Meshless Meshless Methods Meshless Meshless Methods Methods - - PowerPoint PPT Presentation

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Meshless Meshless Methods Methods Meshless Meshless Methods Methods

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Contents

• Introduction
• Mess Free Methods

Element Free Galerkin Method Element Free Galerkin Method Moving Particle Semi-Implicit Method C l i

• Conclusion
• Reference

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Introduction

• Numerical methods can be classified in two groups
• (1) Mesh-based method like FEM, FD, BEM, FVM
• (2) Mesh-less methods/Mesh Free Methods (MFM)

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Difficulty in applying FEM

Th t i l f bl f hi h th FEM

• There are certain classes of problems for which the FEM

is difficult, or even impossible to be applied.

• The FEM usually requires remeshing in order to insure

equality between finite element boundaries and the moving discontinuities.

• The main objective for the development of meshless

The main objective for the development of meshless methods(also called mesh free method, MFM) is making approximation based on nodes, not elements.

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Some major advantages of MFMs

(i) problems with moving discontinuities such as crack propagation. (ii) l d f ti b h dl d b tl (ii) large deformation can be handled more robustly, (iii) higher-order continuous shape functions, (iv) non-local interpolation character and (iv) non local interpolation character and (v) no mesh alignment sensitivity.

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History of meshless methods

• The Smoothed Particle Hydrodynamics (SPH)

The advent of the mesh free idea dates back from 1977, with Monaghan , Gingold and Lucy developing a Lagrangian method based on the Kernel Estimates method to model astrophysics problems.

• The Diffuse Element Method (DEM)

DEM was introduced by Nayroles and Touzot in 1991. The idea behind the DEM was to replace the FEM interpolation within an element by the Moving Least Square (MLS) local interpolation. p

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• The Element Free Galerkin (EFG)

Continued

• The Element-Free Galerkin (EFG)

In 1994 Belytschko and colleagues introduced the Element-Free Galerkin Method (EFG), an extended version of Nayroles’s Method EFG is one of the most popular mesh-free methods and extended version of Nayroles s Method. EFG is one of the most popular mesh-free methods and its application has been extended to different classes of problems such as fracture and crack propagation, wave propagation ,acoustics and fluid flow.

• Reproducing Kernel Particle Method

In 1995 Liu proposed the (RKPM) in an attempt to construct a procedure to correct the lack of i i h SPH h d Th RKPM h b f ll d i l i l h i consistency in the SPH method. The RKPM has been successfully used in multiscale techniques, vibration analysis, fluid dynamics and many other applications.

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Continued

• Finite Point Method
• Finite Point Method

The Finite Point method was proposed by On˜ate and colleagues in 1996. It was

• riginally introduced to model fluid flow problems and later applied to model many
• riginally introduced to model fluid flow problems and later applied to model many
• ther mechanics problems such as elasticity and plate bending.
• Meshless Local Petrov-Galerkin

The Meshless Local Petrov-Galerkin introduced by Atluri and Zhu in 1998 presents a different approach in constructing a mesh-free method different approach in constructing a mesh free method

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Continued

• Radial Basis F nctions (RBFs)
• Radial Basis Functions (RBFs)

RBF were first applied to solve partial differential equations in 1991 by Kansa, when a technique based on the direct Collocation method and the Multiquadric RBF was used to model fluid dynamics.

• Point Interpolation Method

The Point Interpolation method (PIM) uses the Polynomial Interpolation technique to construct the approximation. It was introduced by Liu in 2001 as an alternative to the Moving Least Square method Moving Least Square method.

• Moving Particle Semi-Implicit Method (MPS)

MPS method was developed by Koshizuka and applied to fluid flow problems.

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Introduction to EFGM

• An element free Galerkin method which is

applicable to arbitrary shapes but requires only nodal data nodal data.

• In this method moving least-square interpolants

are used to construct the trial and test functions are used to construct the trial and test functions for the variational principle (weak form); the dependant variable and its gradient are continuous p g in the entire domain.

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Continued

I i i l b d di i

• Imposing essential boundary conditions

In the EFG formulation, Lagrange Multipliers are used in the weak form to enforce the essential used in the weak form to enforce the essential boundary conditions.

• Process for Numerical Integration

Process for Numerical Integration

An auxiliary cell structure,is used in order to create a “structure” to define the quadrature q points.

