IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
Crack shape sensitivity by the adjoint variable method using a boundary-only formula G. Rus Carlborg1 Department of Structural Mechanics, University of Granada, Spain M. Bonnet Laboratoire de M´ecanique des Solides (UMR CNRS 7649), Ecole Polytechnique, 91128 Palaiseau cedex, France R. Gallego Department of Structural Mechanics, University of Granada, Spain Abstract. This communication presents, in the framework of two-dimensional linear elastodynamics in the frequency domain, a method for evaluating crack shape sensitivities of integral functionals, based on the adjoint state approach and resulting in a sensitivity formula expressed in terms of integrals on the crack curve and crack tip contributions involving the direct and adjoint stress intensity factors. This method is well-suited to boundary element treatments of e.g. crack reconstruction inverse problems. Crack shape sensitivity values obtained by this method are seen to compare well with finite difference estimations. An example of crack identification from synthetic data is also presented.
1
Introduction
The need to compute the sensitivity of integral functionals with respect to shape parameters arises in many situations where a geometrical domain plays a primary role; shape optimization and inverse problems are the most obvious, as well as possibly the most important, of such instances. In addition to numerical differentiation techniques, shape sensitivity evaluation can be based on either direct differentiation or the adjoint variable approach. this communication being focused on the latter. This communication, focused on the latter approach, aims at presenting and demonstrating the implementation of a method for evaluating crack shape sensitivities of integral functionals, in the framework of two-dimensional linear elastodynamics in the frequency domain. This method is based on the adjoint state approach and results in a sensitivity formula expressed in terms of integrals on the crack and contributions from the crack tips involving the direct and adjoint stress intensity factors. This sensitivity formula is thus well-suited to the use of boundary element methods (BEMs), which are quite often used for solving e.g. crack reconstruction inverse problems, see e.g. Mellings and Aliabadi (1995); Nishimura (1995). Consider an elastic body Ω ∈ R2 of finite extension, externally bounded by the closed curve S and containing a crack Γ. Temporarily considering the more general time-domain framework, the displacement u, strain ε and stress σ are related by the field equations: div σ + ρω 2 u = 0 1
σ = C :ε
1 ε = (∇u + ∇T u) 2
For questions please email:
[email protected]
1
in Ω
(1)
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
(C: fourth-order elasticity tensor). Besides, displacements and tractions are prescribed on the portions Su and Sp = S \ Su of S, the crack surface Γ is stress-free and initial rest is assumed: ¯ (on Su ), u=u
¯ (on Sp ), p=p
u = u˙ = 0 (in Ω, at t = 0) (2)
p = 0 (on Γ),
where p ≡ σ.n is the traction vector, defined in terms of the outward unit normal n to Ω. The above conditions define the direct problem. Let us introduce the following generic objective function: Z TZ Z TZ Z J (Γ) = J(uΓ , pΓ , Γ) = ϕu (uΓ , x, t) dS dt + ϕp (pΓ , x, t) dS dt + ψ(x) dS (3) 0
Sp
0
Su
Γ
which is encountered for instance in minimization-based algorithms for solving the inverse problem of crack detection ((uΓ , pΓ ) refer to the solution of problem (1, 2) for a given crack configuration). A boundary-only expression for the derivative of J (Γ) with respect to crack perturbations is sought.
2 Adjoint variable method Our problem is to express the derivative of the derivative of a cost functional with respect to some geometrical parameters in an effective way. A detailed description of the full mathematical process for a continuous problem is given in Bonnet (1999). Any sufficiently small perturbation of Γ can be described by means of a domain transformation which does not affect the external boundary S, i.e. of the form xη = x + ηθ(x) where η is a time-like parameter and the transformation velocity field θ(x) is such that θ = 0 on S. Denoting ?
by f = f,η + ∇f.θ the Lagrangian derivative of some field variable f , the derivatives at η = 0 of integrals over generic domains V ⊂ R2 and curves Σ ⊂ R2 have the well-known form: Z Z ? Z Z ? d d d f dV = (f +f div θ) dV f ds = (f +f θ) ds (4) dη V dη Σ ds V Σ ?
