An inverse problem in linear crack media under heat transfer using boundary element method pps

The boundary element method is used to determine the heat transfer in two-dimensional problems, with and without crack.. Basically finite element method FEM require the subdivision of th

Abstract

The purpose of this work presents an application of the inverse problem to fracture mechanics under heat transfer The boundary element method is used to determine the heat transfer in two-dimensional problems, with and without crack Crack location in the material is detected based on the estimation of the energy loss at two tips of a crack Numerical results show that the presented method obtains high reliability

1 Introduction

The boundary element method (BEM) is a numerical method for obtaining approximate solutions by solving boundary integral equations Equations provide

a determined formulation of boundary-value problems in different fields of engineering, elasticity, plasticity, fracture mechanics, wave propagation and

electromagnetic field problems [6]

Basically finite element method (FEM) require the subdivision of the region into small elements but BEM on the other hand only requires the subdivision of the boundary of the region (Figure 1a,b) BEM consists of two different approaches, the indirect and the direct approach (Figure 2a,b) For D-BEM at least one closed boundary is required and one side of the surface can be regarded Using I-BEM, both sides of a surface can be considered I-BEM can also deal with open boundary problems [12]

An inverse problem in linear crack media under heat transfer using

boundary element method

Toan Cao-Duc* and Hung Nguyen-Dang++

* European Master in Engineering Science of Mechanics of Construction(EMMC), Centre d'Excellence Belgo-Vietnamien en Sciences Appliquées, Programme de Coopération Universitaire Institutionnelle, L’université de Polytechnique de Ho Chi

Minh ville – L’universités Liege de Belgique

++

Full professor of Fracture Mechanics Department, University of Liege, Chemin

des chevreuils 1, B-4000 liege, Belgium

Figure 1: a) Finite element mesh b) Boundary element mesh

Figure 2 : a) Direct BEM b) Indirect BEM

Nowadays, BEM is used in many fields of fracture mechanics as it calculates stresses and displacement at crack tips An important advantage of BEM determines discontinuous geometry in mechanical, medical or soil problems

We have known that non destruction testing is concerned with the detection of hidden structural flaws, without damaging the surface or function of structure Some of the most common non destruction testing methods are based on visualization of flaws by means of various techniques, such as dye penetrates, magnetic particles, eddy current, ultrasonic and radiographic [2, 11].Therefore, In this paper, we only research BEM for heat transfer in plate problems, and apply this in inverse problems to detect discontinuous geometry in material

2 Boundary element formulation of Laplace's equation

The statement of the differential equation for Laplace's equation ∇ u2 =0 is as follows

Ω

2uωd

with ω a test function

(1)

The transformation on the boundary is treated on 2-D by using Green's integral theorem The transformation of the differential operator to the boundary is done

by applying Green's theorem twice to the weighted residual

Ω

Ω

wd

u ii

Ω Γ

d x x

w u wd

n u

i i i

= ∫ u,i w−w,i u n i dΓ+∫w,ii udΩ (2) The fundamental solution of the Green's function for the unbounded space is obtained by solving the differential equation

) ( )

( ∗ =∇2 ∗ =−δ −ξ

∗

x u

u

The minus sign of the Dirac distribution is introduced for convenience so that the obtained system matrices become positive In 2-D the fundamental solution is given by:

) , (x ξ

π

ξ

1 2

1

−

=

−

) , (x ξ

q n

=

∂

∂

x n

r

ξ π

∂

2

1 2

1

(5)

with the common shortening r of the Euclidean distance x−ξ = (x i−ξi)2 The functions u and ∗ q are presented as single layer and double layer potentials, ∗

respectively After selecting w = u∗, Eq (2) and∇ u2 =0 associated with considering the property of the Dirac distribution, they lead to

Ω

−

ξ) ( ( , ) ( ) ( , ) ( )) ,

Where

n

x u x q

∂

∂

= ( )

)

The common notations x for the field point (marked by the vectorx) and ξfor the load point (marked by the vectorξ) have been used The definition (7) of q deviates from the physical definition of the heat flux vector

n

x u k

q w

∂

∂

−

In a heat transfer problem, physical constants such as the heat conductivity k need

to be taken into account Eq (6) leads to the integral equation

Γ Γ

Γ Γ

d x q x u d

x u x q u

c

) ( ) , ( )

( ) , ( )

( ) (

) (

ξ ξ

ξ π α

ξ

∫

= +

− 4 42

1 2

1

(9) The factor c(ξ)is called boundary factor determining as follows

⎪

⎪

⎩

⎪

⎪

⎨

⎧

−

=

, , ,

0 1 2

1

π α

In discretization of the boundary, every element has one or a lot of nodes At node m of element e: the value of u is u m e and the value of q is q m e Shape functions Φ describe the spatial distribution on the element With M nodes in m element e, the shapes of u e (x)and )q e (x are interpolated by

