abstract: The goal of this paper was to do an analysis of two-dimensional unsplit mass and momentum conserving Finite Volume Methods for Advection for Volume of Fluid Fields with interfaces and validating their rates of convergence. Specifically three unsplit transport methods and one split transport method were amalgamated individually with four Piece-wise Linear Reconstruction Schemes (PLIC) i.e. Unsplit Eulerian Advection (UEA) by Owkes and Desjardins (2014), Unsplit Lagrangian Advection (ULA) by Yang et al. (2010), Split Lagrangian Advection (SLA) by Scardovelli and Zaleski (2003) and Unsplit Averaged Eulerian-Lagrangian Advection (UAELA) with two Finite Difference Methods by Parker and Youngs (1992) and two Error Minimization Methods by Pilliod Jr and Puckett (2004). The observed order of accuracy was first order in all cases except when unsplit methods and error minimization methods were used consecutively in each iteration, which resulted in second-order accuracy on the shape error convergence. The Averaged Unsplit Eulerian-Lagrangian Advection (AUELA) did produce first-order accuracy but that was due to a temporal error in the numerical setup. The main unsplit methods, Unsplit Eulerian Advection (UEA) and Unsplit Lagrangian Advection (ULA), preserve mass and momentum and require geometric clipping to solve two-phase fluid flows. The Unsplit Lagrangian Advection (ULA) can allow for small divergence in the velocity field perhaps saving time on the iterative solver of the variable coefficient Poisson System. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2019

Time splitting is a numerical procedure used in solution of partial differential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiffness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly. In the present work a nonlinear advective equation is solved numerically using different techniques based on a generalised framework for splitting methods. The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their influence on the split solutions has been made. An understanding of influence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made. In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods. Three time-level schemes also need explicit filtering to remove the computational mode. This filtering can have a significant impact on the obtained numerical solutions, and hence three-time level schemes appear to be unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation.

Ce travail porte sur l'étude d'une méthode d'éléments finis discontinus (ou méthode de type Galerkin Discontinu, DGTD) basée sur l'utilisation d'un maillage héxaédrique régulier, proposée pour la résolution des équations de Maxwell dans le domaine temporel, afin de l'adapter à la caractérisation de structures rayonnantes et de l'associer à des techniques de domaines fictifs.<br />On présente tout d'abord une méthode Galerkin Discontinu s'appuyant sur une formulation centrée pour approcher les flux numériques aux interfaces du maillage et sur un schéma en temps explicite de type saute-mouton. Ainsi, le schéma obtenu est non-diffusif, stable, peu dispersif, parfaitement adapté à l'utilisation de maillages localement raffinés de manière non-conforme. La méthode a été dotée de parois absorbantes performantes (modèle Unsplit-PML), permettant de prendre en compte facilement des objets à cheval entre le domaine de calcul et la couche absorbante. Nous avons ensuite utilisé la méthode pour effectuer des calculs d'impédances, de paramètres S et de T.O.S. sur des structures rayonnantes planaires. La comparaison entre la simulation et la mesure de ces structures montre le bon fonctionnement de la méthode.<br />Nous avons alors couplé une méthode de domaines fictifs avec la méthode DGTD afin de prendre en compte la présence d'objets métalliques à géométries complexes. La méthode des domaines fictifs utilise deux maillages de manière indépendante: un maillage cartésien, pour faire évoluer le champ électromagnétique dans l'espace libre, et un maillage surfacique qui permet de prendre en compte l'objet métallique. La convergence de la méthode (pour la méthode FDTD) est liée à une relation de compatibilité entre le maillage volumique et le maillage surfacique: le plus petit élément du maillage surfacique impose la taille des éléments du maillage volumique. Ainsi, pour des objets présentant de tout petits détails, cette condition n'est assurée que si le maillage volumique est de l'ordre du plus petit élément du maillage surfacique, ce qui peut devenir extrêmement contraignant sans le recours à des techniques de raffinement local, telle que celle rendue possible par la méthode Galerkin Discontinu et utilisée ici.

We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems. / text