Ergebnisberichte des Instituts für Angewandte Mathematik

Permanent URI for this collection

http://hdl.handle.net/2003/35666

Browse

Now showing 1 - 20 of 131

Effective Viscosity Closures for Dense Suspensions in CSP Systems via Lubrication-Enhanced DNS and Numerical Viscometry
(2025-07) Münster, Raphael; Mierka, Otto; Kuzmin, Dmitri; Turek, Stefan
Dense particle suspensions are promising candidates for next-generation Concentrated Solar Power (CSP) receivers, enabling operating temperatures above 800 °C. However, accurate modeling of the rheological behavior of granular flows is essential for reliable computational fluid dynamics (CFD) simulations. In this study, we develop and assess numerical methodologies for simulating dense suspensions pertinent to CSP applications. Our computational framework is based on Direct Numerical Simulation (DNS), augmented by lubrication force models to resolve detailed particle-particle and particle-wall interactions at volume fractions exceeding 50%. We conducted a systematic series of simulations across a range of volume fractions to establish a robust reference dataset. Validation was performed via a numerical viscometer configuration, permitting direct comparison with theoretical predictions and established benchmark results. Subsequently, the viscometer arrangement was generalized to a periodic cubic domain, serving as a representative volume element for CSP systems. Within this framework, effective viscosities were quantified independently through wall force measurements and energy dissipation analyses. The close agreement between these two approaches substantiates the reliability of the results. Based on these findings, effective viscosity tables were constructed and fitted using polynomial and piecewise-smooth approximations. These high-accuracy closure relations are suitable for incorporation into large-scale, non-Newtonian CFD models for CSP plant design and analysis.
A Chimera domain decomposition method with weak Dirichlet-Robin coupling for finite element simulation of particulate flows
(2025-07) Münster, Raphael; Mierka, Otto; Kuzmin, Dmitri; Turek, Stefan
We introduce a new multimesh finite element method for direct numerical simulation of incompressible particulate flows. The proposed approach falls into the category of overlapping domain decomposition / Chimera / overset grid meshes. In addition to calculating the velocity and pressure of the fictitious fluid on a fixed background mesh, we solve the incompressible Navier-Stokes equations on body-fitted submeshes that are attached to moving particles. The submesh velocity and pressure are used to calculate the hydrodynamic forces and torques acting on the particles. The coupling with the background velocity and pressure is enforced via (i) Robin-type boundary conditions for an Arbitrary-Lagrangian-Eulerian (ALE) formulation of the submesh problems and (ii) a Dirichlet-type distributed interior penalty term in the weak form of the background mesh problem. The implementation of the weak Dirichlet-Robin coupling is discussed in the context of discrete projection methods. Detailed numerical studies are performed for standard test problems involving fixed and moving immersed objects. A comparison with fictitious boundary methods illustrates significant gains in the accuracy of drag and lift approximation.
A geometric multigrid solver for the incompressible Navier-Stokes equations using discretely divergence-free finite elements in 3D
(2025-03) Lohmann, Christoph
A geometric multigrid solution technique for the incompressible Navier-Stokes equations in three dimensions is presented, utilizing the concept of discretely divergence-free finite elements without requiring the explicit construction of a basis on each mesh level. For this purpose, functions are constructed in an a priori manner spanning the subspace of discretely divergence-free functions for the Rannacher-Turek finite element pair under consideration. Compared to mixed formulations, this approach yields smaller system matrices with no saddle point structure. This prevents the use of complex Schur complement solution techniques and more general preconditioners can be employed. While constructing a basis for discretely divergence-free finite elements may pose significant challenges and its use prevents a structured assembly routine, a basis is utilized only on the coarsest mesh level of the multigrid algorithm. On finer grids, this information is extrapolated to prescribe boundary conditions efficiently. Here, special attention is required for geometries introducing bifurcations in the flow. In such cases, so called ‘global’ functions with an extended support are defined, which can be used to prescribe the net flux through different branches. Various numerical examples for meshes with different shapes and boundary conditions illustrate the strengths, limitations, and future challenges of this solution concept.
A posteriori error analysis for optimization with PDE constraints
(2025-03) Gaspoz, Fernando; Kreuzer, Christian; Veeser, Andreas; Wollner, Winnifried
We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control.
Strictly equivalent a posteriori error estimators for quasi-optimal nonconforming methods
(2025-02-04) Kreuzer, Christian; Rott, Matthias; Veeser, Andreas; Zanotti, Pietro
