test on mls

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dataname: 11-18-11 w1BF s1 t7 z139

bead1(0.6229, 0.3963, 0.3388)

(7.7

104,

2.2

104,

0.0035)

bead2(0.6249, 0.4593, 0.3262) (4.7 104,0.0022,0.0056)directly estimated normal strain: ex = 0.150, ey = 0.031, ez = 0.167maximum MLS estimated normal strain: e = 0.048

= underestimated strain components.

Figure 1: Displacement reconstruction of a set of real data.

Figure 2: MLS-calculated strain component e11 at t=16 hours.

J(x) =

NI=1

w(x, xI)(pT(xI)a(x) uI)2

w(x, xI) = w(dI, dmI), dI =

||x

xI

||uh(x) = pT(x)a(x) =

I(x)uI

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minimization ofJ(x) in moving least square regression leads to

a(x) = A1BU, where

A = PTWP =

N=1

w(x, x)p(x)pT(x),

B = PTW =w(x, x1)p(x1), , w(x, xN)p(xN)

a,i(x) = (A

1

,i B + A1B,i(x))U, where

A1,i = A1A,iA1.

Figure 3: [2]

Visibility criterion ( [1] ): include MLS node P only if the evaluation point

is visible from it discontinuous shape function.[4] proves that the discontinuous approximation generated by the visibility cri-terion converge at the same rate as continuous EFG approximation.

Figure 4: Left: [3]; Right: [2]

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Ways to construct smooth shape function for non-concave bodies( [3] )

A. diffraction method: wrap the domain of influence around a concave boundary.

modified parameter:

dI(x) = (s1 + s2(x)

s0(x))s0(x)

s0(x) = ||x xI||2, s1(x) = ||xw xI||2, s2(x) = ||x xw||2,

derivative calculation:

dw

dxi=

w

dI

dI

xidI

xi= (

s1 + s2s0

)s2

xi+ (1 )(s1 + s2

s0)s0

xi

s0xi

= xi xIis0

, s2xi

= xi xcis2

Figure 5: Top: [4]; Bottom: [3]

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B. transparency method: endow boundaries within the domain of influence

with a varying measure of transparency.

modified parameter:

dI(x) = s0(x) + dmI (sc(x)

sc), 2

Figure 6: [3]

Numerical results on infinite plate with a hole:

Figure 7: [3]

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method efficiency accuracy implementationplain

visibilitydiffraction

Table 1: Interpolation time for 20 nodes.

Numerical experiment:

Test A)

u1 = (x2

1 + x2

2) 0.1,u2 = (x1 + x2 sin(x3)) 0.1,u3 = x1x2x3

0.1;

Test B) Infinite 3D specimen with a hole under internal pressure

Hole radius: a, internal pressure: p.

(r) =a3p

4

1

r3

ui = (r)xi

Tr = pa3

r3

T =pa3

2r3

Coupled FE-EFG method: allows for the use of the EFG method in thecrack region and the finite element method to handle complex geometries and

essential boundary conditions.

Enriched MLS basis representative for crack tip fields( [3]):

pT(x) =

1,x,y,z,

r cos

2,r sin

2,r sin sin

2,r cos cos

2

References

[1] T. Belytschko. 1994 Belytschoko etal Element free galerkin methods.pdf.

[2] Petr Krysl and Ted Belytschko. Element-free Galerkin method: Convergence

of the continuous and discontinuous shape functions. Computer Methods in

Applied Mechanics and Engineering, 148(3-4):257277, September 1997.

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Figure 8: Short term results for test A. Data number: 10000.

[3] D Organ, M Fleming, T Terry, T Belytschko, and T Terry. Continuous mesh-

less approximations for nonconvex bodies by diffracti0n and transparency.

Computational Mechanics, 18, 1996.

[4] N. Sukumar, B. Moran, T. Black, and T. Belytschko. An element-free

Galerkin method for three-dimensional fracture mechanics. Computational

Mechanics, 20(1-2):170175, July 1997.

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Figure 9: Long term results for test A. Data number: 10000.

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Figure 10: Short term relative L2 error for test B. Data number: 10000. Meshdistance to the sphere: 0.01.

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Figure 11: Short term relative L2 error for test B. Data number: 20000. Meshdistance to the sphere: 0.01.

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Figure 12: Long term relative error for test B. Left: Data number: 50000.Right: Data number: 10000. Point distance to the sphere: 0.001.

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Figure 13: Long term relative error for test B. Left: Data number: 50000.Right: Data number: 10000. Point distance to the sphere: 0.001. (Cont)

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Figure 14: Long term relative error for test A. Data number: 10000. Pointdistance to the sphere: 0.001.

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Figure 15: Relative error for test B. dI=0.1. Left: Data number: 100000. Right:Data number: 10000.

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Figure 16: Relative error for test B. dI=0.1. Left: Data number: 100000. Right:Data number: 10000(Cont).

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Figure 17: Convergence test for test B. Point distance to the sphere: 0.001.dI = 0.1

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