Meshfree Particle Methods for Thin Plates

Transcription

1 Meshfree Partice Methods for Thin Pates Hae-Soo Oh, Christopher Davis Department of Mathematics and Statistics, University of North Caroina at Charotte, Charotte, NC Jae Woo Jeong Department of Mathematics, Miami University, Hamiton, OH 450 Abstract In this paper, we are concerned with meshfree partice methods for the soutions of the cassica pate mode The vertica dispacement of a thin pate is governed by a fourth order eiptic equation and thus the approximation functions for numerica soutions are required to have continuous partia derivatives Hence, the conventiona finite eement method has difficuties to sove the fourth order probems Meshfree methods have the advantage of constructing smooth approximation functions, however, most of the earier works on meshfree methods for pate probems used either moving east squares method with penaty method or couping FEM with meshfree method to dea with essentia boundary conditions In this paper, by using generaized product partition of unity, introduced by Oh et a, we introduce meshfree partice methods in which approximation functions have high order poynomia reproducing property and the Kronecker deta property We aso prove error estimates for the proposed meshfree methods Moreover, to demonstrate the effectiveness of our method, resuts of the proposed method are compared with existing resuts for various shapes of pates with variety of boundary conditions and oads Keywords: Meshfree methods, generaized product partition of unity; partition of unity function with fat-top; reproducing poynomia partice shape functions; Kirchhoff pate mode Introduction A arge number of structura components in engineering can be cassified as pates Typica exampes in civi engineering structures are foor and foundation sabs, ock-gates, thin retaining was, bridge decks and sab bridges Pates are indispensabe in ship buiding, automobie, and aerospace industries The stress resutants of a thin pate, such as membrane forces, shear forces and moments, can be cacuated through the 3-dimensiona easticity Under certain hypotheses such as Kirchhoff Corresponding author Te: ; fax: ; E-mai: hso@unccedu Supported in part by NSF grant DMS , DMS

2 and Reissner-Mindin, the 3-dimensiona easticity for a pate is reduced to a two-dimensiona probem The Kirchhoff pate mode obtained through the Kirchhoff hypothesis is we suited for very thin pates A imitation of this mode is that the governing equation of the dispacement vector of this pate mode is a fourth order differentia equation Thus, for the conventiona finite eement soution of this probem, the Argyris and Be trianges, the Bogner-Fox-Schmit rectange ([6],[7]) are suggested to construct C -continuous finite eements These Hermite-type finite eements are difficut to impement By introducing two additiona unknown functions, the Kirchhoff hypothesis is reaxed to have the first (or the third) order shear deformation pate mode However, these pate modes have ocking probems and boundary ayer probems The boundary ayer probem worsens as the pate gets thinner In this paper, we are concerned with the cassica pate mode Numerous papers and books have been pubished on the cassica pate mode for thin pates ([33],[37] and the papers referenced therein) that have no boundary ayer probems Moreover, instead of using C -finite eements, for the Kirchhoff pate mode, non conforming finite eement methods ([6],[7],[40]) and mortar finite eement methods ([24]) have aso been appied Meshess methods ([],[3],[4],[5],[4],[8],[34],[35]) have severa advantages over the conventiona finite eement method ([6],[7],[36]) Their fexibiity and wide appicabiity have gained attention from scientists and engineers to this very dynamic research area ([0]-[2]) Meshess methods use fexibe smooth base functions and use no mesh or use background mesh minimay Actuay, meshess methods have been referred to as meshfree methods ([],[3], [4]), Reproducing Kerne Partice Methods(RKPM) ([3],[8],[2],[22],[23]), Reproducing Kerne Eement Methods (RKEM) ([8],[9],[20]), GFEM (PUFEM)([25],[34],[35]), h-p Coud Method([8]) and Eement Free Gaerkin Method (EFGM) ([]) Even though these approaches are appicabe in soving many difficut science and engineering probems, they are imited by their arge matrix condition numbers ([34],[35]), inefficiency in handing essentia boundary conditions ([2],[5]), compexity in constructing a partition of unity ([9]), engthy numerica integration ([9]), and so on To aeviate difficuties encountered in meshess methods, Oh et a introduced three cosedform partition of unity (PU) that have fat-top: (i) Convoution partition of unity ([29]) for background meshes of arbitrary convex poygona shapes of partition of a given domain; Using convoution partition of unity, Oh et a introduced severa meshess methods that are caed uniform RPPM (Reproducing Poynomia Partice Method) ([30]), patchwise RPPM, adaptive RPPM, and RSPM (Reproducing Singuarity Partice Method) in ([28],[29],[3],[32]) Note that RPPM is simiar to RKPM ([3],[3],[8], [9],[20],[2],[22],[23]) (ii) Amost everywhere partition of unity ([27]) for background meshes on non convex domains and imposing essentia boundary condition on convex as we as non convex domains (iii) Generaized product partition of unity ([26]) for construction of non-uniformy distributed partice shape functions Using PU functions with fat-top gives reativey sma matrix condition numbers In this paper, for the thin pate probems, the generaized product partition of unity is appied to construct smooth oca approximation functions that have the reproducing poynomia property and the Kronecker deta property In section 2, we briefy review the generaized product partition of unity Definitions and terminoogies that are used in this paper are aso introduced In section 3, reproducing poynomia partice methods (RPPM) for pate probems are introduced We estimate the error 2

3 bounds of numerica soutions of the proposed RPPM in the L 2 -norm, the -seminorm and the 2-seminorm, respectivey In section 4, the variationa formuation of thin pate is described In section 5, the reference shape functions that satisfy the camped boundary conditions of thin pates are constructed In section 6, effectiveness of the proposed meshfree method (RPPM) is demonstrated with various shapes of pates Finay, the concuding remarks are given in section 7 2 Cosed-form-smooth-partition of unity functions with fat-top Let Ω be a connected open subset of R d We define the vector space C m (Ω) to consist of a those functions φ which, together with a their partia derivatives α φ(= α α d d φ) of orders α = α + + α d m, are continuous on Ω In the foowing, a function φ C m (Ω) is said to be a C m -function If Ψ is a function defined on Ω, we define the support of Ψ as suppψ = {x Ω Ψ(x) 0} For an integer k 0, we aso use the usua Soboev space denoted by H k (Ω) For u H k (Ω), the norm and the semi-norm, respectivey, are u k,ω = α k u k,ω = α =k Ω Ω α u 2 dx α u 2 dx /2 /2, u k,,ω = max α k {esssup α u(x) : x Ω}; (), u k,,ω = max α =k {esssup α u(x) : x Ω} (2) A famiy {U k : k D} of open subsets of R d is said to be a point finite open covering of Ω R d if there is an integer M such that any x Ω ies in at most M of the open sets U k and Ω k U k For a point finite open covering {U k : k D} of a domain Ω, suppose there is a famiy {φ k : k D} of Lipschitz functions on Ω satisfying the foowing conditions: For k D, 0 φ k (x), x R d 2 The support of φ k is contained in U k, for each k D 3 k D φ k(x) = for each x Ω Then {φ k : k D} is caed a partition of unity (PU) subordinate to the covering {U k : k D} The covering sets {U k } are caed patches By amost everywhere partition of unity, we mean {φ k : k D} such that the condition 3 of a partition of unity is not satisfied ony at finitey many points (2D) or ines (3D) on a part of the boundary Let ω = supp(φ) Then ω ft = {x ω : φ(x) = } and ω n-ft = {x ω : 0 < φ(x) < } are caed the fat-top part and the non fat-top part of ω, respectivey The function φ is said to 3

