Two-dimensional numerical modelling of wave propagation in soil media

Wave propagation in soil media is encountered in many engineering applications. Given that the soil is unbounded, any numerical model of finite size must include absorbing boundary conditions implemented at the artificial boundaries of the domain to allow waves to radiate away to infinity. In this work, a finite element model is developed under plane strain conditions to simulate the effects of harmonic loading induced waves. The soil can be homogeneous or multi-layered where the soil properties are linear elastic. It may overlay rigid bedrock or half-space. It may also incorporate various discontinuities such as foundations, wave barriers, embankments, tunnels or any other structure. For the case of soil media over rigid bedrock, lateral wave radiation is ensured through the implementation of the consistent transmitting boundaries, using the Thin Layer Method (TLM), which allow replacing the two semi-infinite media, on the left and right of a central domain of interest, by equivalent nodal forces simulating their effect. Those are deduced from an eigenvalue problem formulated in the two semi-infinite lateral media. In the case of soil media over half-space, the Thin Layer Method is combined to the Paraxial Boundary Conditions to allow the incoming waves to radiate away to infinity laterally and in-depth. The performance of this coupled model is enhanced by incorporating a buffer layer between the soil medium and the underlain half-space. For extensive analyses, the eigenvalue problem related to the TLM may become computationally demanding, especially for soil media with multi-wavelength depths. As the TLM involves thin sub-layers, in comparison to the wavelength, the size of the eigenvalue problem increases with increasing depth. A modified version of the TLM is proposed in this work to reduce the computational effort of the related eigenvalue problem. This dissertation work led to the development of a Fortran computer code capable of simulating wave propagation in two-dimensional soil media models with either structured or unstructured triangular mesh grids. This latter option allows considering soil-structure problems with geometrical complexities, different soil layering configurations and various loading conditions. The pre- and post-processing as well as the analysis stages are all user friendly and easy

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges

Boundary Integral Equation Methods for Simulation and Design of Photonic Devices

This thesis presents novel boundary integral equation (BIE) and associated optimization methodologies for photonic devices. The simulation and optimization of such structures is a vast and rapidly growing engineering area, which impacts on design of optical devices such as waveguide splitters, tapers, grating couplers, and metamaterial structures, all of which are commonly used as elements in the field of integrated photonics. The design process has been significantly facilitated in recent years on the basis of a variety of methods in computational electromagnetic (EM) simulation and design. Unfortunately, however, the expense required by previous simulation tools has limited the extent and complexity of the structures that can be treated. The methods presented in this thesis represent the results of our efforts towards accomplishing the dual goals of 1) Accurate and efficient EM simulation for general, highly-complex three-dimensional problems, and 2) Development of effective optimization methods leading to an improved state of the art in EM design. One of the main proposed elements utilizes BIE in conjunction with a modified-search algorithm to obtain the modes of uniform waveguides with arbitrary cross sections. This method avoids spurious solutions by means of a certain normalization procedure for the fields within the waveguides. In order to handle problems including nonuniform waveguide structures, we introduce the windowed Green function (WGF) method, which used in conjunction with auxiliary integral representations for bound mode excitations, has enabled accurate simulation of a wide variety of waveguide problems on the basis of highly accurate and efficient BIE, in two and three spatial dimensions. The "rectangular-polar" method provides the basic high-order singular-integration engine. Based on non-overlapping Chebyshev-discretized patches, the rectangular-polar method underlies the accuracy and efficiency of the proposed general-geometry three-dimensional BIE approach. Finally, we introduce a three-dimensional BIE framework for the efficient computation of sensitivities — i.e. gradients with respect to design parameters — via adjoint techniques. This methodology is then applied to the design of metalenses including up to a thousand parameters, where the overall optimization process takes in the order of three hours using five hundred computing cores. Forthcoming work along the lines of this effort seeks to extend and apply these methodologies to some of the most challenging and exciting design problems in electromagnetics in general, and photonics in particular.</p

Modelling and Characterization of Guiding Micro-structured Devices for Integrated Optics

In this thesis we show several modelling tools which are used to study nonlinear photonic band-gap structures and microcavities. First of all a nonlinear CMT and BPM were implemented to test the propagation of spatial solitons in a periodic device, composed by an array of parallel straight waveguides. In addition to noteworthy theoretical considerations, active functionalities are possible by exploiting these nonlinear regimes. Another algorithm was developed for the three-dimensional modelling of photonic cavities with cylindrical symmetry, such as microdisks. This method is validated by comparison with FDTD. We also show the opportunity to confine a field in a region of low refractive index lying in the centre of a silicon microdisk. High Q-factor and small mode volumes are achieved. Finally the characterization of microdisks in SOI with Q-factor larger than 50000 is presente

Bessel beams: a novel approach to periodic structures

Bessel and Bessel-like beams in Kerr-like nonlinear materials are numerically investigated. This is conducted with a view to exploiting the behaviour of such profiles for the direct laser writing of periodic structures in highly nonlinear glasses. A highly efficient numerical model is developed for the propagation of radially symmetric profiles based on the quasi-discrete Hankel transform (QDHT), making use of a reconstruction relation to allow the field to be sampled at arbitrary positions that do not coincide with the numerical grid. This Hankel-based Adaptive Radial Propagator (HARP) is shown to be up to 1000 times faster than standard FFT-based methods. The critical self-focusing of the Gaussian beam is reproduced to confirm the accuracy of HARP. Following this the critical self-focusing behaviour of a Bessel-Gauss beam is investigated. It is observed that, for certain parameters, increasing the beam power may prevent blowup in the Bessel-Gauss beam. Below the threshold for self-focusing the Bessel-Gauss beam exhibits periodic modulation in the direction of propagation. The existing equation describing this behaviour is shown to be inaccurate and a modification is proposed based on a power dependent beat-length. This modified beat-length equation is demonstrated to be accurate in both the paraxial and quasi-nonparaxial regime. As the beam decays, the intensity modulation appears negatively chirped. It is demonstrated that this chirp may be controlled through careful shaping of the window. It is also shown that a small Gaussian seed beam may be used to control the positions of the maxima. It is demonstrated that a set of nonlinear Bessel functions exist that exhibit a similar quasi-stationary behaviour in a nonlinear medium to the linear Bessel beam in a linear medium. Furthermore it is shown for the first time that higher-order, Bessel-like, stationary solutions exist for beams with azimuthal phase, and boundary conditions for these functions are derived