = φ([−3,3];0;5)(x− 7) = φ(x7)(x), for all x ∈ [4, 6].
Similarly, using the relation φ(x5)|[3,8](x) = φ([−3,3];0;5)|[−2,3](x − x5), we can show ψx8(x) =
φ(x8)(x), for all x ∈ [4, 6]. Thus, we have
8∑
j=0
(x− xj)αψxj(x) =
8∑
j=0
(x− xj)αφ(xj)(x) = xα, for all x ∈ [0, 6], α = 0, 1, 2, 3, 4, 5.
On the other hand, Lemma 4.1 leads us to
12∑
j=4
(x− xj)αψxj(x) = xα, for all x ∈ [6, 12], α = 0, 1, 2, 3, 4, 5.
Lemma 4.1. Let φ(xj) be the RPP shape functions of reproducing order k associated with theparticles xj ∈ [α, β], j = 1, 2, · · · , n. Let T (x) = ax+ b, a 6= 0, be a linear transformation on R.Then the functions defined by
ψξj(ξ) := (φ(xj) T−1)(ξ) , j = 1, 2, · · · , n,
become RPP shape functions associated with the particles ξj = T (xj) on T ([α, β]).
Proof. Using a0 = 1, we have
n∑
j=1
(ξ − ξj)αψξj(ξ) =
n∑
j=1
(T (x)− T (xj))α φ(xj)(T−1(T (x)))
= aαn∑
j=1
(x− xj)α φ(xj)(x) = aαδα0 = δα0 , for α = 0, 1, 2, · · · , k.
31
4.4 Numerical Examples
Using similar arguments of the interpolation error estimates in ([2],[9]), one can prove the fol-lowing theorem.
Theorem 4.1. Suppose u ∈ Hm+[n/2]+1(Ω)(or u ∈ Cm(Ω)), 4K − 2 > d, and uR is an RPP-approximate solution of an elliptic boundary value problem on a convex domain Ω ⊂ R
n. Then,we have
‖u− uR‖1,Ω ≤ Chl−1|u|l,Ω.where l = minm, 2K or minm, 2K−1 according as the RPP shape functions are C0 or Cr, r >0, respectively. The diameters of supports of shape functions are ≤ √nKh. The constant C isindependent of u and h.
We now demonstrate the effectiveness of the RPP shape functions in dealing with ellipticboundary value problems. Without loss of generality, we use the RPP shape functions corre-sponding to the particles that are uniformly distributed in [0,∞).
Example 4.1. Consider one-dimensional boundary value problem − d2udx2 = f in Ω = (0, 1), that
satisfy the boundary conditions: u(0) = 0 and u(1) = 1.
The C1-piecewise polynomial RPP shape functions of reproducing order 4, listed in appendixB-II, are applied for the approximate solutions of this problem with respect to various regularitiesof f(x). In this case, all particles are on the right hand side of the left boundary node x1 = 0.
The relative errors in percent with respect to the energy norm are depicted in Fig. 5 whenthe true solutions are, respectively, x0.7, x1.7, x2.7, and x4.7.
Theorem 4.1 imply that the errors versus the mesh sizes in log-log scale is below the liney = αx+ b, where α = mink+1,m− 1, k := the polynomial reproducing order of RPP shapefunctions, and m := the regularity of the true solution.
In Fig. 5, this example strongly supports the theory. Indeed, since the polynomial repro-ducing order is k = 4, and the true solutions are in C1(Ω), C2(Ω), C3(Ω), and C5(Ω), respectively,the slope α becomes 0, 1, 2, and 4, according as the true solution is x0.7, x1.7, x2.7, and x4.7,respectively. On the other hand, the slope of the computed relative errors are 0.2, 1.2, 2.2, 4.1,respectively.
AcknowledgementsWe are very grateful to referees who provide various constructive suggestions and comments
to improve this paper.