We devise a posteriori error estimators for quasi-optimal nonconforming finite element methods approximating symmetric elliptic problems of second and fourth order. These estimators are defined for all source terms that are admissible to the underlying weak formulations. More importantly, they are equivalent to the error in a strict sense. In particular, their data oscillation part is bounded by the error and, furthermore, can be designed to be bounded by classical data oscillations. The estimators are computable, except for the data oscillation part. Since even the computation of some bound of the oscillation part is not possible in general, we advocate to handle it on a case-by-case basis. We illustrate the practical use of two estimators obtained for the Crouzeix-Raviart method applied to the Poisson problem with a source term that is not a function and its singular part with respect to the Lebesgues measure is not aligned with the mesh.
Pressure robust Finite Element discretizations of the nonlinear Stokes Equations
(2025-01-27) Diening, Lars; Hirn, Adrian; Kreuzer, Christian; Zanotti, Pietro
We present first-order nonconforming Crouzeix-Raviart discretizations for the nonlinear generalized Stokes equations with (r, ε)-structure. Thereby the velocity-errors are independent of the pressure-error; i.e., the method is pressure robust. This improves suboptimal rates previously experienced for non pressure robust methods.
Finite Element Simulation for Elastic and Plastic Fluids
(2024-11) Saghir, Muhammad Tayyab Bin; Damanik, Hogenrich; Turek, Stefan
In this study, we present the development of a 2D finite element solver for simulating fluids exhibiting both elastic and plastic constitutive properties. We achieve this by combining the constitutive models of the Oldroyd-B model for viscoelastic fluids and the Papanastasiou model for Bingham fluids within a single Eulerian numerical framework. Our aim within this approach is to approximate the given velocity, pressure, and elastic stresses. We employ a higher-order finite element method for the velocity-stress approximation and a discontinuous pressure element. This specific element pair has proven highly effective for accurately capturing the behavior of both Oldroyd-B and Bingham fluids, including nonlinear viscosity functions. Our study consists of two main steps. Firstly, we validate the numerical results for each module component of the constitutives to ensure the accuracy of the approximations and calculations. This step is crucial for establishing the reliability and robustness of our approach. Subsequently, in the second step, we apply the solver to simulate elastoviscoplastic fluid behavior in a porous medium. By investigating fluid flow and deformation within this specific context, we aim to demonstrate the capabilities and potential of our methodology.
Numerical simulation and mixing characterization of Taylor bubble flows in coiled flow inverters
(2024-08) Mierka, Otto; Münster, Raphael; Surkamp, Julia; Kockmann, Norbert; Turek, Stefan
The here presented work is dedicated to the development of a software analysis tool specialized for Taylor bubble flows in a wide range of applications and covering a variety of geometrical realizations. Accordingly, a higher order FEM based interface tracking simulation software has been designed which due to the underlying isoparametric discretization and interface aligned mesh construction allows a semi-implicit treatment of the surface tension force term. The such designed numerical framework guarantees only a negligible amount of mass loss rate resulting in suitability of the tool for long time-scale simulations. Exploiting these numerical advantages the software has been applied for the system of Coiled Flow Inverter (CFI) capillaries characterized by a range of coil diameters and Reynolds numbers for which the pseudo-periodic flowfield has been obtained and extracted for further mixing quantification studies by the help of particle tracking based analysis. In the framework of this postprocessing analysis such a suitable transformation of the results has been applied which reveals the most important features of the characteristic flow patterns and makes it possible to qualitatively but even quantitatively characterize the behavior of the established flow patterns. Therefore, such a combination of robust CFD techniques together with the respective process performance quantification makes this approach suitable for tailored process design.
Direct numerical simulation of dispersion and mixing in gas-liquid Dean-Taylor flow with influence of a 90° bend
(2024-08) Mierka, Otto; Münster, Raphael; Surkamp, Julia; Turek, Stefan; Kockmann, Norbert
Gas-liquid capillary flow finds widespread applications in reaction engineering, owing to its ability to facilitate precise control and efficient mixing. Incorporating compact and regular design with Coiled Flow Inverter (CFI) enhances process efficiency due to improved mixing as well as heat and mass transfer leading to a narrow residence time distribution. The impact of Dean and Taylor flow phenomena on mixing and dispersion within these systems underscores their significance, but is still not yet fully understood. Direct numerical simulation based on finite element method enables full 3D resolution of the flow field and detailed examination of laminar flow profiles, providing valuable insights into flow dynamics. Notably, the deflection of flow velocity from the center axis contributes is followed by tracking of particle with defined starting positions, aiding in flow visualization and dispersion characterization. In this CFD study, the helical flow with the influence of the centrifugal force and pitch (Dean flow) as well as the capillary two-phase flow (Taylor bubble) is described and characterized by particle dispersion and related histograms. Future prospects in this field include advancements in imaging techniques to capture intricate flow paterns, as well as refined particle tracking methods to beter understand complex flow behavior.
