Lehrstuhl III Angewandte Mathematik und Numerik

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Efficient numerical and algorithmic realization of a time-simultaneous Pressure-Schur-Complement solver for the incompressible Navier-Stokes equations in FEAT3
(2024) Arndt, Mirco; Turek, Stefan; Sokolov, Andriy
The aim of this thesis is to develop, implement and validate a fast time-simultaneous solver for computational fluid dynamic (CFD) problems, which is optimized on high-performance computing (HPC) architectures to efficiently exploit numerous cores through a parallelization in space and time.The sequential-in-time and parallel-in-space projection method approach, which decouples velocity and pressure, serves as the basis for the development of the new solver approach. The Crank-Nicolson scheme is used as time stepping scheme. Since the structure and settings of the new solver approach are not clear, especially with respect to a convective term such as in the Navier-Stokes equations, fundamental investigations must first be carried out. In order to explore the necessary requirements and characteristics of the new solver approach, the first step is to look at the theory for general convergence criteria. From the theory it was determined that a divergence-free velocity is a necessary condition for convergence. Two different divergence-free approaches have been implemented and tested for the new time-simultaneous solver: The augmented Lagrangian method and a newly developed update similar to the projection method update, which only approximately guarantees a divergence-free velocity. Both approaches lead to a reduced number of iterations until convergence, but in terms of computational time, the approximate divergence-free update is significantly more efficient and leads to a considerable reduction in computational time. With the newly developed approximate update approach, thousands of time steps of the Navier-Stokes equations can be computed parallel-in-space and -time. In the time-simultaneous case, the choice of the right-hand side for the pressure Poisson problem is not clear and has a significant importance. Numerical results verify that a divergence-free velocity component from the previous time step with a non-divergence-free component from the current time step ensures the most optimal convergence behavior. By ensuring a approximate divergence-free velocity, even in the case of the incompressible Navier-Stokes equations, it is possible to compute large time intervals time-simultaneously. A multigrid in time approach leads to a faster achievement of a divergence-free velocity for large time-intervals, since the approximate divergence-free velocity is not lost as quickly with larger time steps. It has been investigated that the multigrid in time approach only achieves a speed up in combination with a multigrid in space solver. To compute a large number of time steps, especially in 3D, the memory usage must be taken into account. An embedding of the solver in a memory-optimized local recoupling approach using FGMRES has been investigated, which can further enforce convergence. A local recoupling with FGMRES is not recommended because the performance in terms of the computational time is insufficient. To enforce convergence, a global recoupling seems to be the better approach. Although it ensures fewer iterations and can speed up the computation, it uses too much memory. Embedding the solver in a recoupling approach is not necessary and can be neglected, as it also requires additional programming effort, such as new data types. Another question was whether a fast existing solver for the pressure Poisson problem using prehandling could be integrated in the solver. The fast solver needs the Q2/Q1 finite element with the true Laplacian matrix. The theory already provides that the Q2/Q1 finite element is more prone to problems than the Q2/P1 finite element. Therefore, it is not recommended to integrate the solver, as in this case a recoupling approach using GMRES is necessary to achieve convergence. In conclusion, the newly developed solver is decoupled from velocity and pressure, as well as from time and space. However, to achieve convergence, a divergence-free velocity must be ensured at least approximately at each time point. A time-simultaneous solver is much faster than a sequential-in-time solver, which can only compute parallel-in-space. To have direct access to the different cores on a computer, an implementation in C++ (FEAT3) is recommended.