4 be a function with fat-top if the interior of ω ft is non-void Moreover, {φ k : k D} is caed a partition of unity with fat-top whenever it is a partition of unity and φ k is a function with fat-top for each k D ω ft is aso denoted by Q ft whenever φ is associated with a patch Q Notice that if f,, f n are ineary independent on ω ft and φ is a function with fat-top, the product functions, φ f,, φ f n, are aso ineary independent on ω However, if φ is not a function with fat-top, the product functions, φ f,, φ f n, coud be ineary dependent The hat functions of the conventiona finite eement are PU functions with no fat-top and hence PUFEM using the hat functions as PU yieds a arge matrix condition number in genera A weight function (or window function) is a non-negative continuous function with compact support and is denoted by w(x) Consider the foowing conica window function: For x R, w(x) = { ( x 2 ) if x, 0 if x >, (3) where is an integer Then w(x) is a C -function In R d, the weight function w(x,, x d ) can be constructed from a one-dimensiona weight function as w(x,, x d ) = d i= w(x i) In this paper, we use the normaized window function defined by wδ (x) = Aw(x/δ), (4) where A = [(2 + )!]/[2 2+ (!) 2 δ] ([3]) is the constant that makes R w δ (x)dx = Let Λ be a finite index set and Ω denotes a bounded domain in R d Let {x j : j Λ} be a set of a finite number of uniformy or non-uniformy spaced points in R d, that are caed partices Definition 2 Let k be a non-negative integer Then the functions φ j (x) corresponding to the partices x j, j Λ are caed the RPP(reproducing poynomia partice) shape functions with the reproducing property of order k (or simpy, of reproducing order k ) if and ony if they satisfy the foowing condition: (x j ) α φ j (x) = x α, for x Ω R d and for 0 α k (5) j Λ Note that the RPP shape functions φ j, j Λ, of reproducing order k can exacty interpoate poynomias of degree ess than or equa to k 2 One-dimensiona partition of unity functions without fat-top For any positive integer n, C n -piecewise poynomia basic PU functions are constructed as foows: For integers n, we define a piecewise poynomia function by φ (pp) g n (x) = φ L g n (x) := ( + x) n g n (x) if x [, 0], φ R g n (x) := ( x) n g n ( x) if x [0, ], 0 if x, (6) 4

5 where g n (x) = a (n) 0 +a (n) ( x)+a(n) 2 ( x)2 + +a (n) n ( x)n whose coefficients are inductivey constructed by the foowing recursion formua: if k = 0, k a (n) k = a (n ) j if 0 < k n 2, (7) j=0 ) if k = n For exampe, 2(a (n) n 2 g (x) = ; g 2 (x) = 2x; g 3 (x) = 3x + 6x 2 ; g 4 (x) = 4x + 0x 2 20x 3 ; g 5 (x) = 5x + 5x 2 35x x 4 Then, φ (pp) g n has the foowing properties whose proofs can be found in ([29]) φ (pp) g n (x) + φ (pp) g n (x ) = for a x [0, ] and 0 φ g (pp) n (x), for a x R Hence, {φ (pp) g n (x j) j Z} is a partition of unity on R φ (pp) g n (x) is a C n -function 22 Generaized two-dimensiona product partition of unity with fat-top Using the basic PU function φ (pp) g n defined by (6), we construct a C n -PU function with fat-top whose support is [a δ, b + δ] with (a + δ) < (b δ) in a cosed form as foows: ψ (δ,n ) [a,b] (x) = φ L g n ( x (a+δ) 2δ ) if x [a δ, a + δ], if x [a + δ, b δ], φ R g n ( x (b δ) 2δ ) if x [b δ, b + δ], 0 if x / [a δ, b + δ] Here, in order to make a PU function have a fat-top, we assume δ (b a)/3 Let us note that (x) is actuay the convoution, χ [a,b] (x) w n δ (x), of the characteristic function χ [a,b] ψ (δ,n ) [a,b] and the scaed window function w n δ, defined by (4) (Theorem 35 of [29]) Since the two functions φ R g n, φ L g n, defined by (6), satisfy the foowing reation: if ϕ : [ δ, δ] [0, ] is defined by φ R g n (ξ) + φ L g n (ξ ) =, for ξ [0, ], (9) ϕ(x) = (x + δ)/(2δ), (8) then we have φ R g n (ϕ(x)) + φ L g n (ϕ(x) ) =, for x [ δ, δ] 5

6 ψ x R z y ψ x R = ψ x R = 0 x δ δ Figure : Sketch of Ψ R x ( Ψ x=0 ) Using the atter equation gives two basic one-dimensiona C n -functions if x δ, ψ0 R (x) = φ R g n ( x+δ 2δ ) if x [ δ, δ], 0 if x δ, such that 0 if x δ ψ0 L (x) = φ L g n ( x δ 2δ ) if x [ δ, δ], if x δ (0) () 0 ψ L 0 (x), ψ R 0 (x), ψ R 0 (x) + ψ L 0 (x) =, for a x R These PU functions (0) and () are extended to basic two-dimensiona C n -PU functions on R 2 as foows: Suppose P P 2 is a straight ine connecting two points P (x, y ) and P 2 (x 2, y 2 ) with x x 2 such that y < y 2 if x = x 2 Then the ange between the positive x-axis and P P 2 is determined by the foowing formua ( ) tan θ = y2 y if x 2 x, x 2 x (2) π/2 if x 2 = x Let T P P 2 be an affine transformation on R 2 that transforms the straight ine P P 2 onto the y-axis defined by T P P 2 (x, y) = ( x, ỹ) : [ ] [ ] cos(π/2 θ) sin(π/2 θ) x x ( x, ỹ) = (3) sin(π/2 θ) cos(π/2 θ) y y 6

7 0 y Ψ L x L Ψ P 2 L 0 R Ψ x δ δ 0 x P Ψ R 0 (a) (b) Figure 2: (a) Schematic diagram of basic PU functions, Ψ R x = Ψ x=0 and Ψ L x = Ψ x=0 in dimension two; (b) Transformed basic PU, Ψ R = Ψ P P 2 and Ψ L = Ψ P P 2, by the affine transformation T P P 2 L 4 ψ 0 4 ψ 3 ft Q 6 ft Q 5 0 L 3 Q ft 7 Q ft Q ft 4 ψ 2 L 2 Q ft 8 Q ft 2 Q ft ψ L Figure 3: By the ines L, L 2, L 3, L 4, the Domain Ω is partitioned into eight patches Q,, Q 8 and the fat-top parts of corresponding product PU functions are denoted by Q ft,, Qft 8, respectivey 7

