Modeling by Meshless Method LRPIM (local radial point interpolation method)

Modeling by Meshless Method LRPIM (local radial

point interpolation method)

A. Moussaoui

a

, T. Bouziane

b

a. Department of physics, Faculty of Sciences, Moulay Ismail University

B.P.11201 Meknes, Morocco, moussaoui.physique@gmail

b. Department of physics, Faculty of Sciences, Moulay Ismail University

B.P.11201 Meknes, Morocco, [email protected]

Abstract:

Numerical solutions in physical engineering problems need appropriate numerical approximation methods. Meshless methods have attracted increasing attention in recent years for seeking of approximate solutions of initial boundary value problem governed by partial differential equations. In this paper, we present a study of a 2D problem of an elastic homogenous rectangular plate by using the method LRPIM. We investigate the convergence and accuracy of method LRPIM and numerical values are presented to specifying the convergence domain by precising maximum and minimum values as a function of distribution nodes number and by using two radial basis functions: Gaussian (EXP) and thin plate spline (TPS). It also presents a comparison with numerical results for different materials and the radial basis functions RBF. Finally a comparative study of numerical results with analytical solutions is presented.

Keywords:

Linear Elasticity, rectangular plate, Meshless Method LRPIM, Radial basis function, support domain.

1

Introduction

Meshless method has attracted more and more attention from researchers in recent years, and it is regarded as a potential numerical method in computational mechanics. Several meshless methods, such as smooth particle hydrodynamics (SPH) method [1], element free Galerkin (EFG) method[3], meshless local Petrov-Galerkin (MLPG) method[4], point interpolation method (PIM)[5-6], local point interpolation method (LPIM) [7] and local radial point interpolation method (LRPIM) has been proposed by Liu et al. [6]. In LRPIM, the point interpolation using the radial basis function is utilized to construct shape functions with delta function property. The widely used radial basis functions (RBFs) arer: thin plate spline (TPS) and Gaussian (EXP) [8-9]. Local weak forms are developed using weighted residual method locally from the partial differential equation of linear elasticity of 2D solids

and a number of numerical examples are presented to demonstrate the convergence and accuracy, validity and efficiency of the present methods. The local radial point interpolation method LRPIM is a Meshless method with properties of simple implementation of the essential boundary conditions and with the lower cost than MLS methods.

This paper deals with the effect of sizing parameter of subdomains on the convergence and accuracy of the methods and numerical values are presented to specifying the convergence domain by precising maximum and minimum values as a function of distribution nodes number and by using the radial basis functions TPS (the thin plate spline) and Gaussian (EXP). It also presents a comparison with numerical results for different materials. Numerical results agree with the analytical solution of the deflection.

The LRPIM method will be developed for solving the problem of a thin elastic homogenous plate. The local weak form and numerical implementation are presented in section 3, numerical example for 2D problem are given in section 4. Then, the paper ends with results, discussions and finally the conclusions.

2

RPIM shape functions in meshless method

) x (

uh is composed of two part:Pj(x)Polynomial basis functions and Ri(x)the radial basis

functions RBFs [10-11]:





    n 1 i m 1 j j j i i h (x)b P (x)a R (x) u (1)

n is the number of field nodes in the local support domain and m is the number of polynomial terms.

Radial basis is a function of distance r:

r



(x



x

_i

)

2



(y



y

_i

)

2 (2) The above equation (1) can be expressed in the matrix form [11] :U₁ RaPb (3)

Where U₁The vector of function values:U₁ 



u₁, u₂, u₃,..., u_n



T

R The moment matrix of RBFs, P the moment matrix of Polynomial basis function and a, b the values of unknowns coefficients (Radial and Polynomial)

We note that, to obtain the unique solutions of Eq. (2), the constraint conditions should be applied as

follows [12]:



   n 1 i i ij(x)a 0 j 1, 2, ...,m P (4)

By combining Eqs. (3) and (4) yields a set of equations in the matrix form:

_T

G

a

₀

b

a

P

P

R

U

U

_









































0

0

1 1 (5)

The unknowns vector can be obtained by inversion of the matrix

_













0

T

P

P

R

G

Substitution of the vector obtained by inversion of matrix

G

into Eq. (1) leads to:



   n 1 i i i h u (x) u 1



T (x)U Φ (6)

3

Local weak form method LRPIM

Let us consider a two-dimensional problem of solid mechanics in domain



bounded by



whose strong-form of governing equation and the essential boundary conditions are given by:



_ij,j

(

x

)



b

_i

(

x

)



0

(7)



_ij

n

_j



t

0_i on



t (8)

u

_i



u

0_i on



_u (9) Where in



:

σ

T



[



_xx

,



_{y y}

,



_xy

]

is the stress vector,

b

T



[

b

_x

,

b

_y

]

the body force vector.