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Continued

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A computational model for a A computational model for a meshless method

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Mathematical Formulation

EFG th d M i l t th d f i ti f th

Mathematical Formulation

EFG method uses Moving least square method for approximation of the unknown function u, which encompasses

• a weighted function associated to each node
• a polynomial basis function (Interpolation function)
• a polynomial basis function (Interpolation function)
• a set of coefficients which depend on the position
• The approximate function uh (x) is defined as

(1)

• --------------------------- (1)
• m is the number of terms in the basis function
• pi(x) are the basis function

( ) th k ffi i t hi h f ti f th ti l

• ai(x) are the unknown coefficients which are functions of the spatial

coordinates x.

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Continued

A li d d i b i 1 d 2

• A common linear and quadratic bases in 1D and 2D space are:

Linear Bases: pT

(m=2) = {1 , x}

1-D p (m=2) {1 , x} 1 D pT

(m=3) = {1 , x , y} 2-D

Quadratic Bases: pT

(m=3) = {1 , x , x2} 1-D

pT

(m=6) = {1 , x , y , x2 , xy , y2} 2-D

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Continued

• Define a function
• Define a function

w(I) is a weighted function with compact support. i th t t l b f d i id th d i f i fl n is the total number of nodes inside the domain of influence.

• Using matrix notation

J = (P a-u)T W(x) (P a-u) J (P a u) W(x) (P a u)

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Continued

F th

i ti

h( )

i il ith th di For the approximation uh(x) , very similar with the ordinary finite element method, we have

uh (x) = u (x)

Ø is the shape function given by ø

T( ) A 1( ) B( )

ø = pT (x) A‐1(x) B(x).

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Continued

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Continued

I th EFGM th i t ti i f d t

• In the EFGM the integration is performed at

each integration point of a simple integration cell also called a bucket cell also called a bucket.

• The domain of influence of each integration

point is defined by the radius of influence r. point is defined by the radius of influence r.

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Application of EFG method to Application of EFG method to Structural Engineering Problem

A b f i l h bj d li b d f d fi d

• A bar of unit length subjected to a linear body force and fixed

at point x = 0.

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Comparison between EFG & Exact Comparison between EFG & Exact method for 1D problem

0.3 0.35 0.2 0.25 ment 0 1 0.15 displacem EFG displacement Exact displacement 0.05 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 position (x)

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Bar subjected to a unit force at the free end i i) EFG d ii) FEM using i) EFG and ii) FEM

Displacement in EFG Displacement in FEM 0.0000 0.0000 0.2011 0.2000 0.3995 0.4000 0 6005 0 6000 0.6005 0.6000 0.7989 0.8000 1 0000 1 0000 1.0000 1.0000

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Continued

0 8 0.9 1 FEM Displacement 1.2 1.4 EFG Displacement 0 5 0.6 0.7 0.8 e m e n t 0.8 1 e m e n t 0 2 0.3 0.4 0.5 d is p la c 0.4 0.6 d is p la c e 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 x

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The initial and boundary conditions are

⎫ ) ( ) ( t t ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ = = Γ = = = Γ = = = =

2 2 1 1

, ) (or edge, at the , ) (or edge at the ) , , ( ) , , ( 0, at

,

t v y t u x y x v y x u t

x y

⎪ ⎪ ⎪ ⎭ = = Γ = = = Γ =

4 4 4 3

, )

• r

( edge, at the , ) (or edge, at the v v u W y t t L x

y x

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The The governing governing equation equation of

• f a

a viscous viscous incompressible incompressible fluid fluid squeezed squeezed between between two two long long parallel parallel plates plates are are given given by by Continuity Continuity and and between between two two long long parallel parallel plates plates are are given given by by Continuity Continuity and and Navier Navier-

• Stokes

Stokes equations equations. .

x-momentum equation

2 = − ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − ∂ ∂

x

f x P x v y u y x u x t u μ μ ρ

(1a) (1a)

⎦ ⎣ ⎠ ⎝

2 = − ∂ + ⎥ ⎤ ⎢ ⎡ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ∂ ∂ − ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ ∂ − ∂ f P v u v v μ μ ρ

(1b) (1b) y-momentum equation

2 = ∂ + ⎥ ⎦ ⎢ ⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ∂ ∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ ∂ ∂

y

f y x y x y y t μ μ ρ

∂ ∂

(1b) (1b) Continuity equation

= ∂ ∂ + ∂ ∂ y u x u

(1c) (1c)

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Numerical Implementation Numerical Implementation The weighted integral forms of Eqs. (1a), (1b) and (1c) can written as The weighted integral forms of Eqs. (1a), (1b) and (1c) can written as

