Divyesh_14Me63r02(new)

Under the guidance of

Prof. Manas Chandra Ray

Simple MeshFree Model for Analysis of

Smart Composite Beam

Presented By

Divyesh Mistry (Roll No. : 14ME63R02)

Department of Mechanical EngineeringIndian Institute of Technology, Kharagpur

May 20162 MAY 2016 IIT KHARAGPUR 1

Objectives

To define shape function construction for MeshFree method and Meshfree interpolation/approximation technique.

To derive the solution for smart composite beam with EFG method.

Comparison of Responses of Mfree Method for Three Layered (0/90/0) Composite Beam with Exact Solution Method.

Comparison of Responses of Mfree Method for Four Layered (0/90/0/90) Composite Beam with Exact Solution Method.

2 MAY 2016 IIT KHARAGPUR 2

IIT KHARAGPUR 3

Introduction

The development of the MFree methods can be traced back more than seventy years to the collocation methods (Slater, 1934; Barta, 1937;Frazer et al., 1937; Lanczos, 1938, etc.).

Belytschko Lu and Gu in 1994 refined and modified the method and called their method EFG, element free Galerkin. These methods do not require a mesh at least for the field variable interpolations.

Adaptive analysis and simulations using MFree methods become very efficient and much easier to implement, even for problems which pose difficulties for the traditional FEM.

2 MAY 2016

IIT KHARAGPUR 4

Why Mesh Free method over FEM ?

Low accuracy of stress

• The stresses obtained in FEM are often discontinuous at the

interfaces of the elements due to the piecewise (or elementwise)

continuous nature of the displacement field assumed in the FEM

formulation.

Limitation in the analyses of some problems

• Under large deformations, considerable loss in accuracy in FEM

results can arise from the element distortions.

• It is difficult to simulate crack growth with arbitrary and complex

paths which do not coincide with the original element interfaces.

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IIT KHARAGPUR 5

Why Mesh Free method over FEM ?

Difficulty in adaptive analysis

• In an adaptive analysis using FEM, re-meshing (re-zoning) is

required to ensure proper connectivity, for this remeshing

purpose, complex, robust and adaptive mesh generation

processors have to be developed.

High cost in creating an FEM mesh

• The creation of a mesh for a problem domain is a prerequisite in

using any FEM code and package and the analyst has to spend

most of the time in such a mesh creation, and it becomes the

major component of the cost of a computer aided design (CAD)

project.2 MAY 2016

Flow chart for FEM and Mesh free method

Shape functions based on a pre-defined element

Mesh generation Node generation

Shape functions basedon a nodes in support domain

Post-processing

Discretize system equation

Solution for field variables

Geometry creation


FEM Method Mesh Free Method

IIT KHARAGPUR 7

Meshfree Shape Function Construction

Step 1: Domain Representation

“An MFree method is a method used to establish system

algebraic equations for the whole problem domain without the

use of a predefined mesh for the domain discretization.”

MFree method, the problem domain and its boundary are first

modelled and represented by using sets of nodes scattered in

the problem domain and on its boundary.

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Domain Representation in FEM and MFree. [3]

2 MAY 2016

Step 2: Function interpolation/approximation

Since there is no mesh of elements in an MFree method, the field variable (e.g., a component of the displacement) at any point at = () within the problem domain is interpolated using function values at field nodes within a small local support domain of the point at, i.e.,

() =

Local support domains used in the MFree method to construct shape functions. [ 21]2 MAY 2016 IIT KHARAGPUR 9

(1)

Mesh Free Shape Function Construction

The field variables at an arbitrary point in the problem domain are approximated using a group of field nodes in a local support domain. Hence, a moving domain based interpolation/ approximation technique is necessary to construct the MFree shape function for the approximation of the field variables using a set of arbitrarily distributed nodes.