) n , n ( 1 2 

n _{denotes the vector of unit outward normal at a point on the natural boundaries}

0

t _{is the prescribed effort, [u ,u ]}

2

1 the displacement components in the plan and [u ,u ] 0 2 0

1 on the essential boundaries.

In the local Petrov-Galerkin approaches [3], one may write a weak form over



_Q a local quadrature domain (for node I), which may have an arbitrary shape, and contain the point xQ in question, see Fig. 1. The generalized local weak form of the differential Eq. (7) is obtained by:

(

(

x

)

b

(

x

))

d

0

Q I i j , ij















(10)

Where



Q is the local domain of quadrature for node I and



I is the weight or test function,

) ( CK

I





 [4].

Generally, in meshfree methods, the representation of field nodes in the domain will be associated to other repartitions of problem domain:



_I influence domain for nodes interpolation,



_S is the support domain for accuracy. For each node



_is the weight function domain, and



_Q is the quadrature domain for local integration.

Figure 1. The local sub-domains around point x_Q and boundaries

Using the divergence theorem [4] in Eq. (10), we obtain:

n d d b d 0 Q Q Q I i j , I ij I j ij 

































(11)

Where



Q 



Qi



Qu 



Qt Qi



: The internal boundary of the quadrature domain Qt



: The part of the natural boundary that intersects with the quadrature domain Qu



: The part of the essential boundary that intersects with the quadrature domain We can then change the expression of équ(11):

0 d b d d n d n d n Q Q Qi Qu Qt I i j , I ij I j ij I j ij I j ij   





















































(12)

Using the RPIM shape functions (see sub-section 2), we can approximate the trial function for the displacement at a point x (x



_S ) as eq.(6)

The stress vector is defined by: σCε CL_duh (13) Where

C

is the symmetric elasticity tensor of the material

                ) 1 ( 2 / E 0 0 0 ) 1 /( E ) 1 /( E 0 ) 1 /( E ) 1 /( E 2 2 2 2        C

Eq.(12) can be written:



















Q Qi Qu Qi Q d d d d d T      V σ



tV



tV



t VI



VIb



0 I I I (14) Where            x , I y , I y , I x , I I 0 0    

V is a matrix that contains the derivatives of the weight functions

and

_













I I

0

0





V

is the matrix of weight function.

Substituting the differential operator

                   x / y / y / 0 0 x / d

L into equation (13) we obtain:



  nt 1 I I I u B C σ (15) Where            x , I y , I y , I x , I 0 0     I

B . By using the matrix

           1 2 2 1 n n n 0 0 n n

L , the tractions of a point x can be written as: t LTnσ (16) Substituting Eqs. (15, 16) into Eq. (14), we obtain the discrete systems of linear equations for the node I.





















 Qt Q t Qu Qi Q d d t ] d d d [ _{I I} n 1 I I I T I _{   }  V CB  L CBV  L CBV  u V  Vb  0 I T n I T n I I (17)

The matrix form of Eq. (17) can be written as in matrix form:



  t n 1 I I I Iu f K (18) Where expression of nodal matrix KIis













Qu Qi Q d d d _{I I} T I _{ }  V CB



L CB V



L CBV



K T _I n I T n I I (19)

And nodal force vector with contributions from body forces applied in the problem domains:







 Qt Q d d t _{I I}  V



 V b



f 0 I (20)

Wheren0 denote the set of the nodes in the support domain



S of pointxQ.

Two independent linear equations can be obtained for each node in the entire problem domain and by assembling all these

2

*

n

equations to obtain the final global system equations:

k

_2n*2n

u

_2n*1



f

_2n*1 (21) To solve the precedent system, the standard Gauss quadrature formula is applied with 16 Gauss points

[3, 13] for calculating integrals in Eqs (19, 20) on both boundary and domain.

The size of quadrature domain is specified by setting



Q 2 and a regular distribution of nodes on

the mid-surface of plate in (x, y) plane is employed.

4

Numerical 2D elastostatic example

This section is about numerical results for a cantilever rectangular plate see (Fig. 2). First, were investigated the effects of the size of support and quadrature domains and was examined numerically convergence of LRPIM for several materials; then, comparisons will be made with the analytic solution for several materials [14]{We choose: steel, zinc, aluminium and copper with:

( E3.107N/m2 ,





0

.

3

; E113.105N/m2 ,





0

.

