08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Adaptive higher order discontinuous Galerkin methods for porous-media multi-phase flow with strong heterogeneities(2018) Kane, Birane; Siebert, Kunibert (Prof. Dr.)In this thesis, we develop, analyze, and implement adaptive discontinuous Galerkin (DG) finite element solvers for the efficient simulation of porous-media flow problems. We consider 2d and 3d incompressible, immiscible, two-phase flow in a possibly strongly heterogeneous and anisotropic porous medium. Discontinuous capillarypressure functions and gravity effects are taken into account. The system is written in terms of a phase-pressure/phase-saturation formulation. First and second order Adams-Moulton time discretization methods are combined with various interior penalty DG discretizations in space, such as the symmetric interior penalty Galerkin (SIPG), the nonsymmetric interior penalty Galerkin (NIPG) and the incomplete interior penalty Galerkin (IIPG). These fully implicit space time discretizations lead to fully coupled nonlinear systems requiring to build a Jacobian matrix at each time step and in each iteration of a Newton-Raphson method. We provide a stability estimate of the saturation and the pressure with respect to initial and boundary data. We also derive a-priori error estimates with respect to the L2(H1) norm for the pressure and the L∞(L2)∩L2(H1) norm for the saturation. Moving on to adaptivity, we implement different strategies allowing for a simultaneous variation of the element sizes, the local polynomial degrees and the time step size. These approaches allow to increase the local polynomial degree when the solution is estimated to be smooth and refine locally the mesh otherwise. They also grant more flexibility with respect to the time step size without impeding the convergence of the method. The aforementioned adaptive algorithms are applied in series of homogeneous, heterogeneous and anisotropic test cases. To our knowledge, this is the first time the concept of local hp-adaptivity is incorporated in the study of 2d and 3d incompressible, immiscible, two-phase flow problems. Delving into the issue of efficient linear solvers for the fully-coupled fully-implicit formulations, we implement a constrained pressure residual (CPR) two-stage preconditioner that exploits the algebraic properties of the Jacobian matrices of the systems. Furthermore, we provide an open-source DG two-phase flow simulator, based on the software framework DUNE, accompanied by a set of programs including instructions on how to compile and run them.Item Open Access Modeling and analysis of almost unidirectional flows in porous media(2017) Armiti, Alaa; Rohde, Christian (Prof. Dr.)Item Open Access A two scale model for liquid phase epitaxy with elasticity(2015) Kutter, Michael; Rohde, Christian (Prof. Dr.)Epitaxy, a special form of crystal growth, is a technically relevant process for the production of thin films and layers. It gives the possibility to generate microstructures of different morphologies, such as steps, spirals or pyramids. These microstructures are influenced by elastic effects in the epitaxial layer. There are different epitaxial techniques, one is the so-called liquid phase epitaxy. Thereby, single particles are deposited out of a supersaturated liquid solution on a substrate where they contribute to the growth process. The thesis studies a two scale model including elasticity, introduced in [Ch.Eck, H.Emmerich. Liquid-phase epitaxy with elasticity. Preprint 197, DFG SPP 1095, 2006]. It consists of a macroscopic Navier-Stokes system and a macroscopic convection-diffusion equation for the transport of matter in the liquid, and a microscopic problem that combines a phase field approximation of a Burton-Cabrera-Frank model for the evolution of the epitaxial layer, a Stokes system for the fluid flow near the layer and an elasticity system for the elastic deformation of the solid film. Suitable conditions couple the single parts of the model. As main result, existence and uniqueness of a solution is proven in suitable function spaces. Furthermore, an iterative solving procedure is proposed, which reflects on the one hand the strategy of the proof of the main result via fixed point arguments and, on the other hand, can be a basis for an numerical algorithm.Item Open Access Liquid vapor phase transitions : modeling, Riemann solvers and computation(2016) Zeiler, Christoph; Rohde, Christian (Prof. Dr.)The numerical approximation of liquid vapor flows within the compressible regime is a challenging task because complex physical effects at the phase interfaces govern the global flow behavior. We develop a sharp interface approach which treats the phase boundary like a shock wave discontinuity and takes capillarity effects into account. The approach relies on the solution of Riemann problems across the interface that separates the liquid and the vapor phase. The Riemann solution accounts for the relevant physics by enforcing appropriate jump conditions at the phase boundary. A wide variety of interface effects can be handled in a thermodynamically consistent way. This includes surface tension, as well as, mass and energy transfer by phase transition. Moreover, the local normal speed of the interface, which is needed to calculate the time evolution of the phase boundary, is given by the Riemann solution. The focus of this work is the development of isothermal and non-isothermal two-phase Riemann solvers for the sharp interface approach. To verify the solvers with respect to numerical and thermodynamic requirements, one-dimensional and radially symmetric problems are studied. Furthermore, the Riemann solvers and the sharp interface approach are successfully validated against shock tube experiments of real fluids (alkanes).Item Open Access On a stochastic Camassa-Holm type equation with higher order nonlinearities(2020) Rohde, Christian; Tang, HaoThe subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.Item Open Access General interface problems. 