Ergebnisberichte des Instituts für Angewandte Mathematik

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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.
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.
Numerical study of the RBF-FD method for the Stokes equations
(2023-04) Westermann, Alexander; Davydov, Oleg; Sokolov, Andriy; Turek, Stefan
We study the numerical behavior of the meshless Radial Basis Function Finite Difference method applied to the stationary incompressible Stokes equations in two spatial dimensions, using polyharmonic splines as radial basis functions with a polynomial extension on two separate node sets to discretize the velocity and the pressure. On the one hand,we show that the convergence rates of the method correspond to the known convergence rates of numerical differentiation by the polyharmonic splines. On the other hand, we show that the main condition for the stability of the numerical solution is that the distributions of the pressure nodes has to be coarser than that of the velocity everywhere in the domain. There seems to be no need for any complex assumptions similar to the Ladyzhenskaya-Babuška-Brezzi condition in the finite element method. Numerical results for the benchmark driven cavity problem are in a good agreement with those in the literature.
Robust adaptive discrete Newton method for regularization-free Bingham model
(2023-03) Fatima, Arooj; Afaq, Muhammad Aaqib; Turek, Stefan; Ouazzi, Abderrahim
Developing a numerical and algorithmic tool which accurately detects unyielded regions in yield stress fluid flow is a difficult endeavor. To address these issues, two common approaches are used to handle singular behaviour at the yield surface, i.e. the augmented Lagrangian approach and the regularization approach. Generally, solvers do not operate effectively when the regularization parameter is very small in the regularization approach. In this work, we use a formulation involving a new auxiliary stress tensor, wherein the three-field formulation is equivalent to a regularization-free Bingham formulation. Additionally, a monolithic finite element method is employed to solve the set of equations resulting from the three-field formulation accurately and effciently, where the velocity, pressure fields are discretized by the higherorder stable FEM pair Q2=Pdisc1 and the auxiliary stress is discretized by the Q2 element. Furthermore, this article presents a novel adaptive discrete Newton method for solving highly nonlinear problems, which exploits the divided difference approach for evaluating the Jacobian. The step size of the solver is dynamically adjusted according to the rate of nonlinear reduction, enabling a robust and efficient approach. Numerical studies of several prototypical Bingham fluid configurations ("viscoplastic fluid flow in a channel", "lid driven cavity" and "rotational Bingham flow in a square reservoir") are used to analyse the performance of this method.
Finite Element approximation of data-driven problems in conductivity
(2023-03) Müller, Annika; Meyer, Christian
This paper is concerned with the finite element discretization of the data driven approach according to [18] for the solution of PDEs with a material law arising from measurement data. To simplify the setting, we focus on a scalar diffusion problem instead of a problem in elasticity. It is proven that the data convergence analysis from [9] carries over to the finite element discretization as long as H(div)-conforming finite elements such as the Raviart-Thomas element are used. As a corollary, minimizers of the discretized problems converge in data in the sense of [9], as the mesh size tends to zero and the approximation of the local material data set gets more and more accurate. We moreover present several heuristics for the solution of the discretized data driven problems, which is equivalent to a quadratic semi-assignment problem and therefore NP-hard. We test these heuristics by means of two examples and it turns out that the “classical” alternating projection method according to [18] is superior w.r.t. the ratio of accuracy and computational time.
Efficient numerical solution of the Fokker-Planck equation using physics-conforming finite element methods
(2023-01) Wegener, Katharina; Kuzmin, Dmitri; Turek, Stefan
We consider the Fokker-Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternatingdirection methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier-Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.
Robust Monolithic Multigrid FEM Solver for Three Field Formulation of Incompressible Flow Problems
(2023-01) Afaq, Muhammad Aaqib; Fatima, Arooj; Turek, Stefan; Ouazzi, Abderrahim
Numerical simulation of three ﬁeld formulations of incompressible ﬂow problems is of interest for many industrial applications, for instance macroscopic modeling of Bing-ham, viscoelastic and multiphase ﬂows, which usually consists in supplementing the mass and momentum equations with a diﬀerential constitutive equation for the stress ﬁeld. The variational formulation rising from such continuum mechanics problems leads to a three ﬁeld formulation with saddle point structure. The solvability of the problem requires diﬀerent compatibility conditions (LBB conditions) [1] to be satisﬁed. Moreover, these constraints over the choice of the spaces may conﬂict/challenge the robustness and the eﬃciency of the solver. For illustrating the main points, we will consider the three ﬁeld formulation of the Navier-Stokes problem in terms of velocity, stress, and pressure. Clearly, the weak form imposes the compatibility constraints over the choice of velocity, stress, and pressure spaces. So far, the velocity-pressure combi-nation took much more attention from the numerical analysis and computational ﬂuid dynamic community, which leads to some best interpolation choices for both accuracy and eﬃciency, as for instance the combination Q2/P1disc. To maintain the computational advantages of the Navier-Stokes solver in two ﬁeld formulations, it may be more suitable to have a Q2 interpolation for the stress as well, which is not stable in the absence of pure viscous term [2]. We proceed by adding an edge oriented stabilization to overcome such situation. Furthermore, we show the robustness and the eﬃciency of the resulting discretization in comparison with the Navier-Stokes solver both in two ﬁeld as well as in three ﬁeld formulation in the presence of pure viscous term. Moreover, the beneﬁt of adding the edge oriented ﬁnite element stabilization (EOFEM) [3, 4] in the absence of the pure viscous term is tested. The nonlinearity is treated with a Newton-type solver [5] with divided diﬀerence evaluation of the Jacobian matrices [6, 7]. The resulting linearized system inside of the outer Newton solver is a typical saddle point problem which is solved using a geometrical multigrid method with Vanka-like smoother [8, 9]. The method is implemented into the FeatFlow [10] software package for the numerical simulation. The stability and robustness of the method is numerically investigated for ”ﬂow around cylinder” benchmark [7, 11].
Bilevel Optimization of the Kantorovich problem and its quadratic regularization Part:II Convergence Analysis
(2022-11) Hillbrecht, Sebastian; Manns, Paul; Meyer, Christian
This paper is concerned with an optimization problem that is constrained by the Kantorovich optimal transportation problem. This bilevel optimization problem can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness induced by the complementarity relations, problems of this type are frequently regularized. Here we apply a quadratic regularization of the Kantorovich problem. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem.
FEM simulation of thixo-viscoplastic flow problems: Error analysis
(2022-10) Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
This note is concerned with error analysis of FEM approximations 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 parameter, nonoptimal estimate for microstructure, and extra regularization requirement for velocity, is due to weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with higher order stabilized FEM. Furthermore, we showed the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical simulations of TVP flow problems in a 4:1 contraction configuration.
Bilevel optimization of the Kantorovich problem and it's quadratic regularization part I: existence results
(2022-09) Hillbrecht, Sebastian; Meyer, Christian
This paper is concerned with an optimization problem governed by the Kantorovich optimal transportation problem. This gives rise to a bilevel optimization problem, which can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness induced by the complementarity relations, problems of this type are frequently regularized. Here we apply a quadratic regularization of the Kantorovich problem. As the title indicates, this is the ﬁrst part in a series of three papers. It addresses the existence of optimal solutions to the bilevel Kantorovich problem and its quadratic regularization, whereas part II and III are dedicated to the convergence analysis for vanishing regularization.

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