25

; E1.107N/m2 ,

34

.

0





_; 6 2 m / N 10 . 17

E _,





0

.

33

_{) respectively} Dimension of the plate are denoted: height}

m

12

D



, length

L



48

m

, the thickness: unit and finally for Loading: P103N

Figure 2. Cantilever plate subjected to distributed traction at the free end.

In our numerical calculations, were considered many regular distributions of nodes n_t: 18, 55, 91, 175 and 189. To calculate the error energy, a background cells are required; then, for each value of

t

n the number of cell was varied. To obtain numerical values, the distribution of the deflection through the plates, size of support domain is varied and



Q 2 size of the quadrature domain.

The sizes of support domain



s (quadrature domain



Q resp.) are defined by: ds 



sdc (rQ 



QdcIresp.) wheredc( dcI resp) is the nodal spacing near node I (Fig. 3) and



s(



Q resp) is the size of the support domain



s( local quadrature domains resp) for node I. The sizes of support domain



_s( quadrature domains resp) will be respectively determined in x and y directions. For simplicity



Sx 



Sy 



S(



Qx 



Qy 



Qresp) is used for



s(



Q resp).

5

Results and Discussions

The standard Gaussian quadrature formula with 16 Gauss points and RPIM approximation, linear polynomial basis functions are applied. The cubic or quadratic spline function is used as the test functions in the LRPIM local weak-form.

Throughout this section and for all calculations,



_Q was fixed and the value 2 (



_Q 2).

5.1 Analysis of results numerical of the radial basis functions TPS

(thin plate spline)

The use of radial basis function RBF-TPS have not been extensively studied on literature, we give in these paragraph numerical results for different materials.

Figure 4 shows the energy error as a function of



_S, cubic spline



₁ as the test function with the radial basis RBF-TBS and steel has been chosen. The results which are calculated by the LRPIM method are influenced by different parameters. It is investigated the variation of maximum and minimum values of



S of convergence domain by increasing the distribution regular field number nodesn_t: = 55, 91, 175,189. It can be seen from figure 4 that if the values of



Sis smaller than 1.80, the energy error is large and LRPIM method is not convergent. The domain of convergence reaches the maximum value at



S 5for if nt 55.

For n_t= 91, 175 or 189 the domain convergence is smaller than that obtained with n_t =55. The greater extremity value of the convergence domain is now



_S equal to 3.66 for (n_t=91, 175, 189). The convergence domain is noticed between a small value



S 1.80and the greater value



S 5

1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 0,0 0,2 0,4 0,6 0,8 1,0 _S E n e rg y e rr o r 55 nodes 91 nodes 175 nodes 189 nodes TPS-RBF, =4.001

Figure 4. Influence of the



_S size of



_s on energy error for different distribution nodes numbers (Steel)

0 1 2 3 4 5 6 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 RBF-TPS, 55 nodes E n e rg y e rr o r Steel Zinc Copper Aluminium 

Figure 5. Variation of the energy error as a function of



for different materials and (nt 55and 3

S 

1 2 3 4 5 6 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 E n e rg y e rr o r 

Steel Copper Zinc Alumi, 55 nodes

Steel Copper Zinc Alumi, 91 nodes

Steel Copper Zinc Alumi, 175 nodes

Figure 6.Variation of the energy error as a function of



for different materials and (nt 55,91,175and



S 3). 1 2 3 4 5 6 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 E n e rg y e rr o r

Steel Copper Zinc Alumi, 55 nodes Steel Copper Zinc Alumi, 91 nodes Steel Copper Zinc Alumi, 189 nodes



Figure 7.Variation of the energy error as a function of



for different materials and (n_t 55, 91,189

and



_S 3)

In figure 5-7 shows the variation of the error energy as a function of the shape parameter



for different values of

E

and



ie different materials and for different values of the number

189 , 175 , 91 , 55  t n (



S 3).

We find that all the curves of different materials have similar paces for a fixed value



in the following domains:

For nt 55, the convergence domain is: 3.25



6.25

For nt 91, the convergence domain is: 1.5



6.25

For nt 175, the convergence domain is: 0.5



6.5

For nt 189, the convergence domain is: 1



6.5

The results illustrated in the figures (5-7) to the number of nodes nt 55,91,175and 189 with the

radial basis RBF-TPS, give an the domain of convergence large enough for the different types of materials studied. This shows better convergence than that given in references [6, 15-16], the authors give a single value (



4.001). We show here that the values



can be varied in a broader segment depending on the number ntthat the domain of convergence is larger than that given by Liu et al.