32
MESH SIZE(h)
RE
LA
TIV
EE
RR
OR
(%)
INE
NE
RG
YN
OR
M
10-2 10-1
10-5
10-4
10-3
10-2
10-1
100
101
X^0.7X^1.7X^2.7X^4.7
Slope = 0.2
Slope = 1.2
Slope = 2.2
Slope = 4.1
Figure 5: The relative errors(%), obtained by C1-RPP shape functions of reproducing order 4,versus mesh size in log-log scale. These errors correspond to the true solutions x0.7, x1.7, x2.7
and x4.7, respectively.
A The supplement to section 3
A.1 The proof of Lemma 3.1
Proof. (1) It suffices to show that for l = −(K − 1), · · · ,−1, 0, 1, · · · , (K − 1), the following issatisfied:
limx→1
f(x− l) = limx→0
f(x− (l − 1)).
The determinant of the Vandermonde matrix V (x) is D ≡ ∏2K−1m=0 m! ([22]). For a notational
brevity, we use l for the ordinal column number of V (x). That is, the column number, −(K −1), · · · ,−1, 0, 1, · · · , (K − 1), are the column number, 1, 2, · · · ,K, · · · , (2K − 1), respectively.Let V(l)(x) is the matrix obtained by replacing the l-th column of V (x) with the base vector ei.Then
limx→1
f(x− l) =detV(l)(1)
D, lim
x→0f(x− (l − 1)) =
detV(l−1)(0)
D.
By evaluating the Vandermonde matrix V (x) at x = 0, 1, we have the following.
33
V (0) =
1 · · · 1 1 1 1 · · · 1 1 1(K − 1)1 · · · 21 1 0 −1 · · · (2−K)1 (1−K)1 (−K)1
(K − 1)2 · · · 22 1 0 1 · · · (2−K)2 (1−K)2 (−K)2
......
......
......
......
......
,
V (1) =
1 1 · · · 1 1 1 1 · · · 1 1K1 (K − 1)1 · · · 21 1 0 −1 · · · (2−K)1 (1−K)1
K2 (K − 1)2 · · · 22 1 0 1 · · · (2−K)2 (1−K)2
......
......
......
......
......
.
Then the l-th column of V (0) is the same as (l+1)-th column of V (1), for l = 1, 2, · · · , 2K − 1.Moreover, for l > 1, V(l)(1) and V(l−1)(0), respectively, can be obtained by replacing the l-thcolumn of V (1) and the (l − 1)-th column of V (0) with the right side vector ei.
(a) If the right hand side of (6) is e1, the continuity of the extension F (x) is straight forward.Suppose the right had side of (6) is e1 and l − 1 6= K, then l 6= K + 1 and hence two columnsof V(l−1)(0) and two columns of V(l)(1), respectively, are the same as e1. Hence, detV(l−1)(0) =detV(l)(1) = 0.
If l−1 = K, then detV(K)(0) = detV (0) and detV(K+1)(1) = detV (1). Thus detV(K)(0)/D =1 = detV(K+1)(1)/D.
Hence, f(x) determined by (6) can be extended to a continuous function F (x) which satisfythe Kronecker delta property.
This argument is independent of the dimension of Vandermonde matrix V (x). Hence, thisconclusion holds even when Ω = [−K,K + 1] \ Z or Ω = [−K − 1,K] \ Z.
(b) Suppose the right hand side of (6) is ej for an odd number j > 0. Without loss ofgenerality, let us suppose the right hand side is e3 = [0, 0, 1, 0, · · · , 0]T . By the column opera-tions(adding the m-th column to the (2K −m)-th column, m = 1, 2, · · · ,K − 1), and N timesrow or column interchanges (depends on the location of l) to V(l−1)(0), we get
(K − 1)1 · · · 2 1 (−K)1 0 0 0 · · · 0 0(K − 1)3 · · · 23 1 (−K)3 0 0 0 · · · 0 0
......
......
......