Also, recall that (∇u)? = ∇ u −∇u.∇θ. Adjoint problem. Following Bonnet (1999), we define the following Lagrangian, involving the test functions u0 , p0 which act as Lagrange multipliers. Z TZ L=J +A=J + [σij u0i,j + ρ¨ ui u0i ] dΩ dt 0 Ω Z TZ Z TZ Z TZ 0 0 − (ui − u¯i )pi ds dt − pi ui ds dt − p¯i u0i ds dt (5) 0
Su
0
Su
0
Γq
The adjoint problem is then found to be defined now by the same governing equations 1 and the boundary and initial conditions: p0 = −
∂ϕp ∂ϕu (on Sp ) u0 = (on Su ) p0 = 0 (on Γ± ) ∂u ∂p u0 = u˙ 0 = 0 (in Ω, at t = T ) 2
(6)
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
The total derivative of L with respect to a domain perturbation is deduced in Bonnet (1999), Z T Z δL = (θk nk ) [[σij u0i,j − ρu˙ i u˙0 i ]] dt ds Γ+ c
0
Z T 1 0 − (θk νk ) {(1 − ν)[KI KI0 + KII KII0 ] + KIII KIII } dt ds µ ∂Γ 0 Z Z T 1−ν − (θk Nk ) [KI KII0 + KII KI0 ] dt ds µ ∂Γc 0 Z + [ψi θi + ψ(θi,i − ni θi,j nj )] ds Z
(7)
Γ
where ∂Γ is the crack tip, [[f ]] = f (x+ ) − f (x− ) (discontinuity of f across Γ), n is the normal on Γ, ν is the unit tangent to Γ at the crack tip pointing outside of Γ (see figure 1). N n=n+
ν tip
Γc tip
Figure 1: Some definitions of the geometry
Reduction to bidimensional cracks in frequency domain. The problem is going to be solved for an harmonic load at a single frequency ω. The static special case will be obtained immediately by obviating any complex components and the leading Πω factor. The values of displacements, stress vectors and stress intensity factors will therefore be represented with complex numbers, so that u(x, t) = Re [u(x)eiωt ], K(t) = Re [Keiωt ] and so on. Using standard arguments, one thus finds that Z T π cijkl ui,j (x, t)u0k,l (x, t)dt = Re cijkl ui,j u¯0k,l ω 0 Z T π ρu˙ i (x, t)u˙0 i (x, t)dt = Re ρω 2 ui u¯0i ω 0 Z T π ¯ n0 Kn (t)Kn0 (t)dt = Re Kn K ω 0 and we reach the expression, Z π h δL = Re (θk nk )[[cijkl ui,j u¯0k,l − ρω 2 ui u¯0i ]] ds ω Γc oi Xn 1 1−ν 0 0 0 0 0 ¯ ¯ ¯ ¯ ¯ − (θk νk ) (1 − ν)[KI KI + KII KII ] + KIII KIII + (θk Nk )[KI KII + KII KI ] µ µ tips Z + [ψi θi + ψ(θi,i − ni θi,j nj )] ds (8) Γ
3
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
where a ¯ stands for the conjugate of the complex value a. Here, we consider the cost function (3) defined by ϕu =
1 |u − uexp |2 2
(on Spexp ),
ϕp =
1 |p − pexp |2 2
(on Suexp )
(9)
where Spexp ⊆ Sp and Suexp ⊆ Su are the measurement subsets, so that the boundary conditions (6) defining the adjoint problem become: p0 = −(u − uexp ) (on Spexp )
p0 = 0 (on Sp \ Spexp )
u0 = −(p − pexp ) (on Suexp ) u0 = 0 (on Su \ Suexp ) p0 = 0 (on Γ± )
(10)
The term involving the gradient of the displacements can be calculated in terms of boundary values only with the help of the expression (11), derived below. ∇s means the surface gradient, which is the projection of the gradient onto the surface. divs is also the projection of the divergence on the surface. n 2ν o 1 σ : ∇¯ u0 = µ divs udivs u¯0 + (∇S u + ∇TS u) : (∇S u¯0 + ∇TS u¯0 ) − (n∇S u)(n∇S u¯0 ) 1−ν 2 divS u = ut,t 0 0 ∇S u = un,t ut,t 1 ut,t = ui,ξ ti g n 2ν o 1 2µ 0 0 0 0 0 σij u¯i,j = µ ut,t u¯t,t + (2un,t u¯n,t + 2ut,t u¯t,t ) − un,t u¯n,t = 2 ui,ξ ti u¯0j,ξ tj (11) 1−ν 2 g (1 − ν) where g is the jacobian of the boundary element mapping ξ → x(ξ)
3 Numerical comparisons: methodology The remainder of this communication is devoted to the study of the reliability of the gradient of the cost function J with respect to the geometrical parameters. The influence of several factors will be considered: discretization, frequency, size of experimental data set, chioce of crack parametrization. Boundary element calculations. The calculation of the direct problems that arise in both methods is made with bidimensional quadratic boundary elements. The original code, that has been modified conveniently for this study, was developed by F. Chirino Godoy (1998). The discretization is made with interior collocation points placed at 0.8L/2 from the center for all the elements, in our examples. The crack is represented by the dual formulation or mixed boundary element method, a combination of equations corresponding to the integral representations of displacements and tractions - one for each lip. The crack tip is modeled by a quarter point straight element, and the stress intensity factors are computed upon the crack opening displacement measured at 4
