Análisis y simulaciones numéricas en mecánica de fluidos y campos de fase = Numerical analysis and simulations for fluid mechanics and phase-field models

• Autores: Giordano Tierra Chica
• Directores de la Tesis: Francisco Manuel Guillén González (dir. tes.)
• Lectura: En la Universidad de Sevilla ( España ) en 2012
• Idioma: inglés
• Número de páginas: 208
• Tribunal Calificador de la Tesis: Inmaculada Higueras Sanz (presid.) , Blanca Climent Ezquerra (secret.) , Franck Boyer (voc.) , Héctor Gómez Díaz (voc.) , Santiago I. Badía Rodríguez (voc.)
• Enlaces
  • Tesis en acceso abierto en: Idus
• Resumen

  • Applied Mathematics research is expanding to encompass quantitative physical phenomena of growing importance. These quantitative phenomena include many interesting applica- tions, such as, biomathematics and industrial applications. The study of the mathematical problems that arise from these fields naturally requires the combination of ideas from PDE theory, numerical analysis and computer science. The synergy of these areas will allow math- ematicians to provide efficient numerical schemes and to achieve a better understanding of the problems.

    The main purpose of this thesis strives to bring us closer to this goal of a better under- standing of these physically motivated models from an applied mathematical point of view. My focus is on topics related to numerical analysis and computational simulations of fluid flows models, phase field models and the couple between them.

    In the first part of the thesis, we study error estimates in fluid flows, for the time- dependent Navier-Stokes equations. The rest of the work, when the Phase-Field models appear, our task has concentrated on designing and studying efficient numerical schemes to approximate the models under a feasible computational cost. Therefore, issues as the convergence of these schemes towards the continuous problem, derivation of error estimates or the long time behaviour of the schemes has remained out of the scopes of this work.

    The thesis is organized in five chapters, which we expect to correspond to five different papers. In fact, the first chapter is already published in SeMA Journal while the second one is submitted to J. Comput. Physics. The other three chapters are going to be submitted soon. This thesis is organized as follows.

    Chapter 1 focuses on a linear Euler Semi-Implicit time scheme with Inf-Sup-stable Finite Element (FE) approximation in space for solving the Navier-Stokes equations for incompressible fluids filling a 3D domain ¿ during a finite time interval (0, T ). We recall the stability of this scheme, we prove some superconvergence in space error estimates between the Stokes projection of the exact solution and the scheme. These error estimates for the velocity are with respect to the energy-norm, and for a weaker norm of L2(0, T ; L2) type (this latter holds only for the case of Taylor-Hood approximation). On the other hand, we also obtain optimal error estimates for the pressure without imposing constraints on the time and spatial discrete parameters, arriving at superconvergence in a norm of L2(0,T;H1) type again by using Taylor-Hood approximations. These results are numerically verified by several computational experiments, where two different splitting in time schemes have been also considered.

    Chapter 2 is devoted to study linear numerical schemes to approximate the Cahn- Hilliard problem (with the Ginzburg-Landau double-well potential), looking at their ap- proximation order and, overall, their long-time stability properties. Then, we study some already known linear schemes and we propose two new ways of approximating in a linear way the potential term.

    The first idea allows us to design a linear approximation of the potential term which is optimal with respect to the numerical dissipation introduced in the discrete energy law, although we have not been able to assure the energy-stability of the corresponding scheme. In fact, the question about the existence of a linear second order unconditionally energy-stable scheme remains as an open problem.

    The second idea allows us to design unconditionally energy-stable linear schemes, but with respect to a modified energy.

    Using both type of approaches, we design first and second order in time linear schemes to approximate the Cahn-Hilliard problem. Furthermore, we compare numerically the effective approximation to the solution (at finite and infinite time) and the energy stability of all schemes through several computational experiments.

    Chapter 3 is focused on efficient second order in time approximations of the Allen- Cahn and Cahn-Hilliard equations. First of all, we present the equations, generic second order schemes (based on a mid-point approximation of the diffusion term) and some schemes already introduced in the literature. Then, we propose new ways of deriving second order in time approximations of the potential term (starting from the main schemes introduced in Chapter 2), yielding to new second order schemes. For these schemes and other second order schemes previously introduced in the literature, we study the constraints on the physical and discrete parameters that can appear to assure the energy-stability, unique solvability and, in the case of nonlinear schemes, the convergence of Newton¿s method to the nonlinear schemes. Moreover, in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, we compare the behaviour of the schemes and the effectiveness of the adapt-time algorithm through several computational experiments.

    In Chapter 4 we extend the ideas used to design efficient numerical schemes for the Allen-Cahn and Cahn-Hilliard equations in Chapter 2 and Chapter 3 in order to design efficient numerical schemes to some physically motivated models. In particular, we focus on models concerning the behaviour of some Liquid Crystals (LC) and in one approach of the deformation of vesicles membranes coupled with incompressible flow fields. The first part of the chapter is devoted to LCs, where Nematic and Smectic-A models are consid- ered, presenting generic second order in time numerical schemes. Moreover, second order in time approximations of the vectorial potential term are described and several numerical experiments have been carried out to show the effectiveness of each scheme. In the second part of this chapter, we study a Vesicle Membrane model and we analyze two different ideas to impose the volume and surface constraints (either Lagrange multipliers or penalization), arriving at two different problems. We provide one energy-stable numerical scheme for each problem, with a linear approximation for the Lagrange multipliers case and a non-linear approach of the penalized problem.

    At last, in Chapter 5 we focus on designing efficient numerical schemes to approximate a thermodynamically consistent Navier-Stokes/Cahn-Hilliard problem modeling the mixture of two incompressible fluids with different densities. The model is based on a diffuse interface approach that is able to describe topological transitions like droplet coalescense or droplet break-up in a natural way. We present a splitting scheme to approximate the model de- coupling the computations of the Navier-Stokes part from the Cahn-Hilliard one, which is unconditionally energy-stable up to the choice of the potential term. It is also detailed the pressure segregation scheme considered in the fluid part during the simulations to save com- putational time. Several numerical experiments are carried out to validate the correctness and the accuracy of the scheme, and to compare the sensitivity of the scheme with respect to different physical parameters.