2 ~

1

= ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − ∂ ∂

∫

x

f x P x v y u y x u x t u w μ μ ρ

(2a) (2a)

⎪ ⎭ ⎪ ⎩ ∂ ⎦ ⎣ ⎠ ⎝ ∂ ∂ ∂ ⎠ ⎝ ∂ ∂ ∂

∫

Ω

x x y y x x t

⎪ ⎫ ⎪ ⎧ ⎤ ⎡ ⎞ ⎛ ⎞ ⎛ 2 ~

2

= ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ∂ ∂ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − ∂ ∂

∫

Ω y

f y P x v y u x y v y t v w μ μ ρ

(2b) (2b)

~

3

= ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ + ∂ ∂

∫

y v x u w

(2c) (2c)

⎭ ⎩ ∂ ∂

∫

Ω

y x

( ) ( )

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Weak form of Eqs. (2a) and (2b) with natural boundary conditions are obtained as Weak form of Eqs. (2a) and (2b) with natural boundary conditions are obtained as

~ ~ ~ ~ ~ 2 ~

1 1 1 1 1 1

= Γ − Ω ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∂ ∂ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂

∫ ∫

d t w d f w P x w x v y u y w x u x w t u w

x x

μ μ ρ

(3a) (3a)

⎦ ⎣ ∂ ⎟ ⎠ ⎜ ⎝ ∂ ∂ ∂ ∂ ∂ ∂

∫ ∫

Γ Ω

x x y y x x t

~ ~ ~ ~ ~ 2 ~

2 2 2 2 2 2

= Γ − Ω ⎥ ⎤ ⎢ ⎡ − ∂ − ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ∂ ∂ + ∂ ∂ + ∂

∫ ∫

d t w d f w P w v u w v w v w μ μ ρ

(3b) (3b)

2

2 2 2

Γ Ω ⎥ ⎦ ⎢ ⎣ ∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ∂ ∂ + ∂ ∂ + ∂

∫ ∫

Γ Ω

d t w d f w P y x y x y y t w

y y

μ μ ρ

Where, Where,

n v u n P u t ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ∂ + ⎟ ⎞ ⎜ ⎛ ∂ μ μ 2

y x x

n x y n P x t ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ∂ + ⎟ ⎠ ⎜ ⎝ − ∂ = μ μ 2 v v u ⎟ ⎞ ⎜ ⎛ ∂ ⎟ ⎞ ⎜ ⎛ ∂ ∂

y x y

n P y v n x v y u t ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = μ μ 2

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Quadratic functional can be written as Quadratic functional can be written as

) , ( v u ∏

⎤ ⎡

2 2 2

∫

Ω

+ Ω ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ =

∏

d x v y u y v x u

v u

μ

2 2 2

2 1

) , (

(4) (4)

( ) ( )

∫ ∫ ∫

Ω Γ Ω

Γ + − Ω + − Ω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ d v t u t d v f u f d t v v t u u

y x y x

ρ ρ

Using penalty function method, the modified functional can be written as Using penalty function method, the modified functional can be written as

) ,

( v

u

p

∏ ⎞ ⎛ ∂ ∂ ⎤ ⎡ ⎞ ⎛ ∂ ∂ ⎞ ⎛∂ ⎞ ⎛

2 2 2

1 ⎞ ⎛ − Ω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + Ω ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

∫ ∫

Ω Ω

∂ ∂

∏

d t v v t u u d x v y u y v

p

x u

v u

2 2

2 1

) , (

ρ ρ μ

(5) (5)

( ) ( )

Ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + Γ + − Ω +

∫ ∫ ∫

Ω Γ Ω

d y v x u d v t u t d v f u f

y x y x 2

2 α

( ) ( )

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Enforcing essential boundary conditions using LMM, the functional Enforcing essential boundary conditions using LMM, the functional b itt b itt can be written as can be written as

) , (

*

v u

P

∏

∫ ∫

∂ ∂

⎟ ⎞ ⎜ ⎛ ⎥ ⎤ ⎢ ⎡ ⎟ ⎞ ⎜ ⎛ ∂ ∂ ⎟ ⎞ ⎜ ⎛∂ ⎟ ⎞ ⎜ ⎛∂

2 2 2 *

1

v u

d d v u v u

∫ ∫

Ω Ω

∂ ∂ + ∂ ∂

− = ∏

⎞ ⎛ Ω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Ω ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂

2 *

2 1

) , (

t v v t u u

v u

d d x v y u y v x u

p

ρ ρ

μ

(6) (6)

( ) ( )

∫ ∫ ∫ ∫ ∫ ∫ ∫

Ω Γ Ω

Γ +

+ Ω + Ω +

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ −

2

2

d v t u t

d d v f u f

y v x u

y x y x

α

( ) ( ) ( ) ( )

∫ ∫ ∫ ∫

Γ Γ Γ Γ

Γ − Γ − Γ − Γ −

+ + +

4 2 4 1

4 2 2 2 4 1 1 1

d v v d v v d u u d u u λ λ λ λ

Taking variation i.e. of Eq. (6), it reduces to

) , (

*

v u

p

∏

δ

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+ Ω ⎥ ⎤ ⎢ ⎡ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ∂ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ∂ + ⎟ ⎟ ⎞ ⎜ ⎜ ⎛∂ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛∂ + ⎟ ⎞ ⎜ ⎛∂ ⎟ ⎞ ⎜ ⎛∂ = ∫

∏

d v u v u v v u u

v u

*

2 2 )

(

δ δ δ μ

δ

( ) ( )

( ) + Γ − + Γ + − Ω + − Ω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Ω ⎥ ⎦ ⎢ ⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ∂ + ⎟ ⎟ ⎠ ⎜ ⎜ ⎝∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝∂ + ⎟ ⎠ ⎜ ⎝∂ ⎟ ⎠ ⎜ ⎝∂

∫ ∫ ∫ ∫ ∫

Ω

∂ + ∂

∏

d u u d v t u t d v f u f d d x y x y y y x x

y x y x p

v v u u

v u

1 1

2 2 )

, (

λ δ δ δ δ δ δ δ δ μ

δ ρ δ ρ

δ

( ) ( )

( ) ( ) ( ) + Γ − + Γ + Γ − + Γ + Ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⎟ ⎠ ⎜ ⎝

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

Γ Γ Ω Ω

∂ ∂

d u u d u d u u d u d v u v u f f

y x y x

t t

4 1 1 1 1 1 1 1

1

λ δ δ λ λ δ δ λ δ α

ρ ρ

(7a)

( ) ( ) ( ) ( ) Γ − + Γ + Γ − + Γ ⎟ ⎠ ⎜ ⎝ ∂ ∂ ⎟ ⎠ ⎜ ⎝ ∂ ∂

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

Γ Γ Γ Γ Γ Γ Γ Γ Ω

d v v d v d v v d v y x y x

4 2 2 2 2 2 4 1 1 1 1 1

4 4 1 1

λ δ δ λ λ δ δ λ

Γ Γ Γ Γ

4 4 2 2

Subject to the constraint Subject to the constraint

∂ ∂ v u δ δ = ∂ + ∂ y x

(7b)

SLIDE 33

( )

∫

=

J T I J I

Φ Φ M ρ

( )

∫

Ω J I J I

ρ

( )

∫ ∫

Γ + Ω = d Φ t d Φ f F

I I I 1

( )

∫ ∫

Γ Ω

Γ + Ω d Φ t d Φ f F

I x I x I 1

( )

∫ ∫

Γ + Ω = d Φ t d Φ f F

I y I y I 2

( )

∫ ∫

Γ Ω

f

I y I y I 2

( )

∫ ∫

Γ + Γ = d N Φ d N Φ G

K I K I IK 1

( )

∫ ∫

Γ + Γ =

4 1 1

d N u d N u q

K K K

and

( )

∫ ∫

Γ Γ

Γ + Γ

4 1

d N Φ d N Φ G

K I K I IK 1

( )

∫ ∫

Γ Γ

Γ + Γ

4 1

4 1 1

d N u d N u q

K K K

( )

∫ ∫

Γ + Γ = d N Φ d N Φ G

K I K I IK 2

( )

∫ ∫

Γ + Γ =

4 2 2

d N v d N v q

K K K

and and

( )

∫ ∫

Γ Γ

4 2

K I K I IK 2

( )

∫ ∫

Γ Γ

4 2

4 2 2

q

K K K

SLIDE 34

Since Since & are arbitrary in the Eq. (7a), following relations are obtained are arbitrary in the Eq. (7a), following relations are obtained from Eq (7) from Eq (7)

1

, , δλ δ δ v u

2

δλ

from Eq. (7) from Eq. (7)

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

1 1 1 12 11

F λ G v K u K u M = + + + &

(8a) (8a)