Adaptive time step control for global-in-time Galerkin-Petrov discretizations of evolution equations in the context of incompressible flows
(2024) Wambach, Lydia Carmen; Turek, Stefan; Schieweck, Friedhelm
The goal of adaptive time step control is to efficiently control the accuracy of a simulation. In terms of accuracy, a higher order time-stepping scheme is advantageous, although it results in a higher computational cost. In particular, High Performance Computing (HPC) focuses on efficient performance by using an ever-increasing number of cores and thus applying numerical schemes in parallel. Therefore, global-in-time adaptive time step control based on a higher order time discretization scheme is of interest, especially in the area of Computational Fluid Dynamics (CFD). In this work, we use the higher order continuous Galerkin-Petrov method as a time discretization scheme in a global-in-time adaptive time step control. The length of the time steps is controlled by so-called controllers based on error estimators. These errors are estimated by two approximations of the solution of different order. Here, we use a linear post-processing step from the continuous Galerkin-Petrov method with low computational cost to obtain a higher order solution. Although this is common in the literature on no-pressure problem types, we present a velocity and a new pressure error estimator for the Navier-Stokes equations. These error estimators approximate the analytic errors suitable, as we show for the heat equation and for incompressible flows as in the Navier- Stokes equations. Especially, for the flow around a cylinder benchmark for Newtonian and non-Newtonian fluids, we present error estimators for the lift and drag coefficients. All numerical tests in this thesis have been implemented in the FEAT3 software. We also introduce a new global-in-time adaptive strategy. This provides a time-parallel approach. In order to deepen this topic, we briefly introduce two existing global-in-time approaches and discuss a possible realization for the cGP(2) adaptive time-stepping, but no parallel numerical investigation is done. The new global-in-time adaptive strategy defines a new time grid in each adaptive iteration and computes the associated solution over the entire time interval. The classic step size controllers described in the literature are adapted to take into account error estimates for pressure, lift, and drag coefficients. Investigations and comparisons follow in the numerical studies for the different model problems, where advisable controllers are highlighted.
Cattaneo–Christov double diffusion model for the entropy analysis of a non-Darcian MHD Williamson nanofluid
(2024-07-31) Sagheer, M:; Sajid, Z.; Hussain, Shafqat; Shahzad, H.
This research is carried out to observe the fluid flow across a stratified sheet in the presence of non-linear thermal radiation. Through the process of similarity transformations, the partial differential equations (PDEs) governing the flow model are transformed into ordinary differential equations (ODEs), which are then numerically solved using the shooting method. This study comprehensively examines the influence of various flow parameters, including the inertial coefficient, magnetic parameter, Brownian motion parameter, radiation parameter, Prandtl number, thermophoresis parameter, and Brinkman number, on key thermophysical characteristics, such as the skin friction coefficient, and the rates of heat, mass, and entropy generation. Notably, the relative difference in the skin-friction coefficient increases with the Weissenberg number, ranging from approximately 1.8 to 2.5. The results indicate that reducing the Cattaneo–Christov temperature parameter decreases the temperature profile while increasing the concentration profile, whereas entropy generation initially rises with increasing Weissenberg number, but decreases near the surface.
Analysis of algebraic flux correction schemes for semi-discrete advection problems
(2023-01-30) Hajduk, Hennes; Rupp, Andreas
Based on recent developments regarding the analysis of algebraic flux correction schemes, we consider a locally bound-preserving discretization of the time-dependent advection equation. Specifically, we analyze a monolithic convex limiting scheme based on piecewise (multi-)linear continuous finite elements in the semi-discrete formulation. To stabilize the discretization, we use low order time derivatives in the definition of raw antidiffusive fluxes. Our analytical investigation reveals that their limited counterparts should satisfy a certain compatibility condition. The conducted numerical experiments suggest that this prerequisite is satisfied unless the size of mesh elements is vastly different.We prove global-in-time existence of semi-discrete approximations and derive an a priori error estimate for finite time intervals with a worst-case convergence rate of 1 2 w. r. t. the L2 error. This rate is optimal in the setting under consideration because we allow all correction factors of the flux-corrected scheme to become zero. In this case, the algorithm reduces to the bound-preserving discrete upwinding method but the limited counterpart of this scheme converges much faster, in practice. Additional numerical experiments are performed to verify the provable convergence rate for a few variants of the scheme.
On the design of global-in-time Newton-multigrid-pressure Schur complement solvers for incompressible flow problems
(2023-06-26) Lohmann, Christoph; Turek, Stefan
In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.
