Lehrstuhl III Angewandte Mathematik und Numerik

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Numerical solution of the Fokker-Planck equation using physics-conforming finite element methods
(2024) Wegener, Katharina Theresa; Kuzmin, Dmitri; Turek, Stefan
In this work, a Fokker-Planck equation (FPE) is used to approximate the orientation distribution of fibers. FPEs combined with the Navier-Stokes equations (NSE) are widely used to predict the motion of the fibers in fiber suspension flows with low Reynolds numbers. The fibers align in response to the flow and randomize in response to fiber-fiber interactions. A precise formulation takes into account that the flow-fiber interaction is bilateral, so that the suspension rheology also depends on the fiber orientation. Various approaches to model fiber suspensions, including the well-known Folgar- Tucker equation, which relies on orientation tensors, are reviewed. We aim to solve the FPE using the continuous Galerkin method. For each point in the 3d physical space, an equation on the surface of a unit sphere representing the orientation states is solved, while for each point on the sphere an advection equation in the 3d physical space has to be solved. We handle this in the framework of an alternating direction approach including subtime stepping. Algebraic flux correction is performed for each equation to ensure positivity preservation as well as the normalization property of the distribution function. Numerical tests are performed for the individual subproblems. Finally, the velocity field is calculated by the incompressible Navier-Stokes equations (NSE), and benchmark problems for the coupled FPE-NSE system are solved. Thus, the relevance of this two-way coupling across the scales can be validated, and the effect of a different number of fibers is examined.
Lagrangian simulation of fiber orientation dynamics using random walk methods
(2023) Ahmadi, Omid; Kuzmin, Dmitri; Turek, Stefan
This thesis focuses on developing a two-way coupled framework for the numerical simulation of fiber suspension flows. The influence of fibers on the flow is accounted for by evaluating a non-Newtonian stress term incorporated into the Navier-Stokes equations. The accuracy of the analysis depends on a second-order tensor field used to approximate the orientation distribution of fibers. In this context, the disperse phase can be treated in the Lagrangian or Eulerian manner. We conduct a comprehensive comparison of these frameworks for one-way coupled scenarios in both two- and three-dimensional homogeneous flows. With a special focus on the Lagrangian approach, the algorithm for solving the two-way coupled fiber suspension flow in a segregated manner is proposed by incorporating the fiber-induced stresses in the finite element formulation of the Navier-Stokes equations. In non-dilute suspensions, fiber-fiber interactions may cause spontaneous changes in the orientation of fibers. Applying the theory of rotary Brownian motion, the effect can be studied using a rotary diffusion term with a Laplace-Beltrami operator. In this work, we develop random walk methodologies to emulate the action of the diffusion term without evolving or reconstructing the so-called orientation distribution function. After deriving simplified forms of Brownian motion generators for rotated reference frames, several practical approaches to generating random walks on the unit sphere are discussed. Among the proposed methods, this research effort presents the projection of Cartesian random walks, as well as polar random walks on the tangential plane. The standard random walks are then projected onto the unit sphere. Moreover, we propose an alternative based on a tabulated approximation of the cumulative distribution function obtained from the exact solution of the spherical heat equation. In the last part of this work, the random walk approaches are compared through several numerical studies, including the study of the orientation distribution of fibers in a three-dimensional homogeneous flow. Then, the two-way coupled solver is validated in a simple geometry, followed by performing a few three-dimensional numerical simulations to study the rheological behavior of the fiber suspension flow through an axisymmetric contraction. The effect of fiber-fiber interactions is also incorporated using the random walk methodology.
Efficient FEM simulation techniques for thixoviscoplastic flow problems
(2023) Begum, Naheed; Turek, Stefan; Sokolov, Andriy
This thesis is concerned with the numerical simulations of thixoviscoplastic (TVP) flow problems. As a nonlinear multifield two-way coupled problem, the analysis of thixoviscoplasticity in complex fluid processes would present a great challenge. Numerical simulations are conceived as economic and credible tools to replicate thixoviscoplastic phenomena in different flow circumstances. This thesis proceeds to provide efficient numerical methods and corresponding algorithmic tools to fulfil this goal. To start with, we generalized the standard FEM settings of Stokes equations, and proceeded with the well-posedness study of the problem to set the foundation for the approximate problem, and for the efficient solver. In the context of solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to develop a monolithic Newton-multigrid TVP solver. Most importantly, we used the numerical simulations of thixoviscoplastic flow problems to understand the complex phenomena of interplay between plasticity and thixotropy. We incorporated thixotropy in well-established academical benchmarks, namely channel flow and lid-driven cavity flow. In addition, we analyzed type of transitions, namely shear localization and shear banding in Couette devices. In the end, we used thixotropy in contraction configuration to highlight the importance of taking in consideration thixotropy of material rheology for an accurate flow simulations.
