CRACK DETECTION IN A THREE DIMENSIONAL BODY. A Thesis by. Steven F. Redpath. BS, Kansas University, May Submitted to the

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1 CRACK DETECTION IN A THREE DIMENSIONAL BODY A Thesis by Steven F. Redpath BS, Kansas University, May 1986 Submitted to the Department of Mathematics and Statistics, College of Liberal Arts and Sciences and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Master of Science May 2006

2 CRACK DETECTION IN A THREE DIMENSIONAL BODY I have examined the final copy of this Thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics. Victor Isakov, Committee Chair We have read this Dissertation and recommend its acceptance: Alan Elcrat, Committee Member Kamran Rokhsaz, Committee Member ii

3 ABSTRACT We propose a method of analyzing a crack in a three dimensional body. We treat the problem as an inverse problem and apply Green s Theorem, Trace Theorem, and the Fredholm Alternative. We model the problem using Helmholtz equation. iii

4 TABLE OF CONTENTS 1 Introduction 1 2 Domains and Spaces Domains Sobolev Spaces Trace Theorem Jump Function Green s Formula Fredholm Alternative Results The Direct and Inverse Problem Weak Solution Calculations Results Bibliography 11 iv

5 CHAPTER 1 INTRODUCTION An important issue in product reliability and cost control is the identification of material defects from overdetermined data. A source of a physical field is applied to a portion of the boundary and measurements of this field are taken at this portion. The inverse problem is to detect and characterize defects such as cracks in the material. Current work in this area includes the investigation of multiple cracks, detection of a large number of small cracks, and the growth of multiple small cracks into one relatively large crack. Some work has been completed in this area. For buried cracks, Friedman and Vogelius [6] proved that two special boundary measurements are needed for unique identification. In 1993, Alessandrini, Beretta, and Vessela [1] proved Lipschitz stability for linear cracks. Andrieux and Abda [3] used a special heat flux to solve the two dimensional inverse crack problem using the Laplace equation in the inverse boundary value problem. Eller [5] models a 3D body with a 2D crack using the Laplace equation. Eller proves the crack can be identified uniquely by boundary measurements. The above methods do not work [9] for the Helmholtz equation. In this paper we will investigate a two-dimensional crack in a three-dimensional body. A body with a simply connected surface will be assumed with some defect (crack) starting at the surface and terminating inside the body. The Helmholtz equation will be used in this inverse boundary value problem. By using the Helmholtz equation, the test function is not limited to static functions but can in fact be a wave function. These wave functions include (but are not limited to) acoustic and electromagnetic fields. 1

6 CHAPTER 2 DOMAINS AND SPACES 2.1 Domains Let R 3 be a simply connected bounded open domain with boundary. Define C 1 () the space of all one time continuously differentiable functions. Let C 1 be piecewise smooth with nonzero interior angles. The outward normal on will be designated ν. Let Figure 2.1: Boundary Definition C 1 [8] be a simply connected surface with no self intersections intersecting at some non-zero angle α. will represent the crack, and c will be a complement of the crack. Define = c and assume that 1 2 =. divides into two domains designated 1 and 2 so that = 1 2 and 1 2 =. The outward normal of the boundary of 1, measured along, is defined ν(1). On of 2, the outward normal is defined as ν(2). Hence along, ν(1) + ν(2) = 0. 2

7 2.2 Sobolev Spaces Let be an open set of R 3. Define H 1 () the set of all square integrable functions, together with all first derivatives in. We introduce a Hilbert norm in H s () : u H s () = D α u 2 L 2 () α s 1 2 (2.1) for all integers s, such that s 0. The space of smooth functions: u : u(1) C ( 1 ) and u(2) C ( 2 ) C (\) = γ(1) α u = γ(2) α u on c for all α The Sobolev space H 1 (\) is defined as the completion of C (\) with respect to the norm (2.1) Trace Theorem Let i have a Lipschitz boundary. Then the mapping γ : u u i which is defined for all u C ( i ) has a unique continuous extension as an operator from H 1 ( i ) onto H 1 2 ( i ). This operator has a continuous right inverse. In particular, there exists a constant C = C() such that u (0) () C u (1) (), u H 1 (). (2.2) 2.4 Jump Function Let γ denote the trace operator on. 1 has Lipschitz boundary, [8] therefore the mapping γ(1) : u u 1 for u C ( 1 ) has a unique continuous extension from H 1 ( 1 ) onto 3

