Eldorado Collection:
http://hdl.handle.net/2003/38230
2024-03-28T16:28:42ZLagrangian simulation of fiber orientation dynamics using random walk methods
http://hdl.handle.net/2003/42385
Title: Lagrangian simulation of fiber orientation dynamics using random walk methods
Authors: Ahmadi, Omid
Abstract: 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.2023-01-01T00:00:00ZEfficient FEM simulation techniques for thixoviscoplastic flow problems
http://hdl.handle.net/2003/42234
Title: Efficient FEM simulation techniques for thixoviscoplastic flow problems
Authors: Begum, Naheed
Abstract: 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.2023-01-01T00:00:00ZMathematical modeling of coolant flow in discontinuous drilling processes with temperature coupling
http://hdl.handle.net/2003/42173
Title: Mathematical modeling of coolant flow in discontinuous drilling processes with temperature coupling
Authors: Fast, Michael; Mierka, Otto; Turek, Stefan; Wolf, Tobias; Biermann, Dirk
Abstract: 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.2023-03-24T00:00:00ZFEM simulation of thixo-viscoplastic flow problems: error analysis
http://hdl.handle.net/2003/42140
Title: FEM simulation of thixo-viscoplastic flow problems: error analysis
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: 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.2023-05-31T00:00:00ZStabilized discontinuous Galerkin methods for solving hyperbolic conservation laws on grids with embedded objects
http://hdl.handle.net/2003/42095
Title: Stabilized discontinuous Galerkin methods for solving hyperbolic conservation laws on grids with embedded objects
Authors: Streitbürger, Florian
Abstract: 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.2023-01-01T00:00:00ZAn adaptive discrete Newton method for a regularization-free Bingham model
http://hdl.handle.net/2003/42055
Title: An adaptive discrete Newton method for a regularization-free Bingham model
Authors: Fatima, Arooj
Abstract: 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/YIC2021.2021.12389.2023-01-01T00:00:00ZMesh optimization based on a Neo-Hookean hyperelasticity model
http://hdl.handle.net/2003/42050
Title: Mesh optimization based on a Neo-Hookean hyperelasticity model
Authors: Schuh, Malte
Abstract: 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.2023-01-01T00:00:00ZVery fast finite element Poisson solvers on lower precision accelerator hardware: A proof of concept study for Nvidia Tesla V100
http://hdl.handle.net/2003/41313
Title: Very fast finite element Poisson solvers on lower precision accelerator hardware: A proof of concept study for Nvidia Tesla V100
Authors: Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Abstract: 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.2022-05-06T00:00:00ZAlgebraically constrained finite element methods for hyperbolic problems with applications in geophysics and gas dynamics
http://hdl.handle.net/2003/41001
Title: Algebraically constrained finite element methods for hyperbolic problems with applications in geophysics and gas dynamics
Authors: Hajduk, Hennes
Abstract: 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.2022-01-01T00:00:00ZMonolithic Newton-multigrid FEM for the simulation of thixotropic flow problems
http://hdl.handle.net/2003/40970
Title: Monolithic Newton-multigrid FEM for the simulation of thixotropic flow problems
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: 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.2021-12-14T00:00:00ZMonolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects
http://hdl.handle.net/2003/40781
Title: Monolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects
Authors: Waseem, Muhammad
Abstract: 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.2020-01-01T00:00:00ZA proof of concept for very fast finite element Poisson solvers on accelerator hardware
http://hdl.handle.net/2003/40773
Title: A proof of concept for very fast finite element Poisson solvers on accelerator hardware
Authors: Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Abstract: 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.2021-12-14T00:00:00ZIsogeometric analysis of Cahn-Hilliard phase field-based Binary-Fluid-Structure Interaction based on an ALE variational formulation
http://hdl.handle.net/2003/40171
Title: Isogeometric analysis of Cahn-Hilliard phase field-based Binary-Fluid-Structure Interaction based on an ALE variational formulation
Authors: Sayyid Hosseini, Babak
Abstract: 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.2020-01-01T00:00:00ZThe Tensor Diffusion approach as a novel technique for simulating viscoelastic fluid flows
http://hdl.handle.net/2003/40131
Title: The Tensor Diffusion approach as a novel technique for simulating viscoelastic fluid flows
Authors: Westervoß, Patrick
Abstract: 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.2021-01-01T00:00:00ZBenchmarking and validation of a combined CFD-optics solver for micro-scale problems
http://hdl.handle.net/2003/39815
Title: Benchmarking and validation of a combined CFD-optics solver for micro-scale problems
Authors: Münster, Raphael; Mierka, Otto; Turek, Stefan; Weigel, Thomas; Ostendorf, Andreas
Abstract: 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.2020-10-27T00:00:00ZEfficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations
http://hdl.handle.net/2003/37872
Title: Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations
Authors: Ul Jabbar, Absaar
Abstract: 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.2018-01-01T00:00:00ZImplementation of linear and non-linear elastic biphasic porous media problems into FEATFLOW and comparison with PANDAS
http://hdl.handle.net/2003/36089
Title: Implementation of linear and non-linear elastic biphasic porous media problems into FEATFLOW and comparison with PANDAS
Authors: Obaid, Abdulrahman Sadeq
Abstract: 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.2017-01-01T00:00:00ZEfficient FEM solver for quasi-Newtonian flow problems with application to granular materials
http://hdl.handle.net/2003/36034
Title: Efficient FEM solver for quasi-Newtonian flow problems with application to granular materials
Authors: Mandal, Saptarshi
Abstract: 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.2016-01-01T00:00:00ZOptimization-based finite element methods for evolving interfaces
http://hdl.handle.net/2003/36033
Title: Optimization-based finite element methods for evolving interfaces
Authors: Basting, Christopher
Abstract: 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.2017-01-01T00:00:00ZNonlinear hyperelasticity-based mesh optimisation
http://hdl.handle.net/2003/35917
Title: Nonlinear hyperelasticity-based mesh optimisation
Authors: Paul, Jordi
Abstract: In this work, various aspects of PDE-based mesh optimisation are treated. Different existing methods are presented, with the focus on a class of nonlinear mesh quality functionals that can guarantee the orientation preserving property. This class is extended from simplex to hypercube meshes in 2d and 3d.
The robustness of the resulting mesh optimisation method allows the incorporation of unilateral boundary conditions of place and r-adaptivity with direct control over the resulting cell sizes. Also, alignment to (implicit) surfaces is possible, but in all cases, the resulting functional is hard to treat analytically and numerically. Using theoretical results from mathematical elasticity for hyperelastic materials, the existence and non-uniqueness of minimisers can be established. This carries over to the discrete case, for the solution of which tools from nonlinear optimisation are used. Because of the considerable numerical effort, a class of linear preconditioners is developed that helps to speed up the solution process.2016-01-01T00:00:00ZCFD-Techniken zur Visualisierung von Strömungen und Bildbearbeitung
http://hdl.handle.net/2003/35774
Title: CFD-Techniken zur Visualisierung von Strömungen und Bildbearbeitung
Authors: Acker, Jens Friedrich
Abstract: Es wurden verschiedene Themenkomplexe, wie Bildentrauschung und Visualisierung von Strömungen unter einem gemeinsamen Aspekt zusammengeführt: der Anwendung von partiellen Differentialgleichungen, finiten Elementen und modifizierten CFD-Programmen auf diese Probleme. Dabei wurden die Themen jeweils in eine ähnliche Richtung entwickelt, so dass der nichtlineare anisotrope Diffusions-Operator (NAD) in verschiedener Form zur Anwendung kam. Im Falle der Visualisierung von Strömungen allerdings in einer, um einen Transportterm erweiterten Form (NATD).