Mathematical modeling of coolant flow in discontinuous drilling processes with temperature coupling
(2023-03-24) Fast, Michael; Mierka, Otto; Turek, Stefan; Wolf, Tobias; Biermann, Dirk
Nickel-based alloys, like Inconel 718, are widely used in industrial applications due to their high-temperature strength and high toughness. However, machining such alloys is a challenging task because of high thermal loads at the cutting edge and thus extensive tool wear is expected. Consequently, the development of new process strategies is needed. We will consider the discontinuous drilling process with coolant. The main idea is to interrupt the drilling process in order to let the coolant to flow around the cutting edge and to reduce thermal loads. Since measurements inside the borehole are (nearly) impossible, simulations are a key tool to analyze and understand the proposed process. In this paper, a 3D fluid flow simulation model with Q2P1 Finite Elements in combination with the Fictitious Boundary Method is presented to simulate the coolant flow around the drill inside the borehole. The underlying equations are transformed into a rotational frame of reference overcoming the challenges of mesh design for high rotational domains inside the fluid domain. Special treatment of Coriolis forces is developed, that modifies the ‘Pressure Poisson’ Problem in the projection step improving the solver for high angular velocities. To further take high velocities into account, a two-scale artificial diffusion technique is introduced to stabilize the simulation. Finally, Q1 Finite Elements are used to simulate the heating and cooling processes in both the tool and the coolant during the complete discontinuous drilling process. The simulation is split into a ‘contact’ and a ‘no contact’ phase and a coupling strategy between these phases is developed. FBM is utilized to switch between the two configurations, thus only one unified grid for both configurations is needed. The results are used to gain insight into the discontinuous drilling process and to optimize the process design.
FEM simulation of thixo-viscoplastic flow problems: error analysis
(2023-05-31) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
This note is concerned with the essential part of Finite Element Methods (FEM) approximation of error analysis for quasi-Newtonian modelling of thixo-viscoplastic (TVP) flow problems. The developed FEM settings for thixotropic generalized Navier-Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton-multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization, nonoptimal estimate for microstructure, and extra regularity requirement for velocity, is due to the weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with stabilized higher order FEM. Furthermore, we show the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical solutions of TVP flow problems in a 4:1 contraction configuration.
Stabilized discontinuous Galerkin methods for solving hyperbolic conservation laws on grids with embedded objects
(2023) Streitbürger, Florian; May, Sandra; Turek, Stefan
This thesis covers a novel penalty stabilization for solving hyperbolic conservation laws using discontinuous Galerkin methods on grids with embedded objects. We consider cut cell grids, that are constructed by cutting the given object out of a Cartesian background grid. The resulting cut cells require special treatments, e.g., adding stabilization terms. In the context of hyperbolic conservation laws, one has to overcome the small cell problem: standard explicit time stepping becomes unstable on small cut cells when the time step is selected based on larger background cells. This work will present the Domain of Dependence (DoD) stabilization in one and two dimensions. By transferring additional information between the small cut cell and its neighbors, the DoD stabilization restores the correct domains of dependence in the neighborhood of the cut cell. The stabilization is added as penalty terms to the semi-discrete scheme. When combined with a standard explicit time-stepping scheme, the stabilized scheme remains stable for a time-step length based on the Cartesian background cells. Thus, the small cell problem is solved. In the first part of this work, we will consider one-dimensional hyperbolic conservation laws. We will start by explaining the ideas of the stabilization for linear scalar problems before moving to non-linear problems and systems of hyperbolic conservation laws. For scalar problems, we will show that the scheme ensures monotonicity when using its first-order version. Further, we will present an L2 stability result. We will conclude this part with numerical results that confirm stability and good accuracy. These numerical results indicate that for both, linear and non-linear problems, the convergence order in various norms for smooth tests is p+1 when using polynomials of degree p. In the second part, we will present first ideas for extending the DoD stabilization to two dimensions. We will consider different simplified model problems that occur when using two-dimensional cut cell meshes. An essential step for the extension to two dimensions will be the construction of weighting factors that indicate how we couple the multiple cut cell neighbors with each other. The monotonicity and L2 stability of the stabilized system will be confirmed by transferring the ideas of the proof from one to two dimensions. We will conclude by presenting numerical results for advection along a ramp, demonstrating convergence orders of p+1/2 to p+1 for polynomials of degree p. Additionally, we present preliminary results for the two-dimensional Burgers and Euler equations on model meshes.
