Sashikumaar Ganesan

LOCAL PROJECTION STABILIZATION OF EQUAL ORDER INTERPOLATION APPLIED TO THE STOKES PROBLEM

The local projection stabilization allows us to circumvent the Babuska-Brezzi condition and to use equal order interpolation for discretizing the Stokes problem. The projection is usually done in a two-level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements. We propose a new local projection stabilization method based on (possibly) enriched finite element spaces and discontinuous projection spaces defined on the same mesh. Optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf-sup condition. Examples are enriched simplicial finite elements and standard quadrilateral/hexahedral elements. The new approach overcomes the problem of an increasing discretization stencil and, thus, is simple to implement in existing computer codes. Numerical tests confirm the theoretical convergence results which are robust with respect to the user-chosen stabilization parameter.

A coupled arbitrary Lagrangian-Eulerian and Lagrangian method for computation of free surface flows with insoluble surfactants

A finite element scheme to compute the dynamics of insoluble surfactant on a deforming free surface is presented. The free surface is tracked by the arbitrary Lagrangian-Eulerian (ALE) approach, whereas the surfactant concentration transport equation is approximated in a Lagrangian manner. Since boundary resolved moving meshes are used in the ALE approach, the surface tension, which may be a linear or nonlinear function of surfactant concentration (equation of state), and the Marangoni forces can be incorporated directly into the numerical scheme. Further, the Laplace-Beltrami operator technique, which reduces one order of differentiation associated with the curvature, is used to handle the curvature approximation. A number of 3D-axisymmetric computations are performed to validate the proposed numerical scheme. An excellent surfactant mass conservation without any additional mass correction scheme is obtained. The differences in using a linear and a nonlinear equation of state, respectively, on the flow dynamics of a freely oscillating droplet are demonstrated.

Arbitrary Lagrangian-Eulerian finite-element method for computation of two-phase flows with soluble surfactants

A finite-element scheme based on a coupled arbitrary Lagrangian-Eulerian and Lagrangian approach is developed for the computation of interface flows with soluble surfactants. The numerical scheme is designed to solve the time-dependent Navier-Stokes equations and an evolution equation for the surfactant concentration in the bulk phase, and simultaneously, an evolution equation for the surfactant concentration on the interface. Second-order isoparametric finite elements on moving meshes and second-order isoparametric surface finite elements are used to solve these equations. The interface-resolved moving meshes allow the accurate incorporation of surface forces, Marangoni forces and jumps in the material parameters. The lower-dimensional finite-element meshes for solving the surface evolution equation are part of the interface-resolved moving meshes. The numerical scheme is validated for problems with known analytical solutions. A number of computations to study the influence of the surfactants in 3D-axisymmetric rising bubbles have been performed. The proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant.

Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems

We give a survey on recent developments of stabilization methods based on local projection type. The considered class of problems covers scalar convection–diffusion equations, the Stokes problem and the linearized Navier–Stokes equations. A new link of local projection to the streamline diffusion method is shown. Numerical tests for different type of boundary layers arising in convection–diffusion problems illustrate the stabilizing properties of the method.

Oscillations of soap bubbles

Oscillations of droplets or bubbles of a confined fluid in a fluid environment are found in various situations in everyday life, in technological processing and in natural phenomena on different length scales. Air bubbles in liquids or liquid droplets in air are well-known examples. Soap bubbles represent a particularly simple, beautiful and attractive system to study the dynamics of a closed gas volume embedded in the same or a different gas. Their dynamics is governed by the densities and viscosities of the gases and by the film tension. Dynamic equations describing their oscillations under simplifying assumptions have been well known since the beginning of the 20th century. Both analytical description and numerical modeling have made considerable progress since then, but quantitative experiments have been lacking so far. On the other hand, a soap bubble represents an easily manageable paradigm for the study of oscillations of fluid spheres. We use a technique to create axisymmetric initial non-equilibrium states, and we observe damped oscillations into equilibrium by means of a fast video camera. Symmetries of the oscillations, frequencies and damping rates of the eigenmodes as well as the coupling of modes are analyzed. They are compared to analytical models from the literature and to numerical calculations from the literature and this work.