0 5 10 15 20 25 30 35 40 45 50 -1,0x10-2 -8,0x10-3 -6,0x10-3 -4,0x10-3 -2,0x10-3 0,0 Zinc Steel=5 Steel=6 Zinc=6 Zinc=5

Analytical solutuion : Steel Analytical solutuion: Zinc

x₁(m)

u₂(m)

175 nodes Steel

Figure 8. Deflections as a function of x₁at x₂ 0, for the radial basis and analytical solution for two materials (steel and zinc,



5; 6and n_t 175;



_S 3).

Figure 8 shows the variations of displacement as a function of x1 for x2 0 with the shape parameter



of the radial basis function RBF-TPS, for two values of



:



5 and 6: with two studied materials: Steel and Zinc. We note that there is a coincidence between the analytical solution and that obtained by the LRPIM method of the radial basis RBF-TPS, it shows the convergence of LRPIM method (see Figure 7).

We studied the effect of the parameter the radial basis functions RBF-EXP on convergence of the LRPIM method.

In figure 9-10 shows the variation of the error energy as a function of



C of the radial basis RBF-EXP and different numbers of nodes (n_t 55,91,175,189) and



_S



3

. For values ranging between 0 and 0006 of



_C, aluminum, zinc and copper have of curves specific, then the LRPIM method is not convergent, but only that of steel, the method has a good convergence, the domain of convergence of the steel is wide (0.001



_C 0.3) with respect to other materials. But for the other materials, the domain of convergent is 0.006 



_C 0.3.

Finally, we can also say that the domain of the convergence is broader than that given in the references [11, 15] which gave the interval 0.003 



C 0.03for a single material. The method is convergent

when the number n_t is very large on a larger domain (for steel Figure 10).

0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 E n e rg y e rr o r RBF-EXP), Steel RBF-EXP), Zinc RBF-EXP), Copper RBF-EXP), Aluminium _c

Figure 9 Variation of the energy error as a function of



_Cfor different materials for n_t 55

0,0 0,1 0,2 0,3 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 length step 0.1 E n e rg y e rr o r 55 nodes 91 nodes 175 nodes 189 nodes _C Steel, RBF- EXP)

Figure 10 Variation of the energy error as a function of



_Cfor different values of n_t (steel) Figure 11 shows the deflection results are plotted as function ofx₁atx2 0,

for the radial basis RBF-EXP, the number of nodes nt_{= 55 and the size of the support domain}

(



S 5 ) with the cubic spline function and shape parameters:



c



0

.

03

to RBF-EXP. We considered nt= 55 for which the method is convergent and the value of energy error is low. There

is a coincidence between the curves representing the radial basis RBF-EXP of the LRPIM method and the curve of the analytical solution which corresponds to the upper end of the domain of convergence i.e.



_S between 1.80 and 5.

0 5 10 15 20 25 30 35 40 45 50 -1,0x10-2 -8,0x10-3 -6,0x10-3 -4,0x10-3 -2,0x10-3 0,0 U2 (m ) X1(m) EXP-RBF;_C=0.03 Analytical solution

Figure 11 Deflections as a function of

x

₁ at x20 for the radial basis RBF-EXP (nt 55for

03 . 0

C 



) and the analytical solution

Figure 12 Shear stress (



₁₂) as function of x₂ at x₁ L/2 for the radial basis RBF-EXP and

175 

t

n . We find that there is a coincidence between the curves representing the LRPIM method and the analytical solution. The results of the LRPIM method with the radial basis RBF- EXP:(



_C 0.03 with



_S 3.66and



_Q 2) is less effective.

-6 -4 -2 0 2 4 6 -140 -120 -100 -80 -60 -40 -20 0    pa ) RBF-EXP) Analytical solution x₂(m) 175 nodes

Figure 12



₁₂ as a function of x₂ at x1 L/2 for the radial RBF-EXP, (nt 175for



C 0.03 ) and the analytical solution.

6

Conclusion

In this paper the meshless LRPIM method is employed for solving a 2D elastostatic problem. The governing equations depend on the weak form and the partitions of domain. The LRPIM method and its dependency on sizing parameter of



S are associated to different parameters coming out of weak form formulation. We have investigated for



_Q 2 the nature of convergence domain as a function of



_S; the effect of number nodes n_t, by varying nature of material and the radial basis functions RBFs.

The results obtained for the number of nodes n_t 55,91,175and 189 for the radial basis functions give an domain of convergence large enough for the different types of materials studied. This shows better convergence than that given in references [6, 15-16]. we conclude that for small values of

) 55 (

nt lead to the upper extremity of convergence domain which is limited to



S 5.

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