An adaptive discrete Newton method for a regularization-free Bingham model
(2023) Fatima, Arooj; Turek, Stefan; Blum, Heribert
Developing a numerical and algorithmic tool which correctly identifies unyielded regions in the yield stress fluid flow is a challenging task. Two approaches are commonly used to handle the singular behaviour at the yield surface, i.e. the Augmented Lagrangian approach and the regularization approach, respectively. Generally in the regularization approach for the resulting nonlinear and linear problems, solvers do not perform efficiently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress [1]. The three field formulation of yield stress fluids corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three field formulation is treated efficiently and accurately by a monolithic finite element method. The velocity and pressure are discretized by the higher order stable FEM pair Q_2⁄(P_1^disc ) and the auxiliary stress is discretized by the Q_2 element. Furthermore, this problem is highly nonlinear and presents a big challenge to any nonlinear solver. We developed a new adaptive discrete Newton's method, which evaluates the Jacobian with the directional divided difference approach [2]. The step size in this process is an important key: We relate this size to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton's method. The resulting linear subproblems are solved using a geometrical multigrid solver. We analyse the solvability of the problem along with the adaptive Newton method for Bingham fluids by doing numerical studies for different prototypical configurations, i.e. "Viscoplastic fluid flow in a channel" [2], "Lid Driven Cavity", "Flow around cylinder", and "Bingham flow in a square reservoir", respectively. References [1] A. Aposporidis, E. Haber, M. A. Olshanskii, A. Veneziani. A Mixed Formulation of the Bingham Fluid Flow Problem: Analysis and Numerical Solution, Comput. Methods Appl. Mech. Engrg. 1 (2011), 2434–2446. [2] A. Fatima, S. Turek, A. Ouazzi, M. A. Afaq. An Adaptive Discrete Newton Method for Regularization-Free Bingham Model, 6th ECCOMAS Young Investigators Conference 7th-9th July 2021, Valencia, Spain. doi: 10.4995/YIC2021.2021.12389.
Mesh optimization based on a Neo-Hookean hyperelasticity model
(2023) Schuh, Malte; Turek, Stefan; Meyer, Christian
For industrial applications CFD-simulations have become an important addition to experiments. To perform them the underlying geometry has to be created on a computer and embedded into a computational mesh. This mesh needs to meet certain quality criteria. This mesh needs to meet certain quality criteria, which are not commonly met by many conventional mesh-generation tool. However, automated tools are necessary to simulate experiments with a time-dependent geometry, for example a rising bubble or fluid-structure interaction. In this thesis, we study the automatic optimization of a mesh by minimizing a neohookean hyperelasticity model. The aim of our mesh optimization is to either smoothen a mesh, or to adapt a mesh to a given geometry. In the first part of the thesis we propose a specific energy model from this class and investigate if a solution to the minimization of this specific energy exists or not. To solve this minimization problem we need to develop an algorithm. After this we have to investigate if this specific energy function fulfills all requirements of this algorithm. The chosen algorithm is an adaptation of Newton’s method in a function space. To globalize the convergence of Newton’s method we use operator-adaption techniques in a Hilbert space. This makes the algorithm a Quasi-Newton-Method. We proceed to add other elements that are known from optimization so that the algorithm becomes even more robust. Finally we perform several numerical tests to investigate the performance of this method. In our studies we find that for a certain set of parameters the solution of the minimization problem exists. This set of parameters is limited, but the limits are reasonable for most practical use-cases. During our numerical tests we find the method to be stable and robust enough to automatically smoothen a mesh, but to adapt a given mesh to a given geometry our results are unclear: For simulations in two dimensions, the developed method seems to perform well and we get promising results with even just a type of Picard-iteration. For simulations in three dimensions, some adaptations might be necessary and more tests are required.