8 H 1 2 ( 1 ). By similarity, γ(2) : u u 2. When the traces γ(1)u and γ(2)u are well defined, the jump across will be denoted [5] [u] = γ(1)u γ(2)u. 2.5 Green s Formula Let have Lipschitz boundary. For all u, v H 1 () we have: j uv + u j v = γuγvν j. Using this formula we can prove Green s formula. Let be a Lipschitz domain and u H 2 (), v H 1 (). Then v u + v u v = γvγ ν u. 2.6 Fredholm Alternative We use Theorem 5.3 from Gilbarg and Trudinger [7]. Consider the problem L λ (u) : u λu = f, in (2.3) ν u = g 1 on. For each λ R exactly one of the following statements holds: EITHER 1) For each f L 2 (), g 1 L 2 (), there exists a unique weak solution of the boundary value problem (2.3). This is true in particular for all λ < 0. OR 2) There exists a weak solution u 0 0 of the homogeneous problem L λ (u) : u 0 λu 0 = 0, in (2.4) 4

9 ν u 0 = 0 on. 3) A solution to (2.3) exists if and only if: fu 0 + u 0 g 1 = 0 for any u 0 solving (2.4). For example: Let λ = 0. The boundary value problem (2.3) becomes u 0 = f in, ν u = g 1 on. The Fredholm alternative states: Either there is a unique solution, or there is a weak solution to the homogeneous equation: u 0 = 0 in, ν u = g 1 on. 5

10 CHAPTER 3 RESULTS 3.1 The Direct and Inverse Problem Let us consider the following boundary value problem: Lu u + k 2 u = 0 in, (3.1) ν u = 0 on, (3.2) u = g 0 on. (3.3) Equations (3.1), (3.2), and (3.3) model a crack in the domain R 3. The inverse problem we are interested in is to find given the additional boundary data ν u = g 1 on. (3.4) 3.2 Weak Solution Define a weak solution to the boundary value problem (3.1), (3.2), and (3.4) a function u H 1 (\) satisfying: 0 = g 1 vdω ( u v k 2 uv)dx, (3.5) where v H 1 (\) is a test function. Per Ladyzhenskaya and Ural tseva [10], the definition that we have given for a generalized solution is indeed an extension of the classical concept of a solution. Rewrite equation (3.5), ( u v k 2 uv)dx = g 1 vdω. It is known ([10] p.162) that the Fredholm alternatives hold for (3.1), (3.2), and (3.4). There is a unique solution u to equation (3.5) for sufficiently large k 2 k 1,..., k m,..., where lim m k m =. k m are eigenvalues. 6

11 3.3 Calculations Now apply Green s theorem to ( u v)dx. Since Green s formula only applies to [8] bounded Lipschitz domains, use it in 1 and 2. Let v H 2 (). ( u v)dx = ν vudω + vudx (3.6) ( u v)dx = ν vudω + vudx Substitute these back into (3.5): 0 = (v ν u ν vu)dω + u( v + k 2 v)dx (v ν u ν vu)dω + u( v + k 2 v)dx. (3.7) 2 2 Expand the 1 and 2 terms. The three component parts are,, and c. On 1 and 2 we are left with: On the crack boundary, the integral becomes: ( ν(1) uγ 1 v ν(2) uγ 2 v ) dω + (g 1 v ν vu) dω. (3.8) ( ν(2) vγ 2 u ν(1) vγ 1 u ) dω. Since ν(1) u = 0 and ν(2) u = 0 we are left with: ( ν(2) vγ 2 u ν(1) vγ 1 u ) dω. (3.9) Simplifying this with the expressions [u] = γ(1)u γ(2)u, and ν(2) = ν(1) we can write this term using the jump function. [u] ν(1) vdω (3.10) The final boundary part is c and it looks like this: c ( γ2 v ν(2) u + γ 1 v ν(1) u ) ( γ 2 u ν(2) v + γ 1 u ν(1) v ) dω. (3.11) Since v H 1 (), and u H 1 (), we have the identities γ 2 v = γ 1 v, and γ 2 u = γ 1 u on c. By the definition of our trace operator, all terms in the above expression cancel out. 7

