Numerical simulation techniques for the efficient and accurate treatment of local fluidic transport processes together with chemical reactions

This work describes a numerical framework developed for the efficient and accurate simulation of microfluidic applications related to two leading ex-periments of the DFG SPP 1740 research initiative, namely the ‘Superfocus Mi-cromixer’ and the ‘Taylor bubble flow’. Both of these basic experiments are con-sidered in a reactive framework using the SPP 1740 specific chemical reaction systems. A description of the utilized numerical components related to special meshing techniques, discretization methods and decoupling solver strategies is provided and its particular implementation is performed in the open-source CFD package FeatFlow [19]. A demonstration of the developed simulation tool is based on already defined validation cases and on suitable examples being re-sponsible for the determination of the related convergence properties (in the range of targeted process parameter values) of the developed numerical frame-work. The subsequent studies give an insight into a parameter estimation method with the aim of determination of unknown reaction-kinetic parameter values by the help of experimentally measured data

On the design of global-in-time Newton-Multigrid-Pressure Schur complement solvers for incompressible flow problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space-time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES~method and then embedded as a smoother into a space-time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier-Stokes equations by using Newton's method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection-diffusion-reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases

Numerical studies of a multigrid version of the parareal algorithm

In this work, a parallel-in-time method is combined with a multigrid algorithm and further on with a spatial coarsening strategy. The most famous parallel-in-time method is the parareal algorithm. Depending on two different operators, it enables the parallelism of time-dependent problems. The operator with huge effort is carried out in parallel. But despite parallelization this can lead to long run times for long-term problems. Since the parareal algorithm has a two-level structure and the time-parallel multigrid methods are also widespread in the area of parallel time integration, we combine these approaches. We use the parareal algorithm as a smoothing operator in the basic framework of a geometrical multigrid method, where we apply a coarsening strategy in time. So we get a multigrid in time method which is strongly parallelizable. For partial differential equations we add an extra spatial coarsening strategy to our multigrid parareal version. All in all we get a method, which has a high parallel efficiency and converges fast due to the multigrid framework, which is shown in the numerical studies of this work. So we will get a highly accurate solution and can greatly reduce the parallel complexity, which is especially important for long-term problems with a limited number of processors

Augmented Lagrangian acceleration of global-in-time Pressure Schur complement solvers for incompressible Oseen equations

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier-Stokes equations

Natural convection of incompressible viscoelastic fluid flow

We revisit the MIT Benchmark 2001 and introduce a viscoelastic constitutive law into the fluid in motion. Our goal is to study the effect of viscoelasticity into the periodical behavior of the physical quantities of the corresponding benchmark. We use a robust numerical technique in simulating complex fluid flow problems based on higher order Finite Element discretization. While marching in time, an A-stable method of second order is favorable, i.e Crank-Nicolson scheme, to reproduce periodical behaviors. We use a differential form of viscoelastic model, i.e Oldroyd-B type and find out that a small amount of viscoelasticity reduces the oscillatory behavior

Monolithic Newton-multigrid FEM for the simulation of thixotropic flow problems

This contribution is concerned with the application of Finite Element Method (FEM) and Newton-Multigrid solvers to simulate thixotropic flows. The thixotropic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyse the thixotropic viscoplastic flow problem in 4:1 contraction configuration

Improving Convergence of Time-Simultaneous Multigrid Methods for Convection-Dominated Problems using VMS Stabilization Techniques

We present the application of a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation for stabilized convection-diffusion equations in the regime of small diffusion coefficients. We use Galerkin finite elements and the Crank-Nicolson scheme for discretization in space and time. The multigrid method blocks all time steps for each spatial unknown, enhancing parallelization in space. While the number of iterations of the solver is bounded above for the 1D heat equation, convergence issues arise in convection-dominated cases. In singularly perturbed advection-diffusion scenarios, Galerkin FE discretizations are known to show instabilities in the numerical solution.We explore a higher-order variational multiscale stabilization, aiming to enhance solution smoothness and improve convergence without compromising accuracy