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Randwertprobleme der Elastizitätstheorie für Polyeder- Singularitäten und Approximation mit Randelementmethoden. (German) Zbl 0709.73009
Darmstadt: Technische Hochschule, FB Math., Diss. 133 S. (1989).
The dissertation consists of 5 chapters. It deals with the analysis of singularities in the solution of three-dimensional linear elasticity boundary value problems (b.v.p.) and with the numerical solution of these b.v.p. via Galerkin (direct) boundary element methods (BEM) on graduated meshes near the singularities.
In Chapters 1 and 2, the author establishes the decomposition of the solution of linear elasticity b.v.p. in non-smooth 3D domains (with corners and edges) into regular and singular parts. The author’s method is based on a technique developed earlier by M. Dauge [e.g.: C. R. Acad. Sci., Paris, Ser. I 304, 563-566 (1987; Zbl 0609.47064)]. Chapter 3 is devoted to the proof of asymptotic optimal approximation results for functions with singular parts approximated by finite element functions on graduated meshes. In Chapter 4, the boundary integral equations for mixed 3D linear elasticity b.v.p. including problems with inhomogeneous materials and cracks are discussed. Because of the strong ellipticity of the corresponding boundary integral operators, the Galerkin BEM converges and asymptotic optimal, or almost optimal rate estimates can be immediately derived from the approximation results.
Representative numerical experiments complete this excellent mathematical analysis of singularities in 3D elasticity and of their numerical treatment.
Reviewer: U.Langer

74B05 Classical linear elasticity
74S15 Boundary element methods applied to problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations