Eldorado Collection:
http://hdl.handle.net/2003/35666
2023-12-08T16:41:17ZNumerical Analysis of a Time-Simultaneous Multigrid Solver for Stabilized Convection-Dominated Transport Problems in 1D
http://hdl.handle.net/2003/42157
Title: Numerical Analysis of a Time-Simultaneous Multigrid Solver for Stabilized Convection-Dominated Transport Problems in 1D
Authors: Drews, Wiebke; Turek, Stefan; Lohmann, Christoph
Abstract: 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.2023-08-01T00:00:00ZOn a lack of stability of parametrized BV solutions to rate-independent systems with non-convex energies and discontinuous loads
http://hdl.handle.net/2003/42125
Title: On a lack of stability of parametrized BV solutions to rate-independent systems with non-convex energies and discontinuous loads
Authors: Andreia, Merlin; Meyer, Christian
Abstract: 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.2023-08-01T00:00:00ZFEM simulations for nonlinear multifield coupled problems: application to Thixoviscoplastic Flow
http://hdl.handle.net/2003/42042
Title: FEM simulations for nonlinear multifield coupled problems: application to Thixoviscoplastic Flow
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: 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.2023-07-01T00:00:00ZEfficient Newton-multigrid FEM Solver for Multifield Nonlinear Coupled Problems Applied to Thixoviscoplastic Flows
http://hdl.handle.net/2003/41999
Title: Efficient Newton-multigrid FEM Solver for Multifield Nonlinear Coupled Problems Applied to Thixoviscoplastic Flows
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: 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.2023-07-01T00:00:00ZOn the design of global-in-time Newton-Multigrid-Pressure Schur complement solvers for incompressible flow problems
http://hdl.handle.net/2003/41998
Title: On the design of global-in-time Newton-Multigrid-Pressure Schur complement solvers for incompressible flow problems
Authors: Lohmann, Christoph; Turek, Stefan
Abstract: 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.2023-05-01T00:00:00ZFEM Modeling and Simulation of Thixo-viscoplastic Flow Problems
http://hdl.handle.net/2003/41670
Title: FEM Modeling and Simulation of Thixo-viscoplastic Flow Problems
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: 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.2023-05-01T00:00:00ZNumerical study of the RBF-FD method for the Stokes equations
http://hdl.handle.net/2003/41366
Title: Numerical study of the RBF-FD method for the Stokes equations
Authors: Westermann, Alexander; Davydov, Oleg; Sokolov, Andriy; Turek, Stefan
Abstract: 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.2023-04-01T00:00:00ZRobust adaptive discrete Newton method for regularization-free Bingham model
http://hdl.handle.net/2003/41290
Title: Robust adaptive discrete Newton method for regularization-free Bingham model
Authors: Fatima, Arooj; Afaq, Muhammad Aaqib; Turek, Stefan; Ouazzi, Abderrahim
Abstract: 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.2023-03-01T00:00:00ZFinite Element approximation of data-driven problems in conductivity
http://hdl.handle.net/2003/41289
Title: Finite Element approximation of data-driven problems in conductivity
Authors: Müller, Annika; Meyer, Christian
Abstract: 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.2023-03-01T00:00:00ZEfficient numerical solution of the Fokker-Planck equation using physics-conforming finite element methods
http://hdl.handle.net/2003/41213
Title: Efficient numerical solution of the Fokker-Planck equation using physics-conforming finite element methods
Authors: Wegener, Katharina; Kuzmin, Dmitri; Turek, Stefan
Abstract: 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.2023-01-01T00:00:00ZRobust Monolithic Multigrid FEM Solver for Three Field Formulation of Incompressible Flow Problems
http://hdl.handle.net/2003/41212
Title: Robust Monolithic Multigrid FEM Solver for Three Field Formulation of Incompressible Flow Problems
Authors: Afaq, Muhammad Aaqib; Fatima, Arooj; Turek, Stefan; Ouazzi, Abderrahim
Abstract: 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].2023-01-01T00:00:00ZBilevel Optimization of the Kantorovich problem and its quadratic regularization Part:II Convergence Analysis
http://hdl.handle.net/2003/41139
Title: Bilevel Optimization of the Kantorovich problem and its quadratic regularization Part:II Convergence Analysis
Authors: Hillbrecht, Sebastian; Manns, Paul; Meyer, Christian
Abstract: 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.2022-11-01T00:00:00ZFEM simulation of thixo-viscoplastic flow problems: Error analysis
http://hdl.handle.net/2003/41109
Title: FEM simulation of thixo-viscoplastic flow problems: Error analysis
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: 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.2022-10-01T00:00:00ZBilevel optimization of the Kantorovich problem and it's quadratic regularization part I: existence results
http://hdl.handle.net/2003/41087
Title: Bilevel optimization of the Kantorovich problem and it's quadratic regularization part I: existence results
Authors: Hillbrecht, Sebastian; Meyer, Christian
Abstract: 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.2022-09-01T00:00:00ZAn extension of a very fast direct finite element Poisson solver on lower precision accelerator hardware towards semi-structured grids
http://hdl.handle.net/2003/41031
Title: An extension of a very fast direct finite element Poisson solver on lower precision accelerator hardware towards semi-structured grids
Authors: Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Abstract: Graphics cards that are equipped with Tensor Core units designed for AI applica tions, for example the NVIDIA Ampere A100, promise very high peak rates concerning their
computing power (156 TFLOP/s in single and 312 TFLOP/s in half precision in the case of
the A100). This is only achieved when performing arithmetically intensive operations such as
dense matrix multiplications in the aforementioned lower precision, which is an obstacle when
trying to use this hardware for solving linear systems arising from PDEs discretized with the
finite element method. In previous works, we delivered a proof of concept that the predecessor
of the A100, the V100 and its Tensor Cores, can be exploited to a great extent when solving
Poisson’s equation on the unit square if a hardware-oriented direct solver based on prehandling
via hierarchical finite elements and a Schur complement approach is used. In this work, using
numerical results on an A100 graphics card, we show that the method also achieves a very high
performance if Poisson’s equation, which is discretized by linear finite elements, is solved on a
more complex domain corresponding to a flow around a square configuration.2022-07-01T00:00:00ZFEM simulations for thixo-viscoplastic flow problems: wellposedness results
http://hdl.handle.net/2003/41030
Title: FEM simulations for thixo-viscoplastic flow problems: wellposedness results
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: In this contribution, we shall be concerned with the question of wellposedness of thixo viscoplastic flow problems in context of FEM approximations. We restrict our analysis to a quasi Newtonian modeling approach with the aim to set foundations for an efficient monolithic Newton multigrid solver. We present the wellposedness of viscoplastic subproblems and structure subproblems in
parallel/independent fashion showing the possibility for a combined treatment. Then, we use the fixed
point theorem for the coupled problem. For the numerical solutions, we choose 4:1 contraction config uration and use monolithic Newton-multigrid solver. We analyse the effect of taking into consideration
thixotropic phenomena in viscoplastic material and opening up for more different coupling by inclusions
of shear thickening and shear thinning behaviors for plastic viscosity and/or elastic behavior below the
critical yield stress limit in more a general thixotropic models.2022-07-01T00:00:00ZAn adaptive time stepping scheme for rate-independent systems with non-convex energy
http://hdl.handle.net/2003/40855
Title: An adaptive time stepping scheme for rate-independent systems with non-convex energy
Authors: Andreia, Merlin; Meyer, Christian
Abstract: We investigate a local incremental stationary scheme for the numerical solution of
rate-independent systems. Such systems are characterized by a (possibly) non-convex energy
and a dissipation potential, which is positively homogeneous of degree one. Due to the non-convexity of the energy, the system does in general not admit a time-continuous solution. In
order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of
our approach in comparison to existing methods is an adaptive choice of the step size for the
update of the curve parameter depending on a prescribed tolerance for the residua in the energy-dissipation balance and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise aﬃne approximations
generated by the algorithm converge (weakly) to a so-called V-parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical ﬁndings and show that an adaptive
choice of the step size indeed pays oﬀ as they lead to a signiﬁcant increase of the step size
during sticking and in viscous jumps.2022-04-01T00:00:00ZMathematical Modeling of Coolant Flow in Drilling Processes with Temperature Coupling
http://hdl.handle.net/2003/40828
Title: Mathematical Modeling of Coolant Flow in Drilling Processes with Temperature Coupling
Authors: Fast, Michael; Mierka, Otto; Turek, Stefan
Abstract: The paper presents a mathematical modeling approach for a novel drilling
strategy with coolant flow. Numerical tools for efficient simulations of such drilling
applications are explained. We model the fluid flow with the Navier-Stokes equation in a
rotational frame of reference and the solid domain is treated with the Fictitious Boundary
Method (FBM). This enables us to utilize a unified mesh for the solid and fluid part of the
domain and heat transfer between these is treated in an implicit way.2022-03-01T00:00:00ZNumerical studies of a multigrid version of the parareal algorithm
http://hdl.handle.net/2003/40827
Title: Numerical studies of a multigrid version of the parareal algorithm
Authors: Wambach, Lydia; Turek, Stefan
Abstract: In this work, a parallel-in-time method is combined with a multigrid algorithm
and further on with a spatial coarsening strategy. The most famous parallel-in-time
method is the parareal algorithm. Depending on two different operators, it enables the
parallelism of time-dependent problems. The operator with huge effort is carried out
in parallel. But despite parallelization this can lead to long run times for long-term
problems. Since the parareal algorithm has a two-level structure and the time-parallel
multigrid methods are also widespread in the area of parallel time integration, we
combine these approaches. We use the parareal algorithm as a smoothing operator in
the basic framework of a geometrical multigrid method, where we apply a coarsening
strategy in time. So we get a multigrid in time method which is strongly parallelizable.
For partial differential equations we add an extra spatial coarsening strategy to our
multigrid parareal version. All in all we get a method, which has a high parallel
efficiency and converges fast due to the multigrid framework, which is shown in the
numerical studies of this work. So we will get a highly accurate solution and can greatly
reduce the parallel complexity, which is especially important for long-term problems
with a limited number of processors.2022-03-01T00:00:00ZL𝛼-Regularization of the Beckmann Problem
http://hdl.handle.net/2003/40815
Title: L𝛼-Regularization of the Beckmann Problem
Authors: Lorenz, Dirk; Mahler, Hinrich; Meyer, Christian
Abstract: We investigate the problem of optimal transport in the so-called Beckmann form, i.e. given two Radon measures on a compact set, we seek an optimal flow field which is a vector valued Radon measure on the same set that describes a flow between these two measures and minimizes a certain linear cost function.
We consider L𝛼 regularization of the problem, which guarantees uniqueness and forces the solution to be an integrable function rather than a Radon measure. This regularization naturally gives rise to a semi-smooth Newton scheme that can be used to solve the problem numerically. Besides motivating and developing the numerical scheme, we also include approximation results for vanishing regularization in the continuous setting.2022-01-01T00:00:00ZOptimal control of non-convex rate-independent systems via vanishing viscosity – The finite dimensional case
http://hdl.handle.net/2003/40357
Title: Optimal control of non-convex rate-independent systems via vanishing viscosity – The finite dimensional case
Authors: Knees, Dorothee; Meyer, Christian; Sievers, Michael
Abstract: We investigate an optimal control problem governed by the evolution of a rate-independent system in finite dimensions. The rate-independent system is determined by a (possibly) non-convex energy, which contains the controllable, external load, and a dissipation potential, which is assumed to be positively homogenous of degree one. Under the several di˙erent concepts of solutions for these rate-independent systems, we bear on the so-called normalized parametrized BV solutions and prove the existence of a globally optimal solution of the optimal control problem constrained by this notion of solution. Our main result however concerns the approximation of optimal solutions by means of viscous regularization. The crucial issue in this context is that normalized parametrized BV solutions are in general non-unique and lack regularity, whereas the viscous solutions are unique and time-continuous. With the help of an additional regularity assumption on at least one optimal solution and a tailored penalization of the energy, one can nonetheless show that global minimizers of the viscous optimal control problems converge to an optimal solution of the original problem as the viscosity parameter tends to zero.2021-07-01T00:00:00ZVery Fast Finite Element Poisson Solvers on Lower Precision Accelerator Hardware - A “Proof-of-Concept” Study for NVIDIA Tesla V100
http://hdl.handle.net/2003/40355
Title: Very Fast Finite Element Poisson Solvers on Lower Precision Accelerator Hardware - A “Proof-of-Concept” Study for NVIDIA Tesla V100
Authors: Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Abstract: Recently, accelerator hardware in the form of graphics cards including Tensor Cores, specialized for AI, has significantly gained in importance in the domain of high performance computing. For example, NVIDIA’s Tesla V100 promises a com-puting power of up to 125 TFLOP/s achieved by Tensor Cores, but only if half precision floating point format is used. We describe the diÿculties and discrepancy between theoretical and actual computing power if one seeks to use such hardware for numerical simulations, i.e., solving partial di˙erential 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 re-quirements, based on “prehandling” by means of hierarchical 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.2021-07-01T00:00:00ZDoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension
http://hdl.handle.net/2003/40353
Title: DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension
Authors: May, Sandra; Streitbuerger, Florian
Abstract: In this work, we present the Domain of Dependence (DoD) stabilization for sys tems of hyperbolic conservation laws in one space dimension. The base scheme uses a
method of lines approach consisting of a discontinuous Galerkin scheme in space and
an explicit strong stability preserving Runge-Kutta scheme in time. When applied
on a cut cell mesh with a time step length that is appropriate for the size of the
larger background cells, one encounters stability issues. The DoD stabilization con sists of penalty terms that are designed to address these problems by redistributing
mass between the inflow and outflow neighbors of small cut cells in a physical way.
For piecewise constant polynomials in space and explicit Euler in time, the stabi lized scheme is monotone for scalar problems. For higher polynomial degrees p, our
numerical experiments show convergence orders of p + 1 for smooth flow and robust
behavior in the presence of shocks.2021-07-01T00:00:00ZA Proof of Concept for Very Fast Finite Element Poisson Solvers on Accelerator Hardware
http://hdl.handle.net/2003/40294
Title: A Proof of Concept for Very Fast Finite Element Poisson Solvers on Accelerator Hardware
Authors: Ruda, Dustin; Turek, Stefan; Ribbrock, Dirk; Zajac, Peter
Abstract: It is demonstrated that modern accelerator hardware specialized in AI, e.g., “next gen GPUs” equipped with Tensor Cores, can be profitably used in finite element simulations by means of a new hardware-oriented method to solve linear systems arising from Poisson’s equation in 2D. We consider the NVIDIA Tesla V100 Tensor Core GPU with a peak performance of 125 TFLOP/s, that is only achievable in half precision and if operations with high arithmetic intensity, such as dense matrix multiplications, are executed, though. Its computing power can be exploited to a great extent by the new method based on “prehandling” without loss of accuracy. We obtain a significant reduction of computing time compared to a standard geometric multigrid solver on standard x64 hardware.2021-06-01T00:00:00ZAn Adaptive Discrete Newton Method for Regularization-Free Bingham Model
http://hdl.handle.net/2003/40284
Title: An Adaptive Discrete Newton Method for Regularization-Free Bingham Model
Authors: Fatima, Arooj; Turek, Stefan; Ouazzi, Abderrahim; Afaq, Muhammad Aaqib
Abstract: Developing a numerical and algorithmic tool which correctly identiﬁes unyielded regions in 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, solvers do not perform eﬃciently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress. The three ﬁeld formulation of the yield stress ﬂuid corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three ﬁeld formulation is solved 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. Therefore, we developed a new adaptive discrete Newton method, which evaluates the Jacobian with the divided diﬀerence approach. We relate the step length to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton method. We analyse the solvability of the problem along with the adaptive Newton method for Bingham ﬂuids by doing numerical studies for a prototypical conﬁguration ”viscoplastic ﬂuid ﬂow in a channel”.2021-01-01T00:00:00ZMonolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension
http://hdl.handle.net/2003/40283
Title: Monolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension
Authors: Afaq, Muhammad Aaqib; Turek, Stefan; Ouazzi, Abderrahim; Fatima, Arooj
Abstract: We have developed a monolithic Newton-multigrid solver for multiphase ﬂow problems which solves velocity, pressure and interface position simultaneously. The main idea of our work is based on the formulations discussed in [1], where it points out the feasibility of a fully implicit monolithic solver for multiphase ﬂow problems via two formulations, a curvature free level set approach and a curvature free cut-oﬀ material function approach. Both formulations are fully implicit and have the advantages of requiring less regularity, since neither normals nor curvature are explicitly calculated, and no capillary time restriction has to be respected. Furthermore, standard Navier-Stokes solvers might be used, which do not have to take into account inhomogeneous force terms. The reinitialization issue is integrated within the formulations.
The nonlinearity is treated with a Newton-type solver with divided diﬀerence evaluation of the Jacobian matrices. 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 using higher order stable Q_2/P_1^disc FEM for velocity and pressure and Q_2 for all other variables. The method is implemented into an existing software package for the numerical simulation of multiphase ﬂows (FeatFlow). The robustness and accuracy of this solver is tested for two diﬀerent test cases, static bubble and oscillating bubble, respectively.2021-01-01T00:00:00ZFinite element methods for the simulation of thixotropic flow problems
http://hdl.handle.net/2003/40282
Title: Finite element methods for the simulation of thixotropic flow problems
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan
Abstract: This note is concerned with the application of Finite Element Methods (FEM) and Newton-Multigrid solvers for the simulation of thixotropic ﬂow problems.
The thixotropy phenomena are introduced into viscoplastic material by taking into ac-count the internal material micro structure using a scalar structure parameter. Firstly, the viscoplastic stress is modiﬁed to include the thixotropic stress dependent on the structure parameter. Secondly, an evolution equation for the structure parameter is introduced to induce the time-dependent process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. Substantially, this is done sim-ply by introducing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The modiﬁed thixotropic viscoplastic stress w.r.t. the structure parameter is integrated in quasi-Newtonian manner into the generalized Navier-Stokes equations and the evolution equation for the structure parameter constitutes the main core of full set of modeling equations, which are creditable as the privilege answer to incorporate thixotropy phenomena. A fully coupled monolithic ﬁnite element approach has been exercised which manages the material internal micro structure parameter, ve-locity, and pressure ﬁelds simultaneously. The nonlinearity of the corresponding problem, related to the dependency of the diﬀusive stress on the material parameters and the non-linear structure parameter models on the other hand, is treated with generalized Newton’s method w.r.t. the Jacobian’s singularities having a global convergence property. The lin-earized systems inside the outer Newton loops form a typical saddle-point problem which is solved using a geometrical multigrid method with a Vanka-like smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous linear pressure and biquadratic velocity spaces. We examine the accuracy, robustness and eﬃciency of the Newton-Multigrid FEM solver throughout the solution of thixotropic viscoplastic ﬂow problems in Couette device and in 4:1 contraction.2021-06-01T00:00:00ZMonolithic Newton-multigrid FEM for the simulation of thixotropic flow problems
http://hdl.handle.net/2003/40237
Title: Monolithic Newton-multigrid FEM for the simulation of thixotropic flow problems
Authors: Begum, Naheed; Ouazzi, Abderrahim; Turek, Stefan2021-05-01T00:00:00ZConvergence Analysis of a Local Stationarity Scheme for Rate-Independent Systems and Application to Damage
http://hdl.handle.net/2003/40190
Title: Convergence Analysis of a Local Stationarity Scheme for Rate-Independent Systems and Application to Damage
Authors: Sievers, Michael
Abstract: This paper is concerned with an approximation scheme for rate-independent systems governed by a non-smooth dissipation and a possibly non-convex energy functional. The scheme is based on the local minimization scheme introduced in [EM06], but relies on local stationarity of the underlying minimization problem. Under the assumption of Mosco-convergence for the dissipation functional, we show that accumulation points exist and are so-called parametrized solutions of the rate-independent system. In particular, this guarantees the existence of parametrized solutions for a rather general setting. Afterwards, we apply the scheme to a model for the evolution of damage.2021-04-01T00:00:00ZFourier analysis of a time-simultaneous two-grid algorithm for the one-dimensional heat equation
http://hdl.handle.net/2003/40189
Title: Fourier analysis of a time-simultaneous two-grid algorithm for the one-dimensional heat equation
Authors: Lohmann, Christoph; Dünnebacke, Jonas; Turek, Stefan
Abstract: In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite di˙erence approximation in space while the semi-discrete problem is discretized in time using the θ-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown.
It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence.2021-04-01T00:00:00ZCommutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation
http://hdl.handle.net/2003/40100
Title: Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation
Authors: Kerkhoff, Xenia; May, Sandra
Abstract: We consider one-dimensional distributed optimal control problems with the state equa-tion being given by the viscous Burgers equation. We discretize using a space-time dis-continuous Galerkin approach. We use upwind ﬂux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our ﬁndings by numerical results.2021-03-01T00:00:00ZNumerical simulation techniques for the efficient and accurate treatment of local fluidic transport processes together with chemical reactions
http://hdl.handle.net/2003/40084
Title: Numerical simulation techniques for the efficient and accurate treatment of local fluidic transport processes together with chemical reactions
Authors: Mierka, Otto; Turek, Stefan
Abstract: This work describes a numerical framework developed for the efficient and accurate simulation of microfluidic applications related to two leading ex-periments of the DFG SPP 1740 research initiative, namely the ‘Superfocus Mi-cromixer’ and the ‘Taylor bubble flow’. Both of these basic experiments are con-sidered in a reactive framework using the SPP 1740 specific chemical reaction systems. A description of the utilized numerical components related to special meshing techniques, discretization methods and decoupling solver strategies is provided and its particular implementation is performed in the open-source CFD package FeatFlow [19]. A demonstration of the developed simulation tool is based on already defined validation cases and on suitable examples being re-sponsible for the determination of the related convergence properties (in the range of targeted process parameter values) of the developed numerical frame-work. The subsequent studies give an insight into a parameter estimation method with the aim of determination of unknown reaction-kinetic parameter values by the help of experimentally measured data.2021-02-01T00:00:00ZNewton-multigrid FEM solver for the simulation of Quasi-Newtonian modeling of thixotropic flows
http://hdl.handle.net/2003/40025
Title: Newton-multigrid FEM solver for the simulation of Quasi-Newtonian modeling of thixotropic flows
Authors: Ouazzi, Abderrahim; Begum, Naheed; Turek, Stefan
Abstract: This paper is concerned with the application of Finite Element Methods (FEM) and Newton-Multigrid solvers to simulate thixotropic flows using quasi-Newtonian modeling.
The thixotropy phenomena are introduced to yield stress material by taking into consideration the in-ternal material microstructure using a structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropy throughout the structure parameter. Secondly, an evolution equation for the struc-ture parameter is introduced to induce the time-dependent process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. This is done simply by introduc-ing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The nonlinearity, related to the dependency of the diffusive term on the material parameters, is treated with generalized Newton’s method w.r.t. the Jacobian’s singularities having a global convergence property. The linearized systems inside the outer Newton loops are solved using the geometrical multigrid with a Vanka-like linear smoother taking into account a stable FEM approximation pair for velocity and pres-sure with discontinuous pressure and biquadratic velocity spaces.
We analyze the application of using the quasi-Newtonian modeling approach for thixotropic flows, and the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of the thixotropic flows using manufactured solutions in a channel and the prototypical configuration of thixotropic flows in Couette device.2021-02-01T00:00:00ZEine Machbarkeitsstudie zu schnellen FEM-Poisson-Lösern auf Beschleunigerhardware als Beispiel für Hardware-orientierte Numerik
http://hdl.handle.net/2003/40024
Title: Eine Machbarkeitsstudie zu schnellen FEM-Poisson-Lösern auf Beschleunigerhardware als Beispiel für Hardware-orientierte Numerik
Authors: Turek, Stefan; Ruda, Dustin; Ribbrock, Dirk; Poelstra, Heiko; Zajac, Peter2021-01-01T00:00:00ZIncreased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems
http://hdl.handle.net/2003/39972
Title: Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems
Authors: Dünnebacke, Jonas; Turek, Stefan; Lohmann, Christoph; Sokolov, Andriy; Zajac, Peter
Abstract: We discuss how ‘parallel-in-space & simultaneous-in-time’ Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypicalapplication, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finitedifference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearizedproblems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.2020-12-01T00:00:00ZAnalysis and numerical treatment of bulk-surface reaction-diﬀusion models of Gierer-Meinhardt type
http://hdl.handle.net/2003/39810
Title: Analysis and numerical treatment of bulk-surface reaction-diﬀusion models of Gierer-Meinhardt type
Authors: Bäcker, Jan-Phillip; Röger, Matthias; Kuzmin, Dmitri
Abstract: We consider a Gierer-Meinhardt system on a surface coupled with aparabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (SIAM J. Appl. Dyn. Syst. 18, 2019).We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The proof uses Schauders fixed point theorem and a splitting in a surface and a bulk part. We also solve a reduced system, corresponding to the assumption of a well mixed bulk solution, numerically. We use operator-splitting methods which combine a finite element discretization of the Laplace-Beltrami operator with a positivity-preserving treatment of the source and sink terms. The proposed methodology is based on the flux-corrected transport (FCT) paradigm. It constrains the space and time discretization of the reduced problem in a manner which provides positivity preservation, conservation of mass, and second-order accuracy in smooth regions. The results of numerical studies for the system on a two-dimensional sphere demonstrate the occurrence of localized steady-state multispike pattern that have also been observed in one-dimensional models.2020-10-01T00:00:00ZEnriched Galerkin method for the shallow-water equations
http://hdl.handle.net/2003/39242
Title: Enriched Galerkin method for the shallow-water equations
Authors: Hauck, Moritz; Aizinger, Vadym; Frank, Florian; Hajduk, Hennes; Rupp, Andreas
Abstract: This work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similarly to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic benchmarks and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG.2020-04-01T00:00:00ZBenchmarking and Validation of a Combined CFD-Optics Solver for Micro-Scale Problems
http://hdl.handle.net/2003/39166
Title: Benchmarking and Validation of a Combined CFD-Optics Solver for Micro-Scale Problems
Authors: Münster, Raphael; Mierka, Otto; Turek, Stefan; Weigel, Thomas; Ostendorf, Andreas
Abstract: In this work we present a new approach for coupled CFD-Optics problems that consists of a combination of a Finite Element Method (FEM) based flow solver with a ray tracing based tool for optic forces that are induced by a laser. This is a setup that is mainly encountered in the field of optical traps. We combined the open-source computational fluid dynamics (CFD) package FEATFLOW with the raytracing software of the LAT-RUB with this task in mind. We benchmark and analyze the solver first based on a configuration with a single spherical particle that is subjected to the laser forces of an optical trap. The setup is based on an experiment which is then compared to the results of our combined CFD-Optics solver. As an extension of the standard procedure to simulate such problems we present a method with a time-stepping scheme that contains a macro step approach. The results showthat this macro time-stepping scheme provides a significant acceleration of the standard procedure while still maintaining good accuracy. A second configuration is analyzed that involves non-spherical geometries such as micro rotors. We describe a procedure that is able to efficiently and accurately calculate optical forces with surface triangulations as input geometries. Then we proceed to compare simulation results of the final angular velocity of the micro rotor with experimental measurements.2020-05-01T00:00:00ZAn algebraic flux correction scheme facilitating the use of Newton-like solution strategies
http://hdl.handle.net/2003/39094
Title: An algebraic flux correction scheme facilitating the use of Newton-like solution strategies
Authors: Lohmann, Christoph
Abstract: Building on recent advances in the analysis and design of algebraic flux correction (AFC) schemes, new differentiable limiter functions are constructed for efficient nonlinear solution strategies. The proposed scaling parameters are used to limit artificial diffusion operators incorporated into the residual of a high order target scheme to produce accurate and bound-preserving finite element approximations to hyperbolic problems. Due to this stabilization procedure, the occurring system becomes highly nonlinear and the efficient computation of corresponding solutions is a challenging task. The presented regularization approach makes the AFC residual twice continuously differentiable so that Newton’s method converges quadratically for suÿciently good initial guesses. Furthermore, the performance of each nonlinear iteration is improved by expressing the Jacobian as the sum and product of matrices having the same sparsity pattern as the Galerkin system matrix. Eventually, the AFC methodology constructed for stationary problems is extended to transient test cases and validated numerically by applying it to several numerical benchmarks.2020-04-01T00:00:00ZOn the threshold condition for Dörfler marking
http://hdl.handle.net/2003/39072
Title: On the threshold condition for Dörfler marking
Authors: Diening, Lars; Kreuzer, Christian
Abstract: It is an open question if the threshold condition θ < θ_* for the Dörﬂer marking parameter is necessary to obtain optimal algebraic rates of adaptive ﬁnite element methods. We present a (non-PDE) example ﬁtting into the common abstract convergence framework (axioms of adaptivity) and which is potentially converging with exponential rates. However, for Dörfler marking θ > θ_* the algebraic converges rate can be made arbitrarily small.2020-03-01T00:00:00ZOptimal control of perfect plasticity Part II: Displacement tracking
http://hdl.handle.net/2003/39071
Title: Optimal control of perfect plasticity Part II: Displacement tracking
Authors: Meyer, Christian; Walther, Stephan
Abstract: The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is optimize the displacement ﬁeld in the domain occupied by the body by means of prescribed Dirichlet boundary data, which serve as control variables. The arising optimization problem is nonsmooth for several reasons, in particular, since the control-to-state mapping is not single-valued. We therefore apply a Yosida regularization to obtain a single-valued control-to-state operator. Beside the existence of optimal solutions, their approximation by means of this regularization approach is the main subject of this work. It turns out that a so-called reverse approximation guaranteeing the existence of a suitable recovery sequence can only be shown under an additional smoothness assumption on at least one optimal solution.2020-03-01T00:00:00ZStrong Stationarity for Optimal Control of Variational Inequalities of the Second Kind
http://hdl.handle.net/2003/39069
Title: Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind
Authors: Christof, Constantin; Meyer, Christian; Schweizer, Ben; Turek, Stefan
Abstract: This paper is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. So-called strong stationarity conditions are derived in an abstract framework. Strong stationarity conditions are regarded as the most rigorous ones, since they imply all other types of stationarity concepts and are equivalent to purely primal optimality conditions. The abstract framework is afterwards applied to four application-driven examples.2020-03-01T00:00:00ZQuasi-optimal and pressure robust discretizations of the stokes equations by moment- and divergence-preserving operators
http://hdl.handle.net/2003/39037
Title: Quasi-optimal and pressure robust discretizations of the stokes equations by moment- and divergence-preserving operators
Authors: Kreuzer, Christian; Verfürth, Rüdiger; Zanotti, Pietro
Abstract: We approximate the solution of the Stokes equations by a new quasi-optimal and pressure robust discontinuous Galerkin discretization of arbitrary order. This means quasi-optimality of the velocity error independent of the pressure. Moreover, the discretization is well-defined for any load which is admissible for the continuous problem and it also provides classical quasioptimal estimates for the sum of velocity and pressure errors. The key design principle is a careful discretization of the load involving a linear operator, which maps discontinuous Galerkin test functions onto conforming ones thereby preserving the discrete divergence and certain moment conditions on faces and elements.2020-02-01T00:00:00ZLocally bound-preserving enriched Galerkin methods for the linear advection equation
http://hdl.handle.net/2003/38536
Title: Locally bound-preserving enriched Galerkin methods for the linear advection equation
Authors: Kuzmin, Dmitri; Hajduk, Hennes; Rupp, Andreas
Abstract: In this work, we introduce algebraic ﬂux correction schemes for enriched (P1 ⊕ P0 and Q1 ⊕ P0) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P1/Q1 component in the admissible range, we limit the ﬂuxes and element contributions, the complete removal of which would correspond to ﬁrst-order upwinding. The ﬁrst limiting procedure that we consider in this paper is based on the ﬂux-corrected transport (FCT) paradigm. It belongs to the family of predictor-corrector algorithms and requires the use of small time steps. The second limiting strategy is monolithic and produces nonlinear problems with well-deﬁned residuals. This kind of limiting is well suited for stationary and time-dependent problems alike. The need for inverting consistent mass matrices in explicit strong stability preserving Runge-Kutta time integrators is avoided by reconstructing nodal time derivatives from cell averages. Numerical studies are performed for standard 2D test problems.2020-01-01T00:00:00ZOptimal control of perfect plasticity Part I: Stress tracking
http://hdl.handle.net/2003/38529
Title: Optimal control of perfect plasticity Part I: Stress tracking
Authors: Meyer, Christian; Walther, Stephan
Abstract: The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress ﬁeld by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulﬁlled so that the system admits a solution, whose stress ﬁeld is unique. This gives rise to a well deﬁned control-to-state operator, which is continuous but not Gˆateaux-differentiable. The control-to-state map is therefore regularized, ﬁrst by means of the Yosida regularization and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach.2020-01-01T00:00:00ZThe Concept of Prehandling as Direct Preconditioning for Poisson-like Problems
http://hdl.handle.net/2003/38465
Title: The Concept of Prehandling as Direct Preconditioning for Poisson-like Problems
Authors: Ruda, Dustin; Turek, Stefan; Zajac, Peter; Ribbrock, Dirk
Abstract: To benefit from current trends in HPC hardware, such as increasing avail-ability of low precision hardware, we present the concept of prehandling as a direct way of preconditioning and the hierarchical finite element method which is exceptionally well-suited to apply prehandling to Poisson-like problems, at least in 1D and 2D. Such problems are known to cause ill-conditioned stiffness matrices and therefore high computational errors due to round-off. We show by means of numerical results that by prehandling via the hierarchical finite element method the condition number can be significantly reduced (while advantageous properties are preserved) which enables us to obtain sufficiently accurate solutions to Poisson-like problems even if lower computing precision (i.e. single or half precision format) is used.2019-12-01T00:00:00ZMatrix-free subcell residual distribution for Bernstein ﬁnite elements: Monolithic limiting
http://hdl.handle.net/2003/38464
Title: Matrix-free subcell residual distribution for Bernstein ﬁnite elements: Monolithic limiting
Authors: Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; Tomov, Vladimir; Tomas, Ignacio; Shadid, John. N.
Abstract: This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order ﬁnite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipula-tions of element contributions to the global nonlinear system. The required modiﬁcations can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vec-tors associated with the standard Galerkin discretization are manipulated di-rectly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for ﬂux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary prob-lems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems2019-12-01T00:00:00ZA time-simultaneous multigrid method for parabolic evolution equations
http://hdl.handle.net/2003/38463
Title: A time-simultaneous multigrid method for parabolic evolution equations
Authors: Dünnebacke, Jonas; Turek, Stefan; Zajac, Peter; Sokolov, Andriy
Abstract: We present a time-simultaneous multigrid scheme for parabolic equations that is motivated by blocking multiple time steps together. The resulting method is closely related to multigrid waveform relaxation and is robust with respect to the spatial and temporal grid size and the number of simultaneously computed time steps. We give an intuitive understanding of the convergence behavior and briefly discuss how the theory for multigrid waveform relaxation can be applied in some special cases. Finally, some numerical results for linear and also nonlinear test cases are shown.2019-12-01T00:00:00ZBasic Machine Learning Approaches for the Acceleration of PDE Simulations and Realization in the FEAT3 Software
http://hdl.handle.net/2003/38462
Title: Basic Machine Learning Approaches for the Acceleration of PDE Simulations and Realization in the FEAT3 Software
Authors: Ruelmann, Hannes; Geveler, Markus; Ribbrock, Dirk; Zajac, Peter; Turek, Stefan
Abstract: In this paper we present a holistic software approach based on the FEAT3 software for solving multidimensional PDEs with the Finite Element Method that is built for a maximum of performance, scalability, maintainability and extensibilty. We introduce basic paradigms how modern computational hardware architectures such as GPUs are exploited in a numerically scalable fashion. We show, how the framework is extended to make even the most recent advances on the hardware market accessible to the framework, exemplified by the ubiquitous trend to customize chips for Machine Learning. We can demonstrate that for a numerically challenging model problem, artificial neural networks can be used while preserving a classical simulation solution pipeline through the incorporation of a neural network preconditioner in the linear solver2019-12-01T00:00:00ZSimulating two-dimensional viscoelastic fluid flows by means of the “Tensor Diffusion” approach
http://hdl.handle.net/2003/38461
Title: Simulating two-dimensional viscoelastic fluid flows by means of the “Tensor Diffusion” approach
Authors: Westervoß, Patrick; Turek, Stefan
Abstract: In this work, the novel “Tensor Diffusion” approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized “Tensor Stokes” problem, avoiding the need of considering a separate stress tensor in the solution process. Besides fully developed channel flows, the “Tensor Diffusion” approach is evaluated as well in the context of general two-dimensional flow configurations, which are simulated by a suitable four-field formulation of the viscoelastic model respecting the “Tensor Diffusion”.2019-12-01T00:00:00ZThe "Tensor Diffusion" approach for simulating viscoelastic fluids
http://hdl.handle.net/2003/38432
Title: The "Tensor Diffusion" approach for simulating viscoelastic fluids
Authors: Westervoß, Patrick; Turek, Stefan; Damanik, Hogenrich; Ouazzi, Abderrahim
Abstract: In this work, the novel "Tensor Diffusion" approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized "Tensor Stokes" problem, avoiding the need of considering a separate stress tensor in the solution process. Besides fully developed channel flows, the “Tensor Diffusion” approach is evaluated as well in the context of general two-dimensional flow configurations, which are simulated by a suitable four-field formulation of the viscoelastic model respecting the "Tensor Diffusion". However, substituting the extra-stress tensor by the "Tensor Diffusion" in the complete flow model is desired for general two-dimensional flows as well, again to be able to reduce the full nonlinear viscoelastic model to a Stokes-like problem.2019-11-01T00:00:00ZNumerical Simulation and Benchmarking of Drops and Bubbles
http://hdl.handle.net/2003/38378
Title: Numerical Simulation and Benchmarking of Drops and Bubbles
Authors: Turek, Stefan; Mierka, Otto2019-11-01T00:00:00ZOptimal control of an abstract evolution variational inequality with application to homogenized plasticity
http://hdl.handle.net/2003/38261
Title: Optimal control of an abstract evolution variational inequality with application to homogenized plasticity
Authors: Meinlschmidt, Hannes; Meyer, Christian; Walther, Stephan
Abstract: The paper is concerned with an optimal control problem governed by a state equa-tion in form of a generalized abstract operator differential equation involving a maximal monotoneoperator. The state equation is uniquely solvable, but the associated solution operator is in generalnot Gˆateaux-differentiable. In order to derive optimality conditions, we therefore regularize the stateequation and its solution operator, respectively, by means of a (smoothed) Yosida approximation.We show convergence of global minimizers for regularization parameter tending to zero and derivenecessary and sufficient optimality conditions for the regularized problems. The paper ends with anapplication of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.2019-09-01T00:00:00ZAn isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates
http://hdl.handle.net/2003/38257
Title: An isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates
Authors: Dornisch, Wolfgang; Stöckler, Joachim
Abstract: We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patches Ω_k with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces X_h,k on each patch Ω_k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of [12] and highlights several aspects which did not receive full attention before. In particular, by choosing appropriate spaces of polynomial splines as Lagrange multipliers, we obtain a uniform infsup-inequality. Moreover, we provide a new additional condition on the discrete spaces X_h,k which is required for obtaining optimal convergence rates of the mortar method. Our numerical examples demonstrate that the optimal rate is lost if this condition is neglected.2019-09-01T00:00:00ZOn the solvability and iterative solution of algebraic flux correction problems for convection-reaction equations
http://hdl.handle.net/2003/38190
Title: On the solvability and iterative solution of algebraic flux correction problems for convection-reaction equations
Authors: Lohmann, Christoph
Abstract: New results on the theoretical solvability of nonlinear algebraic flux correction (AFC) problems are presented and a Newton-like solution technique exploiting an efficient computation of the Jacobian is introduced. The AFC methodology is a rather new and unconventional approach to algebraically stabilize finite element discretizations of convection-dominated transport problems in a bound-preserving manner. Besides investigations concerning the theoretical solvability, the development of efficient iterative solvers seems
to be one of the most challenging problems. The purpose of this paper is to take the next step to remove such obstacles: For the linear convection-reaction equation, the existence of a unique solution is shown under a mild coercivity condition and some restriction on the limiter. Additionally, the numerical effort for solving such problems is drastically reduced by the use of a highly customized implementation of the Jacobian. The benefit of this approach is illustrated by in-depth numerical studies.2019-08-01T00:00:00ZA monolithic operator-adaptive Newton-Multigrid solver for Navier-Stokes Equations in 3D
http://hdl.handle.net/2003/38189
Title: A monolithic operator-adaptive Newton-Multigrid solver for Navier-Stokes Equations in 3D
Authors: Jendrny, Robert; Mierka, Otto; Münster, Raphael; Turek, Stefan
Abstract: The aim of this paper is to describe a new, fast and robust solver for 3D flow problems which are described by the incompressible Navier-Stokes equations.
The correspondig simulations are done by a monolithic 3D flow solver, i.e. velocity and pressure are solved at the same time. During these simulations the convective part is linearized using two different methods: Fixpoint method and Newton method. The Fixpoint method is working in a quite robust way, but it has a slow convergence depending on the Reynolds number. In contrast, if the Newton method does not fail, the simulations which are done by this linearization converge typically much faster. In the case of the Newton method quadratical convergence is obtained. The challenging part is to find a method which unites the stability of the Fixpoint method and the fast convergence of the Newton method.
For the resulting operator-adaptive Newton method, several numerical examples are considered: The flow around a sphere and a cylinder is simulated to analyze the behaviour of the used methods. Since the behaviour of the linearization types is different between each of them, the results caused by varying Reynolds numbers and the arised equations are analyzed concerning the efficiency of each method.2019-08-01T00:00:00ZA-priori error analysis of local incremental minimization schemes for rate-independent evolutions
http://hdl.handle.net/2003/38139
Title: A-priori error analysis of local incremental minimization schemes for rate-independent evolutions
Authors: Meyer, Christian; Sievers, Michael
Abstract: This paper is concerned with a priori error estimates for the local incremental
minimization scheme, which is an implicit time discretization method for the approximation of rate-independent
systems with non-convex energies. We first show by means of a counterexample that
one cannot expect global convergence of the scheme without any further assumptions on the energy.
For the class of uniformly convex energies, we derive error estimates of optimal order, provided that
the Lipschitz constant of the load is sufficiently small. Afterwards, we extend this result to the case
of an energy, which is only locally uniformly convex in a neighborhood of a given solution trajectory.
For the latter case, the local incremental minimization scheme turns out to be superior compared to
its global counterpart, as a numerical example demonstrates.2019-07-01T00:00:00ZMonolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws
http://hdl.handle.net/2003/38138
Title: Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws
Authors: Kuzmin, Dmitri
Abstract: Using the theoretical framework of algebraic flux correction and invariant domain
preserving schemes, we introduce a monolithic approach to convex limiting
in continuous finite element schemes for linear advection equations, nonlinear
scalar conservation laws, and hyperbolic systems. In contrast to fluxcorrected
transport (FCT) algorithms that apply limited antidiffusive corrections
to bound-preserving low-order solutions, our new limiting strategy exploits
the fact that these solutions can be expressed as convex combinations
of bar states belonging to a convex invariant set of physically admissible solutions.
Each antidiffusive flux is limited in a way which guarantees that the
associated bar state remains in the invariant set and preserves appropriate local
bounds. There is no free parameter and no need for limit fluxes associated with
the consistent mass matrix of time derivative term separately. Moreover, the
steady-state limit of the nonlinear discrete problem is well defined and independent
of the pseudo-time step. In the case study for the Euler equations, the
components of the bar states are constrained sequentially to satisfy local maximum
principles for the density, velocity, and specific total energy in addition
to positivity preservation for the density and pressure. The results of numerical
experiments for standard test problems illustrate the ability of built-in convex
limiters to resolve steep fronts in a sharp and nonoscillatory manner.2019-06-01T00:00:00ZLimiting and divergence cleaning for continuous finite element discretizations of the MHD equations
http://hdl.handle.net/2003/38107
Title: Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations
Authors: Kuzmin, Dmitri; Klyushnev, Nikita2019-06-01T00:00:00ZRadiale Basisfunktionen
http://hdl.handle.net/2003/38102
Title: Radiale Basisfunktionen
Authors: Karras, Samuel; Linz, Lucia; Buchatz, David; Symann, Paul; Schulte Huxel, Linus; Dreyer, Philip Marten; Heise, Christopher; von Lehmden, Johann2019-06-01T00:00:00ZMusik und Mathematik
http://hdl.handle.net/2003/38101
Title: Musik und Mathematik
Authors: Sivalingam, Kiruththika; Kliewer, Viktoria; Köchling, Gerrit2019-06-01T00:00:00ZPP-Löser für die Navier-Stokes Gleichung
http://hdl.handle.net/2003/38100
Title: PP-Löser für die Navier-Stokes Gleichung
Authors: Helmich, Thomas; Tiemann, Enno2019-06-01T00:00:00ZA stabilized DG cut cell method for discretizing the linear transport equation
http://hdl.handle.net/2003/38094
Title: A stabilized DG cut cell method for discretizing the linear transport equation
Authors: Engwer, Christian; May, Sandra; Nüßing, Andreas; Streitbürger, Florian2019-06-01T00:00:00ZLineare Inverse Probleme
http://hdl.handle.net/2003/38077
Title: Lineare Inverse Probleme
Authors: Sauer, Katharina; Gierschner, Christine; Sklarzyk, Melanie; Meise, Bianca2019-05-01T00:00:00ZQuasi-best approximation in optimization with PDE constraints
http://hdl.handle.net/2003/38018
Title: Quasi-best approximation in optimization with PDE constraints
Authors: Gaspoz, Fernando; Kreuzer, Christian; Veeser, Andreas; Wollner, Winnifried
Abstract: We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regularization parameter. Furthermore, if the operators of control-action and observation are compact, this quasibest-approximation constant becomes independent of the Tikhonov parameter as the meshsize tends to 0 and we give quantitative relationships between meshsize and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.2019-04-01T00:00:00ZRegularization for optimal control problems associated to nonlinear evolution equations
http://hdl.handle.net/2003/38017
Title: Regularization for optimal control problems associated to nonlinear evolution equations
Authors: Meinlschmidt, Hannes; Meyer, Christian; Rehberg, Joachim2019-04-01T00:00:00ZQuadratically regularized optimal transport
http://hdl.handle.net/2003/37962
Title: Quadratically regularized optimal transport
Authors: Lorenz, Dirk A.; Manns, Paul; Meyer, Christian
Abstract: We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss-Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).2019-03-01T00:00:00ZMatrix-free subcell residual distribution for Bernstein finite elements: Low-order schemes and FCT
http://hdl.handle.net/2003/37961
Title: Matrix-free subcell residual distribution for Bernstein finite elements: Low-order schemes and FCT
Authors: Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; Abgrall, Remi
Abstract: In this work, we introduce a new residual distribution (RD) framework for the design of matrix-free bound-preserving finite element schemes. As a starting point, we consider continuous and discontinuous Galerkin discretizations of the linear advection equation. To construct the corresponding local extremum diminishing (LED) approximation, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high-order Bernstein finite element discretizations involves localization to subcells and definition of bound-preserving weights for subcell contributions. Using strong stability preserving (SSP) Runge-Kutta methods for time integration, we prove the validity of discrete maximum principles under CFL-like time step restrictions. The low-order version of our method has roughly the same accuracy as the one derived from a piecewise (multi)-linear approximation on a submesh with the same nodal points. In high-order extensions, we currently use a flux-corrected transport (FCT) algorithm which can also be interpreted as a nonlinear RD scheme. The properties of the algebraically corrected Galerkin discretizations are illustrated by 1D, 2D, and 3D examples for Bernstein finite elements of different order. The results are as good as those obtained with the best matrix-based approaches. In our numerical studies for multidimensional problems, we use quadrilateral/hexahedral meshes but our methodology is readily applicable to unstructured/simplicial meshes as well.2019-03-01T00:00:00ZFinite Element Discretization of Local Minimization Schemes for Rate-Independent Evolutions