Very fast finite element Poisson solvers on lower precision accelerator hardware: A proof of concept study for Nvidia Tesla V100
(2022-05-06) Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Recently, accelerator hardware in the form of graphics cards including Tensor Cores, specialized for AI, has significantly gained importance in the domain of high-performance computing. For example, NVIDIA’s Tesla V100 promises a computing power of up to 125 TFLOP/s achieved by Tensor Cores, but only if half precision floating point format is used. We describe the difficulties and discrepancy between theoretical and actual computing power if one seeks to use such hardware for numerical simulations, that is, solving partial differential equations with a matrix-based finite element method, with numerical examples. If certain requirements, namely low condition numbers and many dense matrix operations, are met, the indicated high performance can be reached without an excessive loss of accuracy. A new method to solve linear systems arising from Poisson’s equation in 2D that meets these requirements, based on “prehandling” by means of hier-archical finite elements and an additional Schur complement approach, is presented and analyzed. We provide numerical results illustrating the computational performance of this method and compare it to a commonly used (geometric) multigrid solver on standard hardware. It turns out that we can exploit nearly the full computational power of Tensor Cores and achieve a significant speed-up compared to the standard methodology without losing accuracy.
Algebraically constrained finite element methods for hyperbolic problems with applications in geophysics and gas dynamics
(2022) Hajduk, Hennes; Kuzmin, Dmitri; Gassner, Gregor
The research conducted in this thesis is focused on property-preserving discretizations of hyperbolic partial differential equations. Computational methods for solving such problems need to be carefully designed to produce physically meaningful numerical solutions. In particular, approximations to some quantities of interest should satisfy local and global discrete maximum principles. Moreover, numerical methods need to obey certain conservation relations, and convergence of approximations to the physically relevant exact solution should be ensured if multiple solutions may exist. Many algorithms based on the aforementioned design principles fall into the category of algebraic flux correction (AFC) schemes. Modern AFC discretizations of nonlinear hyperbolic systems express approximate solutions as convex combinations of intermediate states and constrain these states to be admissible. The main focus of our work is on monolithic convex limiting (MCL) strategies that modify spatial semi-discretizations in this way. Contrary to limiting approaches of predictor-corrector type, their monolithic counterparts are well suited for transient and steady problems alike. Further benefits of the MCL framework presented in this thesis include the possibility of enforcing entropy stability conditions in addition to discrete maximum principles. Using the AFC methodology, we transform finite element discretizations into property-preserving low order methods and perform flux correction to recover higher orders of accuracy without losing any desirable properties. The presented methods produce physics-compatible approximations, which exhibit excellent shock capturing capabilities. One novelty of this work is the tailor-made extension of monolithic convex limiting to the shallow water equations with a nonconservative topography term. Our generalized MCL schemes are entropy stable, positivity preserving, and well balanced in the sense that lake at rest equilibria are preserved. Another desirable property of numerical methods for the shallow water equations is the capability to handle wet-dry transitions properly. We present two new approaches to dealing with this issue. To corroborate our computational results with theoretical investigations, we perform numerical analysis for property-preserving discretizations of the time-dependent linear advection equation. In this context, we prove stability and derive an a~priori error estimate in the semi-discrete setting. We also compare the monolithic convex limiting strategy to two representatives of related flux-corrected transport algorithms. Another highlight of this thesis is the chapter on MCL schemes for arbitrary order discontinuous Galerkin (DG) discretizations. Building on algorithms developed for continuous Lagrange and Bernstein finite elements, we extend our MCL schemes to the high order DG setting. This research effort involves the design of new AFC tools for numerical fluxes that appear in the DG weak formulation. Our limiting strategy for DG methods exploits the properties of high order Bernstein polynomials to construct sparse discrete operators leading to compact-stencil nonlinear approximations. The proposed numerical methods are applied to various hyperbolic problems. Scalar equations are considered mainly for testing purposes and to simplify numerical analysis. Besides the shallow water system, we study the Euler equations of gas dynamics.
Monolithic Newton-multigrid FEM for the simulation of thixotropic flow problems
(2021-12-14) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
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.
Monolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects
(2020) Waseem, Muhammad; Turek, Stefan; Kuzmin, Dmitri
We study the monolithic finite element method, based on the least-squares minimization principles for the solution of non-Newtonian fluids with non-isothermal effects. The least-squares functionals are balanced by the linear and nonlinear weighted functions and the residuals comprised of L2-norm only. The weighted functions are the function of viscosities and proved significant for optimal results. The lack of mass conservation is an important issue in LSFEM and is studied extensively for the diverse range of weighted functions. Therefore, we consider only inflow/outflow problems. We use the Krylov subspace linear solver, i.e. conjugate gradient method, with a multigrid method as a preconditioner. The SSOR-PCG is used as smoother for the multigrid method. The Gauss-Newton and fixed point methods are employed as nonlinear solvers. The LSFEM is investigated for two main types of fluids, i.e. Newtonian and non-Newtonian fluids. The stress-based first-order systems, named SVP formulations, are employed to investigate the Newtonian fluids. The different types of quadratic finite elements are used for the analysis purposes. The nonlinear Navier-Stokes problem is investigated for two mesh configurations for flow around cylinder problem. The coefficients of lift/drag, pressure difference, global mass conservation are analyzed. The comparison of linear and nonlinear solvers, based on iterations, is presented as well. The analysis of non-Newtonian fluids is divided into two parts, i.e. isothermal and non-isothermal. For the non-Newtonian fluids, we consider only Q2 finite elements for the discretization of unknown variables. The isothermal non-Newtonian fluids are investigated with SVP-based formulations. The power law and Cross law viscosity models are considered for investigations with different nonlinear weighted functions. We study the flow parameters for flow around cylinder problem along with the mass conservation for shear thinning and shear thickening fluids. To study the non-isothermal non-Newtonian fluids, we introduced a new first-order formulation which includes temperature and named it as SVPT formulation. The non-isothermal effects are obtained due to the additional viscous dissipation in the fluid flow and from the preheated source as well. The flow around cylinder problem is analyzed for a variety of flow parameters for Cross law fluids. It is shown that the MPCG solver generates very accurate results for the coupled and highly complex problems.
A proof of concept for very fast finite element Poisson solvers on accelerator hardware
(2021-12-14) Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
It is demonstrated that modern accelerator hardware specialized in AI, e.g., “next gen GPUs” equipped with Tensor Cores, can be profitably used in finite element simulations by means of a new hardware-oriented method to solve linear systems arising from Poisson's equation in 2D. We consider the NVIDIA Tesla V100 Tensor Core GPU with a peak performance of 125 TFLOP/s, that is only achievable in half precision and if operations with high arithmetic intensity, such as dense matrix multiplications, are executed, though. Its computing power can be exploited to a great extent by the new method based on “prehandling” without loss of accuracy. We obtain a significant reduction of computing time compared to a standard geometric multigrid solver on standard x64 hardware.
Isogeometric analysis of Cahn-Hilliard phase field-based Binary-Fluid-Structure Interaction based on an ALE variational formulation
(2020) Sayyid Hosseini, Babak; Turek, Stefan; Möller, Matthias
This thesis is concerned with the development of a computational model and simulation technique capable of capturing the complex physics behind the intriguing phenomena of Elasto-capillarity. Elastocapillarity refers to the ability of capillary forces or surface tensions to deform elastic solids through a complex interplay between the energy of the surfaces (interfaces) and the elastic strain energy in the solid bulk. The described configuration gives rise to a three-phase system featuring a fluid-fluid interface (for instance the interface of a liquid and an ambient fluid such as air) and two additional interfaces separating the elastic solid from the first and second fluids, respectively. This setup is encountered in the wetting of soft substrates which is an emerging young field of research with many potential applications in micro- and nanotechnology and biomechanics. By virtue of the fact that a lot of physical phenomena under the umbrella of the wetting of soft substrates (e.g. Stick-slip motion, Durotaxis, Shuttleworth effect, etc.) are not yet fully understood, numerical analysis and simulation tools may yield invaluable insights when it comes to understanding the underlying processes. The problem tackled in this work – dubbed Elasto-Capillary Fluid-Structure Interaction or Binary-Fluid-Structure Interaction (BFSI) – is of multiphysics nature and poses a tremendous and challenging complexity when it comes to its numerical treatment. The complexity is given by the individual difficulties of the involved Two-phase Flow and Fluid-Structure Interaction (FSI) subproblems and the additional complexity emerging from their complex interplay. The two-phase flow problems considered in this work are immiscible two-component incompressible flow problems which we address with a Cahn-Hilliard phase field-based two-phase flow model through the Navier-Stokes-Cahn-Hilliard (NSCH) equations. The phase