Simulation techniques for viscoelastic fluids with zero solvent viscosity based on three field approaches
(2024) Ahmad, Rida; Turek, Stefan; Sokolov, Andriy
Solving viscoelastic fluid flow problems is challenging task due to their complex behavior, which involves the coupling of elastic and viscous stresses in a highly nonlinear manner. Additionally, numerically simulating pure polymer melts is particularly challenging due to the absence of solvent contributions to viscosity in the standard viscoelastic model. The absence of a diffusive operator in the momentum equation prevents the problem from being addressed in a decoupled manner and imposes limitations on the application of solution methods, generally rendering multigrid solvers impractical with a monolithic approach. This thesis aims to present a finite element method for solving the two-dimensional three-field Stokes flow for pure polymer melts, using the Elastic Viscous Stress Splitting (EVSS) and Tensor Stokes formulation. The formulation is expressed in terms of velocity, pressure, and the stress tensor. Both the EVSS and Tensor Stokes formulations help to reintroduce velocity coupling into the momentum equation by applying a change of variables in the standard viscoelastic formulation. This approach enables the problem to be handled in a decoupled manner and facilitates the application of multigrid solution methods using a monolithic approach. Nevertheless, this change of variables introduces additional terms with second-order velocity derivatives in the convective part of the constitutive equation for stress. To address this, the four-field approach is often employed, which includes the deformation tensor as an additional field to manage higher-order derivatives. The convective term is reformulated by taking into account the divergence-free nature of the velocity field, shifting the higher-order derivatives to the test function in the weak formulation thus maintaining the problem size to three field. The velocity, pressure, and stress are discretized using the higher-order disc stable FEM triplet (Q2,P1 ,Q3). The proposed scheme is evaluated using the Oldroyd-B, Giesekus, and PTT exponential fluids, employing both decoupled and monolithic solution approaches. Numerical results are obtained for a four-to-one curved contraction, for highly viscoelastic fluids with the aim to achieve results at relatively large values of the relaxation parameter λ and observe the shear-thinning effect w.r.t. the relaxation parameter λ.
Efficient Newton-multigrid FEM solver for multifield nonlinear coupled problems applied to thixoviscoplastic flows
(2023-09-06) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
This note is concerned with efficient Newton-multigrid finite element method (FEM) solver for multifield nonlinear flow problems. In our approach, for efficient FEM solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to obtain the numerical solutions. We concretize our ideas on thixoviscoplastic flow problems. It is a two-field coupled nonlinear problem. And beside the integrated nonlinearity within momentum and microstructure equations, thixoviscoplastic problems induce a nonlinear two-way coupling. As far as FEM numerical solutions are concerned, we set the problem in a suitable variational form to use the corresponding wellposedness analysis to develop FEM techniques for the solver. Indeed, the wellposedness study is not an intellectual exercise, rather it is the foundation for the approximate thixoviscoplastic problem as well as for the development of an efficient solver. We base our investigations for the solver on our wellposedness and error analysis results of thixoviscoplastic flow problems published in Proc. Appl. Math. Mech. We continue our series, and proceed to develop a monolithic Newton-multigrid thixoviscoplastic solver. The solver is based on Newton's method and geometric multigrid techniques to treat the coupling of the problem. So, we use Local Pressure Schur Complement (LPSC) concept to solve the coupled problem on mesh's elements, and proceed with outer blocks Gauss-Seidel iteration to update the global solutions. Furthermore, we handle the nonlinearity of the problem with the combined adaptive discrete Newton's and multigrid methods. The adaptivity within discrete Newton's method is based on the adaptive step-length control for the discrete differencing in the Jacobian calculations, while the convergence of linear multigrid solver is made to match the convergence requirement of nonlinear solver, accordingly. And the solver's update parameters are solely dependent on the actual convergence rate of the nonlinear problem. We provide the numerical results of solver performance for thixoviscoplastic lid-driven cavity flow.
