Doctor of Philosophy

dissertationInverse Electrocardiography (ECG) aims to noninvasively estimate the electrophysiological activity of the heart from the voltages measured at the body surface, with promising clinical applications in diagnosis and therapy. The main challenge of this emerging technique lies in its mathematical foundation: an inverse source problem governed by partial differential equations (PDEs) which is severely ill-conditioned. Essential to the success of inverse ECG are computational methods that reliably achieve accurate inverse solutions while harnessing the ever-growing complexity and realism of the bioelectric simulation. This dissertation focuses on the formulation, optimization, and solution of the inverse ECG problem based on finite element methods, consisting of two research thrusts. The first thrust explores the optimal finite element discretization specifically oriented towards the inverse ECG problem. In contrast, most existing discretization strategies are designed for forward problems and may become inappropriate for the corresponding inverse problems. Based on a Fourier analysis of how discretization relates to ill-conditioning, this work proposes refinement strategies that optimize approximation accuracy o f the inverse ECG problem while mitigating its ill-conditioning. To fulfill these strategies, two refinement techniques are developed: one uses hybrid-shaped finite elements whereas the other adapts high-order finite elements. The second research thrust involves a new methodology for inverse ECG solutions called PDE-constrained optimization, an optimization framework that flexibly allows convex objectives and various physically-based constraints. This work features three contributions: (1) fulfilling optimization in the continuous space, (2) formulating rigorous finite element solutions, and (3) fulfilling subsequent numerical optimization by a primal-dual interiorpoint method tailored to the given optimization problem's specific algebraic structure. The efficacy o f this new method is shown by its application to localization o f cardiac ischemic disease, in which the method, under realistic settings, achieves promising solutions to a previously intractable inverse ECG problem involving the bidomain heart model. In summary, this dissertation advances the computational research of inverse ECG, making it evolve toward an image-based, patient-specific modality for biomedical research

A New Singular S-FEM For The Linear Elastic Fracture Mechanics

Ph.DDOCTOR OF PHILOSOPH

Trace Finite Element Methods for PDEs on Surfaces

In this paper we consider a class of unfitted finite element methods for discretization of partial differential equations on surfaces. In this class of methods known as the Trace Finite Element Method (TraceFEM), restrictions or traces of background surface-independent finite element functions are used to approximate the solution of a PDE on a surface. We treat equations on steady and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in detail. We review the error analysis and algebraic properties of the method. The paper navigates through the known variants of the TraceFEM and the literature on the subject

7th International Meshing Roundtable '98

The goal of the 7th International Meshing Roundtable is to bring together researchers and developers from industry, academia, and government labs in a stimulating, open environment for the exchange of technical information related to the meshing process. In the past, the Roundtable has enjoyed significant participation from each of these groups from a wide variety of countries

Simulation of pore-scale flow using finite element-methods

I present a new finite element (FE) simulation method to simulate pore-scale flow. Within the pore-space, I solve a simplified form of the incompressible Navier-Stoke’s equation, yielding the velocity field in a two-step solution approach. First, Poisson’s equation is solved with homogeneous boundary conditions, and then the pore pressure is computed and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field I estimate the effective permeability of porous media samples characterized by thin section micrographs, micro-CT scans and synthetically generated grain packings. This two-step process is much simpler than solving the full Navier Stokes equation and therefore provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier-Stoke’s equation. My numerical model is verified with an analytical solution and validated on samples whose permeabilities and porosities had been measured in laboratory experiments (Akanji and Matthai, 2010). Comparisons were also made with Stokes solver, published experimental, approximate and exact permeability data. Starting with a numerically constructed synthetic grain packings, I also investigated the extent to which the details of pore micro-structure affect the hydraulic permeability (Garcia et al., 2009). I then estimate the hydraulic anisotropy of unconsolidated granular packings. With the future aim to simulate multiphase flow within the pore-space, I also compute the radii and derive capillary pressure from the Young-Laplace equation (Akanji and Matthai,2010

Analysing fibre composite designs for high-solidity ducted tidal turbine blades

This study elaborates a one-way fluid-structure interaction numerical model utilised in investigating the structural mechanics concerning the rotor blades comprising a ducted high-solidity tidal turbine. Coupling hydrodynamic outcomes as structural inputs in effort of acknowledging the most applicable setup, distinct designs are investigated, solid blades and cored blades, implementing fibre-reinforced composite materials, analysed within criteria related to blade axial deformation, induced radial strains, and rotor specific mass

Advanced discontinuous integral-equation schemes for the versatile electromagnetic analysis of complex structures