Im ersten Themenkomplex Bildentrauschung wurden vorherige Ansätze, wie zum Beispiel Faltung mit einem Gaußkern und Perona-Malik angeführt, um die Eigenschaften und Möglichkeiten von nichtlinearer anisotroper Diffusion (NAD mit/ohne Strukturtensor) aufzuzeigen. Nach einer Erweiterung durch einen Transportterm, wurde später der NATD-Operator dazu benutzt, um unter Steuerung durch vorberechnete Strömungsfelder, Rauschtexturen so zu verändern, dass Visualisierungen dieser Strömungsfelder entstehen, die auch für problematische Strömungen geeignet sind, welche Strukturen auf unterschiedlichen Geschwindigkeitsskalen aufweisen oder über die keinerlei Vorabinformationen vorliegen. Die Eignung dieser Visualisierungsmethode für diese Art von Strömungen, wurde anhand von Beispielen gezeigt. Insbesondere für vorher unklare Strömungsstrukturen ist diese Methode besser andere Techniken, wie zum Beispiel Partikelverfolgung, deren Probleme auch in Anhang A besprochen werden. Die Änderung des in den Vorarbeiten von Martin Rumpf und Tobias Preusser gezeigten Ansatzes auf einen variationellen FEM-Ansatz ermöglicht, auch mit unstrukturierten Gittern arbeiten zu können und auch die möglichen Anwendungsbereiche stark zu erweitern.
Zusätzlich wurde bezüglich des Themas Bildwiederherstellung (``Inpainting'') ein interessanter Ansatz zur Verwendung von CFD-Techniken vorgestellt und dessen Vor- und Nachteile an Beispielen getestet. Dabei wurde ein Ansatz von Bayumy A Youssef, der mit einer Wirbelstromformulierung des inkompressiblen Navier-Stokes (iNS) Problems auf finiten Volumen (FV) arbeitet, auf einen FEM-Ansatz mit der Standardformulierung des (iNS) umgestellt und in eine Form gebracht, dass vorhandene Numeriksoftware (FeatFlow) damit arbeiten konnte.
Um die Bildwiederherstellung zu beschleunigen, wurde ein Programm geschrieben (Quellcode im Anhang B), das aus benutzergenerierten Angabe von Schadstellen in Form einer Bildmaske, automatisch optimierte Rechengitter generiert, welche zusammenhängende Segmente des ursprünglichen Tensorproduktgitters darstellen. Dabei traten bei der Erzeugung der Randparametrisierung (wird sowohl für Feast als auch FeatFlow benötigt) diverse Problemfälle auf, die aber programmintern durch zwei zusätzliche Arbeitsschritte behoben werden konnten. Dies reduzierte die Größe der zu rechnenden Probleme stark und führte zu einer starken Beschleunigung der Rechnungen.
Zusätzlich wurde der ursprüngliche Ansatz zur Bildwiederherstellung, durch eine Transformation des Farbraumes und damit verbundener Konzentration der Bildinformation mit höchster Korrelation in einen einzigen Farbkanal, auch auf Farbbilder erweitert. Die vorherige Transformation hilft dabei, eventuelle Farbverfälschungen zu minimieren.2016-01-01T00:00:00ZEfficient numerical methods for the simulation of particulate and liquid-solid flows
http://hdl.handle.net/2003/35169
Title: Efficient numerical methods for the simulation of particulate and liquid-solid flows
Authors: Münster, Raphael
Abstract: In this work a set of efficient numerical methods for the simulation of particulate flows and virtual prototyping applications are proposed. These methods are implemented as modular components in the FEATFLOW software package which is used as the fluid flow solver. In direct particulate flow simulations the calculation of the hydrodynamic forces acting on the particles is of central importance. For this task acceleration techniques are proposed based on hierarchical spatial partitioning. For arbitrary shaped particles the usage of distance maps to rapidly process the needed geometric information is employed and analyzed. In case of collisions between the particles it is shown how these same structures can be used to efficiently handle the collision broad phase and narrow phase. The computation of collision forces in the proposed particulate flow solving scheme can be handled by several collision models. The used models are based on a constrained-based formulation which leads to a linear complementarity problem (LCP). Another approach is added into the particulate flow solver that is based on the discrete element method (DEM). This approach is suited very well to an Implementation on graphic processing units (GPU) as the particles can be handled independently and thus excellent use of the massive parallel computing capabilities of the GPU can be made. In order to extend the DEM to handle non-spherical particles or rigid bodies, an inner sphere representation of such shapes is used. Furthermore, a mesh adaptation technique to increase the numerical efficiency of the CFD-simulations is shown which is based on Laplacian smoothing with special weights. The proposed techniques are validated in various benchmark configurations or comparisons to experimental data.2016-01-01T00:00:00ZNumerical techniques for the simulation of PDEs on surfaces for biomathematical problems
http://hdl.handle.net/2003/35134
Title: Numerical techniques for the simulation of PDEs on surfaces for biomathematical problems
Authors: Ali, Ramzan2016-03-01T00:00:00ZEfficient computations for multiphase flow problems using coupled lattice Boltzmann-level set methods
http://hdl.handle.net/2003/35115
Title: Efficient computations for multiphase flow problems using coupled lattice Boltzmann-level set methods
Authors: Safi, Seyed Mohammad Amin
Abstract: Multiphase flow simulations benefit a variety of applications in science and engineering as for
example in the dynamics of bubble swarms in heat exchangers and chemical reactors or in the
prediction of the effects of droplet or bubble impacts in the design of turbomachinery systems.
Despite all the progress in the modern computational fluid dynamics (CFD), such simulations still
present formidable challenges both from numerical and computational cost point of view.
Emerging as a powerful numerical technique in recent years, the lattice Boltzmann method
(LBM) exhibits unique numerical and computational features in specific problems for its ability
to detect small scale transport phenomena, including those of interparticle forces in multiphase
and multicomponent flows, as well as its inherent advantage to deliver favourable computational
efficiencies on parallel processors.
In this thesis two classes of LB methods for multiphase flow simulations are developed which
are coupled with the level set (LS) interface capturing technique. Both techniques are demonstrated
to provide high resolution realizations of the interface at large density and viscosity differences
within relatively low computational demand and regularity restrictions compared to the
conventional phase-field LB models. The first model represents a sharp interface one-fluid formulation,
where the LB equation is assigned to solve for a single virtual fluid and the interface
is captured through convection of an initially signed distance level set function governed by the
level set equation (LSE). The second scheme proposes a diffuse pressure evolution description
of the LBE, solving for velocity and dynamic pressure only. In contrast to the common kineticbased
solutions of the Cahn-Hilliard equations, the density is then solved via a mass conserving
LS equation which benefits from a fast monolithic reinitialization.