Numerical solution of the Fokker-Planck equation using physics-conforming finite element methods
(2024) Wegener, Katharina Theresa; Kuzmin, Dmitri; Turek, Stefan
In this work, a Fokker-Planck equation (FPE) is used to approximate the orientation distribution of fibers. FPEs combined with the Navier-Stokes equations (NSE) are widely used to predict the motion of the fibers in fiber suspension flows with low Reynolds numbers. The fibers align in response to the flow and randomize in response to fiber-fiber interactions. A precise formulation takes into account that the flow-fiber interaction is bilateral, so that the suspension rheology also depends on the fiber orientation. Various approaches to model fiber suspensions, including the well-known Folgar- Tucker equation, which relies on orientation tensors, are reviewed. We aim to solve the FPE using the continuous Galerkin method. For each point in the 3d physical space, an equation on the surface of a unit sphere representing the orientation states is solved, while for each point on the sphere an advection equation in the 3d physical space has to be solved. We handle this in the framework of an alternating direction approach including subtime stepping. Algebraic flux correction is performed for each equation to ensure positivity preservation as well as the normalization property of the distribution function. Numerical tests are performed for the individual subproblems. Finally, the velocity field is calculated by the incompressible Navier-Stokes equations (NSE), and benchmark problems for the coupled FPE-NSE system are solved. Thus, the relevance of this two-way coupling across the scales can be validated, and the effect of a different number of fibers is examined.
Lagrangian simulation of fiber orientation dynamics using random walk methods
(2023) Ahmadi, Omid; Kuzmin, Dmitri; Turek, Stefan
This thesis focuses on developing a two-way coupled framework for the numerical simulation of fiber suspension flows. The influence of fibers on the flow is accounted for by evaluating a non-Newtonian stress term incorporated into the Navier-Stokes equations. The accuracy of the analysis depends on a second-order tensor field used to approximate the orientation distribution of fibers. In this context, the disperse phase can be treated in the Lagrangian or Eulerian manner. We conduct a comprehensive comparison of these frameworks for one-way coupled scenarios in both two- and three-dimensional homogeneous flows. With a special focus on the Lagrangian approach, the algorithm for solving the two-way coupled fiber suspension flow in a segregated manner is proposed by incorporating the fiber-induced stresses in the finite element formulation of the Navier-Stokes equations. In non-dilute suspensions, fiber-fiber interactions may cause spontaneous changes in the orientation of fibers. Applying the theory of rotary Brownian motion, the effect can be studied using a rotary diffusion term with a Laplace-Beltrami operator. In this work, we develop random walk methodologies to emulate the action of the diffusion term without evolving or reconstructing the so-called orientation distribution function. After deriving simplified forms of Brownian motion generators for rotated reference frames, several practical approaches to generating random walks on the unit sphere are discussed. Among the proposed methods, this research effort presents the projection of Cartesian random walks, as well as polar random walks on the tangential plane. The standard random walks are then projected onto the unit sphere. Moreover, we propose an alternative based on a tabulated approximation of the cumulative distribution function obtained from the exact solution of the spherical heat equation. In the last part of this work, the random walk approaches are compared through several numerical studies, including the study of the orientation distribution of fibers in a three-dimensional homogeneous flow. Then, the two-way coupled solver is validated in a simple geometry, followed by performing a few three-dimensional numerical simulations to study the rheological behavior of the fiber suspension flow through an axisymmetric contraction. The effect of fiber-fiber interactions is also incorporated using the random walk methodology.
Efficient FEM simulation techniques for thixoviscoplastic flow problems
(2023) Begum, Naheed; Turek, Stefan; Sokolov, Andriy
This thesis is concerned with the numerical simulations of thixoviscoplastic (TVP) flow problems. As a nonlinear multifield two-way coupled problem, the analysis of thixoviscoplasticity in complex fluid processes would present a great challenge. Numerical simulations are conceived as economic and credible tools to replicate thixoviscoplastic phenomena in different flow circumstances. This thesis proceeds to provide efficient numerical methods and corresponding algorithmic tools to fulfil this goal. To start with, we generalized the standard FEM settings of Stokes equations, and proceeded with the well-posedness study of the problem to set the foundation for the approximate problem, and for the efficient solver. In the context of solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to develop a monolithic Newton-multigrid TVP solver. Most importantly, we used the numerical simulations of thixoviscoplastic flow problems to understand the complex phenomena of interplay between plasticity and thixotropy. We incorporated thixotropy in well-established academical benchmarks, namely channel flow and lid-driven cavity flow. In addition, we analyzed type of transitions, namely shear localization and shear banding in Couette devices. In the end, we used thixotropy in contraction configuration to highlight the importance of taking in consideration thixotropy of material rheology for an accurate flow simulations.