http://hdl.handle.net/2003/37960
Title: Finite Element Discretization of Local Minimization Schemes for Rate-Independent Evolutions
Authors: Meyer, Christian; Sievers, Michael
Abstract: This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in [EM06], which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions for mesh size tending to zero exist and are so-called parametrized solutions of the continuous problem. The discrete problems are solved by means of a mass lumping scheme for the non-smooth dissipation functional in combination with a semi-smooth Newton method. A numerical test indicates the efficiency of this approach. In addition, we compared the local minimization scheme with a time stepping scheme for global energetic solutions, which shows that both schemes yield different solutions with differing time discontinuities.2019-03-01T00:00:00ZOscillation in a posteriori error estimation
http://hdl.handle.net/2003/37945
Title: Oscillation in a posteriori error estimation
Authors: Kreuzer, Christian; Veeser, Andreas
Abstract: In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the following two properties. First, it is dominated by the error, irrespective of mesh fineness and the regularity of data and the exact solution. Second, it captures in terms of data the part of the residual that, in general, cannot be quantified with finite information. The new twist in our approach
is a locally stable projection onto discretized residuals.2019-03-01T00:00:00ZGoal oriented a posteriori error estimators for problems with modified discrete formulations based on the dual weighted residual method
http://hdl.handle.net/2003/37927
Title: Goal oriented a posteriori error estimators for problems with modified discrete formulations based on the dual weighted residual method
Authors: Kumor, Dustin; Rademacher, Andreas
Abstract: The article at hand focuses on finite element discretizations, where the continuous and the discrete formulations differ. We introduce a general approach based on the dual weighted residual method for estimating on the one hand the discretization error in a user specified quantity of interest and on the other hand the discrete model error induced by using different discrete techniques. Here, the usual error identities are obtained plus some additional terms. Furthermore, the numerical approximation of the error identities is discussed. As a simple example, we consider selective reduced integration for stabilizing the finite element discretization of linear elastic problems with nearly incompressible material behavior. This example fits well in the general setting. However, one has to be very careful in the numerical approximation of the error identities, where different reconstruction techniques have to be used for the additional terms due to the deviating discrete bi-linear form. Numerical examples substantiate the accuracy of the a posteriori error estimators and the efficiency of the adaptive methods based on them.2019-02-01T00:00:00ZRandom walk methods for Monte Carlo simulations of Brownian diffusion on a sphere
http://hdl.handle.net/2003/37920
Title: Random walk methods for Monte Carlo simulations of Brownian diffusion on a sphere
Authors: Novikov, Alexei; Kuzmin, Dmitri; Ahmadi, Omid
Abstract: This paper is focused on efficient Monte Carlo simulations of Brownian diffusion
effects in particle-based numerical methods for solving transport equations
on a sphere (or a circle). Using the heat equation as a model problem,
random walks are designed to emulate the action of the Laplace-Beltrami
operator without evolving or reconstructing the probability density function.
The intensity of perturbations is fitted to the value of the rotary diffusion
coefficient in the deterministic model. Simplified forms of Brownian motion
generators are derived for rotated reference frames, and several practical
approaches to generating random walks on a sphere are discussed. The alternatives
considered in this work include projections of Cartesian random
walks, as well as polar random walks on the tangential plane. In addition,
we explore the possibility of using look-up tables for the exact cumulative
probability of perturbations. Numerical studies are performed to assess the
practical utility of the methods under investigation.2019-02-01T00:00:00ZQuasi-optimal and pressure robust discretizations of the Stokes equations by new augmented Lagrangian Formulations
http://hdl.handle.net/2003/37917
Title: Quasi-optimal and pressure robust discretizations of the Stokes equations by new augmented Lagrangian Formulations
Authors: Kreuzer, Christian; Zanotti, Pietro
Abstract: We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure robust, in the sense that the velocity H^1-error is proportional to the best H^1-error to the analytical velocity. This shows that such a property can be achieved without using conforming and divergence-free pairs. We bound also the pressure L^2-error, only in terms of the best approximation errors to the analytical velocity and the analytical pressure. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order
to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by Discontinuous Galerkin methods.2019-02-01T00:00:00ZConvergence of adaptive C0-interior penalty Galerkin method for the biharmonic problem
http://hdl.handle.net/2003/37889
Title: Convergence of adaptive C0-interior penalty Galerkin method for the biharmonic problem
Authors: Dominicus, Alexander; Gaspoz, Fernando; Kreuzer, Christian
Abstract: We develop a basic convergence analysis for an adaptive C0IPG method for the Biharmonic problem which provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, by Kreuzer and Georgoulis, here we have to deal with the fact that the Lagrange finite element spaces may possibly contain no proper C1-conforming subspace. This prevents from a straight forward generalisation and requires the development of some new key technical tools.2019-01-01T00:00:00ZA quasi-optimal Crouzeix-Raviart discretization of the Stokes equations
http://hdl.handle.net/2003/37838
Title: A quasi-optimal Crouzeix-Raviart discretization of the Stokes equations
Authors: Verfürth, Rüdiger; Zanotti, Pietro2018-12-01T00:00:00ZAlgebraic limiting techniques and hp-adaptivity for continuous finite element discretizations
http://hdl.handle.net/2003/37139
Title: Algebraic limiting techniques and hp-adaptivity for continuous finite element discretizations
Authors: Kuzmin, Dmitri2018-09-01T00:00:00ZA partition of unity approach to adaptivity and limiting in continuous finite element methods
http://hdl.handle.net/2003/37110
Title: A partition of unity approach to adaptivity and limiting in continuous finite element methods
Authors: Kuzmin, Dmitri; Quezada de Luna, Manuel; Kees, Christopher E.
Abstract: The partition of unity finite element method (PUFEM) proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner. The lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures. In contrast to traditional hp-adaptivity for finite elements, the proposed approach preserves discrete conservation properties and the continuity of traces at common boundaries of adjacent mesh cells. In the context of space discretizations, a continuous blending function is used to combine finite element bases corresponding to high-order polynomials and piecewise-linear approximations based on the same set of nodes. In a similar vein, spatially partitioned time discretizations can be designed using weights that depend continuously on the space variable. The design of blending functions may be based on a priori knowledge (e.g., in applications to problems with singularities or boundary layers), local error estimates, smoothness indicators, and/or discrete maximum principles. In adaptive methods, changes of the finite element approximation exhibit continuous dependence on the data. The presented numerical examples illustrate the typical behavior of local H1 and L2 errors.2018-08-01T00:00:00ZGradient-based limiting and stabilization of continuous Galerkin methods
http://hdl.handle.net/2003/37077
Title: Gradient-based limiting and stabilization of continuous Galerkin methods
Authors: Kuzmin, Dmitri
Abstract: In this paper, we stabilize and limit continuous Galerkin discretizations of a linear transport equation using an algebraic approach to derivation of artificial diffusion operators. Building on recent advances in the analysis and design of edge-based algebraic flux correction schemes for singularly perturbed convection-diffusion problems, we derive algebraic stabilization operators that generate nonlinear high-order stabilization in smooth regions and enforce discrete maximum principles everywhere. The correction factors for antidiffusive element or edge contributions are defined in terms of nodal gradients that vanish at local extrema. The proposed limiting strategy is linearity-preserving and provides Lipschitz continuity of constrained terms. Numerical examples are presented for two-dimensional test problems.2018-07-01T00:00:00ZFrame-invariant directional vector limiters for discontinuous Galerkin methods
http://hdl.handle.net/2003/36906
Title: Frame-invariant directional vector limiters for discontinuous Galerkin methods
Authors: Hajduk, Hennes; Kuzmin, Dmitri; Aizinger, Vadym
Abstract: Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertexbased slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics.2018-06-01T00:00:00ZA flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds
http://hdl.handle.net/2003/36905
Title: A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds
Authors: Sokolov, Andriy; Davydov, Oleg; Kuzmin, Dmitri; Westermann, Alexander; Turek, Stefan
Abstract: In this article we introduce a FCT stabilized Radial Basis Function (RBF)-Finite Difference (FD) method for the numerical solution of convection dominated problems. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the solution for an almost random placement of scattered data nodes. The method can be applicable both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable.2018-06-01T00:00:00ZBathymetry reconstruction using inverse shallow water models: Finite element discretization and regularization
http://hdl.handle.net/2003/36876
Title: Bathymetry reconstruction using inverse shallow water models: Finite element discretization and regularization
Authors: Hajduk, Hennes; Kuzmin, Dmitri; Aizinger, Vadym
Abstract: In the present paper, we use modified shallow water equations (SWE) to reconstruct the bottom topography (also called bathymetry) of a flow domain without resorting to traditional inverse modeling techniques such as adjoint methods. The discretization in space is performed using a piecewise linear discontinuous Galerkin (DG) approximation of the free surface elevation and (linear) continuous finite elements for the bathymetry. Our approach guarantees compatibility of the discrete forward and inverse problems: for a given DG solution of the forward SWE problem, the underlying continuous bathymetry can be recovered exactly. To ensure well-posedness of the modified SWE and reduce sensitivity of the results to noisy data, a regularization term is added to the equation for the water height. A numerical study is performed to demonstrate the ability of the proposed method to recover bathymetry in a robust and accurate manner.2018-04-01T00:00:00ZA monolithic conservative level set method with built-in redistancing
http://hdl.handle.net/2003/36873
Title: A monolithic conservative level set method with built-in redistancing
Authors: Quezada de Luna, Manuel; Kuzmin, Dmitri; Kees, Christopher E.