Rigorous comparisons against established numerical solutions of multiphase NS equations for
rising bubble problems are carried out in two and three dimensions, which further provide an
unprecedented basis to evaluate the effect of different numerical and implementation aspects of
the schemes on the overall performance and accuracy. The simulations are eventually applied
to other physically interesting multiphase problems, featuring the flexibility and stability of the
scheme under high Re numbers and very large deformations.
On the computational side, an efficient implementation of the proposed schemes is presented in
particular for manycore general purpose graphical processing units (GPGPU). Various segments
of the solution algorithm are then evaluated with respect to their corresponding computational
workload and efficient implementation outlines are addressed.2016-01-01T00:00:00ZHierarchical finite element methods for compressible flow problems
http://hdl.handle.net/2003/33994
Title: Hierarchical finite element methods for compressible flow problems
Authors: Bittl, Melanie
Abstract: The thesis is concerned with the introduction of the CG1-DG2 method and the design of an hp-adaptive algorithm in the context of convection-dominated problems in 2D. The CG1-DG2 method combines the continuous Galerkin (CG) method with the discontinuous Galerkin (DG) method by enriching the continuous linear finite element (CG1) space with discontinuous quadratic basis functions. The resulting finite element approximation is continuous at the vertices of the mesh but may be discontinuous across edges. Analysis of the CG1-DG2 discretization in the context of a scalar advection equation shows that the use of upwind-biased convective fluxes leads to an approximation which is stable and exhibits the same convergence rates as the quadratic discontinuous (DG2) method. However, the CG1-DG2 space has fewer degrees of freedom than the DG2 space. In the case of Poisson's equation different strategies known from the DG method can be adopted to approximate the numerical fluxes: the symmetric and non-symmetric interior penalty method as well as the Baumann-Oden method. A priori error estimates for the DG2 method can be shown to hold for the CG1-DG2 approximation as well. Numerical studies confirm that the proposed method is stable and converges at the same rate as the fully discontinuous piecewise-quadratic version. We also present an extension of the CG1-DG2 method to solve the Euler equations and show numerical results which indicate that the CG1-DG2 method gives results similar to those obtained by the DG method. The second part of this thesis presents an hp-adaptive framework for convection-dominated problems. The idea of this algorithm is to divide the mesh in smooth and non-smooth parts, where the smoothness refers to the regularity of the approximated solution. In smooth parts the polynomial degree is increased (p-adaptivity) whereas in non-smooth parts h-adaptivity for linear elements is used. Hereby, a parameter-free regularity estimator is used to determine the smoothness of a function and its gradient by comparing those with reconstructed approximations. In smooth elements the CG1-DG2 method is used. In non- smooth elements a flux-corrected transport scheme is applied and combined with h-adaptivity based on the so-called reference solution approach. Numerical experiments are performed for advection and advection-diffusion equations. Those show the advantage of the hp-adaptive algorithm over pure h-refinement in the context of FCT schemes.2014-01-01T00:00:00ZNumerical aspects of population balance equations coupled to computational fluid dynamics
http://hdl.handle.net/2003/33623
Title: Numerical aspects of population balance equations coupled to computational fluid dynamics
Authors: Bayraktar, Evren
Abstract: It can be the motion of clouds, the movement of a smoke plume, or the dynamics of fluids in processes which are interesting to food, petroleum, chemical, pharmaceutical and many other industries; they are all governed by the same physical laws: fluid dynamics and population balances.
Numerical solution of Population Balance Equations (PBE) coupled to Computational Fluid Dynamics (CFD) is a promising approach to simulate liquid/gas-liquid dispersed flows, for which the governing physical phenomena are breakup and coalescence of bubbles/droplets, additional to transport phenomena of fluids.
In the literature, there are many breakup and coalescence models to close the PBE. Unfortunately, there is no unified framework for these closures; and, it is one of our objectives: to determine appropriate coalescence and breakage kernels for liquid/gas-liquid dispersions.
Another objective is to investigate numerical techniques for one-way coupled CFD and PBE, and to develop a computational tool. The developed tool is based on the incompressible flow solver FeatFlow which is extended with Chien's Low-Reynolds number k-epsilon turbulence model and PBE.
The presented implementation ensures strictly conservative treatment of sink and source terms which is enforced even for geometric discretization of the internal coordinate. The validation of our implementation which covers a wide range of computational and experimental problems enables us to proceed into three-dimensional applications as, turbulent flows in a pipe and through static mixers.
Regarding the studies on static mixers, not only we have obtained numerical results; we have conducted comprehensive experimental studies in the Sulzer Chemtech Ltd. laboratories (Winterthur, Switzerland). The inclusive experimental results has offered a good ground for verifying the adopted mathematical models and numerical techniques.
The obtained satisfactory results in the studies for one-way coupled CFD and PBE has motivated us to study two-way coupled CFD-PBE models. The so far developed numerical recipe of which main ingredients are the method of classes, positivity-preserving linearization and the high-order FEM-AFC with FeatFlow including the standard k-epsilon solver has been extended to cover bubble induced turbulence and mixture-model with algebraic slip relation. A smart algorithm is developed, offering a compromise between the computational cost and the accuracy.
Numerical simulation of air-in-water dispersed phase systems in a flat bubble column which is, numerically, a very challenging case-study and is experimentally studied by Becker et al. has been performed with the developed computational tool. The dynamic movement of the bubble swarm which is observed in the experiments have been successfully simulated.
Keywords: computational fluid dynamics (CFD), population balances, coalescence, breakage, numerical solution, method of classes, parallel parent daughter classes, simulation, static mixers, multiphase flows.2014-09-18T00:00:00ZFictitious boundary and penalization methods for treatment of rigid objects in incompressible flows
http://hdl.handle.net/2003/33474
Title: Fictitious boundary and penalization methods for treatment of rigid objects in incompressible flows
Authors: Anca, Dan
Abstract: The Fictitious Boundary Method (FBM) and the Penalty Method (PM) for solving the incompressible Navier-Stokes equations modeling steady or unsteady incompressible flow around solid and rigid, non-deformable objects are presented and numerically analyzed and compared in this thesis. The proposed methods are finite element methods to simulate incompressible flows with small-scale time-(in)dependent geometrical details. The FBM, described and already validated in [1, 43, 48], is based on a finite element method background grid which covers the whole computational domain and is independent of the shape, number and size of any solid obstacle contained inside. The fluid part is computed by a multigrid finite element solver, while the behavior of the solid part is governed by the mechanics principles regarding motion and interactions of type fluid-solid, solid-solid or solid-wall collisions. A new treatment of imposing the Dirichlet boundary conditions for the case of immersed rigid boundary objects is proposed by using the penalization method as a more general framework then the FBM, but containing it as a special case. The new PM approach has a stronger mathematical background. In contrast to FBM, the PM does not imply a direct modification or artificial techniques over the matrix of the system of equations like the fictitious boundary method. A pairing of the penalty method with multigrid solvers is used, while the computational domain is fixed and needs no re-meshing during the simulations. However, the degree of geometrical details that the coarse mesh contains has an impact onto numerical results, a fact which will be investigated/ clarified in this thesis. The presented method is a finite element method, easy to be incorporated into standard CFD codes, for simulating particulate flow or, in general, flows with immersed time-(in)dependent and complicated shaped objects. The aim is to analyze and validate the penalty method and compare, qualitatively and quantitatively, with the already validated FBM regarding the aspects of accuracy of the solution, efficiency, robustness and behavior of the solvers. Different techniques to avoid the numerical difficulties that arise by using penalty method will be particularly described and analyzed.2014-07-08T00:00:00ZKommunikationsvermeidende und asynchrone Verfahren zur Lösung dünnbesetzter linearer Gleichungssysteme auf modernen Höchstleistungsrechnern
http://hdl.handle.net/2003/33103
Title: Kommunikationsvermeidende und asynchrone Verfahren zur Lösung dünnbesetzter linearer Gleichungssysteme auf modernen Höchstleistungsrechnern
Authors: Klinger, Marcel2014-05-09T00:00:00ZEfficient FEM and multigrid solvers for the least squares method with application to non-Newtonian fluid flows
http://hdl.handle.net/2003/32882
Title: Efficient FEM and multigrid solvers for the least squares method with application to non-Newtonian fluid flows
Authors: Nickaeen, Masoud2014-02-21T00:00:00ZNumerical analysis of collision models in 2D particulate flow
http://hdl.handle.net/2003/31135
Title: Numerical analysis of collision models in 2D particulate flow
Authors: Usman, Kamran2013-10-21T00:00:00ZAdvanced numerical treatment of chemotaxis driven PDEs in mathematical biology
http://hdl.handle.net/2003/30452
Title: Advanced numerical treatment of chemotaxis driven PDEs in mathematical biology
Authors: Strehl, Robert
Abstract: From the first formulation of chemotaxis-driven partial differential equations (PDEs) by Keller and Segel in the 1970's up to the present, much effort has been expended in modelling complex chemotaxis re- lated processes. The shear complexity of such resulting PDEs crucially limits the postulation of analytical results. In this context the sup- port by numerical tools are of utmost interest and, thus, render the implementation of a numerically well elaborated solver an undoubt- edly important task. In this work I present different iteration strategies (linear/nonlinear, decoupled/monolithic) for chemotaxis-driven PDEs. The discretiza- tion follows the method of lines, where I employ finite elements to resolve the spatial discretization. I extensively study the numerical efficiency of the iteration strategies by applying them on particular chemotaxis models. Moreover, I demonstrate the need of numerical stabilization of chemotaxis-driven PDEs and apply a exible scalar algebraic ux correction. This methodology preserves the positivity of the fully discretized scheme under mild conditions and renders the numerical solution non-oscillatory at a low level of additional compu- tational costs. This work provides a first detailed study of accurate, efficient and exible finite element schemes for chemotaxis-driven PDEs and the implemented numerical framework provides a valuable basis for fu- ture applications of the solvers to more complex models.2013-08-06T00:00:00ZImplicit, monolithic simulation of Newtonian and non-Newtonian flows on unstructured grids based on the Lattice Boltzmann equations
http://hdl.handle.net/2003/29736
Title: Implicit, monolithic simulation of Newtonian and non-Newtonian flows on unstructured grids based on the Lattice Boltzmann equations
Authors: Mahmood, Rashid2012-11-02T00:00:00ZNumerical analysis of new class of higher order Galerkin time discretization schemes for nonstationary incompressible flow problems
http://hdl.handle.net/2003/29660
Title: Numerical analysis of new class of higher order Galerkin time discretization schemes for nonstationary incompressible flow problems
Authors: Hussain, Shafqat2012-10-08T00:00:00ZNumerical simulation for viscoplastic fluids via finite element methods
http://hdl.handle.net/2003/29415
Title: Numerical simulation for viscoplastic fluids via finite element methods
Authors: El-Borhamy, Mohamed
Abstract: The design of efficient, robust and flexible numerical schemes to cope with nonlinear CFD problems has become the main nerve in the field of numerical simulation. This work has developed and analyzed the Newton-Multigrid process in the frame of monolithic approaches to solve stationary and nonstationary viscoplastic fluid problems. From the mathematical point of view, the viscoplastic problem exhibits several severe problems which might be arisen to draw the mathematical challenges. The major difficulty is the unbounded value of the viscosity which needs regularization. Several regularization techniques have been proposed to cope with this problem yet, while the accuracy is still not even close to be compared to the real model. Herein, two methods are used for the treatment of the non-differentiability, namely Bercovier-Engelman and modified bi-viscous models regularizations. To compute the solution at very small values of the regularization parameter which can be considered numerically as zero, we use the continuation technique.
Other difficulties would be addressed in the circle of the nonlinearity, the solenoidal velocity field, as well as the convection dominated problem which are typically involved in the standard Navier-Stokes equation. The use of mixed higher order finite element methods for flow problems is advantageous, since one can partially avoid the addition of stabilization terms to handle for instance the lack of coercivity, the domination of the convective part as well as the incompressibility. In the case of mixed lower order finite element methods, edge oriented stabilization has been introduced to provide results in the case of the lack of coercivity and convection dominated problems.
The main drawback of this stabilizer is to optimize or choose appropriately the free parameters to maintain high accuracy results from the scheme. Viscoplastic fluids are involved in many industrial applications which require numerical simulation to get a big mathematical insight and to predict the fluids behavior. The dependence of pressure on the viscoplastic constitutive law is confirmed as much as the dependence of velocity. Moreover, the behavior of the pressure is strongly related to the yield property for the unyielded regimes. In the case of a constant yield stress value together with the absence of the external densities, the field of pressure is prescribed by the null value wherever the null value of the deformation tensor is considered.
Real life examples to prescribe the behavior of the viscoplastic fluids might be described in case of standard benchmarks: viscoplastic flow in channel, viscoplastic flow in a lid driven cavity and viscoplastic flow around a cylinder. In each case we confirm the experimental and theoretical results which are used to analyze viscoplastic problems for the physical behavior with respect to the unyielded regimes and the cessation of time.2012-04-12T00:00:00ZA Hierarchical Flow Solver for Optimisation with PDE Constraints
http://hdl.handle.net/2003/29239
Title: A Hierarchical Flow Solver for Optimisation with PDE Constraints
Authors: Köster, Michael
Abstract: Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations.
The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes-- and Navier--Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level.
Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement.2011-12-21T00:00:00ZFinite element simulation techniques for incompressible fluid structure interaction with applications to bioengineering and optimization
http://hdl.handle.net/2003/28956
Title: Finite element simulation techniques for incompressible fluid structure interaction with applications to bioengineering and optimization
Authors: Razzaq, Mudassar
Abstract: Numerical techniques for solving the problem of fluid structure interaction with an
elastic material in a laminar incompressible viscous flow are described. An Arbitrary
Lagrangian-Eulerian (ALE) formulation is employed in a fully coupled monolithic
way, considering the problem as one continuum. The mathematical description and
the numerical schemes are designed in such a way that more complicated constitutive
relations (and more realistic for biomechanics applications) for the fluid as well as the
structural part can be easily incorporated. We utilize the well-known Q2P1 finite element
pair for discretization in space to gain high accuracy and perform as time-stepping
the 2nd order Crank-Nicholson, resp., Fractional-Step-q -scheme for both solid and
fluid parts. The resulting nonlinear discretized algebraic system is solved by a Newton
method which approximates the Jacobian matrices by a divided differences approach,
and the resulting linear systems are solved by iterative solvers, preferably of Krylovmultigrid
type.