8 Then we define two PU functions by that satisfy Ψ P P 2 (x, y) = ψ R 0 ( x), Ψ P P 2 (x, y) = ψ L 0 ( x) = Ψ P P 2 (x, y), (4) Ψ P P 2 (x, y) + Ψ P P 2 (x, y) =, for a (x, y) R 2 For exampe, if the ine P P 2 is the y-axis (denoted by x = 0 in (5) and (6)), then the two-dimensiona C n -functions are Ψ x=0 (x, y) = ψ R 0 (x) and Ψ x=0(x, y) = ψ L 0 (x), for a (x, y) R 2 (5) In other words, two step-ike-functions are the composition of the coordinate projection, (x, y) x, and ψ0 R, ψl 0, respectivey The graph of Ψ x=0 (simpy denoted by Ψ R x ) is sketched in Fig The schematic diagram for Ψ x=0 and Ψ x=0 is shown in Fig 2 That is, Ψ x=0 (x, y) = if x δ Ψ x=0 (x, y) = 0 if x δ Ψ x=0 (x, y) = 0 if x δ and Ψ x=0 (x, y) = if x δ (6) 0 Ψ x=0 (x, y) if x δ 0 Ψ x=0 (x, y) if x δ 23 Generaized product partition of unity Suppose the given domain Ω is partitioned into patches Q j, j =,, n (background mesh) by ines and rays as shown in Fig 3 Then the cosed form partition of unity functions Ψ P j, j =,, n with fat-top, caed generaized product partition of unity, are introduced in [26] In the foowing, we briefy review generaized product partition of unity for those patches shown in Fig 3 Here δ is a sma number (usuay in [00, 0]) that depends on the sizes of patches We describe the generaized product PU functions with a specific exampe in the foowing: The trianguar patch Q of Fig 3 is surrounded by ines L, L 2, L 4 Using (4), the stepike-basic PU functions of (5) on R 2 are transformed onto ines L, L 2, L 4 to get three pairs of PU functions Ψ L, Ψ L := Ψ L ; Ψ L2, Ψ L 2 := Ψ L2 ; Ψ L4, Ψ L 4 := Ψ L4 (7) 2 The fat-top part of each patch Q j that is outside the dotted ines is denoted by Q ft j on which ony one of each pairs of basic PU functions is one 3 Among the six functions in (7) reated to ines encosing Q, those which are one on Q ft are Ψ L, Ψ L2, and Ψ L 4 A cosed form PU function corresponding to the patch Q is the product of these basic PU functions, that is, Ψ P = Ψ L Ψ L2 Ψ L 4 4 Simiary, the cosed form PU functions with wide fat-top corresponding to patches Q j, j = 2,, 8, respectivey, are Ψ P 2 = Ψ L Ψ L 2 Ψ L 4, Ψ P 3 = Ψ L Ψ L 2, Ψ P 4 = Ψ L Ψ L2 Ψ L 3, Ψ P 5 = Ψ L Ψ L3 Ψ L 4, Ψ P 6 = Ψ L Ψ L4, Ψ P 7 = Ψ L Ψ L2 Ψ L4, Ψ P 8 = Ψ L 2 Ψ L4 8

9 Then, using the arguments simiar to [26], one can show that 8 Ψ P j (x, y) =, for a (x, y) Ω j= These functions with fat-top are caed the generaized product PU functions (we refer to [26] for the proof and the constructions for genera cases) It was shown in [26] that if a patch Q j is a rectange [a, b] [c, d], then Ψ P j ψ (δ,n ) [a,b] ψ (δ,n ) [c,d], of one-dimensiona functions defined by (8) is the tensor product 3 Reproducing poynomia partice methods 3 Construction of RPP shape functions Unike existing meshfree partice methods such as the moving east squares methods and the reproducing kerne partice methods, we construct cosed-form piecewise-poynomia partice shape functions with use of generaized product partition of unity (i) Suppose a domain Ω is divided into quadranguar and trianguar patches Let ˆR = [0, ] [0, ](Fig 6) be the reference rectanguar patch and ˆT (Fig 5) be the reference trianguar patch Let Q be a quadrange with vertices (a i, b i ), i =, 2, 3, 4 and et T be a triange with vertices (a i, b i ), i =, 2, 3 (Note that Q and T coud be arger than physica patches as shown in Exampe 3 of Section 6) We define a biinear mapping T (q) : ˆR Q by T (q) (ξ, η) = (x, y), where x = a ( ξ)( η) + a 2 ξ( η) + a 3 ξη + a 4 ( ξ)η = A + A 2 ξ + A 3 η + A 4 ξη, y = b ( ξ)( η) + b 2 ξ( η) + b 3 ξη + b 4 ( ξ)η = B + B 2 ξ + B 3 η + B 4 ξη and a inear mapping T (t) : ˆT T by (8) T (t) : { x = a ( ξ η) + a 2 ξ + a 3 η, y = b ( ξ η) + b 2 ξ + b 3 η (9) (ii) (a) Partices and partice shape functions on ˆR: suppose ˆφ ij (ξ, η) = L i (ξ) L j (η), 0 i, j (k + ), where L i (ξ) is the Lagrange interpoating poynomia corresponding to the node ξ i for i =,, (k + ) and L j (η) is aso the Lagrange poynomia corresponding to the nodes η j, for j =,, (k + ) Then we have k+ k+ ξ α i η α 2 j i= j= ˆφ ij (ξ, η) = ξ α η α 2, for integers α, α 2 such that 0 α, α 2 k (20) 9

10 (b) Partices and partice shape functions on ˆT : Let n = (k + )(k + 2)/2 Then for s =, 2,, n, partices (ξ s, η s ) ˆT, and partice shape functions ˆφ s are those shape functions for the conventiona FEM Then we have n ξ α ˆφ s (ξ, η) = ξ α η α 2, for integers α, α 2 such that 0 α + α 2 k (2) s= s η α 2 s Then the RPP orders after these reference RPP shape functions are panted into the physica domain become as foows: Lemma 3 Through the patch mappings, T (q) or T (t), partices as we as partice shape functions are panted in the physica domain Ω Then there are some restrictions on the RPP order of the transformed partice shape functions on physica quadranguar patches: Let (x i, y j ) = T (q) (ξ i, η j ) and φ ij = ˆφ ij T (q), for 0 i, j k Then k+ k+ x α i y α 2 j φ ij (x, y) = x α y α 2, for 0 α + α 2 k (22) i= j= For exampe, even though ˆφ ij, 0 i, j k, generate the monomia ξ k η k, the transformed partice shape functions φ ij, 0 i, j k, are not abe to generate the monomia x k y k 2 Let (x s, y s ) = T (t) (ξ s, η s ) and φ s = ˆφ s T (t), for s =, 2,, n = (k + )(k + 2)/2 Then n x α s y α 2 s φ s (x, y) = x α y α 2, for 0 α + α 2 k (23) Proof () s= k+ k+ x α i y α 2 j φ ij (x, y) i= j= k+ k+ = [A + A 2 ξ i + A 3 η j + A 4 ξ i η j ] α [B + B 2 ξ i + B 3 η j + B 4 ξ i η j ] α 2 ˆφij (ξ, η) i= j= If 0 α + α 2 k, by the reation (20), each term of [A + A 2 ξ i + A 3 η j + A 4 ξ i η j ] α [B + B 2 ξ i + B 3 η j + B 4 ξ i η j ] α 2 is generated by ˆφ ij (ξ, η), 0 i, j k Hence the ast equation equas to [A + A 2 ξ + A 3 η + A 4 ξη] α [B + B 2 ξ + B 3 η + B 4 ξη] α 2 = x α y α 2 (2) Since T (t) is a inear mapping, T (t) can be constructed expicity and hence the reation (23) is obvious 0

11 (iii) Let ω δ be the support of the generaized product PU function Ψ P corresponding to the -th patch in the domain Ω for each =,, N Let φ () ij be partice shape functions corresponding to the partices that fa into ω δ by the patch mapping Then, for each, =,, N, the approximation functions Ψ P φ () s, s (k + )(k + 2)/2 φ () ij, i, j (k + )(when the -th patch is rectanguar) and Ψ P (when the -th patch is trianguar) are smooth cosed-form piecewise-poynomias and have the compact support ω δ Reproducing Poynomia Partice Methods (RPPM) proposed in this paper is a meshfree partice method and it is actuay the Gaerkin approximation methods using the oca approximation functions constructed above for goba basis functions Let us note that these basis functions satisfy the Kronecker deta property and have the poynomia reproducing property of order k 32 Error estimates Since the oca approximation functions corresponding the partices panted through the patch mappings have the reproducing poynomia property whenever no essentia boundary conditions are imposed, the proposed method is a meshfree partice method Whereas, since the method empoys the generaized product partition of unity for the construction of approximation functions, it is aso a PUFEM Thus, we can estimate the errors of approximate soutions in two different approaches: an estimate for meshfree methods simiar to [3] and an estimate for PUFEM simiar to [25] In this section, modifying the error estimates for PUFEM due to [3] and [25], we have the foowing extended error estimates for our method Theorem 3 Let Ω be a two-dimensiona poygona domain Let Ψ P be a generaized product partition of unity corresponding to a patch Q, for =,, N, and ω be the support of Ψ P such that 0 Ψ P on ω n-fat (non-fat-top part); Ψ P = on ω ft (fat-top part); (24) ω = ω ft ω n-fat (note: ω ft is aso denoted by Q ft in Figs 3, 8, 9) We assume that card({i x ω i }) M for a x Ω Let a coection of oca approximation spaces V H 2 (Ω ω ) be given Let u H 2 (Ω) be the function to be approximated Assume that the oca approximation spaces V have the foowing approximation properties: on each patch Ω ω, the function u can be approximated by a function v V such that u v m,ω ε (m), and u v m,,ω n-fat ε (m),, for m = 0,, 2 hod for a Let v = N function k= c kφ () k u app := V, (φ () k is either φ () ij N (Ψ P v ) V RP P, = in (22) or φ () s in (23)) Then the