Abstract: We introduce a new level set method for representing evolving interfaces. In the case of divergence-free velocity fields, the new method satisfies a conservation principle. Conservation is important for many applications such as modeling two-phase incompressible flow. In the present implementation, the conserved quantity is defined as the integral of a smoothed characteristic function. The new approach embeds level sets into a volume of fluid formulation. The evolution of an approximate signed distance function is governed by a conservation law for its (smoothed) sign. The non-linear level set transport equation is regularized by adding a flux correction term that assures a non-singular Jacobian and penalizes deviations from a distance function. The result is a locally conservative level set method with built-in elliptic redistancing. The continuous model is monolithic in the sense that the level set transport model, the volume of fluid law of mass conservation, and the minimization problem that preserves the approximate distance function property are incorporated into a single equation. There is no need for any extra stabilization, artificial compression, flux limiting, redistancing, mass correction, and other numerical fixes which are commonly used in level set or volume of fluid methods. In addition, there is just one free parameter that controls the strength of regularization and penalization in the model. The accuracy and conservation properties of the monolithic finite element / level set method are illustrated by the results of numerical studies for passive advection of free interfaces.2018-04-01T00:00:00ZAlgebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields
http://hdl.handle.net/2003/36864
Title: Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields
Authors: Lohmann, Christoph
Abstract: This work extends the algebraic flux correction (AFC) paradigm to finite element discretizations of conservation laws for symmetric tensor fields. The proposed algorithms are designed to enforce discrete maximum principles and preserve the eigenvalue range of evolving tensors. To that end, a continuous Galerkin approximation is modified by adding a linear artificial diffusion operator and a nonlinear antidiffusive correction. The latter is decomposed into edge-based fluxes and constrained to prevent violations of local bounds for the minimal and maximal eigenvalues. In contrast to the flux-corrected transport (FCT) algorithm developed previously by the author and existing slope limiting techniques for stress tensors , the admissible eigenvalue range is defined implicitly and the limited antidiffusive terms are incorporated into the residual of the nonlinear system. In addition to scalar limiters that use a common correction factor for all components of a tensor-valued antidiffusive flux, tensor limiters are designed using spectral decompositions. The new limiter functions are analyzed using tensorial extensions of the existing AFC theory for scalar convection-diffusion equations. The proposed methodology is backed by rigorous proofs of eigenvalue range preservation and Lipschitz continuity. Convergence of pseudo time-stepping methods to stationary solutions is demonstrated in numerical studies.2018-04-01T00:00:00ZConvergence of adaptive finite element methods with error-dominated oscillation
http://hdl.handle.net/2003/36843
Title: Convergence of adaptive finite element methods with error-dominated oscillation
Authors: Kreuzer, Christian; Veeser, Andreas
Abstract: Recently, we devised an approach to a posteriori error analysis, which
clarifies the role of oscillation and where oscillation is bounded in terms of the current
approximation error. Basing upon this approach, we derive plain convergence
of adaptive linear finite elements approximating the Poisson problem. The result
covers arbritray H^-1-data and characterizes convergent marking strategies.2018-03-01T00:00:00ZAn Investigation on Separation Points of Power-Law model Along a Rotating Round-Nosed Body
http://hdl.handle.net/2003/36781
Title: An Investigation on Separation Points of Power-Law model Along a Rotating Round-Nosed Body
Authors: Begum, Naheed; Siddiqa, Sadia; Ouazzi, Abderrahim; Hossain, Md. Anwar
Abstract: The purpose of present study is to numerically investigate the natural convection flow of Ostwalde-de Waele type power law non-Newtonian fluid along the surface of rotating axi-symmetric round-nosed body. For computational purpose rotating hemisphere is used as a case study in order to examine the heat transfer mechanism near such transverse curvature geometries. The numerical scheme is applied after converting the dimensionless system of equations into primitive variable formulations. Implicit finite difference method is used to integrate the equations numerically. Its worth mentioning that all the numerical simulations performed here are valid particularly for the class of shear thickening fluid with wide range of Prandtl number, i.e. (10:0 ≤ Pr ≤ 1500:0). A detailed discussion is done to understand the effects of buoyant forces and power-law exponents on the rate of heat transfer and skin friction coefficient at the surface of the hemisphere. Comparison of present numerical results for different values of buoyancy ratio parameter λ with other published data has been shown in graphical form. For the first time the velocity profiles are plotted at the point of separation, which occurs when the portion of the boundary layer closest to the wall or leading edge reverses in flow direction. It is recorded that an increase in the power-law index n and Prandtl number Pr leads to an increase in the friction factor as well as in the rate of heat transfer.2018-02-01T00:00:00ZOn the Prospects of Using Machine Learning for the Numerical Simulation of PDEs: Training Neural Networks to Assemble Approximate Inverses
http://hdl.handle.net/2003/36777
Title: On the Prospects of Using Machine Learning for the Numerical Simulation of PDEs: Training Neural Networks to Assemble Approximate Inverses
Authors: Ruelmann, Hannes; Geveler, Markus; Turek, Stefan
Abstract: In an unconventional approach to combining the very successful Finite Element Methods (FEM) for PDE-based simulation with techniques evolved from the domain of Machine Learning (ML) we employ approximate inverses of the system matrices generated by neural networks in the linear solver. We demonstrate the success of this solver technique on the basis of the Poisson equation which can be seen as a fundamental PDE for many practically relevant simulations [Turek 1999]. We use a basic Richardson iteration applying the approximate inverses generated by fully connected feedforward multilayer perceptrons as preconditioners.2018-02-01T00:00:00ZNumerical Benchmarking for 3D Multiphase Flow: New Results for a Rising Bubble
http://hdl.handle.net/2003/36307
Title: Numerical Benchmarking for 3D Multiphase Flow: New Results for a Rising Bubble
Authors: Turek, Stefan; Mierka, Otto; Bäumler, Kathrin
Abstract: Based on the benchmark results in [1] for a 2D rising bubble, we present the extension towards 3D providing test cases with corresponding reference results, following the suggestions in [2]. Additionally, we include also an axisymmetric configuration which allows 2.5D simulations and which provides further possibilities for validation and evaluation of numerical multiphase flow components and software tools in 3D.2017-12-01T00:00:00ZNumerical study of the RBF-FD level set based method for partial differential equations on evolving-in-time surfaces
http://hdl.handle.net/2003/36231
Title: Numerical study of the RBF-FD level set based method for partial differential equations on evolving-in-time surfaces
Authors: Sokolov, Andriy; Davydov, Oleg; Turek, Stefan
Abstract: In this article we present a Radial Basis Function (RBF)-Finite Difference (FD) level
set based method for numerical solution of partial differential equations (PDEs) of
the reaction-diffusion-convection type on an evolving-in-time hypersurface Γ (t). In a
series of numerical experiments we study the accuracy and robustness of the proposed
scheme and demonstrate that the method is applicable to practical models.2017-11-01T00:00:00ZTwo-phase Natural Convection Dusty Nanofluid Flow
http://hdl.handle.net/2003/36223
Title: Two-phase Natural Convection Dusty Nanofluid Flow
Authors: Siddiqa, Sadia; Begum, Naheed; Hossain, M. A.; Gorla, Rama Subba Reddy; Al-Rashed, Abdullah A. A. A.