Consider an unknown scalar function of a field variable () in the domain: The MLS approximation of () is defined at as,

The basis function is often built using monomials from the Pascal triangle to ensure minimum completeness. is a vector of coefficients given by,


(2)


Approximate function and the nodal parameters in the MLS approximation. [ 2 ]

𝑱=∑𝒊=1

𝒏¿¿


(3)

IIT KHARAGPUR 12


The stationarity of with respect to gives,

Which leads to the following set of linear relations,

(4)

where, is the vector that collects the nodal parameters of field function for all the nodes in the support domain.

and is called the weighted moment matrix defined by

(5)

The matrix H in Equation is defined as,

H=

2 MAY 2016


Solving Equation (4) for we have

Substituting the above equation back into Equation (2), we obtain.

(6)

Where is the vector of MLS shape functions corresponding n nodes in the support domain of the point, and can be written as,

= (7)

In the MLS, the coefficient is the function of which makes the approximation of weighted least squares move continuously. Therefore, the MLS shape function will be continuous in the entire global domain, as long as the weight functions are chosen properly.


IIT KHARAGPUR 14

Choice of the weight function

is always chosen to have the following properties,

1) within the support domain. 2) outside the support domain.3) monotonically decreases from the point of interest at .4) is sufficient smooth, especially on the boundary of

Domains of influence and local node numbering at point ‘x’ .[ 3 ]

2 MAY 2016

Choice of the weight function

The cubic spline function (W1) has the following form of

The quartic spline function (W2) is given by

The exponential function (W3) is expressed as


IIT KHARAGPUR 16

Derivatives of weight function and shape function

For the cubic spline weight function is chosen as follows:

Here,

is distance between node and sampling point and is size of

influence of node.

¿¿ 𝒊𝒇 𝒓 ≤ 𝟏𝟐¿

𝒓=‖𝒙− 𝒙𝒊 ‖ /𝒅𝒎𝒊

2 MAY 2016

Weight functions and their first order derivatives. W1: cubic spline; W2: quartic spline;W3: exponential function (=0.3) [3].


IIT KHARAGPUR 18

Smart Materials

Definition:

Smart materials can be defined as the materials which possess the inherent capability of converting some form of input energy into another form and have the means and imperative to achieve some definite purpose set by the designer in a controlled manner.

Examples: • Piezoelectric materials• Magnetostrictive materials• Electrostrictive materials• Shape memory alloys (SMA)

Applications :• Noise and vibration control• Sensors and actuators• Structural health monitoring

2 MAY 2016

Piezoelectric Materials

Direct Effect : Sensor

Mechanical Input

Electrical Output

Electrical input

Mechanical Output

Converse Effect : Actuator

Constitutive equation for Converse Effect [13]:

Stress vector : { }T

x y z xy xz yz

Strain vector : { }T

x y z xy xz yz

The most commonly used piezoelectric monolithic piezoelectricmaterials are PZT (lead zirconate titanate) and PVDF (Poly-vinyledyne fluoride)

{ }T

x y zE E E E Electric field vector :

{ } [ ]{ } [ ]{ }C e E


Piezoelectric Materials

Elastic [C] and Piezoelectric [e] Constant Matrices

[𝐶 ]=[𝐶11 𝐶12 𝐶13 0 0 0𝐶12 𝐶22 𝐶23 0 0 0𝐶13 𝐶23 𝐶33 0 0 00 0 0 𝐶44 0 00 0 0 0 𝐶55 00 0 0 0 0 𝐶66

] [𝑒 ]=[0 0 𝑒31

0 0 𝑒32

0 0 𝑒33

0 𝑒24 0𝑒15 0 00 0 0

]Constitutive equation for Direct Effect :

{𝑫 }=[𝒆 ]𝑻 {∈ }+[𝜺 ]{𝑬 }{𝐷 }=[𝐷𝑥 𝐷 𝑦 𝐷𝑧 ]

𝑇

[𝜀 ]=[𝜀11 0 00 𝜀22 00 0 𝜀33

]Electric Displacement Vector :

Dielectric constant matrix :


Smart Composite Beam Using MFree Method

Schematic diagrams of smart (a) simply supported and (b) cantilevered laminated composite beams.2 MAY 2016 21

The governing stress equilibrium equations and the charge equilibrium equation [13] for an electro-mechanical solid are

(7)

Where is the component of Cauchy stress tensor at any point in the solid, and is the electric displacement at the point along the direction denoted by i .

