A cut finite element method for coupled bulk-surface problems on time-dependent domains

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh

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

WAVEx: Stabilized Finite Elements for Spectral Wind Wave Models Using FEniCSx

Several potential FEM discretizations of the Wave Action Balance Equation are discussed. The methods, which include streamline upwind Petrov-Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented using the open source finite element library FEniCSx for simplified 2-D cases. Open source finite element libraries, such as FEniCSx, typically only support geometries up to dimension of 3. The Wave Action Balance Equation is 4 dimensions in space so this presents difficulties. A method to use a FEM library, such as FEniCSx, to solve problems in domains with dimension larger than 4 using the product basis is discussed. A new spectral wind wave model, WAVEx, is formulated and implemented using the new finite element library FEniCSx. WAVEx is designed to allow for construction of multiple FEM discretizations with relatively small modifications in the Python code base. An example implementation is then demonstrated with WAVEx using continuous finite elements and SUPG stabilization in geographic/spectral space. For propagation in time, a generalized one step implicit finite difference method is used. When source terms are active, the second order operator splitting scheme known as Strang splitting is used. In the splitting scheme, propagation is solved using the aforementioned implicit method and the nonlinear source terms are treated explicitly using second order Runge-Kutta. Several test cases which are part of the Office for Naval Research Test Bed (ONR Test Bed) are demonstrated both with and without 3rd generation source terms and results are compared to analytic solutions, observations, and SWAN output

Final report for research project with title: Dynamics of Electrically-Induced Flow of Viscoelastic Fluids (grant number: PE8/906)

The interaction of an externally applied electric _eld with a liquid can give rise to interesting ow instabilities and pattern formation. For example, it has been demonstrated that the application of an electric _eld to an initially at polymer-air or polymer-polymer interface may result in an electrohydrodynamic instability which leads to the formation of columnar structures. This phenomenon could be exploited in order to form well-controlled patterns at the microscale and nanoscale with many practical engineering applications. The scope of the present research project is to achieve fundamental understanding of the electrically-induced ow of viscoelastic liquid _lms and to investigate the e_ect of various factors (e.g. the complex uid rheology, the presence of surface active materials or free charge along the liquid-air or liquid-liquid interfaces, geometric con_guration, etc) that may play an important role in such a process. It is well known that the dynamics and stability of liquid _lms can be very rich and it is characteristic that despite the fact that the _rst attempts to address the stability of a simple system such as a clean (without surfactants) Newtonian liquid _lm under the e_ect of gravity appear in the literature in the late 50's full understanding of the underlying mechanisms was not achieved until recently. One of the goals of the present study was to expand our understanding on the stability of the liquid _lms in the presence of surface active materials (surfactants). The reason for this is threefold. On one hand, the interaction of a surfactant-ladden _lm with an electric _eld is of interest for controlled pattern formation at the micro- and nano-scale. For example, ionic surfactants may interact with the electric _eld thereby a_ecting interfacial concentration and imposing speci_c patterns in the liquid. On the other hand, surfactants attribute non-Newtonian properties to the liquid, because the free surface attains surface elasticity and surface viscosity. Also, at high surfactant concentrations, micelles may form in the bulk and complicate its rheological behavior, rendering the solution viscoelastic. Finally, the governing equation that describes the conservation of surfactant concentration along the interface is identical to the equation that describes the conservation of free charge in the case of dielectric materials. These systems share many similar characteristics and it is possible to draw conclusions from the analogy between them. To this end, we formulated the Orr-Sommerfeld equation for a surfactant-laden _lm with appropriate boundary conditions, and solved it numerically for arbitrary disturbances and analytically for long-wave disturbances. The results from our analysis demonstrate the signi_cant e_ect of surfactant solubility and sorption kinetics on the stability characteristics and provided useful insight in the non-linear dynamics of the ow. The results from this this work have been published to the Journal of Fluid Mechanics. In a subsequent paper that has also been submitted for publication to the Journal of Fluid Mechanics we have investigated the role of surfactants on the mechanism of the long-wave instability in liquid _lm ows. We have also made announcements to several local and international conferences. A second goal of this research project was to develop a robust numerical algorithm capable of handling the ow of viscoelastic material with large interfacial deformations….

Computational analysis of single rising bubbles influenced by soluble surfactant

This paper presents novel insights about the influence of soluble surfactants on bubble flows obtained by Direct Numerical Simulation (DNS). Surfactants are amphiphilic compounds which accumulate at fluid interfaces and significantly modify the respective interfacial properties, influencing also the overall dynamics of the flow. With the aid of DNS local quantities like the surfactant distribution on the bubble surface can be accessed for a better understanding of the physical phenomena occurring close to the interface. The core part of the physical model consists in the description of the surfactant transport in the bulk and on the deformable interface. The solution procedure is based on an Arbitrary Lagrangian-Eulerian (ALE) Interface-Tracking method. The existing methodology was enhanced to describe a wider range of physical phenomena. A subgrid-scale (SGS) model is employed in the cases where a fully resolved DNS for the species transport is not feasible due to high mesh resolution requirements and, therefore, high computational costs. After an exhaustive validation of the latest numerical developments, the DNS of single rising bubbles in contaminated solutions is compared to experimental results. The full velocity transients of the rising bubbles, especially the contaminated ones, are correctly reproduced by the DNS. The simulation results are then studied to gain a better understanding of the local bubble dynamics under the effect of soluble surfactant. One of the main insights is that the quasi-steady state of the rise velocity is reached without ad- and desorption being necessarily in local equilibrium