Mathematical modeling of coolant flow in discontinuous drilling processes with temperature coupling
(2023-03-24) Fast, Michael; Mierka, Otto; Turek, Stefan; Wolf, Tobias; Biermann, Dirk
Nickel-based alloys, like Inconel 718, are widely used in industrial applications due to their high-temperature strength and high toughness. However, machining such alloys is a challenging task because of high thermal loads at the cutting edge and thus extensive tool wear is expected. Consequently, the development of new process strategies is needed. We will consider the discontinuous drilling process with coolant. The main idea is to interrupt the drilling process in order to let the coolant to flow around the cutting edge and to reduce thermal loads. Since measurements inside the borehole are (nearly) impossible, simulations are a key tool to analyze and understand the proposed process. In this paper, a 3D fluid flow simulation model with Q2P1 Finite Elements in combination with the Fictitious Boundary Method is presented to simulate the coolant flow around the drill inside the borehole. The underlying equations are transformed into a rotational frame of reference overcoming the challenges of mesh design for high rotational domains inside the fluid domain. Special treatment of Coriolis forces is developed, that modifies the ‘Pressure Poisson’ Problem in the projection step improving the solver for high angular velocities. To further take high velocities into account, a two-scale artificial diffusion technique is introduced to stabilize the simulation. Finally, Q1 Finite Elements are used to simulate the heating and cooling processes in both the tool and the coolant during the complete discontinuous drilling process. The simulation is split into a ‘contact’ and a ‘no contact’ phase and a coupling strategy between these phases is developed. FBM is utilized to switch between the two configurations, thus only one unified grid for both configurations is needed. The results are used to gain insight into the discontinuous drilling process and to optimize the process design.
FEM simulation of thixo-viscoplastic flow problems: error analysis
(2023-05-31) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
This note is concerned with the essential part of Finite Element Methods (FEM) approximation of error analysis for quasi-Newtonian modelling of thixo-viscoplastic (TVP) flow problems. The developed FEM settings for thixotropic generalized Navier-Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton-multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization, nonoptimal estimate for microstructure, and extra regularity requirement for velocity, is due to the weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with stabilized higher order FEM. Furthermore, we show the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical solutions of TVP flow problems in a 4:1 contraction configuration.
Stabilized discontinuous Galerkin methods for solving hyperbolic conservation laws on grids with embedded objects
(2023) Streitbürger, Florian; May, Sandra; Turek, Stefan
This thesis covers a novel penalty stabilization for solving hyperbolic conservation laws using discontinuous Galerkin methods on grids with embedded objects. We consider cut cell grids, that are constructed by cutting the given object out of a Cartesian background grid. The resulting cut cells require special treatments, e.g., adding stabilization terms. In the context of hyperbolic conservation laws, one has to overcome the small cell problem: standard explicit time stepping becomes unstable on small cut cells when the time step is selected based on larger background cells. This work will present the Domain of Dependence (DoD) stabilization in one and two dimensions. By transferring additional information between the small cut cell and its neighbors, the DoD stabilization restores the correct domains of dependence in the neighborhood of the cut cell. The stabilization is added as penalty terms to the semi-discrete scheme. When combined with a standard explicit time-stepping scheme, the stabilized scheme remains stable for a time-step length based on the Cartesian background cells. Thus, the small cell problem is solved. In the first part of this work, we will consider one-dimensional hyperbolic conservation laws. We will start by explaining the ideas of the stabilization for linear scalar problems before moving to non-linear problems and systems of hyperbolic conservation laws. For scalar problems, we will show that the scheme ensures monotonicity when using its first-order version. Further, we will present an L2 stability result. We will conclude this part with numerical results that confirm stability and good accuracy. These numerical results indicate that for both, linear and non-linear problems, the convergence order in various norms for smooth tests is p+1 when using polynomials of degree p. In the second part, we will present first ideas for extending the DoD stabilization to two dimensions. We will consider different simplified model problems that occur when using two-dimensional cut cell meshes. An essential step for the extension to two dimensions will be the construction of weighting factors that indicate how we couple the multiple cut cell neighbors with each other. The monotonicity and L2 stability of the stabilized system will be confirmed by transferring the ideas of the proof from one to two dimensions. We will conclude by presenting numerical results for advection along a ramp, demonstrating convergence orders of p+1/2 to p+1 for polynomials of degree p. Additionally, we present preliminary results for the two-dimensional Burgers and Euler equations on model meshes.