Inf-Sup stable discretization of the quasi-static Biot's equations in poroelasticity
(2024-07)
We propose a new full discretization of the Biot’s equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters. We further construct an interpolant showing how the error decays for smooth solutions.
Simulation techniques for the viscoelastic fluids with pure polymer melts based on EVSS approach
(2024-05) Ahmad, Rida; Zajac, Peter; Turek, Stefan
To obtain the solution of the viscoelastic ﬂuid simulation with pure polymer melts is a highly challenging task due to the lack of the solvent contribution to the viscosity in the standard viscoelastic formulation. The aim of this paper is to present a mixed ﬁnite element method for solving the three ﬁeld Stokes ﬂow with zero solvent viscosity employing the Elastic Viscous Stress Splitting (EVSS) formulation. On one hand, the EVSS formulation helps to recover the velocity coupling back into the momentum equation by the application of the change of variables in the standard viscoelastic formulation. On the other hand, additional terms containing the second order velocity derivatives appear in the convective part of the constitutive equation for stress. We have reformulated the convective term by considering the divergence-free nature of the velocity ﬁeld and shifted the higher order derivatives to the test function in the weak formulation. The velocity, pressure and stress are discretized by the higher order stable FEM triplet Q2/P1disc/Q3. The proposed scheme is tested for Oldroyd-B, Giesekus and PTT exponential ﬂuids employing both the decoupled and the monolithic solution approaches. The numerical results are obtained on four to one contraction for highly viscoelastic ﬂuids with the aim to observe the shear-thinning eﬀect as the relaxation parameter increases.
Fast Semi-Iterative Finite Element Poisson Solvers for Tensor Core GPUs Based on Prehandling
(2024-01) Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk
The impetus for the research presented in this work is provided by recent developments in the field of GPU computing. Nvidia GPUs that are equipped with Tensor Cores, such as the A100 or the latest H100, promise an immense computing power of 156 and 495 TFLOPS, respectively, but only for dense matrix operations carried out in single precision (with even higher rates in half precision), since this serves their actual purpose of accelerating AI training. It is shown that this performance can also be exploited to a large extent in the domain of matrix-based finite element methods for solving PDEs, if specially tailored, hardware-oriented methods are used. Such methods need to preserve sufficient accuracy, even if single precision is used, and mostly consist of dense matrix operations. A semi-iterative method for solving Poisson’s equation in 2D and 3D based on prehandling, i.e., explicit preconditioning, by means of hierarchical finite elements or generating systems, that satisfies these requirements, is derived and analyzed.Actual benchmark results on an H100 allow the determination of optimal solver configurations in terms of performance, which ultimately exceeds that of a standard geometric multigrid solver on CPU.
Improving Convergence of Time-Simultaneous Multigrid Methods for Convection-Dominated Problems using VMS Stabilization Techniques
(2024-01) Drews, Wiebke; Turek, Stefan; Lohmann, Christoph
We present the application of a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation for stabilized convection-diffusion equations in the regime of small diffusion coefficients. We use Galerkin finite elements and the Crank-Nicolson scheme for discretization in space and time. The multigrid method blocks all time steps for each spatial unknown, enhancing parallelization in space. While the number of iterations of the solver is bounded above for the 1D heat equation, convergence issues arise in convection-dominated cases. In singularly perturbed advection-diffusion scenarios, Galerkin FE discretizations are known to show instabilities in the numerical solution.We explore a higher-order variational multiscale stabilization, aiming to enhance solution smoothness and improve convergence without compromising accuracy.
Augmented Lagrangian acceleration of global-in-time Pressure Schur complement solvers for incompressible Oseen equations
(2023-12) Lohmann, Christoph; Turek, Stefan
This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier-Stokes equations.
Numerical Analysis of a Time-Simultaneous Multigrid Solver for Stabilized Convection-Dominated Transport Problems in 1D
(2023-08) Drews, Wiebke; Turek, Stefan; Lohmann, Christoph