1(1994) Nicaise, Serge; Sändig, Anna-MargareteWe study transmission problems for elliptic operators of order 2m with general boundary and interface conditions, introducing new covering conditions. This allows to prove solvability, regularity and asymptotics of solutions in weighted Sobolev spaces. We give some numerical examples for the location of the singular exponents.Item Open Access Robust numerical algorithms for dynamic frictional contact problems with different time and space scales(2010) Hager, Corinna; Wohlmuth, Barbara (Prof. Dr.)In many technical and engineering applications, numerical simulation is becoming more and more important for the design of products or the optimization of industrial production lines. However, the simulation of complex processes like the forming of sheet metal or the rolling of a car tire is still a very challenging task, as nonlinear elastic or elastoplastic material behaviour needs to be combined with frictional contact and dynamic effects. In addition, these processes often feature a small mobile contact zone which needs to be resolved very accurately to get a good picture of the evolution of the contact stress. In order to be able to perform an accurate simulation of such intricate systems, there is a huge demand for a robust numerical scheme that combines a suitable multiscale discretization of the geometry with an efficient solution algorithm capable of dealing with the material and contact nonlinearities. The aim of this thesis is to design such an algorithm by combining several different methods which are described in the following. Our main field of application is structural mechanics. Here, we base the implementation on finite element methods in space and implicit finite difference schemes in time. The conditions for both plasticity and frictional contact are given in terms of a set of local inequality constraints which are formulated by introducing additional inner or dual degrees of freedom. As the meshes are generally non-matching at the contact interface, we employ mortar techniques to incorporate the contact constraints in a variationally consistent way. By using biorthogonal basis functions for the discrete multiplier space, the contact conditions can be enforced node-wise, and a two-body contact problem can be solved in the same way as a one-body problem. The next step in the construction of an efficient solution algorithm is to reformulate the local inequality conditions for plasticity and contact in terms of nondifferentiable equalities. These nonlinear complementarity functions can be combined with the equations for the bulk material to form a set of nonlinear semismooth equations which are then solved by means of a generalized form of the Newton method. Due to the local structure of the inequality constraints, this iterative scheme can be implemented as an active set strategy where the active sets are updated in each Newton iteration. Further, the additional dual degrees of freedom can easily be eliminated using local static condensation. We remark that the well-known radial return method is a special case of this general framework if the plastic hardening laws are linear. However, the convergence properties of the Newton iteration strongly depend on the choice of the NCP function. In this context, we show that the function corresponding to the radial return method is not optimal, and we present a family of modified NCP functions which allow for better convergence results. Another important issue for the robust simulation of dynamic contact problems is related to the inertia terms. If standard time discretization schemes like the trapezoidal rule are used, the contact stress often shows spurious oscillations in time that become worse when the time step is refined. In order to avoid this effect, we employ a modified mass matrix where no mass is associated with the contact nodes. By this, the original semi-discrete system decouples into an algebraic equation in time for the contact nodes and an ordinary differential equation in time for the other nodes. This in turn leads to much smoother results for the contact stress. We present an efficient way of obtaining the modified mass matrix by means of non-standard quadrature formulas used only for the elements near the contact boundary. Furthermore, we prove optimal a priori error estimates for the modified semi-discrete as well as for the fully discrete system, provided that the contact stress is given and that the solution is sufficiently regular. In the last part of the thesis, we deal with the situation that the body features fine local structures near the contact zone by incorporating the multiscale aspect into the discretization. For this, the domain is decomposed into several overlapping subdomains which have