12 satisfies the goba estimates (i) u u app 2 0,Ω M N = (ii) (u u app ) 2,Ω 2M (ε (0) ) 2, (25) N = { (ε (0) (iii) (u u app ) 2 2,Ω N { 4M (ε (0), )2 ( Ψ P 2,ω n-fat =, )2 ( Ψ P,ω n-fat ) 2 + (ε (2) ) 2 + (ε () ) 2}, (26) ) 2 + 5( Ψ P,ω n-fat ) 2} (27) Proof The proof proceeds by using simiar arguments as those of Theorem 62 of [3] For brevity, we use the foowing notations: x u = u, x, (u v ) = e, and Ψ P = Ψ Using the fact that the derivatives of Ψ are zero except for a sma 2δ-width strip around the support of Ψ and the inequaities, (a + + a n ) 2 n(a a2 n), we have the foowing error estimates: (ii) The approximation error in the semi -norm is bounded as foows: (u u app ) 2,Ω = 2 Ω { 2M 2M 2M Ω Ω N Ψ (u v ) 2,Ω = = Ω { [( Ψ e ), x ] 2 + [( [ (Ψ, x e )] 2 + [ (Ψ e, x )] 2 + [ (Ψ, y e )] 2 + [ { [(Ψ, x e ) 2 + (Ψ e, x ) 2 ] + [(Ψ, y e ) 2 + (Ψ e, y ) 2 ] } { [(Ψ, x ) 2 + (Ψ, y ) 2 ](e ) 2 + (Ψ ) 2 [(e, x ) 2 + (e, y ) 2 ] } { ( e 0,,ω n-fat Ψ e ), y ] 2 } (Ψ e, y )] 2 } ) 2 ( Ψ,ω n-fat) 2 + ( e,ω ) 2} (since 0 Ψ on Ω) (iii) The approximation error in the semi 2-norm is bounded as foows: 2

13 N (u u app ) 2 2,Ω = Ψ (u v ) 2 2,Ω = 3 Ω{[ = Ψ, xx e ] 2 + [2 Ω { [( Ψ, x e, x ] 2 + [ Ψ e ), xx ] 2 + [( Ψ e, xx ] 2 } +4 Ω{( Ψ, xy e ) 2 + ( Ψ, y e, x ) 2 + ( Ψ, x e, y ) 2 + ( +3 Ω{[ Ψ, yy e ] 2 + [2 Ψ, y e, y ] 2 + [ Ψ e, yy ] 2 } 3M {[Ψ, xx e ] 2 + [2Ψ, x e, x ] 2 + [Ψ e, xx ] 2 } +4M +3M Ω Ω Ω 4M Ω {(Ψ, xy e ) 2 + (Ψ, y e, x ) 2 + (Ψ, x e, y ) 2 + (Ψ e, xy ) 2 } {[Ψ, yy e ] 2 + [2Ψ, y e, y ] 2 + [Ψ e, yy ] 2 } { e 2 (Ψ 2, xx +Ψ 2, yy +Ψ 2, xy ) + Ψ 2 (e, 2 xx +e, 2 yy +e, 2 xy ) Ψ e ), xy ] 2 + [( Ψ e, xy ) 2 } +4((Ψ, x e, x ) 2 + (Ψ, y e, y ) 2 ) + (Ψ, y e, x ) 2 + (Ψ, x e, y ) 2} 4M { ( e 0,,ω n-fat) 2 ( Ψ 2,ω n-fat) 2 + ( e 2,ω ) 2 + 5( Ψ,ω n-fat) 2 ( e,,ω n-fat) 2} (using 0 Ψ at the ast step) The proof of part (i) is simiar to that of part (ii) Ψ e ), yy ] 2 } For a sma δ, the absoute vaue of the derivatives of Ψ P become arge (O(/δ)), however, since they are non-zero ony aong the 2δ-width strip aong the boundary of its support ω, Ψ P m,ω n-fat, m =, 2, are not too arge as shown in Tabe For exampe, in order to get [ C -generaized product PU functions, suppose φ R g 2 ( x+δ 2δ ) = x + δ ] 2 [ + 2( x + δ ] 2δ 2δ ) is used in the definition (0) Then δ [ x φ R g 2 ( x + δ ] 2 2δ ) dx = 3 δ [ 5δ, xx φ R g 2 ( x + δ ] 2 2δ ) dx = 3 2δ 3 δ Hence, for this choice of PU functions, we have ( Ψ P,ω n-fat L 3 5δ δ ) /2 ; Ψ P 2,ω n-fat ( L 3 ) /2 2δ 3, (28) where L is the ength of the perimeter of the patch Q corresponding to Ψ P The upper bounds are in Tabe for various δ 3

14 Tabe : Bound for Ψ P,ω n-fat [3L/(5δ)] /2 and Ψ P 2,ω n-fat [3L/(2δ 3 )] /2 for various δ, where L is the ength of the perimeter of patch Q δ [3/(5δ)] / [3/(2δ 3 )] / d Ω z t x α n y Figure 4: The 3-dimensiona pate ˆΩ and 2-dimensiona midpane Ω 4

15 4 Modes for eastic pates Let Ω be a bounded domain in R 2 with piecewise smooth boundary, which represents the midpane of a pate, which we assume to be of thickness d (d << diam(ω))(see, Fig 4) We represent the 3-dimensiona pate as ˆΩ = {(x, y, z) R 3 (x, y) Ω, z < d/2} Under some hypothesis, the 3-dimensiona easticity equations for pates are reduced to 2- dimensiona equations (pate modes) The popuar pate modes are the Kirchhoff pate mode for thin pates, and for moderatey thick pates, first order shear deformation pate mode (known as the Reissner-Mindin pate mode), third order shear deformation pate mode, and higher order pate modes for higher accuracy 4 Cassica pate theory The cassica (Kirchhoff) pate theory is one in which the dispacement fied is based on the Kirchhoff hypothesis: () Straight ines perpendicuar to the midde surface before deformation remain straight after deformation; (2) The transverse normas do not experience eongation (ie they are inextensibe); (3) The transverse normas remain perpendicuar to the midde surface after deformation ([33],[36],[37]) From this hypothesis, ɛ zz = 0 and the transverse shear strains are zero: ɛ xz = ɛ yz = 0 Suppose (u, v, w) denote the tota dispacement of a point aong the xyz-coordinate system and (u 0, v 0, w 0 ) denote the vaues of u, v and w at the point (x, y, 0) Then these conditions impy u(x, y, z) = u 0 z w x, v(x, y, z) = v 0 z w y, w(x, y, z) = w 0 (29) Substituting the dispacement functions of (29) into the virtua work formuation, we have ([36],[37]) B(w, v) = F(v), for w, v H 2 (Ω), (30) where B(w, v) = D F(v) = Ω Ω 2 w x 2 2 w y 2 2 w x y T p(x, y)vdxdy + ν 0 ν ( ν) Γ 2 v x 2 2 v y 2 2 v x y ( v M n n dt Q n + M nt Γ t dxdy, (3) ) vdt (32) 5