An adaptive discrete Newton method for a regularization-free Bingham model
(2023) Fatima, Arooj; Turek, Stefan; Blum, Heribert
Developing a numerical and algorithmic tool which correctly identifies unyielded regions in the yield stress fluid flow is a challenging task. Two approaches are commonly used to handle the singular behaviour at the yield surface, i.e. the Augmented Lagrangian approach and the regularization approach, respectively. Generally in the regularization approach for the resulting nonlinear and linear problems, solvers do not perform efficiently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress [1]. The three field formulation of yield stress fluids corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three field formulation is treated efficiently and accurately by a monolithic finite element method. The velocity and pressure are discretized by the higher order stable FEM pair Q_2⁄(P_1^disc ) and the auxiliary stress is discretized by the Q_2 element. Furthermore, this problem is highly nonlinear and presents a big challenge to any nonlinear solver. We developed a new adaptive discrete Newton's method, which evaluates the Jacobian with the directional divided difference approach [2]. The step size in this process is an important key: We relate this size to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton's method. The resulting linear subproblems are solved using a geometrical multigrid solver. We analyse the solvability of the problem along with the adaptive Newton method for Bingham fluids by doing numerical studies for different prototypical configurations, i.e. "Viscoplastic fluid flow in a channel" [2], "Lid Driven Cavity", "Flow around cylinder", and "Bingham flow in a square reservoir", respectively. References [1] A. Aposporidis, E. Haber, M. A. Olshanskii, A. Veneziani. A Mixed Formulation of the Bingham Fluid Flow Problem: Analysis and Numerical Solution, Comput. Methods Appl. Mech. Engrg. 1 (2011), 2434–2446. [2] A. Fatima, S. Turek, A. Ouazzi, M. A. Afaq. An Adaptive Discrete Newton Method for Regularization-Free Bingham Model, 6th ECCOMAS Young Investigators Conference 7th-9th July 2021, Valencia, Spain. doi: 10.4995/YIC2021.2021.12389.
Mesh optimization based on a Neo-Hookean hyperelasticity model
(2023) Schuh, Malte; Turek, Stefan; Meyer, Christian
For industrial applications CFD-simulations have become an important addition to experiments. To perform them the underlying geometry has to be created on a computer and embedded into a computational mesh. This mesh needs to meet certain quality criteria. This mesh needs to meet certain quality criteria, which are not commonly met by many conventional mesh-generation tool. However, automated tools are necessary to simulate experiments with a time-dependent geometry, for example a rising bubble or fluid-structure interaction. In this thesis, we study the automatic optimization of a mesh by minimizing a neohookean hyperelasticity model. The aim of our mesh optimization is to either smoothen a mesh, or to adapt a mesh to a given geometry. In the first part of the thesis we propose a specific energy model from this class and investigate if a solution to the minimization of this specific energy exists or not. To solve this minimization problem we need to develop an algorithm. After this we have to investigate if this specific energy function fulfills all requirements of this algorithm. The chosen algorithm is an adaptation of Newton’s method in a function space. To globalize the convergence of Newton’s method we use operator-adaption techniques in a Hilbert space. This makes the algorithm a Quasi-Newton-Method. We proceed to add other elements that are known from optimization so that the algorithm becomes even more robust. Finally we perform several numerical tests to investigate the performance of this method. In our studies we find that for a certain set of parameters the solution of the minimization problem exists. This set of parameters is limited, but the limits are reasonable for most practical use-cases. During our numerical tests we find the method to be stable and robust enough to automatically smoothen a mesh, but to adapt a given mesh to a given geometry our results are unclear: For simulations in two dimensions, the developed method seems to perform well and we get promising results with even just a type of Picard-iteration. For simulations in three dimensions, some adaptations might be necessary and more tests are required.