For validation and evaluation of the accuracy of the proposed methodology, we present
corresponding results for a new set of FSI benchmarking configurations which describe
the self-induced elastic deformation of a beam attached to a cylinder in laminar channel
flow, allowing stationary as well as periodically oscillating deformations. Then, as an
example for fluid-structure interaction (FSI) in biomedical problems, the influence of
endovascular stent implantation onto cerebral aneurysm hemodynamics is numerically
investigated. The aim is to study the interaction of the elastic walls of the aneurysm
with the geometrical shape of the implanted stent structure for prototypical 2D configurations.
This study can be seen as a basic step towards the understanding of the
resulting complex flow phenomena so that in future aneurysm rupture shall be suppressed
by an optimal setting for the implanted stent geometry.
Keywords: Fluid-structure interaction (FSI), monolithic FEM, ALE, multigrid, incompressible
laminar flow, bio-engineering, optimization, benchmarking.2011-08-05T00:00:00ZFEM simulation of non-isothermal viscoelastic fluids
http://hdl.handle.net/2003/27788
Title: FEM simulation of non-isothermal viscoelastic fluids
Authors: Damanik, Hogenrich
Abstract: Thermo-mechanically coupled transport processes of viscoelastic fluids are important components in many applications in mechanical and chemical engineering. The aim of this thesis is the development of efficient numerical techniques for incompressible, non-isothermal, viscoelastic fluids which take into account the multiscale behaviour in space and time, the multiphase character and significant geometrical changes. Based on special CFD techniques including adaptivity/local grid alignment in space/time and fast hierarchical FEM techniques, the result shall be a new CFD tool which has to be evaluated w.r.t. well-known benchmarks and experimental results. The advantages of such a method are numerous and lead to efficient and accurate solution with respect to different rheological models and type of nonlinearities. In addition, it gives us a great deal of flexibility not only in dealing with well known
difficulties such as High Weissenberg Number Problem (HWNP) but also in treating any new rheological models in the coming future. Several benchmark problems of interest to industrial purposes are also proposed in validating the state of the art of the numerical method.2011-05-27T00:00:00ZA monolithic, off-lattice approach to the discrete Boltzmann equation with fast and accurate numerical methods
http://hdl.handle.net/2003/27715
Title: A monolithic, off-lattice approach to the discrete Boltzmann equation with fast and accurate numerical methods
Authors: Hübner, Thomas2011-04-28T00:00:00ZFast and accurate finite-element multigrid solvers for PDE simulations on GPU clusters
http://hdl.handle.net/2003/27243
Title: Fast and accurate finite-element multigrid solvers for PDE simulations on GPU clusters
Authors: Göddeke, Dominik
Abstract: Der wichtigste Beitrag dieser Dissertation ist es aufzuzeigen, dass Grafikprozessoren (GPUs) als Repräsentanten der Entwicklung hin zu Vielkern-Architekturen sehr gut geeignet sind zur schnellen und genauen Lösung großer, dünn besetzter linearer Gleichungssysteme, insbesondere mit parallelen Mehrgittermethoden auf heterogenen Rechenclustern. Solche Systeme treten bspw. bei der Diskretisierung (elliptischer) partieller Differentialgleichungen mittels finiter Elemente auf. Wir demonstrieren Beschleunigungsfaktoren von mindestens einer Größenordnung gegenüber konventionellen, hochoptimierten CPU-Implementierungen, ohne Verlust von Genauigkeit und Funktionsumfang. Im Detail liefert diese Dissertation die folgenden Beiträge:
Berechnungen in einfach genauer Fließkommadarstellung können für die hier betrachteten Problemklassen nicht ausreichen. Wir greifen die Methode gemischt genauer iterativer Verfeinerung (Nachiteration) wieder auf, um nicht nur die Genauigkeit von berechneten Lösungen zu verbessern, sondern vielmehr die Effizienz des Lösungsprozesses als ganzes zu steigern. Sowohl auf CPUs als auch auf GPUs demonstrieren wir eine deutliche Leistungssteigerung ohne Genauigkeitsverlust im Vergleich zur Berechnung in höherer Fliesskomma-Genauigkeit.
Wir präsentieren effiziente Parallelisierungstechniken für Mehrgitter-Löser auf Grafik-Hardware, insbesondere für numerisch starke Glätter und Vorkonditionierer, die für stark anisotrope Gitter und Operatoren geeignet sind. Ein Beispiel ist die Entwicklung einer effizienten Reformulierung des Verfahrens der zyklischen Reduktion für die Lösung tridiagonaler Gleichungssysteme. Im Hinblick auf Hardware-orientierte Numerik analysieren wir sorgfältig den Kompromiss zwischen numerischer und Laufzeit-Effizienz für inexakte Parallelisierungstechniken, die einige der inhärent sequentiellen Charakteristiken solcher starker Glätter zugunsten besserer Parallelisierungseigenschaften entkoppeln.
Die Reimplementierung großer, etablierter Softwarepakete zur Anpassung auf neue Hardwareplattformen ist oft inakzeptabel teuer. Wir entwickeln einen "minimalinvasiven" Zugang zur Integration von Co-Prozessoren wie GPUs in FEAST, einem exemplarischen finite Elemente Diskretisierungs- und Löserpaket. Der Hauptvorteil unserer Technik ist, dass Applikationen, die auf FEAST aufsetzen, nicht geändert werden müssen um von der Beschleunigung durch solche Co-Prozessoren zu profitieren. Wir evaluieren unseren Zugang auf großen GPU-beschleunigten Rechenclustern für klassische Benchmarkprobleme aus der linearisierten Elastizität und der Simulation stationärer laminarer Strömungsvorgänge, und beobachten gute Beschleunigungsfaktoren und gute schwache Skalierbarkeit. Die maximal erreichbare Beschleunigung wird zudem analysiert und theoretisch modelliert, um bspw. Vorhersagen treffen zu können.
Weiterhin fassen wir die historische Entwicklung des Forschungsgebiets "wissenschaftliches Rechnen auf Grafikhardware" seit 2001/2002 zusammen, d.h. die Entwicklung von GPGPU als obskures Nischenthema hin zum fachübergreifenden Einsatz heute. Die Darstellung umfasst gleichermaßen die Hardware und das Programmiermodell und beinhaltet eine ausgiebige Bibliografie von Veröffentlichungen im Bereich der Simulation von PDE-Problemen auf GPUs.; The main contribution of this thesis is to demonstrate that graphics processors (GPUs) as representatives of emerging many-core architectures are very well-suited for the fast and accurate solution of large sparse linear systems of equations, using parallel multigrid methods on heterogeneous compute clusters. Such systems arise for instance in the discretisation of (elliptic) partial differential equations with finite elements. We report on at least one order of magnitude speedup over highly-tuned conventional CPU implementations, without sacrificing neither accuracy nor functionality. In more detail, this thesis includes the following contributions:
Single precision floating point computations may be insufficient for the class of problems considered in this thesis. We revisit mixed precision iterative refinement techniques to not only increase the accuracy of computed results, but also to increase the efficiency of the solution process. Both on CPUs and on GPUs, we demonstrate a significant performance improvement without loss of accuracy compared to computing in high precision only.
