A new numerical approach is presented to calculate the Green's function for an anisotropic multi-layered half space The formulation is explicit and unconditionally stable It imposes no limit to the thickness of the layered medium and the magnitude of the frequency In the analysis, the Fourier transform and the precise integration method (PIM) are employed Here, the Fourier transform is employed to transform the wave motion equation from the spatial domain to the wavenumber domain A second order ordinary differential equation (ODE) is observed Then, the dual vector representation of the wave motion equation is used to reduce the second order ODE to first order It is solved by the PIM Finally, the Green's function in the wavenumber domain is obtained For the evaluation of the Green's function in the spatial domain, the double inverse Fourier transform over the wavenumber is employed to derive the solutions Especially, for the transversely isotropic medium, the double inverse Fourier transform can be further reduced to a single integral by the cylindrical polar coordinate transform Numerical examples are provided Comparisons with other methods are done Very promising results are obtained

Three-dimensional Green's function for an anisotropic multi-layered half-space

Based on the fundamental solutions due to Dirac's function for transversely isotropic media, a numerical scheme for wave propagation in 2-D orthotropic media is developed within the boundary element method in the time domain. Criteria for selection of the time and space discretizations for convergent and accurate results are established. The applicability and accuracy of the scheme are verified by a number of benchmark problems including dynamic responses of an orthotropic half plane under uniform harmonic surface loadings and a circular cavern in slightly orthotropic media under the incidence of pseudo-P and -SV waves. The procedure is then applied to examine wave propagation for an underground tunnel with different ratios of orthotropy in the media. Effects of different wave parameters and ratios of orthotropy on dynamic response of the tunnel are given.

Time-Domain Boundary Element Method for Underground Structures in Orthotropic Media