∑

∑

=

=

=

=

M

m

m e m e

M

m

m e m e

x q

x q

x u

x u

1

1

) ( )

(

) ( )

(

Φ

Φ

(11)

Or in matrix notation

) ( )

(

) ( )

(

x q

x q

x u

x u

e m e

e m e

∑

∑

=

=

Φ

Φ

(12)

where u and e u are 1 x M row vectors and e Φ is an M x 1 column vector The simplest shape functions are constant and linear shape functions

In constant shape function, only one node exists per element and the values of

e

m

u and q are constant throughout the elements The constant shape function has m e

the value at the node This means that Φ x1( )=1 and

e e

e

e e

e

q x q

x q

u x u

x u

=

=

=

=

) ( )

(

) ( )

(

1 1

1 1

Φ

Φ

(13) Discretization of Eq (10) in 2-D leads to

∑∫ ∑

= +

E

e

x M

m

m e m x

E

e M

m

m e

u u

c

1 1

ξ ξ

ξ ξ

Γ Γ

Γ Φ

Γ

( )

( )

) )

, ( (

) )

, ( (

) ( )

= +

ε ε

ξ ξ

ξ ξ

Γ Γ

Γ Φ Γ

E

e M

m

e m x

m E

e M

m

e

u u

c

1 1

1 1

(15)

For ξ∈Γ

If ξ∉ ,Γ∈Ω For ξ∉ ,Γ∉Ω

If constant elements are used, the node is usually located in the middle of the constant element However, α =π and Eq (11) lead to

2

1 2

=

π

α

ξ) (

0

=

− n

x ).

The collocation method allows us to calculate the unknown boundary data from Eq (14) The simplest approach is to establish a system of equations with many variable unknowns of equations The principle of collocation is to locate the load point sequentially at all nodes of the discretization such as the domain change at the load point u(ξ) that coincides with the nodal value Because linear and higher order polynomial shape functions cause nodes to belong to more than one element, it is to introduce a global node numbering (n=1, ,N) which does not depend on the element {[1] [6]}and it can be written as follows

Gq

f

Equation (20) can be solved and all the boundary values will be known It is possible to calculate internal values of u or its derivatives The values of u is calculated at any internal point i by formulation (6) written in condensed form as:

∫

=

Γ Γ

Γ

Γ uq d d

qu

The values of the internal fluxes in the two Cartesian directions, q and x q are y

calculated by derivatives on Eq (21):

Γ Γ

Γ Γ

d x

q u d x

u q x

u q

i i

∂

∂

=

⎟

⎠

⎞

⎜

⎝

⎛

∂

∂

=

∗

∗

(21)

Γ Γ

Γ Γ

d y

q u d y

u q y

u q

i i

∂

∂

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

=

∗

∗

(22)

3 Example for heat transfer in a plate problem

Laplace's equation of heat transfer considered in a 2-D rectangular domain is illustrated by Figure 3

Figure 3: Element, nodes and double nodes indices There are eight equal elements used implying eight nodes for the temperature and sixteen for the interpolation of the flux according to the double nodes concept

First, the boundary condition is:

50 2

50 0

50 100 2

50 0

=

−

=

+

=

=

) , (

) , (

) , (

) , (

y q

y q

x x

u

x x

u

(23)

Result of problem (BEM)

(4a) (4b)

Figure 4a: Temperature field 4b: Temperature and flux fields

In figure 4a, we have heat transfer in plate that is linear as the function y=ax+b This problem is compatible with the above boundary condition This is reliable In figure 4b, we have temperature and flux fields which are compatible with above theory and problem conditions

4 Inverse problems to detect discontinuous geometry

To determine that material has a crack We practice to measure total energy of

a lot of arbitrary points in plate, with and without crack

Figure 5a : Non-crack Figure 5b: Crack

12

1

≠

−

=∑U non crack U crack

problem is crack => The material has the discontinuous geometry

Considered the above problem (figure 6a, b, 7) to determine how material have location, length and angle of a crack , we interpolate all energy of points inside area which depends on subdivision of mesh (or mesh size) for plate having crack

or without crack Consider heat energy transfer in the plate which has a crack figure 7

Figure 6a Figure 6b

Figure 7 Energy loss at four points in the above circles is lowest in comparison with heat energy transfer in the plate without crack (with a few conditions that one point or two points have the same energy of problem with no crack), basing on above theory, we determine the location of crack as follows:

Ü Coordinate of X1, Y1

4 3 2 1

4

4 3 2 1

1

1

y y y y Y

x x x x X

+ + +

=

+ + +

=

(24)