A weak form of the variational principle [13] which would yield the above equilibrium equations is given by,

𝝈𝒊𝒋 , 𝒋=𝟎𝑫 𝒊 ,𝒊=𝟎

𝛔𝐢𝐣

(8)


IIT KHARAGPUR 23

For the substrate beam, an equivalent single layer first order shear deformation theory (FSDT) describing the axial displacements and at any point in the substrate beam along the length and directions, respectively is considered as,

The axial displacements and at any point in the piezoelectric layer are considered as follows:

and 𝒘=𝒘𝟎+𝒛 𝜽𝒛

𝒖𝒑=𝒖𝟎+𝒉𝟐 𝜽𝒙+(𝒛−𝒉 /𝟐)𝜸 𝒙

𝒘 𝒑=𝒘𝟎+𝒉𝟐 𝜽 𝒛+(𝒛−𝒉 /𝟐)𝜸 𝒛

2 MAY 2016

IIT KHARAGPUR 24

The variation of electric potential across the thickness of the piezoelectric layer is linear . Thus the electric potential function which is zero at the interface between the piezoelectric layer and the substrate beam may be assumed as

Where is the electric potential distribution at the top surface of the piezoelectric layer.

(11)

2 MAY 2016

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(12)

(13)

(14)

The states of strains in the substrate beam and the piezoelectric layer and the electric potential field in the piezoelectric layer can expressed as,

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IIT KHARAGPUR 26

[𝑍¿¿1 ]=[1 0 0 𝑧 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 1 1 0 0 𝑧 0 0 0 0 ]¿

[𝑍¿¿ 2]=[1 0 0 h2

0 0 0 𝑧− h2

0 0

0 0 0 0 1 0 0 0 1 0

0 1 0 0 0 h2

1 0 0 𝑧− h2]¿

[𝑍¿¿𝑝]=−[ 𝑧 −h /2h𝑝0

0 1]¿2 MAY 2016

27

{∈ }=[𝑳]{𝒅 } {𝑬 }={𝑳𝒑 }𝝋𝟎

[𝐿 ]𝑇=[𝜕𝜕 𝑥 0 0 0 0 0 0 0 0 0

0 𝜕𝜕 𝑥 0 0 0 0 0 0 0 0

0 0 1 𝜕𝜕𝑥 0 0 0 0 0 0

0 0 0 0 1 𝜕𝜕 𝑥 0 0 0 0

0 0 0 0 0 0 1 𝜕𝜕 𝑥 0 0

0 0 0 0 0 0 0 0 1 𝜕𝜕 𝑥

]{𝐿𝑝 }=[ 𝜕𝜕 𝑥 1

h𝑝 ]𝑇

(15)

2 MAY 2016

Utilizing Eqs. (5), (6), (7) and (8), the weak form of the variational principle given by Equation (2) can be written for the present overall smart composite beams as follows:

0 00 0 0

0 00 0 01 1 1( ) ( )1 1 10

( ) (/2

pN u N u M M Mx x x x x x xxx x L x x L xp pM Q w Q w R R Rx x x x z x z zx xx x L x x Lx L x

LNp k k k kR u w dxz xz xz z zx z h z hx L k k k

pN Nuxz z zxz z h

1) (/2 /2 /2

0 0

1( { } [ ]{ }0 0/2 /2 00 0

{ } [ ]{ } { } [ ] { } { } [ ] { }0 0 0

0

L Lp pw dx u u dxxz xzz h z h z h hp

L Lp Tw w dx D D D dxz x xzz h z h hp x x LL L LT T T T TD E dx E D dx E D E dxd d