......
......
...· · · · · · · · · · · · 0 0 · · · 0 01 · · · 1 1 1 0 1 2 · · · 2 2
(K − 1)2 · · · 22 1 (−K)2 1 0 2 · · · 2(2−K)2 2(1−K)2
(K − 1)4 · · · 24 1 (−K)4 0 0 2 · · · 2(2−K)4 2(1−K)4
......
......
......
......
......
...
.
34
By applying Laplace expansion (generalized cofactor expansion), we have
detV(l−1)(0) = (−1)N detA · detB, where
A =
(K − 1)1 · · · 2 1 (−K)1
(K − 1)3 · · · 23 1 (−K)3
(K − 1)5 · · · 25 1 (−K)5
......
......
...
B =
0 1 2 2 · · · 2 21 0 2 2(2)2 · · · 2(2−K)2 2(1−K)2
0 0 2 2(2)4 · · · 2(2−K)4 2(1−K)4
......
......
......
...
.
(Let us note that if the right hand side is ej for an even number j, for example e2 = [0, 1, 0, · · · , 0]T ,then this decomposition of determinant of V(l−1)(0) is impossible. One can easily construct acounterexample).
Let V(l)(1) be the matrix resulted by (2K − 1) column interchanges to V(l)(1). Then we have
detV(l)(1) = − det V(l)(1),
and V(l)(1) is the same as V(l−1)(0) except the fact that the 2K-th column has an opposite sign.
If one apply the same column operations and Laplace expansion to V(l)(1), the columns ofthe generalized cofactor matrix corresponding to A are the same as A except that all of the lastcolumn have opposite sign. Therefore, we have
detV(l−1)(0) = − det V(l)(1) = detV(l)(1).
(2) Suppose the right side of (6) is the even-th base vector e2j .Without loss of generality, weassume e2j = e2. Then by the column operations(adding (−1) timesm-th column to (2K−m)-thcolumn, m = 1, 2, · · · ,K − 1), and N ′ times row and column interchanges to V(l−1)(0), we get
(K − 1)2 · · · 22 1 (−K)2 0 0 0 · · · 0 0(K − 1)4 · · · 24 1 (−K)4 0 0 0 · · · 0 0
......
......
......
......
......
...· · · · · · · · · · · · · · · · · · 0 0 · · · 0 01 · · · 1 1 1 0 1 0 · · · 0 0
(K − 1)1 · · · 21 1 (−K)1 1 0 −2 · · · −2(2−K)1 −2(1−K)1
(K − 1)3 · · · 23 1 (−K)3 0 0 −2 · · · −2(2−K)3 −2(1−K)3
......
......
......
......
......
...
.
By generalized cofactor expansion, we get the following
35
detV(l−1)(0) = (−1)N ′
detA′ · detB′, where
A′ =
(K − 1)2 · · · 22 1 (−K)2
(K − 1)4 · · · 24 1 (−K)4
(K − 1)8 · · · 28 1 (−K)8
......
......
...
B′ =
0 1 0 0 · · · 0 01 0 −2 −2(2)1 · · · −2(2−K)1 −2(1−K)1
0 0 −2 −2(2)3 · · · −2(2−K)3 −2(1−K)3
......
......
......
...
.
Hence, we havedetV(l−1)(0) = det V(l)(1) = − detV(l)(1).
Therefore, f(x) cannot be made to be continuous at integers in (−K,K).