16 Tabe 2: Boundary conditions in the cassica theory of pate Type of support Essentia (Geometric) BC Natura (Force) BC camped w = 0, w/ n = 0 None simpe support w = 0 M n = 0 free None M n = M nt = Q n = 0 symmetry w/ n = 0 Q n + M nt / t = 0 antisymmetry w = 0 M n = 0 where ν is the Poisson s ratio and, E is the Young s moduus of an isotropic eastic materia, and Ed 3 D = 2( ν 2 ), ( 2 w M x = D x 2 + ν 2 w y 2 ( 3 w Q x = D x w x y 2 ) (, M y = D ν 2 w ), Q y = D x w y 2 ( 3 w y w x 2 y M n = [M x, M y, M xy M yx ][cos 2 α, sin 2 α, sin α cos α] T, M nt = [ M x + M y, M xy ][sin α cos α, (cos 2 α sin 2 α)] T, p = Q x x + Q y y ), M xy = D( ν) 2 w x y = M yx, ), Q n = Q x cos α + Q y sin α, Here α is the ange between the x-axis and the norma axis of the norma-tangentia coordinate system as shown in Fig 4 The conventiona boundary conditions in the cassica theory of pate are isted in Tabe 2 Let V RP P be an approximation space constructed by use of the generaized partition of unity functions Ψ P and the reference shape functions ˆφ k constructed in the previous sections That is, V RP P = span{ψ P [ ˆφ k T ] : =, 2,, N; k =, 2,, N }, (33) where Ω is partitioned into the N numbers of patches Q,, Q N and T is the patch mapping from a reference patch into ω δ, the support of the generaized product PU function ΨP, which is associated with the physica patch Q Then the approximation space V RP P has the foowing properties: () high reguarity of each member; (2) the Kronecker deta property at amost a partices; (3) reproducing poynomia property of high order on the patches with no intersections with camped boundaries Now, the proposed meshfree partice method (RPPM) for pate probems is the Gaerkin method with use of V RP P as foows: Find w V RP P such that B(w, v) = F(v), for a v V RP P (34) 6

17 η η (0, ) (0, ) (0, 2/3) (/2, /2) (0, /3) (0, 0) ^ T (/2, 0) (, 0) ξ (0,0) ^ T (/3,0) (2/3,0) (,0) ξ d/dξ d/dξ Figure 5: Reference patches and the degree of freedom inear forms on P 3 ( ˆT ) and P 4 ( ˆT ) whose dua basis functions camp one edge denotes an evauation at the point and <- - - indicates an evauation of the ξ-derivative at the point η η (0, ) (0, ) (0, 2/3) (0, /3) (0, 0) ^ R ξ (/2, 0) (, 0) (0, 0) (/3, 0)(2/3, 0) (, 0) ^ R ξ d/d ξ d/d ξ Figure 6: The degree of freedom inear forms on Q 3 ( ˆR) and Q 4 ( ˆR) whose dua basis functions camp one edge denotes an evauation at the point and <- - - indicates an evauation of the ξ-derivative at the point 7

18 η (0, ) η (0, ) (0, 2/3) ^ R d/d η ^ R d/d η (0, /3) (0, 0) (/2, 0) (, 0) ξ (0, 0) (/3, 0) (2/3, 0) (, 0) ξ d/d ξ d/d ξ Figure 7: The degree of freedom inear forms on Q 3 ( ˆR) (Left) and on Q 4 ( ˆR) (Right) whose dua basis camp two edges 5 Partice shape functions with Kronecker deta property to dea with essentia boundary conditions of pates In this section, ˆT and ˆR, respectivey, denote the reference triange with vertices (0, 0), (, 0), (0, ), and the reference rectange [0, ] [0, ] shown in Figs 5 and 6 P n ( ˆT ) = span{ξ i η j 0 i, j n, 0 i + j n} is the space of a poynomias of tota degree ess than or equa to n Q m ( ˆR) = span{ξ i η j 0 i, j m} is the space of a poynomias of degree ess than or equa to m in each variabe Hence, dim(p n ( ˆT )) = (n + )(n + 2)/2 and dim(q m ( ˆR)) = (m + ) 2 V denotes the dua vector space of a vector space V 5 Shape functions for imposing simpe support boundary condition Without oss of generaity, we consider the construction of patchwise smooth RPP shape functions with compact support that have the poynomia reproducing order 3 [RPP functions on the reference Trianguar patch ˆT ] The noda shape functions on the reference triange ˆT corresponding to vertices, (0, 0), (, 0), (0, ), respectivey, are L (ξ, η) = ξ η, L 2 (ξ, η) = ξ, L 3 (ξ, η) = η The foowing ten functions are the Lagrange interpoating shape functions of RPP order 3 that correspond to three vertices, six atera nodes, and one interior node, respectivey: ˆφ = L (3L )(3L 2)/2; ˆφ 2 = L 2 (3L 2 )(3L 2 2)/2; ˆφ 3 = L 3 (3L 3 )(3L 3 2)/2; ˆφ 4 = (9/2)L L 2 (3L ); ˆφ 5 = (9/2)L L 3 (3L ); ˆφ 6 = (9/2)L 2 L (3L 2 ); ˆφ 7 = (9/2)L 2 L 3 (3L 2 ); ˆφ 8 = (9/2)L 3 L (3L 3 ); ˆφ 9 = (9/2)L 3 L 2 (3L 3 ); ˆφ 0 = 27L L 2 L 3 (35) In what foows, the k-th Lagrange interpoating poynomia of degree n associated with n-distinct nodes ξ,, ξ n is denoted by L n,k (ξ) = n ξ ξ i i=,i k ξ k ξ i [RPP functions on the reference rectanguar patch ˆR shown in Fig 6] 8

19 Let L 4,j, j =, 2, 3, 4, be the Lagrange interpoating poynomias of order 3 corresponding to nodes 0, /3, 2/3,, respectivey Then, we have 6 RPP shape functions of order 3: ˆφ k (ξ, η) = L 4,i (ξ) L 4,j (η), k = 4(i ) + j, i, j 4, (36) [Loca approximation functions defined on the domain Ω] Those RPP shape functions defined by (35) and (36) can be panted on a physica domain Ω by proper patch mappings to make smooth oca approximation functions on trianguar or quadranguar patches in Ω For exampe, et T : ˆR Q be a smooth patch mapping and Ψ P (x, y) be the generaized product PU corresponding to Q Then Ψ P (x, y) ˆφ k (T (x, y)), k =,, 6 are smooth functions with compact support ω = suppψ P Q 52 Shape functions for impementing camped boundary conditions The proofs of three technica Lemmas presented in this section can be found in appendix 52 Trianguar reference patches Lemma 5 Suppose the inear functionas (caed the degree of freedom inear forms) N i on P 3 ( ˆT ) are defined by (37) N (f) = f(0, 0), N 2 (f) = f(0, /3), N 3 (f) = f(0, 2/3), N 4 (f) = f(0, ), N 5 (f) = f(/2, 0), N 6 (f) = f(/2, /2), N 7 (f) = f(, 0), N 8 (f) = f (0, 0), N 9(f) = f (0, /3), N 0(f) = f (0, ), for f P 3 ( ˆT ) Then we have the foowing: (i) {N i : i =,, 0} is a basis of the dua space [P 3 ( ˆT )] (ii) If a poynomia f P 3 ( ˆT ) satisfies N j (f) = 0, for j =, 2, 3, 4, 8, 9, 0, then f, f are identicay zero aong the ine ξ = 0 f and η Let { ˆφ j, j =,, 0} be the dua basis on P 3 ( ˆT ) of the basis {N j, j =,, 0} such that for each i =,, 0, N i ( ˆφ j ) = δ j i for j =,, 0 Then it foows from Lemma 5 that for k = 5, 6, 7, ˆφ k = 0, ˆφ k / = 0, ˆφ k / η = 0 (38) aong the η-axis Suppose T is a patch mapping from the reference patch ˆT onto a physica patch Ω T Then we have xy ˆφk (T (x, y)) = ξη ˆφk (T (x, y)) J(T )(x, y) (39) Since ξη ˆφk (0, η) = 0 for k = 5, 6, 7, φ k ˆφ k T and φ k / x, φ k / y, are zero aong the side of Ω T connecting the vertices T (0, 0) and T (0, ) 9