Very fast finite element Poisson solvers on lower precision accelerator hardware: A proof of concept study for Nvidia Tesla V100
(2022-05-06) Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Recently, accelerator hardware in the form of graphics cards including Tensor Cores, specialized for AI, has significantly gained importance in the domain of high-performance computing. For example, NVIDIA’s Tesla V100 promises a computing power of up to 125 TFLOP/s achieved by Tensor Cores, but only if half precision floating point format is used. We describe the difficulties and discrepancy between theoretical and actual computing power if one seeks to use such hardware for numerical simulations, that is, solving partial differential equations with a matrix-based finite element method, with numerical examples. If certain requirements, namely low condition numbers and many dense matrix operations, are met, the indicated high performance can be reached without an excessive loss of accuracy. A new method to solve linear systems arising from Poisson’s equation in 2D that meets these requirements, based on “prehandling” by means of hier-archical finite elements and an additional Schur complement approach, is presented and analyzed. We provide numerical results illustrating the computational performance of this method and compare it to a commonly used (geometric) multigrid solver on standard hardware. It turns out that we can exploit nearly the full computational power of Tensor Cores and achieve a significant speed-up compared to the standard methodology without losing accuracy.
Algebraically constrained finite element methods for hyperbolic problems with applications in geophysics and gas dynamics
(2022) Hajduk, Hennes; Kuzmin, Dmitri; Gassner, Gregor
The research conducted in this thesis is focused on property-preserving discretizations of hyperbolic partial differential equations. Computational methods for solving such problems need to be carefully designed to produce physically meaningful numerical solutions. In particular, approximations to some quantities of interest should satisfy local and global discrete maximum principles. Moreover, numerical methods need to obey certain conservation relations, and convergence of approximations to the physically relevant exact solution should be ensured if multiple solutions may exist. Many algorithms based on the aforementioned design principles fall into the category of algebraic flux correction (AFC) schemes. Modern AFC discretizations of nonlinear hyperbolic systems express approximate solutions as convex combinations of intermediate states and constrain these states to be admissible. The main focus of our work is on monolithic convex limiting (MCL) strategies that modify spatial semi-discretizations in this way. Contrary to limiting approaches of predictor-corrector type, their monolithic counterparts are well suited for transient and steady problems alike. Further benefits of the MCL framework presented in this thesis include the possibility of enforcing entropy stability conditions in addition to discrete maximum principles. Using the AFC methodology, we transform finite element discretizations into property-preserving low order methods and perform flux correction to recover higher orders of accuracy without losing any desirable properties. The presented methods produce physics-compatible approximations, which exhibit excellent shock capturing capabilities. One novelty of this work is the tailor-made extension of monolithic convex limiting to the shallow water equations with a nonconservative topography term. Our generalized MCL schemes are entropy stable, positivity preserving, and well balanced in the sense that lake at rest equilibria are preserved. Another desirable property of numerical methods for the shallow water equations is the capability to handle wet-dry transitions properly. We present two new approaches to dealing with this issue. To corroborate our computational results with theoretical investigations, we perform numerical analysis for property-preserving discretizations of the time-dependent linear advection equation. In this context, we prove stability and derive an a~priori error estimate in the semi-discrete setting. We also compare the monolithic convex limiting strategy to two representatives of related flux-corrected transport algorithms. Another highlight of this thesis is the chapter on MCL schemes for arbitrary order discontinuous Galerkin (DG) discretizations. Building on algorithms developed for continuous Lagrange and Bernstein finite elements, we extend our MCL schemes to the high order DG setting. This research effort involves the design of new AFC tools for numerical fluxes that appear in the DG weak formulation. Our limiting strategy for DG methods exploits the properties of high order Bernstein polynomials to construct sparse discrete operators leading to compact-stencil nonlinear approximations. The proposed numerical methods are applied to various hyperbolic problems. Scalar equations are considered mainly for testing purposes and to simplify numerical analysis. Besides the shallow water system, we study the Euler equations of gas dynamics.