field method – also known as the diffuse interface method – is based on models of fluid free energy and has a solid theoretical foundation in thermodynamics and statistical mechanics. It may therefore be perceived as a physically motivated extension of the level-set or volume-of-fluid methods. It differs from other Eulerian interface motion techniques by virtue of the fact that it does not feature a sharp, but a diffuse interface of finite width whose dynamics are governed by the joint minimization of a double well chemical energy and a gradientsquared surface energy – both being constituents of the fluid free energy. Particularly for two-phase flows, diffuse interface models have gained a lot of attention due to their ability to handle complex interface dynamics such moving contact lines on wetted surfaces, and droplet coalescence or segregation without any special procedures. Our computational model for the FSI subproblem is based on a hyperelastic material model for the solid. When modeling the coupled dynamics of FSI, one is confronted with the dilemma that the fluid model is naturally based on an Eulerian perspective while it is very natural to express the solid problem in Lagrangian formulation. The monolithic approach we take, uses a fully coupled Arbitrary Lagrangian– Eulerian (ALE) variational formulation of the FSI problem and applies Galerkin-based Isogeometric Analysis for the discretization of the partial differential equations involved. This approach solves the difficulty of a common variational description and facilitates a consistent Galerkin discretization of the FSI problem. Besides, the monolithic approach avoids any instability issues that are associated with partitioned FSI approaches when the fluid and solid densities approach each other. The BFSI computational model presented in this work is obtained through the combination of the above described phase field-based two-phase flow and the monolithic fluid-structure interaction models and yields a very robust and powerful method for the simulation of elasto-capillary fluid-structure interaction problems. In addition, we also show that it may also be used for the modeling of FSI with free surfaces, involving totally or partially submerged solids. Our BFSI model may be classified as “quasi monolithic” as we employ a two-step solution algorithm, where in the first step we solve the pure Cahn-Hilliard phase field problem and use its results in a second step in which the binary-fluid-flow, the solid deformation and the mesh regularization problems are solved monolithically.
The Tensor Diffusion approach as a novel technique for simulating viscoelastic fluid flows
(2021) Westervoß, Patrick; Turek, Stefan; Kreuzer, Christian
In this thesis, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluid flows is introduced. Therefore, it is assumed that the extra-stress tensor can be decomposed into a product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity function. As a main potential advantage, which can be demonstrated for fully developed channel flows, the underlying complex material behaviour can be explicitly described by means of the so-called Diffusion Tensor. Consequently, this approach offers the possibility to reduce the complete nonlinear viscoelastic three-field model to a generalised Stokes-like problem regarding the velocity and pressure fields, only. This is enabled by inserting the Diffusion Tensor into the momentum equation of the flow model, while the extra-stress tensor or constitutive equation can be neglected. As a result, flow simulations of viscoelastic fluids could be performed by applying techniques particularly designed for solving the (Navier-)Stokes equations, which leads to a way more robust and efficient numerical approach. But, a conceptually improved behaviour of the numerical scheme concerning viscoelastic fluid flow simulations may be exploited with respect to discretisation and solution techniques of typical three- or four-field formulations as well. In detail, an (artificial) diffusive operator, which is closely related to the nature of the underlying material behaviour, is inserted into the (discrete) problem by means of the Diffusion Tensor. In this way, certain issues particularly regarding the flow simulation of viscoelastic fluids without a Newtonian viscosity contribution, possibly including realistic material and model parameters, can be resolved. In a first step, the potential benefits of the Tensor Diffusion approach are illustrated in the framework of channel flow configurations, where several linear and nonlinear material models are considered for characterising the viscoelastic material behaviour. In doing so, typical viscoelastic flow phenomena can be obtained by simply solving a symmetrised Tensor Stokes problem including a suitable choice of the Diffusion Tensor arising from both, differential as well as integral constitutive laws. The validation of the novel approach is complemented by simulating the Flow around cylinder benchmark by means of a four-field formulation of the Tensor Stokes problem. In this context, corresponding reference results are reproduced quite well, despite the applied lower-order approximation of the tensor-valued viscosity. A further evaluation of the Tensor Diffusion approach is performed regarding two-dimensional contraction flows, where potential advantages as well as improvements and certain limits of this novel approach are detected. Therefore, suitable stabilisation techniques concerning the Diffusion Tensor variable plus the behaviour of deduced monolithic and segregated solution methods are investigated.