Premi Extraordinari de Doctorat, promoció 2018-2019. Àmbit de les TICLes Equacions Integrals superficials més importants són l'Equació de Camp Elèctric (EFIE), per a l'anàlisi de la dispersió electromagnètica d'objectes conductors perfectes (PEC), i la formulació Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT), orientada a l'anàlisi d'objectes homogenis penetrables. Ambdues són normalment discretitzades, amb el Mètode dels Moments (MoM), amb funcions base div-conformes, dependents de les arestes del mallat. Les discretitzacions div-conformes de les formulacions EFIE i PMCHWT representen esquemes conformes; és a dir, amb solucions convergents a dins de l'espai físic de corrents. Tanmateix, les implementations MoM div-conformes requereixen que el mallat sigui conforme geomètricament, amb cada parell de triangles adjacents compartint només una aresta. El desenvolupament d'esquemes div-conformes per a objectes compostos amb línies al llarg de les quals tres o més regions hi intersecten, esdevé molt incòmoda perquè cal definir condicions de continuïtat especials en aquestes línies d'intersecció. A més, els mallats que resulten de la juxtaposició de subdominis independentment mallats són típicament no-conformes geomètricament i per tant no aptes per a l'anàlisi div-conforme convencional en Mètode dels Moments. En aquesta Tesi, es tracta l'anàlisi robusta, precisa i versàtil de la dispersió electromagnètica d'objectes conductors o penetrables amb forma arbitrària i d'objectes compostos amb línies d'intersecció entre differents regions, ja sigui amb mallats conformes com no-conformes. Amb aquest objectiu, fem ús de la formulació d'equació integral EFIE–PMCHWT, la qual resulta de l'aplicació de les formulacions EFIE o PMCHWTal llarg de superfícies de contorn, respectivament, incloent regions conductores o separant regions penetrables. Els esquemes proposats en aquesta Tesi es basen en el desenvolupament dels corrents amb conjunts de funcions base discontínues a través de les arestes del mallat i dependents només dels triangles del mallat. Aquesta estratègia dóna lloc a integrals de contorn amb Kernels hypersingulars, que maneguem mitjançant el testeig de les equacions amb funcions de testeig especialment dissenyades, definides fora de les triangulacions de la superfície de contorn, a dins de la regió a on els camps són zero d'acord amb al Teorema d'Equivalència superficial. Les nostres implementacions de la formulació EFIE-PMCHWT, dependents només de triangles, mostren millor precisió respecte dels esquemes continus convencionals en l'anàlisi d'objectes angulosos a on el modelatge acurat del comportament dels camps singulars és d'importància cabdal. A més, els nostres esquemes mostren en general una gran flexibilitat en l'anàlisi d'objectes compostos amb línies d'intersecció entre regions ja que no hi cal el modelatge especial dels corrents. Finalment, les implementacions proposades poden abordar l'anàlisi d'objectes amb forma arbitrària, totalment homogenis o homogenis a trossos, i amb mallats geomètricament no-conformes.The most prominent surface integral equations, the electric field integral equation (EFIE) used for the scattering analysis of perfectly electrically conducting (PEC) targets and the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation commonly utilized for the analysis of homogeneous penetrable objects, are usually discretized, in the context of method of moments (MoM), with edge-based divergence-conforming basis functions. Divergence-conforming discretizations of the EFIE and PMCHWT formulations excel asconforming schemes, hence with converging solutions in the physical space of currents. However, the divergence-conforming MoM implementations require the underlying mesh to be geometrically conformal, with pairs of adjacent facets sharing a single edge. Thedevelopment of divergence-conforming schemes for composite objects with junctions, viz.boundary lines where more than two regions intersect, becomes somewhat awkward because of the definition of special continuity conditions at junctions. Moreover, the meshes arising from the juxtaposition of independently meshed subdomains in the modular design of complex objects are typically nonconformal and thus not suitable for conventional divergence-conforming MoM schemes. In this thesis, we address the robust, accurate and versatile scattering analysis of PEC and penetrable objects with arbitrary shape and composite objects with junctions meshed with conformal or nonconformal meshes. For this purpose, we employ the EFIE–PMCHWT integral-equation formulation, which follows from the application of the EFIE or PMCHWT formulations over boundary surfaces, respectively, enclosing PEC regions or separating penetrable regions. The proposed schemes rely on the expansion of the corrents with the facet-based, discontinuous-across-edges basis functions. This choice gives rise to boundary integrals with hypersingular kernels, which we handle by testing the equations with well-suited testing functions defined off the boundary tessellation, inside the region where, in light of the surface equivalence principle, the fields must be zero. Our facet-based EFIE-PMCHWT implementations exhibit improved accuracy when compared with the conventional continuous schemes in the analysis of sharp-edged targets where the accurate modelling of singular fields is of great importance. Moreover, our schemes manifest in general great flexibility in the analysis of composite objects with junctions as the special modelling of currents at junctions is not required. Finally, the proposed implementations can handle geometrically nonconformal meshes when applied to piecewise (or fully) homogeneous arbitrarily shaped objects.Postprint (published version