The scaled boundary finite-element method – a fundamental solution-less boundary-element method Original Research Article

The scaled boundary finite-element method – a fundamental solution-less boundary-element method Original Research Article Pages 5551-5568 John P. Wolf, Chongmin Song Close preview | Related articles | Related reference work articles Abstract | Figures/Tables | References Abstract In this boundary-element method based on finite elements only the boundary is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary and thus no singular integrals must be evaluated and general anisotropic material can be analysed. For an unbounded (semi-infinite or infinite) medium the radiation condition at infinity is satisfied exactly. No discretization of free and fixed boundaries and interfaces between different materials is required. The semi-analytical solution inside the domain leads to an efficient procedure to calculate the stress intensity factors accurately without any discretization in the vicinity of the crack tip. Body loads are included without discretization of the domain. Thus, the scaled boundary finite-element method not only combines the advantages of the finite-element and boundary-element methods but also presents appealing features of its own. After discretizing the boundary with finite elements the governing partial differential equations of linear elastodynamics are transformed to the scaled boundary finite-element equation in displacement, a system of linear second-order ordinary differential equations with the radial coordinate as independent variable, which can be solved analytically. Introducing the definition of the dynamic stiffness, a system of nonlinear first-order ordinary differential equations in dynamic stiffness with the frequency as independent variable is obtained. Besides the displacements in the interior the static-stiffness and mass matrices of a bounded medium and the dynamic-stiffness and unit-impulse response matrices of an unbounded medium are calculated. Article Outline 1. Introduction 2. Fundamental equations 2.1. Scaled boundary transformation of geometry 2.2. Governing equations of elastodynamics 2.3. Boundary discretization with finite elements 2.4. Dynamic stiffness on boundary 3. Statics 3.1. Vanishing body loads 3.2. Body loads 4. Mass matrix of bounded medium 5. Numerical solution of dynamic stiffness and unit-impulse response of unbounded medium 5.1. High-frequency asymptotic expansion of dynamic stiffness 5.2. Dynamic stiffness in frequency domain 5.3. Unit-impulse response in time domain 6. Analytical solution in frequency domain 6.1. Displacements 6.2. Dynamic stiffness of bounded medium 6.3. Dynamic stiffness of unbounded medium 7. Implementation 8. Benchmark examples 8.1. Edge-cracked orthotropic plate 8.2. Orthotropic bimaterial plate with crack 8.3. Tunnel in inhomogeneous transversely isotropic unbounded rock 8.4. Solid sphere with spherical symmetry 8.5. Spherical cavity in full space with spherical symmetry 8.6. Cylinder embedded in half-space 8.7. Prism embedded in half-space 9. Conclusions References