A robust inversion method for quantitative 3D shape reconstruction from coaxial eddy-current measurements

This work is motivated by the monitoring of conductive clogging deposits in steam generator at the level of support plates. One would like to use monoaxial coils measurements to obtain estimates on the clogging volume. We propose a 3D shape optimization technique based on simplified parametrization of the geometry adapted to the measurement nature and resolution. The direct problem is modeled by the eddy current approximation of time-harmonic Maxwell's equations in the low frequency regime. A potential formulation is adopted in order to easily handle the complex topology of the industrial problem setting. We first characterize the shape derivatives of the deposit impedance signal using an adjoint field technique. For the inversion procedure, the direct and adjoint problems have to be solved for each coil vertical position which is excessively time and memory consuming. To overcome this difficulty, we propose and discuss a steepest descent method based on a fixed and invariant triangulation. Numerical experiments are presented to illustrate the convergence and the efficiency of the method

A Mixed Method for Axisymmetric Div-Curl Systems

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties

An LpL^p- Primal-Dual Weak Galerkin method for div-curl Systems

This paper presents a new $L^p$-primal-dual weak Galerkin (PDWG) finite element method for the div-curl system with the normal boundary condition for $p>1$. Two crucial features for the proposed $L^p$-PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div-curl system under the low $W^{\alpha, p}$-regularity ($\alpha>0$) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the $L^q$-norm for the primal variable where $\frac{1}{p}+\frac{1}{q}=1$. A series of numerical experiments are presented to demonstrate the performance of the proposed $L^p$-PDWG algorithm.Comment: 22 pages, 2 figures, 8 tables. arXiv admin note: text overlap with arXiv:2101.0346

Negative-norm least-squares methods for axisymmetric Maxwell equations

We develop negative-norm least-squares methods to solve the three-dimensional Maxwell equations for static and time-harmonic electromagnetic fields in the case of axial symmetry. The methods compute solutions in a two-dimensional cross section of the domain, thereby reducing the dimension of the problem from three to two. To achieve this dimension reduction, we work with weighted spaces in cylindrical coordinates. In this setting, approximation spaces consisting of low order finite element functions and bubble functions are analyzed. In contrast to other methods for axisymmetric Maxwell equations, our leastsquares methods allow for discontinuous coefficients with large jumps and non-convex, irregular polygonal domains discretized by unstructured meshes. The resulting linear systems are of modest size, are symmetric positive definite, and can be solved very efficiently. Computations demonstrate the robustness of the methods with respect to the coefficients and domain shape

A general method to determine the stability of compressible flows

Several problems were studied using two completely different approaches. The initial method was to use the standard linearized perturbation theory by finding the value of the individual small disturbance quantities based on the equations of motion. These were serially eliminated from the equations of motion to derive a single equation that governs the stability of fluid dynamic system. These equations could not be reduced unless the steady state variable depends only on one coordinate. The stability equation based on one dependent variable was found and was examined to determine the stability of a compressible swirling jet. The second method applied a Lagrangian approach to the problem. Since the equations developed were based on different assumptions, the condition of stability was compared only for the Rayleigh problem of a swirling flow, both examples reduce to the Rayleigh criterion. This technique allows including the viscous shear terms which is not possible in the first method. The same problem was then examined to see what effect shear has on stability

Multilevel Schwarz Methods for Porous Media Problems

In this thesis, efficient overlapping multilevel Schwarz preconditioners are used to iteratively solve Hdiv-conforming finite element discretizations of models in poroelasticity, and an innovative two-scale multilevel Schwarz method is developed for the solution of pore-scale porous media models. The convergence of two-level Schwarz methods is rigorously proven for Biot’s consolidation model, as well as a Biot-Brinkman model by utilizing the conservation property of the discretization. The numerical performance of the proposed multiplicative and hybrid two-level Schwarz methods is tested in different problem settings by covering broad ranges of the parameter regimes, showing robust results in variations of the parameters in the system that are uniform in the mesh size. For extreme parameters a scaling of the system yields robustness of the iteration counts. Optimality of the relaxation factor of the hybrid method is investigated and the performance of the multilevel methods is shown to be nearly identical to the two-level case. The additional diffusion term in the Biot-Brinkman model yields a stabilization for high permeabilities. Additionally, a homogenizing two-scale multilevel Schwarz preconditioner is developed for the iterative solution of high-resolution computations of flow in porous media at the pore scale, i.e., a Stokes problem in a periodically perforated domain. Different homogenized operators known from the literature are used as coarse-scale operators within a multilevel Schwarz preconditioner applied to Hdiv-conforming discretizations of an extended model problem. A comparison in the numerical performance tests shows that an operator of Brinkman type with optimized effective tensor yields the best performance results in an axisymmetric configuration and a moderately anisotropic geometry of the obstacles, outperforming Darcy and Stokes as coarse-scale operators, as well as a standard multigrid method, that serves as a benchmark test