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
the quarter node of the tip element. Finally, the singular integrals are evaluated by dividing them into an analytically solved part, which only involves static terms, and a regular part solved with constant standard 10-point Gauss quadrature. The discretization of the crack, in opposition to the uniform mesh on the boundary, is graded −1 (sχ) so as to concentrate the elements towards the tips according to the rule ξ = tan , where ξ gives tan−1 (s) the non-dimensional coordinates of the nodes when χ is divided uniformly. Both coordinates are understood between −1 and 1. Although any value between s = 0.0 (uniform mesh) and s = 3.0 gives reasonable results, our choice for a good improvement was s = 2.0. In order to later integrate the expressions along the crack, we should ensure that the data at the crack tips are exactly of the right √ order of singularity, eliminating numerical alterations of it. This can be done by obliging the r terms to be identical in the upper and lower lip of the tip. If we represent the behavior of the data as, √ √ f + − f0 = α+ r + β + rf − − f0 = α− r + β − r +
−
we will force them to behave as α = α +α by a slight and convenient modification of value at the 2 extreme crack tip node. Apart from the boundary element software written in fortran 77, which has eventually been used as a black box, we have extensively checked the subroutines, written in both fortran 77 and 90, for the computation of the cost functional and all the formulae for the adjoint variable method. All the calculations are made with double precision. Definition of the examples. The implemented sensitivity analysis is tested here on two models. Model 1 is based on a circular body of unity radius R = 1.0 with an embedded crack. The material constants are, in consistent units: Young modulus E = 1.0, Poisson coefficient ν = 0.2 and mass density ρ = 1.0. The straight crack of half-length a = 0.3 is placed as shown in figure 2, with its center at the coordinates (0.3, 0.2). Part of the boundary (an arc of adjustable length) is constrained. The non-constrained arc Sp is loaded by an unit normal traction. The measurements are the displacements on an arc Spexp ⊆ Sp of adjustable length, corresponding to the same problem but with the crack center lying at the coordinates (0.1, 0.1). p=1.0
a=0.3
p=1.0 crack center: (0.3,0.2)
measurement of the displacements in X and Y directions
2.0
crack center: (0.3,0.2)
a=0.3
constrained boundary
Stress p=1.0
p=1.0
Figure 2: Geometry definitions 5
measurement of the displacements in X and Y directions
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
Model 2 consists of a square (side length 2 units) with an embedded crack in the same location as in model 1. The measured displacement is defined in the same way as before, on a fraction of the upper side, and the boundary conditions are defined as shown in figure 2. These definitions ensure an unsymmetric solution with contribution of both fracture modes I and II. Method of comparison. Since the exact value of the sensitivity is not known, the results of this formulation must be compared with another approximate one, namely the finite central difference estimator. The crack configuration is defined by a set of N parameters: Γ = Γ(η) with η = {η1 , . . . , ηN }. Conventionally, the derivatives are formulated at η = 0. 1. Sensitivity by finite central differences (FD): J (Γ(η g )) − J (Γ(0)) J (Γ(η g )) − J (Γ(−η g )) d J (Γ(η)) = lim ' ηg →0 dηg ηg 2ηg
(1 ≤ g ≤ N ) (12)
where J is of course defined by (3) and (9) and η g = {0, . . . , 0, ηg , 0, . . . , 0}. 2. Sensitivity by the adjoint variable method (AVM): d J (Γ(η)) using equation (8). dηg The step size used for the finite difference calculations is chosen so as to minimize the sum of the numerical error (low step values) and the nonlinearity of the cost functional (high step values). This resulted in a choice of ηg = 10−6 in (12), as figure 3 suggests (calculated with 36 + 12 + 12 elements, crack + each lip).
gradient value
27.15
27.1
Simple finite differences Centered finite differences 27.05
−8
10
−6
10 finite difference distance
−4
10
Figure 3: Effect of distance finite differences calculation.
