2,022 research outputs found

    Numerical methods for electromagnetic inversion

    Get PDF
    The aim of electromagnetic (EM) sounding methods in geophysics is to obtain information about the subsurface of the earth by recorded measurements taken at the surface. In particular, the goal is to determine variations in the electrical conductivity of the earth with depth by employing an inversion procedure. In this work we focus on one technique, that consists of placing a magnetic dipole above the surface, composed of a transmitter coil and different couples of adjacent receiver coils. The receiver couples are placed at different distances (offsets) from the transmitter coil. In this setting, the electromagnetic induction effect, encoded in the first-order linear Maxwell’s differential equations, produce eddy alterning currents in the soil which induce response electromagnetic fields, that can be used to determine the conductivity profile of the ground by meaning of an inversion algorithm. A typical inversion strategy consists in an iterative procedure involving the computation of the EM response of a layered model (forward modelling) and the solution of the inverse problem. Then, the algorithm attempts to minimize the mismatch between the measured data and the predicted data, by updating the model parameters at each iteration. By assuming that the local subsurface structures are composed by horizontal and homogeneous layers, general integral solutions of Maxwell equations (i.e., the EM fields) for vertical and horizontal magnetic dipoles, can be derived and represented as Hankel transforms, which contain the subsurface model parameters, i.e., the conductivity and the thickness of each layer. By a mathematical point of view, in general, these Hankel transforms are not analytically computable and therefore it is necessary to employ a numerical scheme. Anyway, the slowly decay of the oscillations determined by the Bessel function makes the problem very difficult to handle, because traditional quadrature rules typically fail to converge. In this work we consider two different approaches. The first one is based on the decomposition of the integrand function in a first function for which the corresponding Hankel transform is known exactly, and an oscillating function decays exponentially. For realistic sets of parameters, the oscillations are quite rapidly damped, and the corresponding integral can be accurately computed by a classical quadrature rule on finite intervals. The second approach consists in the application of a Gaussian quadrature formula. We develop a Gaussian rule for weight functions involving fractional powers, exponentials and Bessel functions of the first kind. Moreover, we derive an analytical approximation of these integrals that has a general validity and allows to overcome the limits of common methods based on the modelling of apparent conductivity in the low induction number (LIN) approximation. Having at disposal a reliable method for evaluating the Hankel transforms, by assuming as forward model a homogeneous layered earth, here we also consider the inverse problem of computing the model parameters (i.e., conductivity and thickness of the layers) from a set of measured field values at different offsets. We focus on the specific case of the DUALEM system. We employ two optimization algorithms. The first one is based on the BFGS line-search method and, in order to reduce as much as possible the number of integral evaluations, the analytic approximation of these integrals is used to have a first estimate of the solution. For the second approach we employ the damped Gauss-Newton method. To avoid the dependence on the initial guess of the iterative procedure, we consider a set of different initial models, and we use each one to solve the optimization problem. The numerical experiments, carried out for the study of river-levees integrity, are obtained by employing a virtual machine equipped with the NVIDIA A100 Tensor Core GPU.The aim of electromagnetic (EM) sounding methods in geophysics is to obtain information about the subsurface of the earth by recorded measurements taken at the surface. In particular, the goal is to determine variations in the electrical conductivity of the earth with depth by employing an inversion procedure. In this work we focus on one technique, that consists of placing a magnetic dipole above the surface, composed of a transmitter coil and different couples of adjacent receiver coils. The receiver couples are placed at different distances (offsets) from the transmitter coil. In this setting, the electromagnetic induction effect, encoded in the first-order linear Maxwell’s differential equations, produce eddy alterning currents in the soil which induce response electromagnetic fields, that can be used to determine the conductivity profile of the ground by meaning of an inversion algorithm. A typical inversion strategy consists in an iterative procedure involving the computation of the EM response of a layered model (forward modelling) and the solution of the inverse problem. Then, the algorithm attempts to minimize the mismatch between the measured data and the predicted data, by updating the model parameters at each iteration. By assuming that the local subsurface structures are composed by horizontal and homogeneous layers, general integral solutions of Maxwell equations (i.e., the EM fields) for vertical and horizontal magnetic dipoles, can be derived and represented as Hankel transforms, which contain the subsurface model parameters, i.e., the conductivity and the thickness of each layer. By a mathematical point of view, in general, these Hankel transforms are not analytically computable and therefore it is necessary to employ a numerical scheme. Anyway, the slowly decay of the oscillations determined by the Bessel function makes the problem very difficult to handle, because traditional quadrature rules typically fail to converge. In this work we consider two different approaches. The first one is based on the decomposition of the integrand function in a first function for which the corresponding Hankel transform is known exactly, and an oscillating function decays exponentially. For realistic sets of parameters, the oscillations are quite rapidly damped, and the corresponding integral can be accurately computed by a classical quadrature rule on finite intervals. The second approach consists in the application of a Gaussian quadrature formula. We develop a Gaussian rule for weight functions involving fractional powers, exponentials and Bessel functions of the first kind. Moreover, we derive an analytical approximation of these integrals that has a general validity and allows to overcome the limits of common methods based on the modelling of apparent conductivity in the low induction number (LIN) approximation. Having at disposal a reliable method for evaluating the Hankel transforms, by assuming as forward model a homogeneous layered earth, here we also consider the inverse problem of computing the model parameters (i.e., conductivity and thickness of the layers) from a set of measured field values at different offsets. We focus on the specific case of the DUALEM system. We employ two optimization algorithms. The first one is based on the BFGS line-search method and, in order to reduce as much as possible the number of integral evaluations, the analytic approximation of these integrals is used to have a first estimate of the solution. For the second approach we employ the damped Gauss-Newton method. To avoid the dependence on the initial guess of the iterative procedure, we consider a set of different initial models, and we use each one to solve the optimization problem. The numerical experiments, carried out for the study of river-levees integrity, are obtained by employing a virtual machine equipped with the NVIDIA A100 Tensor Core GPU