Benchmarking and validation of a combined CFD-optics solver for micro-scale problems
(2020-10-27) Münster, Raphael; Mierka, Otto; Turek, Stefan; Weigel, Thomas; Ostendorf, Andreas
In this work, we present a new approach for coupled CFD-optics problems that consists of a combination of a finite element method (FEM) based flow solver with a ray tracing based tool for optic forces that are induced by a laser. We combined the open-source computational fluid dynamics (CFD) package FEATFLOW with the ray tracing software of the LAT-RUB to simulate optical trap configurations. We benchmark and analyze the solver first based on a configuration with a single spherical particle that is subjected to the laser forces of an optical trap. The setup is based on an experiment that is then compared to the results of our combined CFD-optics solver. As an extension of the standard procedure, we present a method with a time-stepping scheme that contains a macro step approach. The results show that this macro time-stepping scheme provides a significant acceleration while still maintaining good accuracy. A second configuration is analyzed that involves non-spherical geometries such as micro rotors. We proceed to compare simulation results of the final angular velocity of the micro rotor with experimental measurements.
Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations
(2018) Ul Jabbar, Absaar; Turek, Stefan; Blum, Heribert
Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid methods is that they have an asymptotically meshindependent convergence behavior. Multigrid with Vanka (or local multilevel pressure Schur complement method) as smoother have been frequently used for the construction of very effcient coupled monolithic solvers for the solution of the stationary incompressible Navier-Stokes equations in 2D and 3D. However, due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence of the underlying mesh, and therefore, coupled multigrid solvers with Vanka smoothing very frequently face convergence issues on meshes with high aspect ratios. Moreover, even on very nice regular grids, these solvers may fail when the anisotropies are introduced from the differential operator. In this thesis, we develop a new class of robust and efficient monolithic finite element multilevel Krylov subspace methods (MLKM) for the solution of the stationary incompressible Navier-Stokes equations as an alternative to the coupled multigrid-based solvers. Different from multigrid, the MLKM utilizes a Krylov method as the basis in the error reduction process. The solver is based on the multilevel projection-based method of Erlangga and Nabben, which accelerates the convergence of the Krylov subspace methods by shifting the small eigenvalues of the system matrix, responsible for the slow convergence of the Krylov iteration, to the largest eigenvalue. Before embarking on the Navier-Stokes equations, we first test our implementation of the MLKM solver by solving scalar model problems, namely the convection-diffusion problem and the anisotropic diffusion problem. We validate the method by solving several standard benchmark problems. Next, we present the numerical results for the solution of the incompressible Navier-Stokes equations in two dimensions. The results show that the MLKM solvers produce asymptotically mesh-size independent, as well as Reynolds number independent convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical simulations also show that the coupled MLKM solvers can handle (both mesh and operator based) anisotropies better than the coupled multigrid solvers.
Implementation of linear and non-linear elastic biphasic porous media problems into FEATFLOW and comparison with PANDAS
(2017) Obaid, Abdulrahman Sadeq; Turek, Stefan; Markert, Bernd
This dissertation presents a fully implicit, monolithic finite element solution scheme to effectively solve the governing set of differential algebraic equations of incompressible poroelastodynamics. Thereby, a two-dimensional, biphasic, saturated porous medium model with intrinsically coupled and incompressible solid and fluid constituents is considered. Our schemes are based on some well-accepted CFD techniques, originally developed for the efficient simulation of incompressible flow problems, and characterized by the following aspects: (1) a special treatment of the algebraically coupled volume balance equation leading to a reduced form of the boundary conditions; (2) usage of a higher-order accurate mixed LBBstable finite element pair with piecewise discontinuous pressure for the spatial discretization; (3) application of the fully implicit 2nd-order Crank-Nicolson scheme for the time discretization; (4) use of a special fast multigrid solver of Vanka-type smoother available in FEATFLOW to solve the resulting discrete linear equation system. Furthermore, a new adaptive time stepping scheme combined with Picard iteration method is introduced to solve a non-linear elastic problem with special hyper-elastic model. For the purpose of validation and to expose themerits and benefits of our new solution strategies in comparison to other established approaches, canonical one- and two-dimensional wave propagation problems are solved and a large-scale, dynamic soil-structure interaction problem serves to reveal the efficiency of the special multigrid solver and to evaluate its performance for different formulations.