different grid spacing; one global mesh that does not resolve the details and overlapping local patches with a fine triangulation. Based on a surface coupling by means of the mortar method, we construct an iterative solution scheme for the coupled problem whose convergence rate is bounded independently of the mesh size or the Lame parameters. Finally, we employ the subdomain decomposition for introducing a finer time step size on the patch. We present suitable interface conditions with no numerical dissipation and prove a priori error estimates with respect to time for the resulting coupled energy-conserving system. The latter can efficiently be solved by the iterative procedure presented before.Item Open Access Efficient simulation of challenging PDE problems on CPU and GPU clusters(2021) Schirwon, Malte; Göddeke, Dominik (Prof. Dr.)The main contribution of this dissertation is to show how efficient parallelization techniques for numerical simulations of partial differential equations (PDEs) can be developed and which aspects have to be considered in order to obtain the best possible performance. For this purpose, the target platforms range from high-performance workstations to small clusters and up to supercomputers. In particular, we focus on platforms accelerated by graphics cards. We emphasize that the efficient numerical simulation of PDE problems comprises and combines, in novel ways, aspects from numerical analysis, numerical methods (algorithmics, data structures and other areas more related to computer science) and hardware details. Many models in science, engineering and economics are based on systems of PDEs. The choice of modeling techniques, the implementation of numerical solution techniques, as well as the chosen target platform limit the accuracy and the duration of the simulation. Increasing the accuracy and/or reducing the duration of the simulation is usually not possible without efficient software. Based on three application scenarios, we adapt already existing methodologies and algorithms to the target platforms or change the way they are implemented in order to achieve optimal efficiency. As a guiding scheme, we consider the challenging case of unstructured data and schemes. The first application is the wave propagation in optical fibers. We present an MPI-parallel implementation that is particularly suitable for small clusters. %Here, we change the numerical method and the implementation technique to increase efficiency and decrease runtime. The second application scenario is the flow in porous media. Based on both applications, we develop implementation techniques that increase their efficiency. Furthermore, we present an adapted version of a neighborhood algorithm that further increases the efficiency for current graphics cards. The increased efficiency and reduced runtime allows to perform more complex simulations. %For example, higher resolutions can be simulated or more physical parameters can be included. One of theses applications is considered to be the third application, which is seismic wave propagation and waveform inversion. The feasibility of developing efficient implementations for computationally powerful target platforms permits us to consider the inversion of seismic waves in viscoelastic materials. In particular, we present an inversion scheme that also allows us to determine the damping parameters of the viscoelastic material. In addition, regularization methods and a modified solver method are presented, which can be used for a more efficient solution of such problems.Item Open Access Analysis of target data-dependent greedy kernel algorithms : convergence rates for f-, f· P- and f/P-greedy(2022) Wenzel, Tizian; Santin, Gabriele; Haasdonk, BernardData-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.Item Open Access Resilience and fault tolerance in high-performance computing for numerical weather and climate prediction(2021) Benacchio, Tommaso; Bonaventura, Luca; Altenbernd, Mirco; Cantwell, Chris D.; Düben, Peter D.; Gillard, Mike; Giraud, Luc; Göddeke, Dominik; Raffin, Erwan; Teranishi, Keita; Wedi, NilsProgress in numerical weather and climate prediction accuracy greatly depends on the growth of the available computing power. As the number of cores in top computing facilities pushes into the millions, increased average frequency of hardware and software failures forces users to review their algorithms and systems in order to protect simulations from breakdown. This report surveys hardware, application-level and algorithm-level resilience approaches of particular relevance to time-critical numerical weather and climate prediction systems. A selection of applicable existing strategies is analysed, featuring interpolation-restart and compressed checkpointing for the numerical schemes, in-memory checkpointing, user-level failure mitigation and backup-based methods for the systems. Numerical examples showcase the performance of the techniques in addressing faults, with particular emphasis on iterative solvers for linear systems, a staple of atmospheric fluid flow solvers. The potential impact of these strategies is discussed in relation to current development of numerical weather prediction algorithms and systems towards the exascale. Trade-offs between performance, efficiency and effectiveness of resiliency strategies are analysed and some recommendations outlined for future developments.