A.2 The RPP shape function with the reproducing property of high order
φ([−4,4];0;7)(x) =
15040(1 + x)(2 + x)(3 + x)(4 + x)(5 + x)(6 + x)(7 + x) x ∈ [−4,−3],− 1720(−1 + x)(1 + x)(2 + x)(3 + x)(4 + x)(5 + x)(6 + x) x ∈ [−3,−2],1240(−2 + x)(−1 + x)(1 + x)(2 + x)(3 + x)(4 + x)(5 + x) x ∈ [−2,−1],− 1144(−3 + x)(−2 + x)(−1 + x)(1 + x)(2 + x)(3 + x)(4 + x) x ∈ [−1, 0],1144(−4 + x)(−3 + x)(−2 + x)(−1 + x)(1 + x)(2 + x)(3 + x) x ∈ [0, 1],− 1240(−5 + x)(−4 + x)(−3 + x)(−2 + x)(−1 + x)(1 + x)(2 + x) x ∈ [1, 2],1720(−6 + x)(−5 + x)(−4 + x)(−3 + x)(−2 + x)(−1 + x)(1 + x) x ∈ [2, 3],− 15040(−7 + x)(−6 + x)(−5 + x)(−4 + x)(−3 + x)(−2 + x)(−1 + x) x ∈ [3, 4],
0 x ∈ R \ [−4, 4]φ([−4,4];1;6)(x) =
15040(3 + x)(4 + x)2(168 + 187x+ 95x2 + 17x3 + x4) x ∈ [−4,−3],− 1720(2 + x)(3 + x)(60− 2x+ 105x2 + 73x3 + 15x4 + x5) x ∈ [−3,−2],1240(1 + x)(2 + x)(228− 14x− 57x2 + 13x3 + 9x4 + x5) x ∈ [−2,−1],− 1144(1 + x)(−144 + 144x+ 160x2 − 39x3 − 17x4 + 3x5 + x6) x ∈ [−1, 0],1144(−1 + x)(−144− 144x+ 160x2 + 39x3 − 17x4 − 3x5 + x6) x ∈ [0, 1],− 1240(−2 + x)(−1 + x)(−228− 14x+ 57x2 + 13x3 − 9x4 + x5) x ∈ [1, 2],1720(−3 + x)(−2 + x)(−60− 2x− 105x2 + 73x3 − 15x4 + x5) x ∈ [2, 3],− 15040(−4 + x)2(−3 + x)(168− 187x+ 95x2 − 17x3 + x4) x ∈ [3, 4],
0 x ∈ R \ [−4, 4]
36
φ([−4,4];2;6)(x) =
15040(4 + x)3(3 + x)(−525− 173x+ 13x2 + x3) x ∈ [−4,−3],− 1720(3 + x)(2 + x)(−3180− 3998x− 1515x2 − 143x3 + 15x4 + x5) x ∈ [−3,−2],1240(2 + x)(1 + x)(−420− 1418x− 1029x2 − 203x3 + 9x4 + x5) x ∈ [−2,−1],− 1144(1 + x)(−144 + 144x+ 52x2 − 363x3 − 233x4 + 3x5 + x6) x ∈ [−1, 0],1144(−1 + x)(−144− 144x+ 52x2 + 363x3 − 233x4 − 3x5 + x6) x ∈ [0, 1],− 1240(−2 + x)(−1 + x)(420− 1418x+ 1029x2 − 203x3 − 9x4 + x5) x ∈ [1, 2],1720(−3 + x)(−2 + x)(3180− 3998x+ 1515x2 − 143x3 − 15x4 + x5) x ∈ [2, 3],− 15040(−4 + x)3(−3 + x)(525− 173x− 13x2 + x3) x ∈ [3, 4],
0 x ∈ R \ [−4, 4]
φ([−3,4];0;6)(x) =