12 From equation (3.1), v + k 2 v = 0. Substituting expressions (3.8) and (3.10) back into (3.7) leaves us with: 0 = (g 1 v ν vu) dω + [u] ν vdω, (3.12) provided v + k 2 v = 0 in, and v H 2 (). 3.4 Results Let 0 be that portion of where the test function is applied and ν u is measured. Let v + k 2 v = 0 in, v H 2 (), ν v = 0 on \ 0. Equation (3.12) becomes: Let 0 =. Case 1: v = sin(kξ x) where ξ =1. [u] ν vdω = ( ν vu g 1 v)dω. 0 (3.13) We compute the following: ν v = v ν = k cos(kξ x)ξ ν Equation (3.13) becomes: k [u] cos(kξ x)ξ νdω = [k cos(kξ x)ξ νu g 1 sin(kξ x)] dω. (3.14) We see that for a plane crack, ξ ν = 0 when ξ is parallel to. (3.15) Case 2: v = cos(kξ x) where ξ =1. We compute the following: ν v = v ν = k sin(kξ x)ξ ν Equation (3.13) becomes: k [u] sin(kξ x)ξ νdω = [k cos(kξ x)ξ νu + g 1 sin(kξ x)] dω. 0 (3.16) 8

13 Case 3: k = 0 The boundary value problem, equations (3.1), (3.2), (3.4) become: u = 0, ν u = 0 on, ν u = g 1 on. Use test function v such that v = ξ where ξ = 1. Equation (3.13) becomes: [u] (ξ ν)dω = [ξu g 1 (ξ ν)] dω. (3.17) The right hand side is comprised of known functions and can be computed. Again, if is a plane crack and parallel to ξ, [ξu g 1 (ξ ν)] dω = 0. 9

14 BIBLIOGRAPHY 10

15 BIBLIOGRAPHY [1] Alessandrini, G., Beretta, E., and Vessela, S., Determining Linear Cracks by Boundary Measurements: Lipschitz Stability, Istituto per le applicazione del calcolo Moro Picone, CNR, 1993 [2] Alessandrini, G., DiBenedetto, E., Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability, Indiana University Math Journal, 46,(1997): 1 83 [3] Andrieux, S., and Ben Abda, A., Identification de fissures planes par une donnée au bord unique; un procédé direct de localisation et d identification, Comptes Rendus à l ingémie des Sciences, series 1, 315, (1992): [4] Elcrat, A., Isakov, V., Necoloiu, A., On finding a surface crack from boundary measurements, Inverse Problems, 11,(1995): [5] Eller, M., Identification of cracks in three-dimensional bodies by many boundary measurements, Inverse Problems, 12,(1996): [6] Friedman, A., Vogelius M., Determining cracks by boundary measurements, Ind. Univ, Math J., 38,(1989): [7] Gilbarg, D., Trudinger, N., Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998 [8] Grisvard, P, Elliptic Problems in Nonsmooth domains, Pitman, Boston, [9] Isakov, V, Inverse Problems for Partial Differential Equations, Springer, New York, [10] Ladyzhenskaya O. and Ural tseva N., Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968 [11] Lions, J.L., Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Springer,

Approximation of inverse boundary value problems by phase-field methods

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