1 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 usin g 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. Vietnam National Commission On Large Dams www.vncold.vn 2 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 0 2 =∇ u is as follows ∫ =∇ Ω Ω 0 2 du ω 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 wdu ii ∫ , = ΩΓ ΩΓ d xx wu wdnu ii ii ∫ ∫ ∂∂ ∂∂ −, Vietnam National Commission On Large Dams www.vncold.vn 3 = ∫∫ +− ΩΓ udwdnuwwu iiiii ,,, (2) The fundamental solution of the Green's function for the unbounded space is obtained by solving the differential equation )()( 2 ξδ −−=∇= ∗∗∗ xuuL (3) 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: ),( ξ xu ∗ = rx lnln π ξ π 2 1 2 1 − =−− (4) ),( ξ xq n u ∗ ∗ = ∂ ∂ = nx x n r r . ξ ξπ π − − −= ∂ ∂ − 2 2 1 2 1 (5) with the common shortening r of the Euclidean distance 2 )( ii xx ξξ −=− .The functions ∗ u and ∗ q are presented as single layer and double layer potentials, respectively. After selecting ∗ = uw , Eq. (2) and 0 2 =∇ u associated with considering the property of the Dirac distribution, they lead to ΩΓ ∈ −= ∗∗ ∫ ξξξξ ,))(),()(),(()( x dxuxqxqxuu (6) Where n xu xq ∂ ∂ = )( )( (7) The common notations x for the field point (marked by the vector x ) and ξ for the load point (marked by the vector ξ ) have been used. The definition (7) of q deviates from the physical definition of the heat flux vector n xu kq w ∂ ∂ −= )( (8) 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 Vietnam National Commission On Large Dams www.vncold.vn 4 ΓΓ ΓΓ dxqxudxuxqu c )(),()(),()()( )( ξξξ π α ξ ∫∫ ∗∗ =+− 43421 2 1 (9) The factor )( ξ c is called boundary factor determining as follows ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − = , , , 0 1 2 1 π α c (10) 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 e m u and the value of q is e m q . Shape functions m Φ describe the spatial distribution on the element. With M nodes in element e, the shapes of )(xu e and )(xq e are interpolated by ∑ ∑ = = = = M m m e m e M m m e m e xqxq xuxu 1 1 )()( )()( Φ Φ (11) Or in matrix notation )()( )()( xqxq xuxu e m e e m e ∑ ∑ = = Φ Φ (12) where e u and e u are 1 x M row vectors and 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 e m q are constant throughout the elements. The constant shape function has the value at the node. This means that 1)( 1 =Φ x and e ee e ee qxqxq uxuxu = = == )()( )()( 11 11 Φ Φ (13) Discretization of Eq. (10) in 2-D leads to ∑ ∫ ∑∑ ∫ ∑ ∗∗ =+ E e x M m m e mx E e M m m e m dxuqdxquuc 1111 εε ξξξξ ΓΓ ΓΦΓΦ ),()(),()()()( (14) )),(()),(()()( ∫ ∫ ∗∗ =+ εε ξξξξ ΓΓ ΓΦΓΦ xm E e M m e mxm E e M m e m dxuqdxquuc 1111 (15) For Γ ∈ ξ If Ω∈ Γ ∉ , ξ For Ω∉ Γ ∉ , ξ Vietnam National Commission On Large Dams www.vncold.vn 5 If constant elements are used, the node is usually located in the middle of the constant element. However, π α = and Eq. (11) lead to 2 1 2 1 = −= π α ξ )(c (16) 0=− nx ).( ξ (17) 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 )( ξ u 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 ( Nn , ,1= ) which does not depend on the element {[1] [6]} and it can be written as follows GqHu = (18) fAx = (19) 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: ∫ ∫ ∗∗ −= ΓΓ ΓΓ duqdquu i (20) The values of the internal fluxes in the two Cartesian directions, x q and y q are calculated by derivatives on Eq. (21): ΓΓ ΓΓ d x q ud x u q x u q i i x ∫ ∫ ∂ ∂ − ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ∗∗ (21) ΓΓ ΓΓ d y q ud y u q y u q i i y ∫ ∫ ∂ ∂ − ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ∗∗ (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 Vietnam National Commission On Large Dams www.vncold.vn 6 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: 502 500 501002 500 = −= += = ),( ),( ),( ),( yq yq xxu xxu (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. Vietnam National Commission On Large Dams www.vncold.vn 7 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 If we have that 0 12 1 ≠−= ∑ crackcracknon UUU _ Δ , this can affirm that above 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 X 1 , Y 1 Vietnam National Commission On Large Dams www.vncold.vn 8 4 4321 4 4321 1 1 yyyy Y xxxx X +++ = +++ = (24) If coordinate of one crack tip in the plate lies on boundary as Figure 8a, b, we will determine coordinate of b X 1 and b Y 1 as follows Figure 8a Figure 8b 2 41 41 2 41 1 1 yy Y xx xx X b b + = == + = (25) Ü Coordinate of X 2 , Y 2 4 4321 4 4321 2 2 '''' '''' yyyy Y xxxx X + ++ = +++ = (26) If coordinate of one crack tip in the plate lies on boundary as figure 9a, b, we will determine coordinate of b X 2 and b Y 2 as follows Figure 9a Figure 9b 2 32 32 2 32 2 2 '' '' '' yy Y xx xx X b b + = == + = (27) Ü We will specify angle α of the crack as follows: Vietnam National Commission On Large Dams www.vncold.vn 9 21 21 XX YY − − = arctan α (28) Ü We will specify length L of the crack as follows: 2 21 2 21 YYXXL − +−= (29) 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) Vietnam National Commission On Large Dams www.vncold.vn 10 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 502 500 501002 500 = −= += = ),( ),( ),( ),( yq yq xxu xxu (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,( = −= = yq yq xxu (31) Step 1: To determine that material has a crack. Points Energy (u i ) Non_crack Energy (u i ) crack UΔ 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 6 114.9978 105.9234 9.0744 Vietnam National Commission On Large Dams www.vncold.vn [...]... University, Belgium Nishimura N and Kobayashi S, “A Boundary Integral Equation Method for an Inverse Problem related to Crack Detection”, Int J Numer Meth Engng, 1991; 32: 1371-87 Friedman A and Vogelius M, “Determining Cracks by Boundary measurements”, Indiana Univ Math J 1989; 38(3): 527-55 Mellings SC and Aliabadi MH, “Flaw Identification Using the Boundary Element method , Int J Numer Meth Engng 1995;... 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... Ltd, 2001 A.A.Becker, “The Boundary Element Method In Engineering”, Department of Mechanical Engineering, University of Nottingham, Published By McGRAW-HILL Book Company Europe, 1992 Costas.Pozrikidis, “A Practical Guide to BOUNDARY ELEMENT METHODS with the software library BEMLIB”, Published By Chapman & Hall/CRC A.TARATOLA, Inverse Problem Theory”, Published By Elsevier, England 1987 I take it at library... Laurence WEGRIA, “Thesis master for Detection de Discontinuities Geometriques par une Methode Inverse University of Liege, Belgium, 1995-1996 Keith A Woodbury, Inverse Engineering Handbook”, University of Alabama, Published By CRC Press LLC 2003 Gernot Beer “Programming the Boundary Element Method an Introduction for Engineers”, Institute for Structural Analysis, University of Technology Graz, Austria,... 1995; 38: 399-419 R Lazarovitch, D Rittel* and I Bucher, “Experimental Crack Identification Using Electrical Impedance Tomography”, Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Technion City, 32000 Haifa, Israel NDT&E International 35(2002)301316 G.Beer and J.O.Watson, “Introduction to Finite and Boundary Element Methods for Engineers”, Published By John Wiley & Sons Ltd,... 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 12 Vietnam National Commission On Large Dams www.vncold.vn References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] NGUYEN DANG HUNG, “Cours de Introduction a la Method des Element aux Frontieres”,... Coordinate X (Point B) Figure 13b: Illustrative Diagram for Convergence of Coordinate Y (Point B) 11 Vietnam National Commission On Large Dams www.vncold.vn 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. .. 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 Mesh 12x12 15x15 19x19 21x21 6 Point A Point B 7.50000 9.16667 7.83333 9.94444 7.86842 9.97368 7.86591 9.97201 12.50000 10.83333... 82.9768 54.8296 985,8800 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 36.4212 ΔU = 36.4212 ≠ 0 so we can affirm the above problem has a We see that crack => The material has the discontinuous geometry Step 2: To determine location, length and angle of the crack 7.90000 10.20000 10.00000 7.80000 coordinate y coordinate x 9.80000 7.70000 7.60000 7.50000 9.60000 9.40000 9.20000 9.00000 7.40000 8.80000 8.60000... 1 4 2 3 4 iterative steps 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) 12.60000 11.40000 12.50000 11.30000 11.20000 co o rd in ate y coo rdinate x 12.40000 12.30000 12.20000 12.10000 11.10000 11.00000 10.90000 10.80000 12.00000 10.70000 10.60000 11.90000 1 2 . An inverse problem in linear crack media under heat transfer usin g boundary element method Toan Cao-Duc* and Hung Nguyen-Dang ++ * European Master in Engineering Science of Mechanics of. 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. 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