1720(1 + x)(2 + x)(3 + x)(4 + x)(5 + x)(6 + x) x ∈ [−3,−2],− 1120(−1 + x)(1 + x)(2 + x)(3 + x)(4 + x)(5 + x) x ∈ [−2,−1],
148(−2 + x)(−1 + x)(1 + x)(2 + x)(3 + x)(4 + x) x ∈ [−1, 0],− 136(−3 + x)(−2 + x)(−1 + x)(1 + x)(2 + x)(3 + x) x ∈ [0, 1],
148(−4 + x)(−3 + x)(−2 + x)(−1 + x)(1 + x)(2 + x) x ∈ [1, 2],− 1120(−5 + x)(−4 + x)(−3 + x)(−2 + x)(−1 + x)(1 + x) x ∈ [2, 3],1720(−6 + x)(−5 + x)(−4 + x)(−3 + x)(−2 + x)(−1 + x) x ∈ [3, 4],0 x ∈ R \ [−3, 4]
φ([−3,4];1;5)(x) =
1720(2 + x)(3 + x)(60 + 170x+ 89x2 + 16x3 + x4) x ∈ [−3,−2],− 1120(1 + x)(2 + x)(−96− 11x+ 35x2 + 11x3 + x4) x ∈ [−2,−1],
148(1 + x)(48− 32x− 60x2 + x3 + 6x4 + x5) x ∈ [−1, 0],− 136(−1 + x)(36 + 48x− 37x2 − 13x3 + x4 + x5) x ∈ [0, 1],
148(−2 + x)(−1 + x)(60− 2x− 7x2 − 4x3 + x4) x ∈ [1, 2],− 1120(−3 + x)(−2 + x)(40− 15x+ 19x2 − 9x3 + x4) x ∈ [2, 3],1720(−4 + x)(−3 + x)(144− 136x+ 65x2 − 14x3 + x4) x ∈ [3, 4],0 x ∈ R \ [−3, 4]
φ([−3,4];2;5)(x) =
1720(2 + x)(3 + x)(1116 + 1482x+ 625x2 + 88x3 + x4) x ∈ [−3,−2],− 1120(1 + x)(2 + x)(112 + 445x+ 355x2 + 83x3 + x4) x ∈ [−2,−1],
148(1 + x)(48− 32x− 28x2 + 105x3 + 78x4 + x5) x ∈ [−1, 0],− 136(−1 + x)(36 + 48x+ 3x2 − 125x3 + 73x4 + x5) x ∈ [0, 1],
148(−2 + x)(−1 + x)(−164 + 478x− 335x2 + 68x3 + x4) x ∈ [1, 2],− 1120(−3 + x)(−2 + x)(−1064 + 1337x− 525x2 + 63x3 + x4) x ∈ [2, 3],1720(−4 + x)(−3 + x)(−2928 + 2520x− 695x2 + 58x3 + x4) x ∈ [3, 4],0 x ∈ R \ [−3, 4]
37
B The supplement to section 4
We construct the piecewise polynomial RPP shape functions associate with uniformly distributedparticles in [0,∞) as follows: xk = k, k = 0, 1, 2, 3, · · · , . The corresponding shape functions arewritten in the reference coordinate system(that is, φ(xk)(x) = φk(x− xk))
[B-I] Adopting those polynomials fi,(3;0;5)(x), i = 1, · · · 6, defined by (19), we have the C0RPP shape functions of reproducing order 5 corresponding to the particles in [0,∞).