Monolithic Newton-multigrid FEM for the simulation of thixotropic flow problems
(2021-12-14) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
This contribution is concerned with the application of Finite Element Method (FEM) and Newton-Multigrid solvers to simulate thixotropic flows. The thixotropic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyse the thixotropic viscoplastic flow problem in 4:1 contraction configuration.
Monolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects
(2020) Waseem, Muhammad; Turek, Stefan; Kuzmin, Dmitri
We study the monolithic finite element method, based on the least-squares minimization principles for the solution of non-Newtonian fluids with non-isothermal effects. The least-squares functionals are balanced by the linear and nonlinear weighted functions and the residuals comprised of L2-norm only. The weighted functions are the function of viscosities and proved significant for optimal results. The lack of mass conservation is an important issue in LSFEM and is studied extensively for the diverse range of weighted functions. Therefore, we consider only inflow/outflow problems. We use the Krylov subspace linear solver, i.e. conjugate gradient method, with a multigrid method as a preconditioner. The SSOR-PCG is used as smoother for the multigrid method. The Gauss-Newton and fixed point methods are employed as nonlinear solvers. The LSFEM is investigated for two main types of fluids, i.e. Newtonian and non-Newtonian fluids. The stress-based first-order systems, named SVP formulations, are employed to investigate the Newtonian fluids. The different types of quadratic finite elements are used for the analysis purposes. The nonlinear Navier-Stokes problem is investigated for two mesh configurations for flow around cylinder problem. The coefficients of lift/drag, pressure difference, global mass conservation are analyzed. The comparison of linear and nonlinear solvers, based on iterations, is presented as well. The analysis of non-Newtonian fluids is divided into two parts, i.e. isothermal and non-isothermal. For the non-Newtonian fluids, we consider only Q2 finite elements for the discretization of unknown variables. The isothermal non-Newtonian fluids are investigated with SVP-based formulations. The power law and Cross law viscosity models are considered for investigations with different nonlinear weighted functions. We study the flow parameters for flow around cylinder problem along with the mass conservation for shear thinning and shear thickening fluids. To study the non-isothermal non-Newtonian fluids, we introduced a new first-order formulation which includes temperature and named it as SVPT formulation. The non-isothermal effects are obtained due to the additional viscous dissipation in the fluid flow and from the preheated source as well. The flow around cylinder problem is analyzed for a variety of flow parameters for Cross law fluids. It is shown that the MPCG solver generates very accurate results for the coupled and highly complex problems.
A proof of concept for very fast finite element Poisson solvers on accelerator hardware
(2021-12-14) Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
It is demonstrated that modern accelerator hardware specialized in AI, e.g., “next gen GPUs” equipped with Tensor Cores, can be profitably used in finite element simulations by means of a new hardware-oriented method to solve linear systems arising from Poisson's equation in 2D. We consider the NVIDIA Tesla V100 Tensor Core GPU with a peak performance of 125 TFLOP/s, that is only achievable in half precision and if operations with high arithmetic intensity, such as dense matrix multiplications, are executed, though. Its computing power can be exploited to a great extent by the new method based on “prehandling” without loss of accuracy. We obtain a significant reduction of computing time compared to a standard geometric multigrid solver on standard x64 hardware.

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