We present efficient parallelisation techniques for multigrid solvers on graphics hardware, in particular for numerically strong smoothers and preconditioners that are suitable for highly anisotropic grids and operators. For instance, an efficient formulation of the cyclic reduction algorithm to solve tridiagonal systems is developed. In view of hardware-oriented numerics, we carefully analyse the trade-off between numerical and runtime performance for inexact parallelisation techniques that decouple some of the inherently sequential characteristics of strong smoothing operators.
For large-scale established software frameworks, the re-implementation tailored to novel hardware platforms is often prohibitively expensive. We develop a 'minimally invasive' approach to integrate support for co-processor hardware like GPUs into FEAST, a finite element discretisation and solver toolbox. Our technique has the major advantage that applications built on top of the toolbox do not have to be changed at all to benefit from co-processor acceleration. The approach is evaluated for benchmark problems in linearised elasticity and stationary laminar flow computed on large-scale GPU-enhanced clusters. Good speedup factors and near-ideal weak scalability are observed. The achievable speedup is analysed and a theoretical speedup model is presented.
Finally, we provide a historical overview of scientific computing on graphics hardware since the early beginnings in 2001/2002, when GPGPU was an obscure research topic pursued by few, to the widespread adoption nowadays. We discuss the evolution of the hardware and the programming model, and provide a comprehensive bibliography of publications related to PDE simulations on GPUs.2010-05-26T10:11:45ZImplicit finite element schemes for compressible gas and particle-laden gas flows
http://hdl.handle.net/2003/27002
Title: Implicit finite element schemes for compressible gas and particle-laden gas flows
Authors: Gurris, Marcel2010-03-29T11:59:23ZEfficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems
http://hdl.handle.net/2003/26998
Title: Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems
Authors: Wobker, Hilmar
Abstract: Bei der Simulation realistischer strukturmechanischer Probleme können Gleichungssysteme mit mehreren hundert Millionen Unbekannten entstehen. Für die effiziente Lösung solcher Systeme sind parallele Multilevel-Methoden unerlässlich, die in der Lage sind, die Leistung moderner Hardware-Technologien auszuschöpfen. Die Finite-Elemente- und Löser-Toolbox FEAST, die auf die Behandlung skalarer Gleichungen ausgelegt ist, verfolgt genau dieses Ziel. FEAST kombiniert Hardware-orientierte Implementierungstechniken mit einer Multilevel-Gebietszerlegungsmethode namens ScaRC. In der vorliegenden Arbeit wird ein Konzept entwickelt, multivariate Elastizitätsprobleme basierend auf der FEAST-Bibliothek zu lösen. Die generelle Herangehensweise besteht darin, die Lösung multivariater Probleme auf die Lösung einer Reihe von skalaren Problemen zurückzuführen. Dieser Ansatz ermöglicht eine strikte Trennung von skalaren "low level" Kernfunktionalitäten (in Form der FEAST-Bibliothek) und multivariatem "high level" Anwendungscode (in Form des Elastizitätsproblems), was aus Sicht der Softwareentwicklungstechnik sehr vorteilhaft ist: Alle Bemühungen zur Verbesserung der Hardware-Effizienz, sowie Anpassungen an zukünftige technologische Entwicklungen können auf skalare Operationen beschränkt werden, während die multivariate Anwendung automatisch von diesen Erweiterungen profitiert. Im ersten Teil der Arbeit werden substantielle Verbesserungen der skalaren ScaRC-Löser entwickelt, die dann als essentielle Bausteine zur Lösung multivariater Elastizitätsprobleme eingesetzt werden. Ausführliche numerische Untersuchungen zeigen, wie sich die Effizienz der skalaren FEAST-Bibliothek auf den multivariaten Lösungsprozess überträgt. Die Löserstrategie wird dann auf nichtlineare Probleme der Elastizität mit finiter Deformation angewandt. Durch Einsatz einer Liniensuche-Methode wird die Robustheit des Newton-Raphson-Verfahrens signifikant erhöht. Es werden verschiedene Strategien miteinander verglichen, wie genau die linearen Probleme innerhalb der nichtlinearen Iteration zu lösen sind. Zur Behandlung der wichtigen Klasse von (fast) inkompressiblen Materialien wird eine gemischte Verschiebung/Druck-Formulierung gewählt, die mit Hilfe von stabilisierten bilinearen finiten Elementen (Q1/Q1) diskretisiert wird. Eine erweiterte Version der klassischen "Druck-Poisson"-Stabilisierung wird präsentiert, die auch für hochgradig irreguläre Gitter geeignet ist. Es werden Vor- und Nachteile der Q1/Q1-Diskretisierung erörtert, insbesondere in Bezug auf zeitabhängige Rechnungen. Zwei Löser-Klassen zur Behandlung der entstehenden Sattelpunkt-Probleme werden beschrieben und miteinander verglichen: einerseits verschiedene Arten von (beschleunigten) entkoppelten Lösern (Uzawa, Druck-Schurkomplement-Methoden, Block-Vorkonditionierer), andererseits gekoppelte Mehrgitter-Verfahren mit Vanka-Glättern. Effiziente Schurkomplement-Vorkonditionierer, die für die erste Löser-Klasse notwendig sind, werden im Rahmen statischer und zeitabhängiger Probleme besprochen. Die zentrale Strategie, die Lösung multivariater Systeme auf die Lösung skalarer Systeme zu reduzieren, ist nur im Falle der entkoppelten Lösungsmethoden anwendbar. Es wird gezeigt, dass für die Klasse der Elastizitätsprobleme, die in dieser Arbeit behandelt werden, die entkoppelten Löser den gekoppelten hinsichtlich numerischer und paralleler Effizienz deutlich überlegen sind.; In the simulation of realistic solid mechanical problems, linear equation systems with hundreds of million unknowns can arise. For the efficient solution of such systems, parallel multilevel methods are mandatory that are able to exploit the capabilities of modern hardware technologies. The finite element and solution toolbox FEAST, which is designed to solve scalar equations, pursues exactly this goal. It combines hardware-oriented implementation techniques with a multilevel domain decomposition method called ScaRC that achieves high numerical and parallel efficiency. In this thesis a concept is developed to solve multivariate elasticity problems based on the FEAST library. The general strategy is to reduce the solution of multivariate problems to the solution of a series of scalar problems. This approach facilitates a strict separation of 'low level' scalar kernel functionalities (in the form of the FEAST library) and 'high level' multivariate application code (in the form of the elasticity problem), which is very attractive from a software-engineering point of view: All efforts to improve hardware-efficiency and adaptations to future technology trends can be restricted