Dz z

( )[0 1 0 /2 0 ] { } 0/20 0

L Ldx q x h h d dxph hp

(16)

28

IIT KHARAGPUR 29

where,1 /2

1 /2

pk

k

h hhNk p

x x xk h h

N dz dz

1 /2

1 /2

pk

k

h hhNk p

x x xk h h

M zdz z dz

/2

/2

( / 2)ph h

p px x

h

M z h dz

1 /2

1 /2

pk

k

h hhNk p

x xz xzk h h

Q dz dz

1 /2

1 /2 2

pk

k

h hhNk p

x xz xzk h h

hR zdz dz

/2

/2

( / 2)ph h

p px xz

h

R z h dz

/2

2 2 2/2

[ ] [ ] [ ][ ]ph h

T p

h

D Z C Z dz

1

1 1 11

[ ] [ ] [ ][ ]k

k

hNT k

k h

D Z C Z dz

/2

2/2

[ ] [ ] [ ][ ]ph h

Tx p

h

D Z e Z dz

/2

/2

[ ] [ ] [ ][ ]ph h

Tp p

h

D Z Z dz

/2

/2

( / 2)ph h

x xh p

z hD D dz

h

1 2[ ] [ ] [ ]xxD D D

2 MAY 2016

IIT KHARAGPUR 30

In order that the left hand side of Eq. (9) becomes zero, the essential boundary conditions of the present overall smart beams can be derived from the variational principle given by Eq. (16) as follows:

simply supported smart beams:

Clamped-free smart beams:

{𝒅𝒃 }∨¿𝒙=𝟎=¿¿ ¿¿ 𝝋𝟎(𝟎)=𝟎

{𝒅𝒃 }∨¿𝒙=𝑳=[𝒘𝟎 𝜽𝒛 𝜸𝒛 ]𝑻∨¿𝒙=𝑳=¿¿¿ 𝛗𝟎(𝐋)=𝟎

{𝒅𝒃 }∨¿𝒙=𝟎=[𝒖𝟎 𝒘𝟎 𝜽𝒙 𝜸 𝒙 𝜽𝒛 𝜸𝒛 ]𝑻∨¿𝒙=𝟎=¿¿¿

2 MAY 2016

IIT KHARAGPUR 31

The variational principle also yields the following interface continuity conditions and prescribed boundary conditions:

¿ ¿

¿ ¿𝝈𝒛𝟏(𝒙 ,−𝒉 /𝟐)=𝝈𝒙𝒛𝟏 (𝒙 ,−𝒉 /𝟐)=𝟎 𝝈𝒙𝒛𝒑 (𝒙 ,𝒉 /𝟐+𝒉𝒑)=𝟎

Here,

2 MAY 2016

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Element Free Galerkin Method

The element-free Galerkin (EFG) method is a MFree method uses the moving least squares (MLS) shape functions, Because the MLS approximation lacks the Kronecker delta function property.

The generalized displacement vector at any point can be expressed in terms of the nodal generalized displacement parameter vector and the nodal electric potential vector, respectively as follows:

{𝒅}=[𝑵 ]{𝑿 } 𝝋𝟎=[𝜳 ]{𝜱 }

{𝑿 }=[ {𝒅𝟏 }𝑻 {𝒅𝟐 }

𝑻 {𝒅𝟑 }𝑻 . . . . {𝒅𝒏 }

𝑻 ]𝑻

[𝑵 ¿¿ [ [𝑵𝟏 ] [𝑵𝟐 ] [𝑵𝟑 ] . . . . [𝑵𝒏 ] ]¿

{𝒅𝒊 }=[𝒖𝟎 𝒊 𝒘 𝟎𝒊 𝜽 𝒙𝒊 𝜽 𝒛𝒊 𝜸 𝒙𝒊 𝜸𝒛𝒊 ]𝑻

[𝑵 𝒊¿¿𝝍𝒊 𝑰2 MAY 2016

IIT KHARAGPUR 33


The generalized strain vector and the generalized electric field vector can be expressed in terms of the nodal generalized displacement parameter vector and the nodal electric potential vector as follows:

Where,

Since the mesh free shape function does not satisfy the property of the Kronecker delta function, the essential boundary conditions are satisfied using Lagrange multipliers.

