Theoretical and numerical studies of chaotic mixing

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity

Numerical methods for drift-diffusion models

The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions

Numerical methods for drift-diffusion models

The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume disretization scheme and recent efforts to generalize this approach to general statistical distribution functions

Theoretical and numerical studies of chaotic mixing

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity

New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

Numerical solution of large-scale linear matrix equations

We are interested in the numerical solution of large-scale linear matrix equations. In particular, due to their occurrence in many applications, we study the so-called Sylvester and Lyapunov equations. A characteristic aspect of the large-scale setting is that although data are sparse, the solution is in general dense so that storing it may be unfeasible. Therefore, it is necessary that the solution allows for a memory-saving approximation that can be cheaply stored. An extensive literature treats the case of the aforementioned equations with low-rank right-hand side. This assumption, together with certain hypotheses on the spectral distribution of the matrix coefficients, is a sufficient condition for proving a fast decay in the singular values of the solution. This decay motivates the search for a low-rank approximation so that only low-rank matrices are actually computed and stored remarkably reducing the storage demand. This is the task of the so-called low-rank methods and a large amount of work in this direction has been carried out in the last years. Projection methods have been shown to be among the most effective low-rank methods and in the first part of this thesis we propose some computational enhanchements of the classical algorithms. The case of equations with not necessarily low rank right-hand side has not been deeply analyzed so far and efficient methods are still lacking in the literature. In this thesis we aim to significantly contribute to this open problem by introducing solution methods for this kind of equations. In particular, we address the case when the coefficient matrices and the right-hand side are banded and we further generalize this structure considering quasiseparable data. In the last part of the thesis we study large-scale generalized Sylvester equations and, under some assumptions on the coefficient matrices, novel approximation spaces for their solution by projection are proposed