Domain sensitivity analysis of the acoustic far‐field pattern

We consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd.     

Linear elliptic boundary value problems in varying domains

The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.     

The effect of high temperature rolling on structure of copper single crystals of different orientation

The work brings the data about the substructure development in single crystals of copper deformed up to about 50% of cross section reduction in rolling at high temperature. No recrystallization process was found. Instead a tendency toward inhomogeneous deformation manifesting itself in crystallographicaly oriented microbands was observed. The results are discussed in terms of the role of dynamical recovery on development of structure and its evolution in the microbands areas.The author wishes to express his thanks to Mr. M. Orkisz for his assistance in carrying out structural observations.     

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

Domain sensitivity analysis of the elastic far-field patterns in scattering from nonsmooth obstacles

We consider elastic scattering problems described by the Dirichlet or the Neumann boundary value problem for the elastodynamic equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far scattered field with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems (K. Dems, Z. Mróz, Internat. J. Solids Structures 20 (1984) 527-552) we obtain a representation for the shape derivative which is well suited for a numerical realization with boundary element methods and which shows in some cases directly the influence of the singularities of the solution on the sensitivity of the far-field patterns.     

Shape sensitivity analysis of stress intensity factors by Generalized J-integrals

We consider an elastic plate with the non-deformed shape Ω_Σ := Ω \ Σ, where Ω is a domain bounded by a smooth closed curve Γ and Σ ⊂ Ω is a curve with the end points {γ₁, γ₂}. If the force g is given on the part Γ_N of Γ, the displacement u is fixed on Γ_D := Γ \ Γ_N and the body force f is given in Ω, then the displacement vector u(x) = (u₁(x), u₂(x)) has unbounded derivatives (stress singularities) near γ_k, k = 1, 2    u(x) = ∑²_{k, l=1} K_l(γ_k)r^1/2_kS^C_kl(θ_k) + u_R(x)     near γ_k. Here (r_k, θ_k) denote local curvilinear polar co-ordinates near γ_k, k = 1, 2, S^C_kl (θ_k) are smooth functions defined on [−π, π] and u_R(x) ∈ {H²(near γ_k)}². The constants K_l(γ_k),   l = 1, 2, which are called the stress intensity factors at γ_k (abbr. SIFs), are important parameters in fracture mechanics. We notice that the stress intensity factors K_l(γ_k) (l = 1, 2;  k = 1, 2) are functionals K_l(γ_k) = K_l(γ_k; ℒ︁, Ω, Σ) depending on the load ℒ︁, the shape of the plate Ω and the shape of the crack Σ. We say that the crack Σ is safe, if K_l(γ_k; Ω)² + K₂(γ_k; Ω)² < RẼ. By a small change of Ω the shape Σ can change to a dangerous one, i.e. we have K₁(γ_k; Ω)² + K₂(γ_k; Ω)² ⩾ RẼ. Therefore it is important to know how K_l(γ_k) depends on the shape of Ω. For this reason, we calculate the Gâteaux derivative of K_l(γ_k) under a class of domain perturbations which includes the approximation of domains by polygonal domains and the Hadamard's parametrization Γ(τ) := {x + τϕ(x)n(x);  x ∈ Γ}, where ϕ is a function on Γ and n is the outward unit normal on Γ. The calculations are quite delicate because of the occurrence of additional stress singularities at the collision points {γ₃, γ₄} = Γ _D ∩ Γ _N. The result is derived by the combination of the weight function method and the Generalized J-integral technique (abbr. GJ-integral technique). The GJ-integrals have been proposed by the first author in order to express the variation of energy (energy release rate) by extension of a crack in a 3D-elastic body. This paper begins with the weak solution of the crack problem, the weight function representation of SIF's, GJ-integral technique and finish with the shape sensitivity analysis of SIF's. GJ-integral J_ω(u; X) is the sum of the P-integral (line integral) P_ω(u, X) and the R-integral (area integral) R_ω(u, X). With the help of the GJ-integral technique we derive an R-integral expression for the shape derivative of the potential energy which is valid for all displacement fields u ∈ H¹. Using the property that the GJ-integral vanishes for all regular fields u ∈ H² we convert the R-integral expression for the shape derivative to a P-integral expression. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.