4
Numerical comparisons: results and discussion
Dependence on the discretization. In order to estimate the global errors and to assign the right degree of importance to the mesh. On the pairs of figures 4, 5 and 6, the horizontal axis represents 6
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
boundary meshes
boundary meshes
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Figure 4: Circular model with no displacement prescriptions. Measurements over the whole circle. Sensitivity to horizontal displacement of the crack. (o) AVM calculation, (x) FD calculation. the number of elements on the external boundary (sweeping from 12 to 192), and each curve has a fixed number of elements in each lip of the crack (ranging from 4 to 64). As seen on the value (left figures) and the estimation of the error (left), the values of the gradient or sensitivity appears to converge according to a bilogaritmic law, as expected. The estimate of the final value has been made from a bilogaritmic extrapolation of the values of the more densely meshed models. The estimated limits for the three examples were 8.76, 4.46 and 1.238 respectively. The extrapolation of the AVM calculation and the finite differences were in reasonable agreement. Regarding the contour and crack meshes, note that the increase in one mesh independently of the other provokes a blockage of the convergence; i.e. both meshes have to be improved to reach the real value. The most interesting fact that these graphics show is that both the AVM and the finite differboundary meshes
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Figure 5: Circular model with no boundary prescriptions. Measurements over a third of the circle. Sensitivity to horizontal displacement of the crack. (o) AVM calculation, (x) FD calculation.
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IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
boundary meshes
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Figure 6: Circular model with no boundary prescriptions. Measurements over the whole circle. Sensitivity to vertical displacement of crack. (o) AVM calculation, (x) FD calculation. ence converge to some value, which moreover appears to be the same. An explanation to why the methods converge with different patterns could be that the AVM is not consistent in the sense that it gives the sensitivity of the continuous problem, not the discretized one, as the finite differences do. The reason is that the AVM equations are valid for a continuous before any discretization. For solving them, we discretize the direct problem, the adjoint problem, and the integration of the AVM formula, introducing errors in all of them. Although the values obtained converge to the gradient of the continuous, they do not coincide with any value. The finite differences do express directly the gradient of the eventually discretized problem. Dependence on the frequency. We now analyze how the frequency of the excitation affects the computation of the gradient, by plotting the agreement between FD and AVM gradient for each circular model 0
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Figure 7: Circular model with no displacement prescriptions. Sensitivity to horizontal displacement of the crack. 8
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
circular model 0
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−51
10
−52
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0
Figure 8: Circular model with a third of the circle constrained and measurements on another third. Top: sensitivity to vertical displacement of crack. Bottom: determinant of the BEM system matrix to identify the eigenfrequencies. frequency. Figures 8, 7 and 9 show the relative gradient error versus the frequency, representing each curve a different discretization in terms of a gradually growing number of elements for the circle and each lip of the crack. The errors are defined as the relative difference between the AVM calculation (avm) and the finite difference (f d) as, |avm − f d| error = q avm2 +f d2 2
The main point to notice is that the error, where the results make sense (outside the high error peaks), regularly decays in accordance with the enhacement of the mesh. The first problem one notes regards the lach of convergence at low frequencies. This is probably due to a bad computation of the FD gradient, since the numerical errors due to the logaritmic dependence of the fundamental solution to the frequency, become too big in comparison with the epsilon chosen for the FD computation. The same logarithmic effect may affect the AVM computation. At certain frequencies, all graphics show a number of peaks in the errors showing invalid solutions. The explanation could be the presence of eigenfrequencies at these points. To support 9
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
square model 0
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Figure 9: Square-shaped model with full measurements. Sensitivity to vertical displacement of crack. this hypothesis we have calculated for the problem in figure 8 the first eigenfrequencies by finite elements (Abaqus, table on the right) and we have plotted the determinant of the system matrix of the boundary element problem for a 24 + 8 + 8 discretization (figure 8), and the same problem by the finite element method, which gives the data in the following table. These data show an agreement in the values of the frequencies of minimum determinant, eigenfrequencies and peaks in the sensitivity errors. The following table confirms the values of the eigenfrequencies computed by the finite element method. eigenfrequencies 0.491