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

2 2 2 22 21

F λ G v K u K v M = + + + &

(8b) (8b)

[ ]{ } { }

1 1

q u G =

T

(8c) (8c)

[ ]

{ } { }

2 2

q v G =

T

(8d) (8d)

SLIDE 35

Ω ⎥ ⎤ ⎢ ⎡ ⎟ ⎞ ⎜ ⎛∂ ⎟ ⎞ ⎜ ⎛ ∂ + ⎟ ⎟ ⎞ ⎜ ⎜ ⎛∂ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ⎟ ⎞ ⎜ ⎛∂ ⎟ ⎞ ⎜ ⎛ ∂

∫

d Φ Φ Φ Φ Φ Φ K

J T I J T I J T I

α μ μ 2 ) ( Ω ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ ⎟ ⎠ ⎜ ⎝ ∂ ⎟ ⎠ ⎜ ⎝ ∂ + ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ⎟ ⎠ ⎜ ⎝ ∂ ⎟ ⎠ ⎜ ⎝ ∂ =∫

Ω

d x x y y x x K

J I J I J I IJ

α μ μ 2 ) (

11

⎤ ⎡ ⎞ ⎛ ∂ ⎞ ⎛ ∂ ⎞ ⎛ ∂ ⎞ ⎛ ∂ Φ Φ Φ Φ

T T

Ω ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ∫

Ω

d x Φ y Φ y Φ x Φ K

J I J I IJ

α μ ) (

12

Ω ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ =∫

Ω

d y Φ x Φ x Φ y Φ K

J T I J T I IJ

α μ ) (

21

⎥ ⎦ ⎢ ⎣ ⎠ ⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎝ ∂

Ω

y x x y

⎥ ⎤ ⎢ ⎡ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛∂ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ⎟ ⎟ ⎞ ⎜ ⎜ ⎛∂ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ ∂ + ⎟ ⎞ ⎜ ⎛∂ ⎟ ⎞ ⎜ ⎛ ∂ = Φ Φ Φ Φ Φ Φ K

J T I J T I J T I IJ

α μ μ 2 ) (

22

⎥ ⎥ ⎦ ⎢ ⎢ ⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ⎟ ⎠ ⎜ ⎝ ∂ ⎟ ⎠ ⎜ ⎝ ∂ y y y y x x K

IJ

α μ μ 2 ) (

22

SLIDE 36

• Eq. (8) can be written as
• Eq. (8) can be written as

(9 )

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

F λ G U K U M = + + &

[ ]

{ } { }

q U G =

T

(9a) (9b)

[ ]

{ } { }

q U G =

Where, Where,

⎤ ⎡M

⎤ ⎡ K K

{ }

⎫ ⎧u & &

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = M M M

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

22 21 12 11

K K K K K

{ }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = v u U & & ⎫ ⎧u

{ }

⎫ ⎧

1

F

{ }

⎬ ⎫ ⎨ ⎧

1

λ λ

{ }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = v u U

{ }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ =

2 1

F F F

{ }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ =

2 1

λ λ

⎫ ⎧

⎤ ⎡

[ ]

⎤ ⎡

T

G

{ }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ =

2 1

q q q

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

2 1

G G G

[ ]

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ =

T T 2 1

G G G

SLIDE 37

Using unconditionally stable implicit backward difference method for time integration Using unconditionally stable implicit backward difference method for time integration Eq.(9) can be written as Eq.(9) can be written as Eq.(9) can be written as Eq.(9) can be written as

[ ]

[ ]

[ ]

⎪ ⎬ ⎫ ⎪ ⎨ ⎧ = ⎬ ⎫ ⎨ ⎧ ⎥ ⎤ ⎢ ⎡ +

+ + +

F U G M K

1 1 1 n n n

) )

(10)

⎪ ⎭ ⎬ ⎪ ⎩ ⎨ = ⎭ ⎬ ⎩ ⎨ ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ q λ G

T

(10) Where, Where,

[ ]

[ ]{ }

{ }

F U M t F

n n

Δ + =

+1

)

[ ]

[ ]

1

[ ]

[ ]

K K t

n

Δ =

+ 1

)

Using Using PM PM to to enforce enforce essential essential boundary boundary conditions, conditions, the the functional functional

*

can can be be written written as as

) , (

*

v u

P

∏

SLIDE 38

Conclusion

• EFG method can be efficiently used for fluid flow

problems and structural mechanic problems with k l d f ti cracks, large deformations.