{∈ }=[𝑩]{𝑿 } {𝑬 }=[𝑩𝝋 ]{𝜱 }

¿ ¿

2 MAY 2016

IIT KHARAGPUR 34

Weak form of variational principle is now given by,

L

T T T T T

xx x pφ φ

0

T T T T T

0 1 L 2 1 0x=L x=0x=0

T T T T T T T T

2 L 0 b L b 3 xx=L x=0 x=L x=0

{X} [B] [D ][B]{X}-[B] [D ][B ]{ }-[N] [ 0 1 0 h/2δ Φ 0 h ] q(x) dx

+ { } [U ][N] {X}+ { } [U ][N] {X}+ {X} [N] [U ] {δ λ δ λ δ }λ

+ {X} [N] [U ] { }- { } {d } - { } {d } - {X} [N] [Uδ λ δ λ δ λ δ ] M

- {X}δ T T T T T T p T T T p

3 x 4 x 4 xx=0 x=Lx=L[N] [U ] M - {X} [N] [U ] M - {X} [N] [U ] M =0δ δ

in which {} and {} are the vectors of Lagrange multipliers for the essential displacement boundary conditions at x =0 and L respectively.


2 MAY 2016

(17)

IIT KHARAGPUR 35


The moment resultants and and the various other matrices appearing in Equation are given by

Where,

𝑴 𝒙=[𝑫𝟑 ][𝑩 ]{𝑿 }𝑴 𝒙𝒑=[𝑫𝟒 ][𝑩 ]{𝑿 }

[𝑫¿¿𝟑]=∑𝒌=𝟏

𝑵

∑𝒉𝒌

𝒉𝒌 +𝟏𝒛 ¿¿

[𝑫𝟒 ]= 𝒉/𝟐

𝒉/𝟐+𝒉𝒑

¿¿

2 MAY 2016

IIT KHARAGPUR 36


1

1 0 0 0 0 00 1 0 0 0 0

[ ]0 0 0 1 0 00 0 0 0 0 1

U

2

0 1 0 0 0 0[ ] 0 0 0 1 0 0

0 0 0 0 0 1U

3[ ] [0 0 1 0 0 0]U 4[ ] [0 0 0 0 1 0]U

2 MAY 2016

IIT KHARAGPUR 37


In order that the left hand side of Eq. (17) be zero, the following discrete governing equations of the overall smart composite beams can be derived:

(18)[𝑲 ]{𝑿 }+[𝑮 ]{𝝀 }=[𝑭𝒑 ]{𝜱 }+{𝑭 }

[𝑮 ]𝑻 {𝑿 }={𝑿𝒃 }

Where, [K] is stiffness matrix, [] is the electro-elastic coupling matrix, [F] is the mechanical load vector, {} is the Lagrange multiplier vector, {} is the vector of essential displacement boundary conditions.

2 MAY 2016

IIT KHARAGPUR 38

3 3 4 4 00

3 3 4 4

[ ] [ ] [ ][ ]{ } [ ] ([ ] [ ] [ ] [ ])[ ]

[ ] ([ ] [ ] [ ] [ ])[ ]

LT T T T

x

T T T

x L

K B D B X dx N U D U D B

N U D U D B

0

{ } [ ] [0 0 0 / 2 0 ]L

T TpF N h h dx

0

[ ] [ ] [ ][ ]L

Tp xF B D B dx

1 0

2

[ ][ ][ ]

[ ][ ]

T

x

x L

U NG

U N

0{ } [{ } { } ]T T TL

0{ } [{ } { } ]T T T

b b bx x LX d d

The sets of discrete governing equations given by Eq. (2) represent the mesh free model (MFM ) of the overall smart composite beams.