Abstract: An analysis is performed to study the two-phase natural convection flow of nano fluid along a
vertical wavy surface. The model includes equations expressing conservation of total mass,
momentum and thermal energy for two-phase nano fluid. Primitive variable formulations
(PVF) are used to transform the dimensionless boundary layer equations into a convenient
coordinate system and the resulting equations are integrated numerically via implicit finite difference iterative scheme. The effect of controlling parameters on the dimensionless quantities such as skin friction coefficient, rate of heat transfer and rate of mass transfer is
explored. It is concluded from the present analysis, that the diffusivity ratio parameter, N_A and particle-density increment number, N_B have pronounced influence on the reduction of heat transfer rate.2017-11-01T00:00:00ZNatural convection of incompressible viscoelastic fluid flow
http://hdl.handle.net/2003/36222
Title: Natural convection of incompressible viscoelastic fluid flow
Authors: Damanik, Hogenrich; Turek, Stefan
Abstract: We revisit the MIT Benchmark 2001 and introduce a viscoelastic constitutive law into the fluid in motion. Our
goal is to study the effect of viscoelasticity into the periodical behavior of the physical quantities of the corresponding
benchmark. We use a robust numerical technique in simulating complex fluid flow problems based on higher order Finite
Element discretization. While marching in time, an A-stable method of second order is favorable, i.e Crank-Nicolson
scheme, to reproduce periodical behaviors. We use a differential form of viscoelastic model, i.e Oldroyd-B type and find out
that a small amount of viscoelasticity reduces the oscillatory behavior.2017-11-01T00:00:00ZDual weighted residual error estimation for the finite cell method
http://hdl.handle.net/2003/36085
Title: Dual weighted residual error estimation for the finite cell method
Authors: Stolfo, Paolo Di; Rademacher, Andreas; Schröder, Andreas
Abstract: The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the ﬁnite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for some two-dimensional examples.2017-09-01T00:00:00ZConvergence of adaptive discontinuous galerkin methods
http://hdl.handle.net/2003/36041
Title: Convergence of adaptive discontinuous galerkin methods
Authors: Kreuzer, Christian; Georgoulis, Emmanuil H.
Abstract: We develop a general convergence theory for adaptive discontinu-
ous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and
LDG schemes as well as all practically relevant marking strategies. Another
key feature of the presented result is, that it holds for penalty parameters only
necessary for the standard analysis of the respective scheme. The analysis
is based on a quasi interpolation into a newly developed limit space of the
adaptively created non-conforming discrete spaces, which enables to generalise
the basic convergence result for conforming adaptive finite element methods by
Morin, Siebert, and Veeser [A basic convergence result for conforming adaptive
finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707–737].2017-08-01T00:00:00ZAn entropy stable spacetime discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations
http://hdl.handle.net/2003/36039
Title: An entropy stable spacetime discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations
Authors: Hiltebrand, Andreas; May, Sandra
Abstract: In this paper, we present an entropy stable scheme for solving the compressible
Navier-Stokes equations in two space dimensions. Our scheme uses entropy variables
as degrees of freedom. It is an extension of an existing spacetime discontinuous
Galerkin method for solving the compressible Euler equations. The physical diffusion
terms are incorporated by means of the symmetric (SIPG) or nonsymmetric
(NIPG) interior penalty method, resulting in the two versions ST-SDSC-SIPG and
ST-SDSC-NIPG. The streamline diffusion and shock-capturing terms from the original
scheme have been kept, but have been adjusted appropriately. This guarantees
that the new scheme essentially reduces to the original scheme for the compressible
Euler equations in regions with underresolved physical diffusion. We show entropy
stability for both versions under suitable assumptions. We also present numerical
results confirming the accuracy and robustness of our schemes.2017-07-01T00:00:00ZPreconditioning for hyperelasticity-based mesh optimisation
http://hdl.handle.net/2003/36014
Title: Preconditioning for hyperelasticity-based mesh optimisation
Authors: Paul, Jordi
Abstract: A robust mesh optimisation method is presented that directly enforces the resulting deformation to be
orientation preserving. Motivated by aspects from mathematical elasticity, the energy functional of
the mesh deformation can be related to a stored energy functional of a hyperelastic material. Formulating the functional in the principal invariants of the deformation gradient allows fine grained control
over the resulting deformation. Solution techniques for the arising nonconvex and highly nonlinear
system are presented. As existing preconditioners are not sufficient, a PDE-based preconditioner is
developed.2017-06-01T00:00:00ZNatural Convection Flow of a Two-Phase Dusty Non-Newtonian Fluid Along a Vertical Surface
http://hdl.handle.net/2003/36004
Title: Natural Convection Flow of a Two-Phase Dusty Non-Newtonian Fluid Along a Vertical Surface
Authors: Siddiqa, Sadia; Begum, Naheed; Hossain, Md. Anwar; Gorla, Rama Subba Reddy
Abstract: The aim of this paper is to present a boundary-layer analysis of two-phase
dusty non-Newtonian fluid flow along a vertical surface by using a modified power-law viscosity
model. This investigation particularly reports the flow behavior of spherical particles
suspended in the non-Newtonian fluid. The governing equations are transformed into nonconserved
form and then solved straightforwardly by implicit finite difference method. The
numerical results of rate of heat transfer, rate of shear stress, velocity and temperature
profiles and streamlines and isotherms are presented for wide range of Prandtl number, i.e,
(0:7 ≤ Pr ≤ 1000:0), with the representative values of the power-law index n. A good
agreement is found between the present and the previous results when compared with some
special cases. The key observation from the present study is that the power-law fluids with
(n > 1) are more likely to promote the rate of heat transfer near the leading edge.2017-06-01T00:00:00ZAdaptive optimal control of the signorini's problem
http://hdl.handle.net/2003/35991
Title: Adaptive optimal control of the signorini's problem
Authors: Rademacher, Andreas; Rosin, Korinna
Abstract: In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini’s problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton’s method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini’s contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini’s contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini’s problem in two and three dimensions.2017-06-01T00:00:00ZNumerical estimation of thermal radiation effects on Marangoni Convection of dusty fluid
http://hdl.handle.net/2003/35981
Title: Numerical estimation of thermal radiation effects on Marangoni Convection of dusty fluid
Authors: Siddiqa, Sadia; Begum, Naheed; Hossain, Anwar; Al-Rashed, Abdullah A. A. A.
Abstract: In this paper, numerical solutions to thermally radiating Marangoni convection of dusty fluid flow along a vertical wavy surface are established. The results are obtained with the understanding that the dust particles are of uniform size and dispersed in optically thick fluid. The numerical solutions of the dimensionless transformed equations are obtained through straightforward implicit finite difference scheme. In order to analyze the influence of various controlling parameters, results are displayed in the form of rate of heat transfer, skin friction coeffcient, velocity and temperature profiles, streamlines and isotherms. It is observed that the variation in thermal radiation parameter significantly alters the corresponding particle pattern and extensively promotes the heat transfer rate.2017-05-01T00:00:00ZSequential limiting in continuous and discontinuous Galerkin methods for the Euler equations
http://hdl.handle.net/2003/35980
Title: Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations
Authors: Dobrev, Veselin A.; Kolev, Tzanio; Kuzmin, Dmitri; Rieben, Robert N.; Tomov, Vladimir Zdravkov
Abstract: We present a new approach to enforcing local maximum principles in finite element schemes for the compressible Euler equations. In contrast to synchronized limiting techniques for systems of conservation laws, the density, momentum, and total energy are constrained in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum are adjusted to incorporate the irreversible effect of density changes. Then the corresponding antidiffusive corrections are limited to satisfy inequality constraints for the total and kinetic energy. The same element-based limiting strategy is employed in the context of continuous and discontinuous Galerkin methods. The sequential nature of the new limiting procedure makes it possible to achieve crisp resolution of contact discontinuities while using sharp local bounds in the energy constraints. A numerical study is performed for piecewise-linear finite element discretizations of 1D and 2D test problems.2017-05-01T00:00:00ZOptimal control of the thermistor problem in three spatial dimensions, Part 2: Optimality conditions
http://hdl.handle.net/2003/35973
Title: Optimal control of the thermistor problem in three spatial dimensions, Part 2: Optimality conditions
Authors: Meinlschmidt, Hannes; Meyer, Christian; Rehberg, Joachim
Abstract: This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion paper of this work. In this part, we address further properties of the set of controls whose associated solutions exist globally such as openness, which includes analysis of the linearized state system via maximal parabolic regularity. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system in which we do not need to refer to the set of controls admitting global solutions. The theoretical findings are illustrated by numerical results. This work is the second of two papers on the three-dimensional thermistor problem.2017-05-01T00:00:00ZNumerical Solutions for Gyrotactic Bioconvection of Dusty Nanofluid along a Vertical Isothermal Surface
http://hdl.handle.net/2003/35966
Title: Numerical Solutions for Gyrotactic Bioconvection of Dusty Nanofluid along a Vertical Isothermal Surface
Authors: Begum, Naheed; Siddiqa, Sadia; Sulaiman, Muhammad; Islam, Saeed; Hossain, Mohammad Anwar; Gorla, Rama Subba Reddy
Abstract: The aim of present paper is to establish the detailed numerical results for bioconvection
boundary-layer flow of two-phase dusty nanofluid. The dusty fluid contains gyrotactic microorganisms
along an isothermally heated vertical wall. The physical mechanisms responsible
for the slip velocity between the dusty fluid and nanoparticles, such as thermophoresis
and Brownian motion, are included in this study. The influence of the dusty nanofluid on
heat transfer and flow characteristics are investigated in this paper. The governing equations
for two-phase model are non-dimensionalized and then solved numerically via twopoint
finite difference method together with the tri-diagonal solver. Results are presented
graphically for wall skin friction coefficient, rate of heat transfer, velocity and temperature
profiles and streamlines and isotherms. To ensure the accuracy, the computational results
are compared with available data and are found in good agreement. The key observation
from present analysis is that the mass concentration parameter, D_ρ, extensively promotes
the rate of heat transfer, Q_w, whereas, the wall skin friction coefficient, τ_w, is reduced by
loading the dust parameters in water based dusty nanofluid.2017-05-01T00:00:00Z