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On boundary integral equations for crack problems. (English) Zbl 0674.73071
Summary: A ubiquitous linear boundary-value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle. One typical example is the scattering of scalar waves by a curved crack or rigid strip (Neumann boundary condition) in two dimensions. This problem can be reduced to an integrodifferential equation, which is often regularized. We adopt a more direct approach, and prove that the problem can be reduced to a hypersingular boundary integral equation. (Similar reductions will obtain in more complicated situations.) Computational schemes for solving this equation are described, with special emphasis on smoothness requirements. Extensions to three-dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed.

MSC:
74R05 Brittle damage
74S30 Other numerical methods in solid mechanics (MSC2010)
65R20 Numerical methods for integral equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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