Where,


2 MAY 2016

IIT KHARAGPUR 39

Computation of axial and transverse shear stresses

The constitutive relation can be used to compute the axial stress at any point in layer of substrate beam as follows,

(19)

Also it used to compute the shear stress at any point in layer of substrate beam as follows,

(20)

where the constants are determined by satisfying the interface continuity conditions and the surface traction at the bottom surface of the beam.

¿

𝝈𝒙𝒛𝒌 =−¿

2 MAY 2016

IIT KHARAGPUR 40

Computation of natural frequencies

A weak form of the variational for the dynamic analysis of problem is given by from equation (8) is

After substitution into the weak variational form, the final discrete system equations for free vibration can be obtained as follows:

[M] is the mass matrix and it is defined as,

Where,

[𝑵 ¿¿ [ [𝑵𝟏 ] [𝑵𝟐 ] [𝑵𝟑 ] . . . . [𝑵𝒏 ] ]¿[𝑵 𝒊¿¿𝝍𝒊 𝑰

2 MAY 2016

IIT KHARAGPUR 41

Computation of natural frequencies

The natural frequencies of the beam vibration can be found by ωsolving the following eigenvalue equations

where is the vector of amplitudes of all nodal displacements or displacement parameters when the MLS shape functions are used.

Where is so called eigenvalue, and q is the eigenvector. This equation can be solved using a standard eigenvalue solver to obtain eigenvalues ( i =1, 2,…, N) and the corresponding q. The natural frequencies of the structures are then given by .

2 MAY 2016

IIT KHARAGPUR 42

Numerical ResultsComparison of Responses of Mfree Method for Three Layered (0°/90°/0°) Composite Beam with Exact Solution Method.

Active static control of shape of a three layered (0°/90°/0°) composite beam (s = 50, h=0.05 m, =250 and =50 N/m2)

2 MAY 2016

Numerical ResultsDistribution of the axial stress across the thickness three layered (0°/90°/0°) composite beam with and without actuated by a piezoelectric layer.

Active static control of shape of a three layered (0/90/0) composite beam (s = 50, h=0.05 m, =250 and =50 N/m2)


Numerical ResultsDistribution of the transverse stress across the thickness three layered (0°/90°/0°) composite beam with and without actuated by a piezoelectric layer.



Numerical ResultsComparison of Responses of MFree Method for four Layered (0°/90°/0°/90°) Composite Beam with Exact Solution Method

Active static control of shape of a four layered (0/90/0/90) composite beam (s = 50, h=0.05 m, =250 and =50 N/m2)

2 MAY 2016 IIT KHARAGPUR45

Numerical ResultsDistribution of the axial stress across the thickness four layered (0°/90°/0°/90°) composite beam with and without actuated by a piezoelectric layer



Numerical ResultsDistribution of the transverse shear stress across the thickness four layered (0°/90°/0°/90°) composite beam with and without actuated by a piezoelectric layer



IIT KHARAGPUR 48

Numerical Results


2 MAY 2016

IIT KHARAGPUR 49

Numerical Results

Natural Frequencies(Hz) for three layered (0/90/0) composite beam

Beam

0/90/0

Source

1st mode

2nd mode

3rd mode

S=50Present [MFree]

32

204

557

FEM[Ref.13]

35

210

564

2 MAY 2016

IIT KHARAGPUR 50

Conclusion

2 MAY 2016

In the present work, a new layer wise laminated beam theory based on element free Galerkin (EFG) is proposed to analyse the numerical simulation of piezoelectric laminated beam have been used to verify the validity and precision of the proposed method.