20 522 Extension to the degree of freedom inear functionas on P n ( ˆT ) for n 4 Lemma 5 can be extended to the construction of the degree of freedom inear functionas for any order n For exampe, suppose the 5 degree of freedom inear functionas on P 4 ( ˆT ) are as shown in Fig 5 That is, the inear functionas are defined as foows: for f = 0 i+j 4 a ijξ i η j P 4 ( ˆT ), N (f) = f(0, 0), N 2 (f) = f(0, /4), N 3 (f) = f(0, 2/4), N 4 (f) = f(0, 3/4), N 5 (f) = f(0, ), N 6 (f) = f(/3, 0), N 7 (f) = f(/3, /3), N 8 (f) = f(/3, 2/3), N 9 (f) = f(2/3, 0), N 0 (f) = f(2/3, /3), N (f) = f(, 0), N 2 (f) = f (0, 0), N 3(f) = f (0, /4), N 4(f) = f (0, 2/4), N 5(f) = f (0, ) By using a simiar argument to Lemma 5, one can show that if N j (f) = 0, for a j =,, 5, then f 0 on R 2 Hence N j defined by (40) are a dua basis of P 4 ( ˆT ) (40) 523 Reference rectanguar patch Lemma 52 (Shape functions camping one side) Suppose the inear forms (the degree of freedom inear functionas) N i on Q 3 ( ˆR) = span{ξ i η j 0 i, j 3} are defined by N (f) = f(0, 0), N 2 (f) = f(0, /3), N 3 (f) = f(0, 2/3), N 4 (f) = f(0, ), N 5 (f) = f(/2, 0), N 6 (f) = f(/2, /3), N 7 (f) = f(/2, 2/3), N 8 (f) = f(/2, ), N 9 (f) = f(, 0), N 0 (f) = f(, /3), N (f) = f(, 2/3), N 2 (f) = f(, ), N 3 (f) = f (0, 0), N 4(f) = f (0, /3), N 5(f) = f (0, 2/3), N 6(f) = f for f Q 3 ( ˆR) Then the 6 inear functionas have the foowing properties: (0, ), (4) (i) {N i : i =,, 6} is a basis of the dua space [Q 3 ( ˆR)] (ii) Suppose f Q 3 ( ˆR) satisfies N j (f) = 0 for j =, 2, 3, 4, 3, 4, 5, 6, then f, f identicay zero aong the ine ξ = 0 and f η are Lemma 53 (Shape functions camping two sides) Suppose the inear forms (the degree of freedom inear functionas) N i on Q 3 ( ˆR) = span{ξ i η j 0 i, j 3} are defined by N (f) = f(0, 0), N 2 (f) = f(0, /3), N 3 (f) = f(0, 2/3), N 4 (f) = f(0, ), N 5 (f) = f(/2, 0), N 6 (f) = f(/2, /3), N 7 (f) = f η (/2, 0), N 8(f) = f(/2, ), N 9 (f) = f(, 0), N 0 (f) = f(, /3), N (f) = f η (, 0), N 2(f) = f(, ), N 3 (f) = f (0, 0), N 4(f) = f (0, /3), N 5(f) = f (0, 2/3), N for f Q 3 ( ˆR) Then the 6 inear functionas have the foowing properties: 6(f) = f (0, ), (42) (i) {N i : i =,, 6} is a basis of the dua space [Q 3 ( ˆR)] (ii) Suppose f Q 3 ( ˆR) satisfies N j (f) = 0 for j =, 2, 3, 4, 5, 7, 9,, 3, 4, 5, 6, then f = = 0 aong the the ines ξ = 0, η = 0 f = f η 20

21 524 Expression of dua basis of order 3 for the camped boundary conditions Let the dua basis of N k be denoted by ˆφ k Then, by using the uniqueness of the dua basis ˆφ k with N k ( ˆφ ) = δk we are abe to specificay determine the basis functions ˆφ k that have the Kronecker deta property However, these functions designed to impose camped boundary conditions no onger have the poynomia reproducing property [A: One-edge-camped partice shape functions of order 3 (Fig 6)] Let { ˆφ j, j =,, 6} be the dua basis on Q 3 ( ˆR) of the basis {N j, j =,, 6} such that for each i =,, 6, N i ( ˆφ j ) = δ j i, for j =,, 6 Then it foows from Lemma 52 that for k = 5, 6, 7, 8, 9, 0,, 2, ˆφ k = 0, ˆφ k / = 0, ˆφ k / η = 0 aong the ine ξ = 0 Now, ˆφk, k = 5, 6, 7, 8, 9, 0,, 2 can be written as foows: Let L 3,2 (ξ), L 3,3 (ξ) be the second and the third Lagrange interpoating poynomias associated with three nodes 0, /2, For k =, 2, 3, 4, et L 4,k (η) be the k-th Lagrange interpoating poynomia associated with nodes 0, /3, 2/3, Since L 3,2 (0) = L 3,3 (0) = 0, ξl 3,2 (ξ) = 4ξ 2 (ξ ), and ξl 3,2 (ξ) = 2ξ 2 (ξ /2) Therefore, ξl 3,2 (ξ), ξl 3,3 (ξ) and their ξ-derivatives are vanishing aong the η-axis Thus, we have { For k = 5, 6, 7, 8, ˆφk (ξ, η) = c k ξl 3,2 (ξ) L 4,k 4 (η), For k = 9, 0,, 2, ˆφk (ξ, η) = c k ξl 3,3 (ξ) L 4,k 8 (η), (43) where c k is the normaizing constant that makes N k ( ˆφ k ) = [A2: Two-edges-camped partice shape functions of order 3 (Fig 7)] ˆφk, k = 6, 8, 0, 2 can be written specificay as foows: For k = 2, 3, et L 3,k (η) be the k-th Lagrange interpoating poynomias of degree 2 associated with nodes 0, /3, Then L 3,2 (η) = (9/2)η(η ) and L 3,2 (η) = (3/2)η(η /3), Hence, ηl 3,2 (η), ηl 3,3 (η) and their η-derivatives are vanishing aong the ξ-axis Thus, we have { For k = 0, 2, ˆφk (ξ, η) = c k ξl 3,2 (ξ) ηl 3,j (η), for j = 2, 3, respectivey; For k = 6, 8, ˆφk (ξ, η) = c k ξl 3,3 (ξ) ηl 3,j (η), for j = 2, 3, respectivey, (44) where c k is the normaizing constant that makes N k ( ˆφ k ) = 525 Expression of the dua basis of higher order for the camped boundary conditions Lemma 52 (one-edge-camped shape functions) and Lemma 53 (two-edges-camped shape functions) can be extended to the constructions of the degree of freedom inear functionas for any order n For exampe, suppose the 25 degree of freedom inear functionas on Q 4 ( ˆR) = span{ξ i η j 0 i, j 4} are as shown in Fig 6 and f = 4 j=0 ( 4 i=0 b ijξ i )η j Q 4 ( ˆR) Suppose the degree of freedom inear functionas N k on Q 4 ( ˆR) are defined by N k (f) = f(0, (k )/4), k =, 2, 3, 4, 5; N k (f) = f(/3, (k 6)/4), k = 6, 7, 8, 9, 0, N k (f) = f(2/3, (k )/4), k =, 2, 3, 4, 5; N k (f) = f(, (k 6)/4), k = 6, 7, 8, 9, 20, N k (f) = f/(0, (k 2)/4), k = 2, 22, 23, 24, 25 2 (45)