If coordinate of one crack tip in the plate lies on boundary as Figure 8a, b, we will determine coordinate of X1b and Y1b as follows

Figure 8a Figure 8b

2

4 1

4 1 2

4 1

1

1

y y Y

x x x x X

b

b

+

=

=

=

+

=

(25)

Ü Coordinate of X2, Y2

4

4 3 2 1

4

4 3 2 1

2

2

' ' ' '

' ' ' '

y y y y Y

x x x x X

+ + +

=

+ + +

=

(26)

If coordinate of one crack tip in the plate lies on boundary as figure 9a, b, we will determine coordinate of X2b and Y2b as follows

Figure 9a Figure 9b

2

3 2

3 2 2

3 2

2

2

' '

' ' ' '

y y Y

x x x x X

b

b

+

=

=

=

+

=

(27)

Ü We will specify angle α of the crack as follows:

2 1

2 1

X X

Y Y

−

−

= arctan

Ü We will specify length L of the crack as follows:

2 2 1 2 2

X

The convergence of problem depends on mesh Solution of problem step by step reaches an exact solution If mesh is small, then convergence of problem is very good Inversely, if mesh is coarse the convergence of problem is not good It will be presented by the following numerical results and error of problem

Diagram Algorithm of Inverse problems to Detect Discontinuous Geometry

Using BEM (Heat Transfer)

5 Numerical results

We consider problem (Figure 10) as follow:

Figure 10 Figure 11

Condition boundary for this problem:

When problem has no crack, we use boundary conditions as follows

50 2

50 0

50 100 2

50 0

=

−

=

+

=

=

) , (

) , (

) , (

) , (

y q

y q

x x

u

x x

u

(30)

When problem has a crack, we use boundary conditions as follows

Part1: Boundary conditions are the same as the above case (30) After that, temperature field is solved; we have temperature values at 1, 2, 3, 4, 5, 6 points

Part2: with temperature points in part1 (1, 2, 3, 4, 5, 6), they will create boundary conditions of above boundary of part2 (as figure 11) The remain of boundary conditions is as follows

50 ) , 2 (

50 ) , 0 (

2 / 50 ) 0 , (

=

−

=

=

y q

y q

x x

u

(31)

Step 1: To determine that material has a crack

Points Energy (ui)

Non_crack

Energy (u i )

1 29.7848 20.4560 9.3288

2 44.9630 39.1256 5.8374

3 55.0354 50.9924 4.0430

4 69.9831 68.2367 1.7464

5 89.9557 83.5645 6.3912

7 149.8768 149.8768 0.0000

10 94.8865 94.8865 0.0000

11 82.9768 82.9768 0.0000

12 54.8296 54.8296 0.0000 Total 1022.3012 985,8800 36.4212

We see that ∑ΔU =36.4212≠0 so we can affirm the above problem has a crack => The material has the discontinuous geometry

Step 2: To determine location, length and angle of the crack

7.30000

7.40000

7.50000

7.60000

7.70000

7.80000

7.90000

iterative steps

8.60000 8.80000 9.00000 9.20000 9.40000 9.60000 9.80000 10.00000 10.20000

iterative steps

Figure 12a Figure12b

Figure 12a: Illustrative Diagram for Convergence of Coordinate X (Point A)

Figure 12b: Illustrative Diagram for Convergence of Coordinate Y (Point A)

11.90000

12.00000

12.10000

12.20000

12.30000

12.40000

12.50000

12.60000

iterative steps

10.60000 10.70000 10.80000 10.90000 11.00000 11.10000 11.20000 11.30000 11.40000

iterative steps

Figure 13a Figure 13b

Figure 13a: Illustrative Diagram for Convergence of Coordinate X (Point B)

In figure 12a,b, Convergence of coordinate X,Y (point A) and figure 13a,b, convergence of coordinate X,Y (point B) step by step reach an exact solution through iterative steps (such as meshes 12x12, 15x15, 19x19, 21x21) We see that convergence of problem is good because through 4 times iterative steps we attain error 10-5 in comparison with supposition of problem Therefore, this result is reliable and is compatible with above theory

Table: To determine the angle and length of slopping crack

7.50000 12.50000

12x12

7.83333 12.16667

15x15

7.86842 12.14912

19x19

7.86591 12.14996

21x21

6 Conclusions

This work presented a special application about using the computing principle

of inverse problem by means of heat transfer and the numerical tool used in the Boundary Element Method The aim of this method is used to detect cracks in two-dimensional solid problem Numerical result of this paper is compared with the experimental method in Reference [11] It proved that presented method gains the fast convergence rate with small error (10-5)

The computing technique of inverse problem can extend to problems such as soil mechanics, medical field, solid mechanics…

Results of above perspectives will be performed in further papers

References

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