The thesis is concerned with the analysis of soil-structure interaction (SSI). In the SSI problem, three linked systems should be considered: the structure, the foundation, and the soil underlying and surrounding the foundation. In order to gain insight into the phenomena associated with this subject, the thesis splits the original field of interest into four parts of research: the soil, the foundation, the structure and the final soilstructure interaction analysis. For the analysis of soil medium the Fourier-Bessel and Fourier transforms as well as precise integration method (PIM) are employed. The former are used to convert the wave motion equation from spatial domain to wavenumber domain, which results in a second order ordinary differential equation (ODE). Then, the dual vector representation of wave motion equation is introduced to reduce the second ODE to first order, which is solved by the PIM. Finally, the dynamic response (Green’s function) of soil medium in the wavenumber domain is obtained. To approach the solutions in the spatial domain, the inverse transform over wavenumber is performed. In investigating the forced vibration of surface foundations, the contact area between the foundation and ground is divided into a number of sub-regions. Here, the forcedisplacement relation of each sub-region is expressed as the newly obtained Green’s function. Then, considering the displacement boundary conditions and force equilibrium of foundations, the desired dynamic impedance functions are obtained. The accuracy of the proposed method is validated by comparing with the solutions in the literature. For the analysis of structure, a boundary oriented formulation is proposed. It is derived within two dimensional (2D) in-plane and three dimensional (3D) frame. In the analysis, the isogeometric approach is adopted to exactly describe the boundary. Then, the boundary scaling technique is employed to represent the solid. To solve the ODE for displacement in the radial scaling direction, the NURBS based collocation method is applied. Numerical examples relative to 2D and 3D problems are presented, all validating the proposed approach. Finally, for the soil-structure interaction analysis, the proposed boundary oriented formulation is employed to model the structure. The geometry of structure is exactly represented. It will allow to handle the structure with an arbitrary number of contour boundaries. For the modeling of soil medium, the boundary element method is applied with the newly derived fundamental solution (Green’s function). These formulations provided in the thesis are applicable for engineering practice. Kurzfassung Diese Thesis beschäftigt sich mit der Analyse der Boden-Bauwerk-Interaktion (BBI). Zur Untersuchung der BBI müssen drei miteinander verbundene Systeme berücksichtigt werden: Die Bauwerksstruktur, die Gründung und der Boden unter und neben der Gründung. Um einen Einblick in die BBI-Effekte zu erhalten, teilt diese Thesis das Arbeitsgebiet in vier Untersuchungsschwerpunkte: Den Boden, die Gründung, das Bauwerk und letztlich die Boden-Bauwerk-Interaktionsanalyse selbst. Zur Analyse des Bodens werden Fourier-Bessel und Fourier Transformationen sowie die Precise-Integration-Method (PIM) genutzt. Erstere werden genutzt, um die Wellenbewegung von der Raumdomäne in die Wellenzahldomäne zu konvertieren, was ein gewöhnliches Differentialgleichungssystem (ODE) zweiter Ordnung erzeugt. Daraufhin wird die duale Vektordarstellung der Wellengleichung eingeführt, um die ODE zweiter Ordnung zur ersten Ordnung zu reduzieren, welche mit der PIM gelöst wird. Anschließend erhält man die dynamische Antwort (Greensche Funktion) des Bodenmediums in der Wellenzahldomäne. Um die Lösungen in der Raumdomäne zu erhalten, wird die inverse Transformation über die Wellenzahl durchgeführt. Zur Untersuchung der aufgebrachten Schwingungen auf Oberflächenfundamente wird die Kontaktfläche zwischen Gründung und Boden in eine Anzahl von Unterflächen unterteilt. Für jede Fläche wird die Beziehung zwischen Last und Verschiebung durch die neu erhaltene Greensche Funktion ausgedrückt. Anschließend wird, unter Berücksichtigung der Verschiebungs-Randbedingungen und des Kräftegleichgewichts am Fundament die gewünschte dynamische Impedanzfunktion ermittelt. Die Genauigkeit dieser Methode wird durch Vergleiche mit Lösungen aus der Literatur validiert. Zur Analyse der Struktur wird eine Methode verwendet, die zur vollständigen Darstellung eines Körpers lediglich dessen Rand beschreibt. Sie wird sowohl für den zweidimensionalen, als auch für den dreidimensionalen Fall hergeleitet. Um den Rand exakt darzustellen wird ein isogeometrischer Ansatz gewählt. Anschließend wird die Scaled Boundary Methode angewendet um ein Volumen-Element darzustellen. Um die gewöhnliche Differentialgleichung der Verschiebung in radialer Richtung zu lösen wird eine NURBS basierte Kollokationsmethode genutzt. Numerische Beispiele für den 2Dund 3D-Fall werden gezeigt, um den Ansatz zu validieren. Um die Boden-Bauwerk-Interaktions-Analyse abzuschließen wird die vorgestellte randorientierte Formulierung eingesetzt, um das Gebäude zu modellieren. Dabei wird die Geometrie des Bauwerks genau dargestellt. Dies erlaubt die Nutzung von Gebäuden mit willkürlichen Geometrien. Zur Modellierung des Bodens wird die Randelementmethode unter Nutzung der neu hergeleiteten Fundamentallösungen (Greensche Funktionen) herangezogen. Diese in dieser Thesis vorgestellten Formulierungen sollen als Richtlinie für anwendende Ingenieure dienen.

Numerical models for the analysis of soil, structure and their interaction

Impact on a transversely anisotropic half-space


 In this paper, the complete formulation of the attenuated orthotropic time-domain half-space boundary element method was proposed to analyze the transient SH-wave scattering problems. By changing the spatial variable, the time-domain half-space Green's functions were obtained in an isotropic-like analytical process by solving the singular form of the scalar wave equation. The stress-free boundary condition of the ground surface was satisfied using the wave source image theory. To include the materials damping, the Barkan approach was used in the formulation to attenuate the half-space by applying a constant logarithmic reduction into the modified boundary integral equation. Finally, the closed-form attenuated orthotropic half-space scalar kernels were obtained in the time-domain for displacement/traction fields by analytical integration of Green's functions. The method was easily implemented in a time-domain computer code to analyze the seismic homogenous orthotropic medium. To elaborate the model of the heterogeneous problems, a sub-structuring approach was presented to satisfy the continuity conditions at the interface depending on the position of the node and normal. Then, several seismic problems including a surface canyon, an underground cavity, a subsurface inclusion, and an alluvial valley subjected to SH-wave was solved and compared with the literature to validate the surface response in the case of isotropic convergence. To evaluate the accuracy and the time of analysis for the proposed method versus the full-space boundary element models, a comparative test was carried out for two examples. A favorite agreement was testified between the responses to verify the proposed method's capability for simple modeling of the orthotropic topographic features. Finally, the ability of the method to analyze the problems with high degrees of freedom was investigated by solving a half-space model including 25 subsurface orthotropic circular inclusions.

T.E. Ricketts

Papers

Wave propagation in an anisotropic half-space

Generalized rayleigh wave propagation in anisotropic rock