to scalar operations, and the multivariate application automatically benefits from these enhancements. In the first part of the thesis, substantial improvements of the scalar ScaRC solvers are developed, which are then used as essential building blocks for the efficient solution of multivariate elasticity problems. Extensive numerical studies demonstrate how the efficiency of the scalar FEAST library transfers to the multivariate solution process. The solver strategy is then applied to treat nonlinear problems of finite deformation elasticity. A line-search method is used to significantly increase the robustness of the Newton-Raphson method, and different strategies are compared how to choose the accuracy of the linear system solves within the nonlinear iteration. In order to treat the important class of (nearly) incompressible material, a mixed displacement/pressure formulation is used which is discretised with stabilised bilinear finite elements (Q1/Q1). An enhanced version of the classical 'pressure Poisson' stabilisation is presented which is suitable for highly irregular meshes. Advantages and disadvantages of the Q1/Q1 discretisation are discussed, especially in the context of transient computations. Two solver classes for the resulting saddle point systems are described and compared: on the one hand various kinds of (accelerated) segregated solvers (Uzawa, pressure Schur complement methods, block preconditioners), and on the other hand coupled multigrid solvers with Vanka-smoothers. Efficient Schur complement preconditioners, which are required for the former class, are discussed for the static and the transient case. The main strategy to reduce the solution of multivariate systems to the solution of scalar systems is only applicable in the case of segregated methods. It is shown that for the class of elasticity problems considered in this thesis, segregated solvers are clearly superior to Vanka-type solvers in terms of numerical and parallel efficiency.2010-03-24T10:08:12ZAnalysis and numerical realisation of discrete projection methods for rotating incompressible flows
http://hdl.handle.net/2003/26005
Title: Analysis and numerical realisation of discrete projection methods for rotating incompressible flows
Authors: Sokolov, Andriy2009-01-22T08:24:51ZAdaptive high-resolution finite element schemes
http://hdl.handle.net/2003/25933
Title: Adaptive high-resolution finite element schemes
Authors: Möller, Matthias
Abstract: The numerical treatment of flow problems by the finite element method
is addressed. An algebraic approach to constructing high-resolution
schemes for scalar conservation laws as well as for the compressible
Euler equations is pursued. Starting from the standard Galerkin
approximation, a diffusive low-order discretization is constructed by
performing conservative matrix manipulations. Flux limiting is
employed to compute the admissible amount of compensating
antidiffusion which is applied in regions, where the solution is
sufficiently smooth, to recover the accuracy of the Galerkin finite
element scheme to the largest extent without generating non-physical
oscillations in the vicinity of steep gradients. A discrete Newton
algorithm is proposed for the solution of nonlinear systems of
equations and it is compared to the standard fixed-point defect
correction approach. The Jacobian operator is approximated by divided
differences and an edge-based procedure for matrix assembly is devised
exploiting the special structure of the underlying algebraic flux
correction (AFC) scheme. Furthermore, a hierarchical mesh adaptation
algorithm is designed for the simulation of steady-state and transient
flow problems alike. Recovery-based error indicators are used to
control local mesh refinement based on the red-green strategy for
element subdivision. A vertex locking algorithm is developed which
leads to an economical re-coarsening of patches of subdivided
cells. Efficient data structures and implementation details are
discussed. Numerical examples for scalar conservation laws and the
compressible Euler equations in two dimensions are presented to assess
the performance of the solution procedure.; In dieser Arbeit wird die numerische Simulation von skalaren
Erhaltungsgleichungen sowie von kompressiblen Eulergleichungen mit
Hilfe der Finite-Elemente Methode behandelt. Dazu werden
hochauflösende Diskretisierungsverfahren eingesetzt, welche auf
algebraischen Konstruktionsprinzipien basieren. Ausgehend von der
Galerkin-Approximation wird eine Methode niedriger Ordnung
konstruiert, indem konservative Matrixmodifikationen durchgeführt
werden. Anschließend kommt ein sog. Flux-Limiter zum Einsatz, der in
Abhängigkeit von der lokalen Glattheit der Lösung den zulässigen
Anteil an Antidiffusion bestimmt, die zur Lösung der Methode niedriger
Ordnung hinzuaddiert werden kann, ohne dass unphysikalische
Oszillationen in der Nähe von steilen Gradienten entstehen. Die
resultierenden nichtlinearen Gleichungssysteme können entweder mit
Hilfe von Fixpunkt-Defektkorrektur-Techniken oder mittels diskreter
Newton-Verfahren gelöst werden. Für letztere wird die Jacobi-Matrix
mit dividierten Differenzen approximiert, wobei ein effizienter,
kantenbasierter Matrixaufbau aufgrund der speziellen Struktur der
zugrunde liegenden Diskretisierung möglich ist. Ferner wird ein
hierarchischer Gitteradaptionsalgorithmus vorgestellt, welcher sowohl
für die Simulation von stationären als auch zeitabhängigen Strömungen
geeignet ist. Die lokale Gitterverfeinerung folgt dem bekannten
Rot-Grün Prinzip, wobei rekonstruktionsbasierte Fehlerindikatoren zur
Markierung von Elementen zum Einsatz kommen. Ferner erlaubt das
sukzessive Sperren von Knoten, die nicht gelöscht werden können, eine
kostengünstige Rückvergröberung von zuvor unterteilten Elementen. In
der Arbeit wird auf verschiedene Aspekte der Implementierung sowie auf
die Wahl von effizienten Datenstrukturen zur Verwaltung der
Gitterinformationen eingegangen. Der Nutzen der vorgestellten
Simulationswerkzeuge wird anhand von zweidimensionalen
Beispielrechnungen für skalare Erhaltungsgleichungen sowie für die
kompressiblen Eulergleichungen analysiert.2008-12-09T12:38:50ZNumerical simulation of immiscible fluids with FEM level set techniques
http://hdl.handle.net/2003/24967
Title: Numerical simulation of immiscible fluids with FEM level set techniques
Authors: Hysing, Shu-Ren
Abstract: Multiphase flows, including free surface and two-phase flows, are commonly encountered in many industrial applications. Effects of wave phenomena are for example important when designing boats, drop formation is essential for ink jet printers, and bubbles may come into play in chemical reactors and heat
exchangers.
Numerical simulation of these phenomena is a complex and challenging task.
The desire to have both high accuracy and computational speed often stands in direct contradiction to each other. The aim of this thesis was to describe a suitable methodology with potential to be both very accurate and also efficient. High resolution benchmarks were also developed in order to validate and quantify the performance of a code (TP2D) developed according to the presented approach. The developed methodology combined a non-conforming finite element discretization with the level set method for tracking the interfaces. A semi-implicit
approach to implementing surface tension forces was also derived, which allowed
for significantly larger time steps in comparison with the traditional explicit approach. The benchmarks were used to compare the developed code with two commercial
CFD codes (Comsol Multiphysics and Ansys Fluent). The commercial codes did not show strong convergence towards the reference solution, in contrast to TP2D which was both faster and significantly more accurate. TP2D
even computed a more accurate solution on the very coarsest grid compared to the best results of the commercial codes.2008-01-29T11:06:42ZFinite element simulation of nonlinear fluids with application to granular material and powder
http://hdl.handle.net/2003/22234
Title: Finite element simulation of nonlinear fluids with application to granular material and powder
Authors: Ouazzi, Abderrahim2006-03-15T07:20:13ZRobuste lineare und nichtlineare Lösungsverfahren für die inkompressiblen Navier-Stokes-Gleichungen
http://hdl.handle.net/2003/2308
Title: Robuste lineare und nichtlineare Lösungsverfahren für die inkompressiblen Navier-Stokes-Gleichungen
Authors: Schmachtel, Rainer
Abstract: Gegenstand der vorliegenden Dissertation ist die Entwicklung von robusten numerischen Verfahren zur Lösung von strömungsmechanischen Problemen für inkompressible Fluide.Bei der Auflösung physikalischer Randschichten verwendet man häufig Gitter mit äußerst langgestreckten Zellen. Durch diese entstehen Probleme bei der Diskretisierung sowie bei der Verwendung von linearen Mehrgitterverfahren. Weiterhin bereiten Viskositäten oder Dichten, die starke Sprünge aufweisen, den numerischen Verfahren Probleme.Ziel der vorliegenden Arbeit ist die Entwicklung eines numerischen Verfahrens, das robust genug ist, um mit diesen Schwierigkeiten fertig zu werden, und dabei nicht auf den Vorteil der hohen Geschwindigkeit linearer Mehrgittermethoden verzichtet. Die Untersuchungen haben gezeigt, daß dafür von der Diskretisierung über den Gittertransfer bis hin zum verwendeten Glättungsverfahren nahezu alle numerischen Komponenten stabilisiert werden müssen. Ein besonderes Augenmerk liegt dabei auf dem Glätter im Mehrgitteralgorithmus, wofür ein geblocktes 'Locales-Druck-Schur-Komplement Verfahren' (LMPSC-Local Multilevel Pressure Schurcomplement)entwickelt worden ist.Ein zweiter Schwerpunkt der Arbeit ist die effektive numerische Behandlung der Nichtlinearität in den Gleichungen. Der Einsatz des klassischen Newton-Verfahrens scheitert häufig daran, daß die entstehende linearisierten Gleichungssysteme nicht mehr numerisch lösbar sind, während das Konvergenzverhalten einfacher Fixpunktiterationen sehr schnell degeneriert. Die hier gefundene Lösung besteht aus einer Art Interpolation der beiden Verfahren, bei der hervorragende (nichtlineare) Konvergenzeigenschaften erzielt werden, die entstehendenlinearen Gleichungssysteme -unter Verwendung des LMPSC-Glätters- jedoch einfach lösbar bleiben.Schließlich wird die Erweiterung der Methoden auf die Boussinesq-Approximation erläutert, undderen Effektivität auch für diese Problemstellungen mit numerischen Tests belegt.; The subject of this thesis is the development of robust numerical methods for the solution of fluidmechanical problems for incompressible fluids.When resolving physical boundary layers, grids with long stretched elements are commonly used. These elements are the cause of problems concerning the discretization as well as the liner multigrid method. Furthermore, strongly jumping viscosity or density lead to problems with the numerical algorithms.Goal of the present work is the development of an effective numerical method, that is robust enoughto deal with these difficulties, but at the same time maintains the high speed of linear multigrid methods.The examinations have shown, that for this purpose, all numerical components have to be stabilized,especially the discretization, the grid transfer and the smoothing algorithms in the multigrid method.Special attention has been turned on the smoothing procedure, where a block-oriented 'Local-Presure-Schur-Complement' (LMPSC) method has been developed.The second central point of this thesis is the effective treatment of the nonlinearity in the equations.More often than not, classical Newton-methods are not applicable, since the resultinglinear systems can not be inverted, while the convergence behavior of classical fixed-point iterations quickly degenerates. The solution to overcome this is a kind of interpolation between the two methods.So, excellent (nonlinear) convergence properties can be achieved, while at the same time the resulting linear systems can be easily solved -using the LMPSC-smoother developed above. Finally, the extension of these methods to the Boussinesq-approximation is explained and the effectivity for this configuration is affirmed by various numerical tests.2003-11-25T00:00:00ZSCARC
http://hdl.handle.net/2003/2307
Title: SCARC
Authors: Kilian, Susanne
Abstract: In der vorliegenden Dissertation wird eine Brücke zwischen effienter Numerik und hardwareorientiertem Software-Design geschlagen, um die präzise Simulation komplexer Probleme aus dem industriellen Anwendungsbereich mit Schwerpunkt Strömungsmechanik zu gewährleisten. Als Prototyp für elliptische Differentialgleichungen 2. Ordnung ist in diesem Zusammenhang die Poisson-Gleichung von großer Bedeutung, deren effiziente und robuste parallele Lösung auf Basis von Diskretisierungen durch finite Elemente im Mittelpunkt der Arbeit steht. Im Rahmen der betrachteten Problemklassen ist man typischerweise mit sehr komplizierten Rechengebieten konfrontiert, deren knotenoptimale Diskretisierung häufig in ausgesprochen großen Gitteranisotropien resultiert. Dies stellt höchste Anforderungen an die Robustheit der verwendeten Löser. Gleichzeitig handelt es sich um äußerst hochdimensionale Probleme mit mehreren Millionen Unbekannten in Ort und Zeit, deren effiziente Lösung nur auf Hochleistungscomputern insbesondere vom Typ Parallelrechner erfolgen kann, was eine entsprechende `parallele' Anpassung der betrachteten Verfahren voraussetzt. Nicht zuletzt ist eine maximale Ausbeute potentieller Rechnerleistung nur durch ein detailliertes Verständnis wesentlicher Hard- und Software-Konzepte möglich. Als wichtige Schlüsseltechniken sind hier Datenlokalität, optimale Cache-Ausnutzung und Parallelisierung zu nennen. Zur Lösung der resultierenden Gleichungssysteme wird das verallgemeinerte Gebietszerlegungs- und Mehrgitterkonzept ScaRC (Scalable Recursive Clustering) vorgestellt, das die Vorteile von Gebietszerlegungsverfahren (hohe parallele Effizienz) und Mehrgitterver- fahren (hohe numerische Effizienz) in geeigneter Weise kombiniert. ScaRC basiert grundlegend auf der Kombination eines globalen, datenparallelisierten Mehrgitterverfahrens mit der blockweisen Glättung durch optimierte lokale Mehrgitterverfahren. Umfangreiche Testreihen belegen die Robustheit dieses Ansatzes, sofern lokale Anisotropien innerhalb einzelner Teilgebiete versteckt werden. Die Qualität der lokalen Lösungsprozesse führt zu einer nachhaltigen Verbesserung des globalen Konvergenzverhaltens. Durch die weitgehende lokale Beschränkung auf verallgemeinerte Tensorprodukt-Gitter in Kombination mit hochregulären Datenstrukturen bzw. optimierter Linearer Algebra werden hohe lokale MFlop/s- Raten erzielt, die durch entsprechende parallele Techniken und Datenstrukturen auch auf das globale (parallele) Problem übertragen werden können. Insgesamt resultiert eine hohe laufzeittechnische Effizienz, was durch praxisrelevante Testreihen untermauert wird.2003-01-08T00:00:00Z