22 Then we have the foowing: N k (f) = 0 for k =, 2, 3, 4, 5 and N k (f) = 0 for k = 2, 22, 23, 24, 25 impy Hence, b 0,j = b,j = 0, for j = 0,, 2, 3, 4 (46) f(ξ, η) = 4 j=0 i=2 4 b i,j ξ i η j N k (f) = 0 for k = 6, 7, 8, 9, 0, N k (f) = 0 for k =, 2, 3, 4, 5, and N k (f) = 0 for k = 6, 7, 8, 9, 20, impy that b 2,0 b 3,0 b 4,0 b 2, b 3, b 4, (/3) 2 (2/3) b 2,2 b 3,2 b 4,2 (/3) 3 (2/3) 3 = b 2,3 b 3,3 b 4,3 (/3) 4 (2/3) (47) b 2,4 b 3,4 b 4, It foows from (46) and (47) that f 0 on R 2 Thus, N k, k =,, 25, are basis for the dua space [Q 4 ( ˆR)] [B: One-edge-camped partice shape functions of order 4 (Fig 6)] Let L 4,k (ξ) be the k-th Lagrange interpoating poynomias of degree 3 associated with nodes 0, /3, 2/3, Let L 5,k (η) be the k-th Lagrange interpoating poynomia of degree 4 associated with η = 0, /4, 2/4, 3/4, Since L 4,k (0) = 0, for k = 2, 3, 4, a four functions, ξl 4,k (ξ), k = 2, 3, 4, and their ξ-derivatives are vanishing aong the η-axis Thus, we have For k = 6, 7, 8, 9, 0, ˆφk (ξ, η) = c k ξl 4,2 (ξ) L 4,k 5 (η), For k =, 2, 3, 4, 5, ˆφk (ξ, η) = c k ξl 4,3 (ξ) L 4,k 0 (η), (48) For k = 6, 7, 8, 9, 20, ˆφk (ξ, η) = c k ξl 4,4 (ξ) L 4,k 5 (η), where c k is the normaizing constant that makes N k ( ˆφ k ) = [B2: Two-edges-camped partice shape functions of order 4 (Fig 7)] Suppose the degree of freedom inear functiona N k on Q 4 ( ˆR) = span{ξ i η j 0 i, j 4} are defined by N k (f) = f(0, (k )/4), k =, 2, 3, 4, 5; N j (f) = f(/3, (k 6)/4), k = 6, 7, 9, 0, N k (f) = f(2/3, (k )/4), k =, 2, 4, 5; N j (f) = f(, (k 6)/4), k = 6, 7, 9, 20, N k (f) = f/(0, (k 2)/4), k = 2, 22, 23, 24, 25, N 8 (f) = f/ η(/3, 0), N 3 (f) = f/ η(2/3, 0), N 8 (f) = f/ η(, 0) For k = 2, 3, 4, et L 4,k (η) be the k-th Lagrange interpoating poynomia of degree 3 associated with four nodes 0, /4, 3/4, Then L 4,k (0) = 0, for k = 2, 3, 4 Hence three functions ηl 4,j (η), j = 2, 3, 4, and their η-derivatives are vanishing aong the ξ-axis Thus, we have For k = 7, 9, 0, ˆφk = c k ξl 4,2 (ξ) ηl 4,j (η), j = 2, 3, 4, respectivey; For k = 2, 4, 5, ˆφk = c k ξl 4,3 (ξ) ηl 4,j (η), j = 2, 3, 4, respectivey; (50) For k = 7, 9, 20, ˆφk = c k ξl 4,4 (ξ) ηl 4,j (η), j = 2, 3, 4, respectivey 22 (49)

23 where c k is the normaizing constant that makes N k ( ˆφ k ) = We define the degree of freedom inear functionas on Q 6 ( ˆR) in a simiar manner Then we have the foowing Lagrange partice shape functions satisfying the camped boundary conditions on one or two sides of the reference rectange ˆR [C: One-edge-camped partice shape functions of order 6] Let L 6,k (ξ) be the k-th Lagrange interpoating poynomias of degree 5 associated with nodes 0, /5, 2/5, 3/5, 4/5, Let L 7,k (η) be the k-th Lagrange interpoating poynomia of degree 6 associated with η = 0, /6, 2/6, 3/6, 4/6, 5/6, Then the foowing 35 partice shape functions satisfy the camped boundary condition aong th η-axis: For k = 8, 9, 0,, 2, 3, 4, For k = 5, 6, 7, 8, 9, 20, 2, For k = 22, 23, 24, 25, 26, 27, 28, For k = 29, 30, 3, 32, 33, 34, 35, For k = 36, 37, 38, 39, 40, 4, 42, ˆφk (ξ, η) = c k ξl 6,2 (ξ) L 7,k 7 (η), ˆφk (ξ, η) = c k ξl 6,3 (ξ) L 7,k 4 (η), ˆφk (ξ, η) = c k ξl 6,4 (ξ) L 7,k 2 (η), ˆφk (ξ, η) = c k ξl 6,5 (ξ) L 7,k 28 (η), ˆφk (ξ, η) = c k ξl 6,6 (ξ) L 7,k 35 (η), where c k is the normaizing constant that makes N k ( ˆφ k ) = [C2: Two-edge-camped partice shape functions of order 6] Let L 6,k (ξ) be the k-th Lagrange interpoating poynomia of degree 5 associated with nodes 0, /5, 2/5, 3/5, 4/5, Let L 6,k (η) be the k-th Lagrange interpoating poynomia of degree 5 associated with η = 0, /6, 2/6, 4/6, 5/6, Then the foowing 25 partice shape functions satisfy the camped boundary condition aong the ξ-axis as we as the η-axis: For k = 9, 0, 2, 3, 4, ˆφk (ξ, η) = c k ξl 6,2 (ξ) ηl 6,j (η), j = 2, 3, 4, 5, 6, respectivey; For k = 6, 7, 9, 20, 2, ˆφk (ξ, η) = c k ξl 6,3 (ξ) ηl 6,j (η), j = 2, 3, 4, 5, 6, respectivey; For k = 23, 24, 26, 27, 28, ˆφk (ξ, η) = c k ξl 6,4 (ξ) ηl 6,j (η), j = 2, 3, 4, 5, 6, respectivey; (52) For k = 30, 3, 33, 34, 35, ˆφk (ξ, η) = c k ξl 6,5 (ξ) ηl 6,j (η), j = 2, 3, 4, 5, 6, respectivey; For k = 37, 38, 40, 4, 42, ˆφk (ξ, η) = c k ξl 6,6 (ξ) ηl 6,j (η), j = 2, 3, 4, 5, 6, respectivey; where c k is the normaizing constant that makes N k ( ˆφ k ) = 6 Numerica exampes 6 Rectanguar pates In this subsection, to demonstrate the effectiveness of the proposed meshfree method (RPPM), we cacuate reiabe true defection coefficients and the true strain energy for a various forms of simpy supported rectanguar pates In this subsection, we use φ R g 2 ( x+δ 2δ ) in Eqn(0) with δ = 005 for the construction of generaized product PU functions so that their first order derivatives are continuous 6 Simpy supported rectanguar pate Let Ω = [ a 2, a 2 ] [ b 2, b 2 ] be the rectanguar pate, where b = ar for some r (5) 23