The work to be presented focuses on the convection-diffusion equation, especially in the regime of small diffusion coefficients, which is solved using a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation. For spatial discretization we use linear finite elements, while the time integrator is given by e.g. the Crank-Nicolson scheme. Blocking all time steps into a global linear system of equations and rearranging the degrees of freedom leads to a space-only problem with vector-valued unknowns for each spatial node. Then, common iterative solution techniques, such as the GMRES method with block Jacobi preconditioning, can be used for the numerical solution of the (spatial) problem and allow a higher degree of parallelization in space. We consider a time-simultaneous multigrid algorithm, which exploits space-only coarsening and the solution techniques mentioned above for smoothing purposes. By treating more time steps simultaneously, the dimension of the system of equations increases significantly and, hence, results in a larger number of degrees of freedom per spatial unknown. This can be used to employ parallel processes more efficiently. In numerical studies, the iterative multigrid solution of a problem with up to thousands of blocked time steps is analyzed in 1D. For the special case of the heat equation, it is well known that the number of iterations is bounded above independently of the number of blocked time steps, the time step size, and the spatial resolution. Unfortunately, convergence issues arise for the multigrid solver in convection-dominated regimes. In the context of the standard Galerkin method if the diffusion coefficient is small compared to the grid size and the magnitude of the velocity field, stabilization techniques are typically used to remove artificial oscillations in the solution. However, in our setting, special higher-order variational multiscale-type stabilization methods are discussed, which simultaneously improve the convergence behavior of the iterative solver as well as the smoothness of the numerical solution without significantly perturbing the accuracy.
On a lack of stability of parametrized BV solutions to rate-independent systems with non-convex energies and discontinuous loads
(2023-08) Andreia, Merlin; Meyer, Christian
We consider a rate-independent system with nonconvex energy under discontinuous external loading. The underlying space is finite dimensional and the loads are functions in BV([0, T]; ℝ^d). We investigate the stability of various solution concepts w.r.t. a sequence of loads converging weakly∗ in BV([0, T]; ℝ^d) with a particular emphasis on the so-called normalized, pparametrized balanced viscosity solutions. By means of two counterexamples, it is shown that common solution concepts are not stable w.r.t. weak∗ convergence of loads in the sense that a limit of a sequence of solutions associated with these loads need not be a solution corresponding to the load in the limit. We moreover introduce a new solution concept, which is stable in this sense, but our examples show that this concept necessarily allows “solutions” that are physically meaningless.
FEM simulations for nonlinear multifield coupled problems: application to Thixoviscoplastic Flow
(2023-07) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
In this note, we are concerned with the solvability of multifield coupled problems with different, often conflictual types of non-linearities. We bring into focus the challenges of getting EFM numerical solutions. As for instance, we share our investigations of the solvability of thixoviscoplastic flow problems in FEM settings. On one hand, nonlinear multifield coupled problems are often lacking unified FEM analysis due to the presence of different nonlinearities. Thus, the importance of treating auxiliary subproblems with different analysis tools to guarantee existence of solutions. Moreover, the nonlinear multifeld problems are extremely sensitive to the coupling. On other hand, monolithic Newton-multigrid FEM solver shows a great success in getting numerical solutions for multifield coupled problems. Thixoviscoplastic flow problem is a perfect example in this regard. It is a two field coupled problem, by means of microstructure dependent plastic-viscosity as well as microstructure dependent yield stress, and microstructure and shear rate dependent buildup and breakdown functions. We adapt different numerical techniques to show the solvability of the problem, and expose the accuracy of FEM numerical solutions via the simulations of thixoviscoplastic flow problems in channel configuration.
Efficient Newton-multigrid FEM Solver for Multifield Nonlinear Coupled Problems Applied to Thixoviscoplastic Flows
(2023-07) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
This note is concerned with efficient Newton-multigrid 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. [1, 2]. 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.
On the design of global-in-time Newton-Multigrid-Pressure Schur complement solvers for incompressible flow problems
(2023-05) 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.
FEM Modeling and Simulation of Thixo-viscoplastic Flow Problems
(2023-05) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
We are concerned, in this work, with Finite Element Method (FEM) for modeling and simulation of thixotropy in viscoplastic materials. We use a quasi-Newtonian approach to integrate the constitutive equation, which results in a new thixo-viscoplastic (TYP) generalized Navier-Stokes (N-8) equations. To solve the corresponding flow fields at once, we developed a FEM TYP solver based on monolithic Newton-multigrid method. The phenomenological process of competition of breakdown and buildup characteristics of thixotropic material is replicated throughout, localization and shear banding for Couette flow on one hand, and induction of more shear rejuvenation layers nearby walls for contraction flow on the other hand.

Browse

Recent Submissions