    Spectroscopic investigations of photon-induced reactions in tin-oxo cage photoresists

    Get PDF
    Molecular compounds such as tin-oxo cages are promising photoresists for Extreme UltraViolet (EUV) photolithography, which is the latest nano-patterning technology for high-end computer chips. Solubility switching of the resist is the key for pattern transfer to the semiconductor substrate. In this thesis, different spectroscopic techniques were used to gain insight into the photochemistry upon exposure, which is crucial for optimizing the resist performance. In one research line, we developed a laser-based high harmonic generation setup as the exposure source in the soft-X-ray (XUV) region to perform broadband absorption spectroscopy on tin-oxo cage samples. Resist-coated thin films were exposed to light with energies of 21 – 70 eV, and the induced changes in the transmission as a function of exposure dose were used to quantify the photoconversion of the resist. The results were compared with those obtained with EUV (92 eV). The resist properties were further investigated using X-ray photoelectron spectroscopy and Total Electron Yield techniques. A synchrotron beamline was used as the exposure source (5-150 eV) to study the low-energy emitted electrons from the resist. Outgassing measurements (residual gas analysis) and ellipsometry techniques were used to investigate the resist’s photoconversion under 92 eV exposure. Outgassing species from the resist were determined to be mainly organic carbon-containing products. The outgassing rate was measured for a few selected masses and the induced resist’s thickness change at different exposure doses was related to the outgassing rate of the resist. The fundamental insight obtained in our studies can help to design improved EUV photoresists

    Boundedness of Operators on Local Hardy Spaces and Periodic Solutions of Stochastic Partial Differential Equations with Regime-Switching

    Get PDF
    In the first part of the thesis, we discuss the boundedness of inhomogeneous singular integral operators suitable for local Hardy spaces as well as their commutators. First, we consider the equivalence of different localizations of a given convolution operator by giving minimal conditions on the localizing functions; in the case of the Riesz transforms this results in equivalent characterizations of h1h^1. Then, we provide weaker integral conditions on the kernel of the operator and sufficient and necessary cancellation conditions to ensure the boundedness on local Hardy spaces for all values of p. Finally, we introduce a new class of atoms and use them to establish the boundedness of the commutators of inhomogeneous singular integral operators with bmo function. In the second part of the thesis, we investigate periodic solutions of a class of stochastic partial differential equations driven by degenerate noises with regime-switching. First, we consider the existence and uniqueness of solutions to the equations. Then, we discuss the existence and uniqueness of periodic measures for the equations. In particular, we establish the uniqueness of periodic measures by proving the strong Feller property and irreducibility of semigroups associated with the equations. Finally, we use the stochastic fractional porous medium equation as an example to illustrate the main results

    On A Saturated Poromechanical Framework and Its Relation to Abaqus Soil Mechanics and Biot Poroelasticity Frameworks

    Full text link
    We introduce a conservational and constitutive framework for a closed and isothermal two-phase material system consisting of a deformable porous solid matrix and a fully saturating, single-phase, and compressible pore fluid without inter-phase mass exchange. We re-derive a generalized fluid mass balance law using fundamental transport rules. We also summarize from the literature a fundamental force balance law for the fluid-solid mixture that does not require any effective stress law a priori. We show that the two conservation laws are coupled naturally to second-order without any constitutive prerequisites. This differs from Biot poroelasticity, which first postulates first-order fluid-solid coupling as two linearized constitutive relationships and then enforces them into simple Eulerian form of conservation laws. Next, we examine a limiting-case unsaturated soil mechanics framework implemented in Abaqus, by assuming isothermal conditions, full saturation, and no adsorption, and then relate it to our framework. We prove that (1) the two mass balance laws are always equivalent regardless of fluid constitutive behaviors, and (2) the two force balance laws are equivalent in their specific forms with a linearly elastic solid skeleton. Finally, taking advantage of a fundamental pore constitutive law, we show how our framework, and by extension the limiting-case Abaqus framework, naturally gives rise to the distinction between drained and undrained settings, and reduces to Biot poroelasticity under simplifying conditions. Notably, our framework indicates the presence of an additional solid-to-fluid coupling term when the solid particle velocity is non-orthogonal to the Darcy velocity.Comment: 20 pages, 68 equations, no figur