0.909
1.149
1.676
circular model
0
2.385
2.597
2.644
10
−1
−1
10
10
−2
relative error
relative error
1.962
square model
0
10
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−3
10
horizontal displacement vertical displacement rotation dilatation test parameter 1 test parameter 2
−4
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1.919
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−3
10
horizontal displacement vertical displacement rotation dilatation test parameter 1 test parameter 2
−4
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−5
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3
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1
10
2
10 number of elements
3
10
Figure 10: Evolution of relative error with the mesh improvement: circular model (left) and square model (right). 10
IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
Comparison of different parametrizations. The sensitivity has been calculated for the four parameters that keep the crack straight (horizontal and vertical translation, rotation about center, dilatation), as well as two other perturbations, defined such that: θ 1 (x1 , x2 ) = (x − xc )e1 + 0.03 + 0.4(x − xc ) − 3(x − xc )2 − 100(x − xc )3 e2 (13) θ 2 (x1 , x2 ) = 0.02 + 0.1(x − xc ) − 3(x − xc )2 e2 (where xc is the abscissa of the crack center). Of course, the latter perturbationstransform the straight initial crack into a curved crack. The relative errors versus mesh size, shown in figure 10 for a fixed frequency of ω = 1.5 and for all six parameters mentioned here, for both the circular and square model, appear to decrease regularly as the mesh is refined.
5 Crack identification Some identification results are summarized in figures 11 (for ω = 0 rd/s, i.e. the static case) and 12 (for ω = 1.5 rd/s), where the number of iterations to convergence (top) and the geometrical fitness (defined as the sum of the squares of the relative differences between six scalar parameters: length, center coordinates and inertia tensor coefficients) between identified crack and real crack (bottom) are shown against the distance of the initial guess center to the true crack center. No simulated measurement errors were introduced, and the straight crack was described via 4 geometrical parameters (center coordinates, angle and half-length).
6 Conclusion A method for the calculation of crack shape sensitivity with a boundary-only formmula based on an adjoint problem has been presented. The main benefit of this approach is its efficiency: once the forward problem is solved, obtaining the adjoint solution entails only setting up a new righthand side and solving a matrix equation which is already in triangular form and thus requires only a small fraction of the computer time used for obtaining the forward solution. Crack shape sensitivity values obtained by this method were seen to compare well with finite difference estimations, and an example of crack identification from synthetic data was also presented.. However, there is certainly scope for improving the accuracy of the sensitivity calculation. On one hand, it is very sensitive to the accuracy of the computation of the stress intensity factors. On the other hand, the adjoint problem is defined for the cost function J associated with the solution of the continuous forward problem, i.e. is not exactly the adjoint problem corresponding to the discretized forward problem, and this slight inconsistency might also play a role in the resulting sensitivity accuracy.
References Bonnet, M., 1999, “A general boundary-only formula for crack shape sensitivity of integral functionals.” C.R. Acad. Sci. Paris, s´erie II, vol. 327, pp. 1215–1221. F. Chirino Godoy, R. A. G., 1998, “C´alculo de factores de intensidad de tensi´on est´aticos y din´amicos mediante el m´etodo de los elementos de contorno con formulaci´on hipersingular,” M´etodos num´ericos para c´alculo y dise˜no en ingenier´ıa, vol. 14 (3), pp. 339–364.
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IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002
Mellings, S. C. and Aliabadi, M. H., 1995, “Flaw identification using the boundary element method,” Int. J. Num. Meth. in Eng., vol. 38, pp. 399–419. Nishimura, N., 1995, “Application of boundary integral equation method to various crack determination problems.” Dynamic fracture mechanics, ed. M. Aliabadi, chap. 7, Comp. Mech. Publ., Southampton. 2
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Figure 11: Crack identification: number of iterations to convergence (top) and geometrical fitness between identified crack and real crack (bottom), for various values of the distance of the initial guess center to the true crack center (ω = 0 rd/s, i.e. static case). 2
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Figure 12: Crack identification: number of iterations to convergence (top) and geometrical fitness between identified crack and real crack (bottom), for various values of the distance of the initial guess center to the true crack center (ω = 1.5 rd/s).
12