The details of the element free Galerkin (EFG) method and its numerical implementation have been presented.

It only needs nodes distributed on the upper and lower surfaces of each layer, which avoids the troublesome in generating mesh and saves much pre-process time MLS approximants, which constitute the back bone of the method were described, including several examples.

The new approach reduces the dimension of approximation of MLS, which also greatly save the computational cost. The convergence characteristic and accuracy of the method are validated with exact solution.

IIT KHARAGPUR 51

Future Scope

2 MAY 2016

With MFree methods like EFG it is possible to review developments and applications for various types of structure mechanics and fracture mechanics applications like bending, buckling, free vibration analysis, sensitivity analysis and topology optimization, single and mixed mode crack problems, fatigue crack growth, and dynamic crack analysis and some typical applications like vibration of cracked structures, thermoselastic crack problems, and failure transition in impact problems.

Meshfree methods is the possibility of simplifying adaptively and problems with moving boundaries and discontinuities, such as phase changes or cracks.

IIT KHARAGPUR 52

References

[ 1 ] Belytschko, T., Y.Y. Lu, and Gu, Element Free Galerkin Methods",

International Journal for Numerical Methods in Engineering, 37, 229-256

(1994).

[ 2 ] Belytschko, T., D. Organ, and Y. Krongauz, A Coupled Finite Element-

Element free Galerkin Method", Computational Mechanics, 17, 186-195

(1995).

[ 3 ] Belytschko, T., Y. Krongauz, M. Fleming, D. Organ and W.K. Liu , Smoothing

and Accelerated Computations in the Element-free Galerkin Method", Journal

of Computational and Applied Mechanics, 74, 111-126 (1996).

[ 4 ] Lancaster, P. and K. Salkauskas, Surfaces Generated by Moving Least

Squares Methods", Mathematics of Computation, 37, 141-158 (1981).

2 MAY 2016

References

[ 5 ] Beissel, S. and Belytschko, T. “Nodal integration of the element-free

Galerkin method,” Computational Methods, Eng.139, 49–74 (1996).

[ 6 ] Duarte, C.A.M. and Oden, J.T. “Hp clouds-A meshless method to solve

boundary value problems,” TICAM Report, 95-05.3 (1995).

[ 7 ] J. Dolbow and T. Belytschko, “An Introduction to Programming the

Meshless Element Free Galerkin Method”, Vol. 5, 3, 207-241 (1998).

[ 8 ] Duarte, C.A. and J.T. Oden, H-p Clouds h-p Meshless Method, Numerical

Methods for Partial Differential Equations, 1-34 (1996).

[ 9 ] Belytschko, T., Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless

Methods: An Overview and Recent Developments", Computer Methods in

Applied Mechanics and Engineering, 139, 3-47 (1996).


54

References

[10] Ray M. C., Mallik N., “Finite Element Solutions for Static Analysis of Smart

Structures With a Layer of Piezoelectric Fiber Reinforced Composites”, AIAA

Journal, vol. 42, no. 7, 2004, pp. 1398-1405, (2004).

[ 11] Anderson EH, Hagood NW. Simultaneous piezoelectric

sensingactuation: analysis and application to controlled structures. Sound

Vibration (1994).

[ 12 ] M.C. Ray , L.Dong and S.N. Atluri.,”Simple efficient finite element for

analysis of Smart composite beam, CMC vol.47,no.3,143-147,(2015).

[ 13] ] Ray M C and Mallik N.Active control of laminated composite beams

using piezoelectric fiber reinforced composite layer Smart Material Structure.

13,146–52 (2004).

[ 14 ] Crawley, E.F. and Luis, J.D., Use of piezoelectric actuators as elements of

intelligent,17 162-169 (1982).2 MAY 2016 IIT KHARAGPUR

IIT KHARAGPUR 552 MAY 2016

Divyesh_14Me63r02(new)

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