Efficient FEM solver for quasi-Newtonian flow problems with application to granular materials
(2016) Mandal, Saptarshi; Turek, Stefan; Richter, Thomas
This thesis is concerned with new numerical and algorithmic tools for flows with pressure and shear dependent viscosity together with the necessary background of the generalized Navier-Stokes equations. In general the viscosity of a material can be constant, e.g. water and this kind of fluid is called as Newtonian fluid. However the flow can be complicated for quasi-Newtonian fluid, where the viscosity can depend on some physical quantity. For example, the viscosity of Bingham fluid is a function of the shear rate. Moreover even further complications can arise when the dependencies of both shear rate and pressure occur for the viscosity as in the case of the granular materials, e.g. Poliquen model. The Navier-Stokes equations in primitive variables (velocity-pressure) are regarded as the privilege answer to incorporate these phenomena. The modification of the viscous stresses leads to generalized Navier-Stokes equations extending the range of their validity to such flow. The resulting equations are mathematically more complex than the Navier-Stokes equations and several problems arise from the numerical point of view. Firstly, the difficulty of approximating incompressible velocity fields and secondly, poor conditioning and possible lack of differentiability of the involved nonlinear functions due to the material laws. The difficulty related to the approximation of incompressible velocity fields is treated by applying the conforming Stokes element Q2/P1 and the lack of differentiability is taken care of by regularization. Then the continuous Newton method as linearization technique is applied and the method consists of working directly on the variational integrals. Next the corresponding continuous Jacobian operators are derived and consequently a convergence rate of the nonlinear iterations independent of the mesh refinement is achieved. This continuous approach is advantageous: Firstly the explicit accessibility of the Jacobian allows a robust method with respect to the starting guess and secondly it avoids the delicate task of choosing the step-length which is required for divided differences approaches. We denote the full Jacobian matrix on the discrete level by A and separate it into two parts: A1 and A2 corresponding to Fixed point and Newton method respectively. A fundamental issue for the continuous Newton method arises when the problem is not ready for it at the initial state due to the poor condition of the 'bad-part' A2 of the Jacobian. Although the Newton method is popular for its local quadratic convergence behavior, however the solver may show unpredictable and undesirable divergent behavior if A2 is poor conditioned. This particular difficulty is handled by our Adaptive Newton method, where we introduce a charateristic function f(Qn), which depends solely on the relative residual change Qn and controls the weighing parameter δn for the 'bad-part' A2 resulting in the swinging back and forth of the solver between Fixed point and Newton state. Finally the new Adaptive Newton method is validated for the Bingham fluid for the benchmark geometry Flow around cylinder and a test case of 2D Couette flow for (modified) Poliquen model having the scope of real world applications is studied to fulfill the objective need of performance.
Optimization-based finite element methods for evolving interfaces
(2017) Basting, Christopher; Kuzmin, Dmitri; Turek, Stefan
This thesis is concerned with the development of new approaches to redistancing and conservation of mass in finite element methods for the level set transport equation. The first proposed method is a PDE- and optimization-based redistancing scheme. In contrast to many other PDE-based redistancing techniques, the variational formulation derived from the minimization problem is elliptic and can be solved efficiently using a simple fixed-point iteration method. Artificial displacements are effectively prevented by introducing a penalty term. The objective functional can easily be extended so as to satisfy further geometric properties. The second redistancing method is based on an optimal control problem. The objective functional is defined in terms of a suitable potential function and aims at minimizing the residual of the Eikonal equation under the constraint of an augmented level set equation. As an inherent property of this approach, the interface cannot be displaced on a continuous level and numerical instabilities are prevented. The third numerical method under investigation is an optimal control approach designed to enforce conservation of mass. A numerical solution to the level set equation is corrected so as to satisfy a conservation law for the corresponding Heaviside function. Two different control approaches are investigated. The potential of the proposed methods is illustrated by a wide range of numerical examples and by numerical studies for the well-known rising bubble benchmark.

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