2,022 research outputs found
Numerical methods for electromagnetic inversion
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
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
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 . 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
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Thermomechanical analysis of geothermal heat exchange systems
Heating and cooling needs have been highly demanded as the extreme weathers become increasingly frequent and global warming becomes well-founded. Because ground temperature keeps relatively constant at 20-30 feet below the surface, using the earth as a thermal mass for temperature conditioning and thermal management creates an energy-efficient and environmentally beneficial approach to surface heating and cooling, which has been used in self-heated pavement, greenhouse, and building integrated photovoltaic thermal systems. Inspired by the human body wherein a blood circulation system keeps skin nearly at a constant temperature under environmental changes, a thermal fluid circulation system is introduced to the geothermal well system.
Through bi-directional heat exchange between surface space with the ground, heat harvested at high temperatures can be stored underground for utilization at low temperatures, so that the surface temperature variations can be significantly reduced for daily and yearly cycles minimizing the heating/cooling needs. Understanding the heat transfer under the ground and thermal stress of the heat exchange systems induced by the temperature changes is critical for system design, performance prediction and optimization, and system control and operation. This dissertation studies heat transfer and thermomechanical problems for different geothermal systems. The temperature field of the earth can be calculated given the heat source and ambient temperature. Due to nonuniform thermal expansion caused by temperature differences or material mismatches, thermal stress will be induced. Its interaction with surface mechanical load and displacement constraint will be investigated for the design and failure analysis of the fluid circulation and heat exchange system.
In the theoretical study, the earth is approximated as a semi-infinite domain. Green's function technique has been used in the analysis of heat conduction, elastic, and thermoelastic problems respectively. The semi-infinite domain with a surface boundary condition can be considered a special case of two semi-infinite domains with a perfectly bonded interface, which forms an infinite bi-material domain. For a Dirichlet boundary value problem with a constant temperature or displacement, the top semi-infinite domain can be considered with infinitely large conductivity or stiffness, respectively; for a Neumann boundary value problem with zero flux or traction, the top semi-infinite domain can be considered with a zero conductivity or stiffness, respectively. The general Green's functions of an infinite bi-material domain can recover the classic solutions for Boussinesq's problem, Mindlin's problem, Kelvin's problem, etc. The three-dimensional (3D) problems can be used to recover the corresponding two-dimensional (2D) problems by an integral of Green's function in one dimension through the Hadamard regularization.
Firstly, the heat transfer problem in an infinite bi-material is introduced and the Green's function is formulated for the temperature change caused by a point heat source in the material. It is used to simulate heat transfer for a spherical heat exchanger embedded underground in geothermal energy applications. The temperature field of the spherical inhomogeneity embedded in an infinite bi-material subjected to a uniform far-field steady-state or sinusoidal heat flux is determined by solving the boundary value problem. Eshelby’s equivalent inclusion method (EIM) is used to consider the mismatch of the thermal conductivities of the particle from the matrix, which is simulated by a prescribed temperature gradient. When the material of one semi-infinite domain exhibits zero or infinite thermal conductivity, the above solution can be used for a semi-infinite domain containing a heat source with heat insulation or constant temperature on the boundary, respectively. The analytical solution has been verified with the finite element method. The formulation is used to simulate a spherical heat source embedded in a semi-infinite domain. The method can be immediately applied to the design of geothermal energy systems for heat storage and harvesting. When the heat exchanger is a long horizontal pipe, a similar procedure can be conducted for the corresponding 2D problem. If the temperature exhibits a cyclic change, such as daily variation, the formulation is extended to the harmonic transient heat conduction problems.
Secondly, a similar formulation has been introduced for the elastic problem of an infinite bi-material. The Green's function is formulated for the displacement caused by a point force in the bi-material. It is used to simulate the stress transfer for a spherical heat exchanger embedded underground in geothermal energy applications. The formulation of the heat transfer problem is extended to the corresponding elastic problem. How a surface mechanical load is transferred to the underground heat exchanger is illustrated. The interactions between a heat exchanger and the surface load are investigated.
Finally, the thermoelastic problem of an infinite bi-material is introduced and the Green's function is formulated for the displacement field caused by a point heat source in the material. It can be straightforwardly used to derive the thermoelastic stress caused by a distributed heat source by volume integrals. However, when the thermal conductivity and elasticity of the heat exchanger are different from the earth in actual geothermal energy applications, the Green's function cannot be directly used. By analogy to Eshelby's equivalent inclusion method, a dual equivalent inclusion method (DEIM) is introduced to address the dual material mismatch in thermal and elastic properties.
The fundamental solutions of a bi-material for thermal, elastic, and thermoelastic problems are versatile and recover the ones of the single material domain for both 2D and 3D problems. The equivalent inclusion method is successfully extended to the thermoelastic problems to simulate the material mismatch. The formulation can be used in designing a geothermal heat exchanger for heat storage and supply for energy-efficient buildings as well as other geothermal applications.
Future work will extend the fundamental solutions to time-dependent thermomechanical load and investigate the daily and seasonal heat exchange with the ground using different configurations of the pipelines. The algorithms will be integrated into the inclusion-based boundary element method (iBEM) for geothermal system design and analysis
On A Saturated Poromechanical Framework and Its Relation to Abaqus Soil Mechanics and Biot Poroelasticity Frameworks
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
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
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
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
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
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