    A Dynamical System View of Langevin-Based Non-Convex Sampling

    Full text link
    Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important challenges: Existing guarantees (1) typically only hold for the averaged iterates rather than the more desirable last iterates, (2) lack convergence metrics that capture the scales of the variables such as Wasserstein distances, and (3) mainly apply to elementary schemes such as stochastic gradient Langevin dynamics. In this paper, we develop a new framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as proximal, randomized mid-point, and Runge-Kutta integrators. Beyond existing methods, our framework also motivates more efficient schemes that enjoy the same rigorous guarantees.Comment: typos corrected, references adde

    Microstructure modeling and crystal plasticity parameter identification for predicting the cyclic mechanical behavior of polycrystalline metals

    Get PDF
    Computational homogenization permits to capture the influence of the microstructure on the cyclic mechanical behavior of polycrystalline metals. In this work we investigate methods to compute Laguerre tessellations as computational cells of polycrystalline microstructures, propose a new method to assign crystallographic orientations to the Laguerre cells and use Bayesian optimization to find suitable parameters for the underlying micromechanical model from macroscopic experiments

    Scattering of elastic waves by an anisotropic sphere with application to polycrystalline materials

    Get PDF
    Scattering of a plane wave by a single spherical obstacle is the archetype of many scattering problems in various branches of physics. Spherical objects can provide a good approximation for many real objects, and the analytic formulation for a single sphere can be used to investigate wave propagation in more complex structures like particulate composites or grainy materials, which may have applications in non-destructive testing, material characterization, medical ultrasound, etc. The main objective of this thesis is to investigate an analytical solution for scattering of elastic waves by an anisotropic sphere with various types of anisotropy. Throughout the thesis a systematic series expansion approach is used to express displacement and traction fields outside and inside the sphere. For the surrounding isotropic medium such an expansion is made in terms of the traditional vector spherical wave functions. However, describing the fields inside the anisotropic sphere is more complicated since the classical methods are not applicable. The first step is to describe the anisotropy in spherical coordinates, then the expansion inside the sphere is made in the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere provide recurrence relations among the unknown expansion coefficients. The remaining expansion coefficients outside and inside the sphere can be found using the boundary conditions on the sphere. Thus, this gives the scattered wave coefficients from which the transition T matrix can be found. This is convenient as the T matrix fully describes the scattering by the sphere and is independent of the incident wave. The expressions of the general T matrix elements are complicated, but in the low frequency limit it is possible to obtain explicit expressions.The T matrices may be used to solve more complex problems like the wave propagation in polycrystalline materials. The attenuation and wave velocity in a polycrystalline material with randomly oriented anisotropic grains are thus investigated. These quantities are calculated analytically using the simple theory of Foldy and show a very good correspondence for low frequencies with previously published results and numerical computations with FEM. This approach is then utilized for an inhomogeneous medium with local anisotropy, incorporating various statistical information regarding the geometrical and elastic properties of the inhomogeneities

    Stable polymer glasses

    Get PDF
    This thesis presents investigations on stable polymer glasses prepared through physical vapour deposition from different perspectives. This is the first time that polymers have been used in simple vapour deposition and made into stable glass. The ability of our lab to create stable polymer glasses with exceptional stability and extremely long lifetimes is demonstrated through the preparation and characterization of ultrastable PS as well as PMMA glasses. Attempts at preparing stable polymer glass with higher molecular weight are reported, including two different methods–using higher molecular weight sources and crosslinking as-deposited glasses with ultraviolet radiation. The surface properties of stable polymer glasses including their surface morphology and surface relaxation are studied. With a slower bulk dynamics in stable glasses as expected, the surface evolution of the as-deposited films and the rejuvenated films are both enhanced compared to the bulk and are not easily distinguishable from each other. Investigations on stable polymer glasses confined to thin films are reported. The results support the existence of a surface mobile layer, and it is found that glass stability decreases with decreasing film thickness, as determined by different measures of stability. By studying stable polymer glasses from different perspectives in this thesis, we hope to provide valuable insights into many fundamental questions about the surface dynamics in thin films, the limit of packing in amorphous materials, and the nature of the complex and fascinating phenomenon–the glass transition
    • …
    corecore