Alexandre M. Roma

Three-dimensional, fully adaptive simulations of phase-field fluid models

We present an efficient numerical methodology for the 3D computation of incompressible multi-phase flows described by conservative phase-field models. We focus here on the case of density matched fluids with different viscosity (Model H). The numerical method employs adaptive mesh refinements (AMR) in concert with an efficient semi-implicit time discretization strategy and a linear, multi-level multigrid to relax high order stability constraints and to capture the flows disparate scales at optimal cost. Only five linear solvers are needed per time-step. Moreover, all the adaptive methodology is constructed from scratch to allow a systematic investigation of the key aspects of AMR in a conservative, phase-field setting. We validate the method and demonstrate its capabilities and efficacy with important examples of drop deformation, Kelvin-Helmholtz instability, and flow-induced drop coalescence.

A nonstiff, adaptive mesh refinement-based method for the Cahn-Hilliard equation

We present a nonstiff, fully adaptive mesh refinement-based method for the Cahn-Hilliard equation. The method is based on a semi-implicit splitting, in which linear leading order terms are extracted and discretized implicitly, combined with a robust adaptive spatial discretization. The fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size. We demonstrate the efficacy of the method with numerical examples. Both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy. Employing this approach, we also identify several stationary solutions of that decomposition on the 2D torus.

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskins lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the systems matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.

Study of the long-time dynamics of a viscous vortex sheet with a fully adaptive nonstiff method

A numerical investigation of the long-time dynamics of two immiscible two-dimensional fluids shearing past one another is presented. The fluids are incompressible and the interface between the bulk phases is subjected to surface tension. The simple case of density and viscosity matched fluids is considered. The two-dimensional Navier–Stokes equations are solved numerically with a fully adaptive nonstiff strategy based on the immersed boundary method. Dynamically adaptive mesh refinements are used to cover at all times the separately tracked fluid interface at the finest grid level. In addition, by combining adaptive front tracking, in the form of continuous interface marker equidistribution, with a predictor–corrector discretization an efficient method is introduced to successfully treat the well-known numerical difficulties associated with surface tension. The resulting numerical method can be used to compute stably and with high resolution the flow for wide-ranging Weber numbers but this study focuses o...

Adaptive mesh refinement for micromagnetics simulations

We present a methodology for efficient micromagnetics simulation. The method combines an unconditionally stable, finite differences scheme with an adaptive mesh refinement technique. It enhances accuracy by covering locally special regions of the domain with a sequence of nested, progressively finer rectangular grid patches that dynamically follow sharp transitions of the magnetization field (e.g., walls and vortices). To illustrate our approach, we consider a rectangular sample of infinite thickness with strong anisotropy in the out-of-plane direction

A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant

Surface active agents play a significant role in interfacial dynamics of multiphase systems.While the understanding of their behavior is crucial to many important practical applications, realistic mathematical modeling and computer simulation represent an extraordinary task. By employing a front-tracking method with Eulerian adaptive mesh refinement capabilities in concert with a finite volume scheme for solving an advection-diffusion equation constrained to a moving and deforming interface, the numerical challenges posed by the full three-dimensional computer simulation of transient, incompressible two-phase flows with an insoluble surfactant are efficiently and accurately tackled in the present work. The individual numerical components forming the resulting methodology are here combined and applied for the first time. Verification tests to check the accuracy and the simulation of the deformation of a droplet in simple shear flow in the presence of an insoluble surfactant are performed, the results being compared to laboratory experiments as well as to other numerical data. In all the cases considered, the methodology presents excellent conservation properties for the total surfactant mass (even to machine precision under certain circumstances).

Numerical study of an inextensible, finite swimmer in Stokesian viscoelastic flow

A numerical investigation of an Immersed Boundary (IB) model of an effectively inextensible, finite swimmer in a Stokesian Oldroyd-B flow is presented. The swimmer model is a two-dimensional sheet of finite extent and its gait is generated by an elastic force which penalizes deviations from a target shape. A non-stiff IB method is employed to remove the impeding time step limitation induced by strong tangential forces on the swimmer. It is found that for a swimmer with a prescribed gait its mean propulsion speed decreases with increasing Deborah number De toward an apparent asymptotic minimal value. However, as the swimmer is allowed to deviate more from the target shape, the monotonic locomotion behavior with De is broken. For a sufficiently flexible swimmer, viscoelasticity can enhance locomotion but the swimmer in the viscoelastic fluid always remains slower than when it is propelling in a Newtonian fluid. Remarkably, the addition of viscoelastic stress diffusion dramatically alters the swimmer propuls...

3D simulations of phase separation with a liquid crystal component

We present three-dimensional numerical simulations of a binary mixture with a nematic liquid crystal and flexible polymer phases using Model B, which is defined by coupling the Cahn-Hilliard equation with the de Gennes-Prost equation. The model is based on the Ginzburg-Landau free energy and the purpose of the work is to analyze in three dimensions how the orientational distortion of the director field induced by interfacial anchoring affects the morphology of the binary mixture.