24 [A: Pate with uniform oad] Suppose the rectanguar pate is uniformy oaded by p(x, y) = p 0 for a (x, y) Ω By M Lévy s method, the defection w(x, y) of the midpane Ω can be expressed in the foowing singe infinite series ([39]): w = 4p 0a 4 π 5 D m= odd ( m 5 α m tanh α m + 2 cosh 2α my + 2 cosh α m b α m 2y 2 cosh α m b sinh 2α ) my b where α m = mπb Here, the maximum defection occurs at the center point (0, 0) and 2a w max = 4p 0a 4 π 5 D m= odd ( ) (m )/2 m 5 sin mπ(x + a 2 ), a (53) ( α ) m tanh α m + 2 (54) 2 cosh α m From (54), the defection coefficient β UL := w maxd p 0 a 4 β UL = 4 π 5 ( ) (m )/2 m= odd = 4 π 5 m= m 5 can be written as ( α ) m tanh α m cosh α m ( ) m ( (2m ) 5 α ) 2m tanh α 2m cosh α 2m (55) Let βn UL be the N-th partia sum of the infinite series (55) Then for each positive integer N, the remainder is estimated as foows: β UL βn UL 4 π 5 (2m ) 5 α 2m tanh α 2m cosh α 2m 4 π 5 = m=n+ m=n+ 2π 5 (2N ) 4 (2m ) 5 4 π 5 N (2x ) 5 dx [B: Pate with point oad] When a rectanguar pate is oaded at center point ony with p(0, 0) = p 0, the defection of the midpane can be expressed in the foowing infinite series([39]): for y 0, w = p 0a 2 2π 3 D m= ( ( + α m tanh α m ) sinh (b 2y)α m b ( sin mπ 2 sin mπ(x+ a 2 ) a m 3 cosh α m ) (b 2y)α m b cosh (b 2y)α ) m b (56), α m = mπb 2a (57) 24

25 Hence, we have w max = w(0, 0) = p 0a 2 2π 3 D = p 0a 2 2π 3 D = p 0a 2 2π 3 D sin 2 mπ 2 (( + α m tanh α m ) sinh α m α m cosh α m ) m 3 cosh α m m= m= odd m= odd The defection coefficient β PL := w maxd p 0 a 2 β PL = 2π 3 m= m 3 ( + α m tanh α m ) sinh α m α m cosh α m cosh α m ( m 3 tanh α m (2m ) 3 α m cosh 2 α m ) now can be expressed in the form ( tanh α 2m α 2m cosh 2 α 2m ) (58) The remainder after the N-th partia sum βn PL of the series (58) is estimated as foows: β UL βn UL 2π 3 (2m ) 3 tanh α α 2m 2m cosh 2 α 2m 2π 3 = m=n+ m=n+ 8π 3 (2N ) 2 (2m ) 3 2π 3 N (2x ) 3 dx [C: Estimated true defections at (0, 0)] Using (56) and (59), we determine N such that the N-th partia sums differ from the true vaues within (with doube precision cacuation in a 32 bit computer) For this purpose, we sove the foowing inequaities for N: 2π and (2N ) 4 8π 3 (2N ) , from which we get N = 2 and N = 44898, respectivey Tabe 3 shows approximated defection coefficients and their effective digits, which means that they are exacty matched with those of the true defection coefficients up to that many digits In the subsequent subsection, we use the numbers in the third and the fifth coumns of Tabe 3 as the true soutions [D: The computed true strain energy] Let us denote the infinite series (53) for a square pate with uniform oad and the infinite series (57) for a square pate with point oad by w UL and w P L, respectivey Using the first 200 terms (the partia sums S200 UL (x, y) and SP L 200 (x, y) ) of each series, we compute the energy of w UL and w P L, respectivey, as foows: U(w UL ) = 2 B(wUL, w UL ) 2 B(SUL 200(x, y), S200(x, UL y)) = , (60) U(w P L ) = 2 B(wP L, w P L ) 2 B(SP L 200(x, y), S P L 200(x, y)) = (6) 25 (59)

26 Tabe 3: For various ratios b/a, we ist the 2-th partia sum of the series (55) and the th partia sum of the infinite series (58) and their effective digits (the digits that are exacty matched with those of the true defection coefficients β UL and β PL, respectivey) Uniform oad Point oad b/a β2 UL effective digits β44898 PL effective digits In subsequent subsection, we compute the reative errors in energy norm by appying these computed true energies to the foowing formua: [ B(utrue, u true ) B(u app, u app ] ) /2 Re Err Eng = (62) B(u true, u true ) The reative error in maximum norm Re Err is defined simiary It is quite invoved to express the dispacement w(x, y) of a camped rectanguar pate in a singe infinite series Thus, we do not compute the true energy of a camped square pate in this paper However, Exampe 63 shows that our method aso effectivey handes camped pate probems 62 Exampes Exampe 6 With materia constants ν = 03, E = 0 9, a = 06, d = 000, p = 000 (These materia constants are the same as those in [5]) We test the proposed meshfree partice methods to the foowing cases: A: Simpy supported rectanguar pate with uniform oad(tabe 4, Tabe 5); B: Simpy supported rectanguar pate with point oad at the center(tabe 4,Tabe 6); C: Camped rectanguar pate with uniform oad(tabe 7); D: Camped rectanguar pate with point oad at the center(tabe 8) [A: Reative errors in energy norm] In Tabe 4, the reative errors in energy norm are computed by using the computed true soutions given by (60) and (6) The anaytic soution for the dispacement w P L of a simpy supported rectanguar pate with point oad has a singuarity (of type r 2 og r) at the center point Thus if a other data reated to the thin pate with point oad are smooth, we have w P L H 2 (Ω), whereas if the pate is uniformy oaded, then the dispacement w UL is highy reguar The effect of the reguarity of w UL and w P L to the accuracy of the computed soutions are we refected 26

27 in Tabe 4 and Fig 8, in which the reative errors in energy norm versus the RPP orders are potted on og og-scae The asymptotic rates of convergence in the energy norm for errors against DOF (ie the increment in og( Re Err eng ) over the increment in og(dof)) are about 05 and 35 for the simpy supported square pate with point oad and with uniform oad, respectivey And the asymptotic rates of convergence in the energy norm for errors against RPP orders are about 0 and 70, respectivey Tabe 4: Reative errors in energy norm (%) for simpy supported square pate rpp order Uinform Load Point Load k DOF Re Err eng (%) Energy Re Err eng (%) Energy E E E E E E E E E E E E (True) Reative Errors in Energy Norm (%) Point oad Uniform oad RPP order Figure 8: Convergence of the proposed method for simpy supported square pate in energy norm [B: Reative errors in maximum norm] In the foowing, we use the foowing notations: β UL = w maxd P a 4 for uniform oad and β P L = w maxd P a 2 27 for point oad, where P is oad, w max

28 0 0 Reative Errors in Max Norm (%) Point oad Uniform oad RPP order Figure 9: Convergence of the proposed method for simpy supported square pate in maximum norm is the vertica dispacement at the center point, a and b are the two side engths of a rectanguar pate shown Fig 0 DOF stands for the degrees of freedom, that is the number of partices empoyed However, those partices ocated aong the boundary with essentia boundary conditions of Tabe 2 were not counted into DOF the resuts in the rows β RPP 4, β RPP 5 and β RPP 6 are those obtained by our method with use of partice shape functions of RPP order 4, RPP order 5 and RPP order 6, respectivey The resuts in the row β Liu are those in ([4],[5]) and the resuts in the row β Timo are the soutions in Timoshenko([39]) βtayor UL are resuts in ([38]) From resuts in Tabes 5, 6, 7, and 8, we observe the foowing Tabes 6 and 8 show that even though our method uses much smaer number of partices (44 in β RPP 6 and 256 in β Liu ), our method yieds better resuts than the moving east squares (MLS) method empoyed in ([4]) Actuay, the resuts in the row β RPP 4 that use 36 partices are simiar to those number in β Liu that use 256(6 6) uniformy spaced partices 2 Impementing RPP shape functions satisfying camped BC and constructing generaized product PU functions for the proposed meshfree partice method is simpe 3 Tabes 5 and 7 show that in the case of uniform oad, the resuts in the row β RPP 4 show that our method using 64 partices for simpy supported BC (36 partices for camped BC) yieds aready enough accuracy In the case of uniform oad, our method yieds amost the same as the computed true soution 28