φ0(x) =
− 1120(−5 + x)(−4 + x)(−3 + x)(−2 + x)(−1 + x) x ∈ [0, 3]
0 x 6∈ [0, 3]
φ1(x) =
124(x− 4)(x− 3)(x− 2)(x− 1)(x+ 1) x ∈ [−1, 2]f6,(3;0;5)(x) x ∈ [2, 3]
0 x 6∈ [−1, 3]
φ2(x) =
− 112(x− 3)(x− 2)(x− 1)(x+ 1)(x+ 2) x ∈ [−2, 1]
f5,(3;0;5)(x) x ∈ [1, 2]
f6,(3;0;5)(x) x ∈ [2, 3]
0 x 6∈ [−2, 3]
φ3(x) =
112(x− 2)(x− 1)(x+ 1)(x+ 2)(x+ 3) x ∈ [−3, 0]f4,(3;0;5)(x) x ∈ [0, 1]
f5,(3;0;5)(x) x ∈ [1, 2]
f6,(3;0;5)(x) x ∈ [2, 3]
0 x 6∈ [−3, 3]
φ4(x) =
− 124(x− 1)(x+ 1)(x+ 2)(x+ 3)(x+ 4) x ∈ [−4,−1]
f3,(3;0;5)(x) x ∈ [−1, 0]f4,(3;0;5)(x) x ∈ [0, 1]
f5,(3;0;5)(x) x ∈ [1, 2]
f6,(3;0;5)(x) x ∈ [2, 3]
0 x 6∈ [−4, 3]
φ5(x) =
1120(x+ 1)(x+ 2)(x+ 3)(x+ 4)(x+ 5) x ∈ [−5,−2]f2,(3;0;5) x ∈ [−2,−1]f3,(3;0;5) x ∈ [−1, 0]f4,(3;0;5) x ∈ [0, 1]
f5,(3;0;5) x ∈ [1, 2]
f6,(3;0;5) x ∈ [2, 3]
0 x 6∈ [−5, 3]
For j ≥ 6, φj(x) = φ([−3,3];0;5)(x) and φ([−3,3];0;5)(x) is defined in section 3.[B-II] Adopting those polynomials gj,(3;1;4)(x), j = 1, · · · 6, defined by (20), we have C1 RPP
shape functions of reproducing order 4 corresponding to the integer particles in [0,∞) as follows:
38
φ0(x) =
1
5400(x− 5)(x− 4)(x− 3)2(x− 2)(x− 1)(8x+ 15) x ∈ [0, 3]
0 x 6∈ [0, 3]
φ1(x) =
− 11080(x− 4)(x− 3)(x− 2)(x− 1)(x+ 1)(8x2 − x− 45) x ∈ [−1, 2]
g6,(3;1;4)(x) x ∈ [2, 3]
0 x 6∈ [−1, 3]
φ2(x) =
1540(x− 3)(x− 2)(x− 1)(x+ 1)(x+ 2)(8x2 + 7x− 45) x ∈ [−2, 1]g5,(3;1;4)(x) x ∈ [1, 2]
g6,(3;1;4)(x) x ∈ [2, 3]
0 x 6∈ [−2, 3]
φ3(x) =
− 1540(x− 2)(x− 1)(x+ 1)(x+ 2)(x+ 3)(8x2 + 15x− 45) x ∈ [−3, 0]
g4,(3;1;4)(x) x ∈ [0, 1]
g5,(3;1;4)(x) x ∈ [1, 2]
g6,(3;1;4)(x) x ∈ [2, 3]
0 x 6∈ [−3, 3]
φ4(x) =
11080(x− 1)x(x+ 2)(x+ 3)(x+ 4)(8x2 + 23x− 45) x ∈ [−4,−1]g3,(3;1;4)(x) x ∈ [−1, 0]g4,(3;1;4)(x) x ∈ [0, 1]
g5,(3;1;4)(x) x ∈ [1, 2]
g6,(3;1;4)(x) x ∈ [2, 3]
0 x 6∈ [−4, 3]
φ5(x) =
− 15400(x+ 1)(x+ 2)(x+ 3)(x+ 4)(x+ 5)2(8x− 9) x ∈ [−5,−2]
g2,(3;1;4)(x) x ∈ [−2,−1]g3,(3;1;4)(x) x ∈ [−1, 0]g4,(3;1;4)(x) x ∈ [0, 1]
g5,(3;1;4)(x) x ∈ [1, 2]
g6,(3;1;4)(x) x ∈ [2, 3]
0 x 6∈ [−5, 3]
For j ≥ 6, φj(x) = φ([−3,3];1;4)(x) and φ([−3,3];1;4)(x) is defined in section 3.By applying a similar method to section 4.2, one can construct piecewise polynomial shape
functions with reproducing property of order 2K − 1(or 2K − 2), associated with non-uniformlydistributed particles in [0,∞).
39
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