Pressure separation––a technique for improving the velocity error in finite element discretisations of the Navier–Stokes equations

Abstract This paper presents a technique to improve the velocity error in finite element solutions of the steady state Navier–Stokes equations. This technique is called pressure separation. It relies upon subtracting the gradient of an appropriate approximation of the pressure on both sides of the Navier–Stokes equations. With this, the finite element error estimate can be improved in the case of higher Reynolds numbers. For practical reasons, the pressure separation can be applied above all for finite element discretisations of the Navier–Stokes equations with piecewise constant pressure. This paper presents a computational study of five ways to compute an appropriate approximation of the pressure. These ways are assessed on two- and three-dimensional examples. They are compared with respect to the error reduction in the discrete velocity and the computational overhead.

Simulations of impinging droplets with surfactant-dependent dynamic contact angle

An arbitrary Lagrangian-Eulerian (ALE) finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface is presented. The numerical scheme solves the time-dependent Navier-Stokes equations for the fluid flow, scalar convection-diffusion equation for the surfactant transport in the bulk phase, and simultaneously, surface evolution equations for the surfactants on the free surface and on the liquid-solid interface. The effects of surfactants on the flow dynamics are included into the model through the surface tension and surfactant-dependent dynamic contact angle. In particular, the dynamic contact angle ( ? d ) of the droplet is defined as a function of the surfactant concentration at the contact line and the equilibrium contact angle ( ? e 0 ) of the clean surface using the nonlinear equation of state for surface tension. Further, the surface forces are included into the model as surface divergence of the surface stress tensor that allows to incorporate the Marangoni effects without calculating the surface gradient of the surfactant concentration on the free surface. In addition to a mesh convergence study and validation of the numerical results with experiments, the effects of adsorption and desorption surfactant coefficients on the flow dynamics in wetting, partially wetting and non-wetting droplets are studied in detail. It is observed that the effects of surfactants are more in wetting droplets than in the non-wetting droplets. Further, the presence of surfactants at the contact line reduces the equilibrium contact angle further when ? e 0 is less than 90?, and increases it further when ? e 0 is greater than 90?. Nevertheless, the presence of surfactants has no effect on the contact angle when ? e 0 = 90 ? . The numerical study clearly demonstrates that the surfactant-dependent contact angle has to be considered, in addition to the Marangoni effect, in order to study the flow dynamics and the equilibrium states of surfactant droplet impingement accurately. The proposed numerical scheme guarantees the conservation of fluid mass and of the surfactant mass accurately.

ParMooNA modernized program package based on mapped finite elements

ParMooN is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor MooNMD (John and Matthies, 2004): strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of ParMooN. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library PETSc. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.

An Object Oriented Parallel Finite Element Scheme for Computations of PDEs: Design and Implementation

Parallel finite element algorithms based on object-oriented concepts are presented. Moreover, the design and implementation of a data structure proposed are utilized in realizing a parallel geometric multigrid method. The ParFEMapper and the ParFECommunicator are the key components of the data structure in the proposed parallel scheme. These classes are constructed based on the type of finite elements (continuous or nonconforming or discontinuous) used. The proposed solver is compared with the open source direct solvers, MUMPS and PasTiX. Further, the performance of the parallel multigrid solver is analyzed up to 1080 processors. The solver shows a very good speedup up to 960 processors and the problem size has to be increased in order to maintain the good speedup when the number of processors are increased further. As a result, the parallel solver is able to handle large scale problems on massively parallel supercomputers. The proposed parallel finite element algorithms and multigrid solver are implemented in our in-house package ParMooN.

A comparative study of a direct discretization and an operator-splitting solver for population balance systems

Abstract A direct discretization approach and an operator-splitting scheme are applied for the numerical simulation of a population balance system which models the synthesis of urea with a uni-variate population. The problem is formulated in axisymmetric form and the setup is chosen such that a steady state is reached. Both solvers are assessed with respect to the accuracy of the results, where experimental data are used for comparison, and the efficiency of the simulations. Depending on the goal of simulations, to track the evolution of the process accurately or to reach the steady state fast, recommendations for the choice of the solver are given.