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 difference 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. © 2022 Walter de Gruyter GmbH, Berlin/Boston.

This chapter is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. The 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 afterward applied to four application-driven examples. © 2022, Springer Nature Switzerland AG.

Recently, accelerator hardware in the form of graphics cards including Tensor Cores, specialized for AI, has significantly gained importance in the domain of high-performance computing. For example, NVIDIA’s Tesla V100 promises a computing power of up to 125 TFLOP/s achieved by Tensor Cores, but only if half precision floating point format is used. We describe the difficulties and discrepancy between theoretical and actual computing power if one seeks to use such hardware for numerical simulations, that is, solving partial differential equations with a matrix-based finite element method, with numerical examples. If certain requirements, namely low condition numbers and many dense matrix operations, are met, the indicated high performance can be reached without an excessive loss of accuracy. A new method to solve linear systems arising from Poisson’s equation in 2D that meets these requirements, based on “prehandling” by means of hier-archical finite elements and an additional Schur complement approach, is presented and analyzed. We provide numerical results illustrating the computational performance of this method and compare it to a commonly used (geometric) multigrid solver on standard hardware. It turns out that we can exploit nearly the full computational power of Tensor Cores and achieve a significant speed-up compared to the standard methodology without losing accuracy. © The Author(s) 2022.

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. © 2021, Springer Nature Switzerland AG.

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 extensibility. 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 solver. © 2021, Springer Nature Switzerland AG.

For the well-defined investigation of reactive bubbly flows it is crucial to know the kinetics of the chemical reaction steps as detailed as possible which means that the hindrance due to mixing should be minimized (intrinsic kinetics). This is especially difficult for fast chemical reactions that are addressed in this book. Therefore, to distinguish between convective mixing and diffusion a tool should be used that enables an optical access to visualize and detect the instant mixing on a microscale and the comparison with numerical simulations. Furthermore, this tool should be easy to handle for chemical engineers and inexpensive. A SuperFocus mixer in a continuous-flow setup is used for this purpose, in which two mixable liquid streams are mixed, with the gas phase dissolved in one of the streams. The results are compared with a classical stopped-flow apparatus. Because chemical reactions are generally following a complex reaction network with many reaction steps and intermediates that are mostly difficult to detect, the measurement of the formation of the relevant products and by-products is a difficult task by its own. Spectroscopic methods including absorption or fluorescence measurements are used here to detect the formation of the relevant products and to determine the gross kinetics for the relevant reaction steps. Finally, the experimental results are used to adapt direct numerical simulations for an estimation of the corresponding diffusion coefficients and reaction constants. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

The Archimedes Tube Crystallizer (ATC) is a small-scale coiled tubular crystallizer operated with air-segmented flow. As individual liquid segments are moved through the apparatus by rotation, the ATC operates as a pump. Thus, the ATC overcomes pressure drop limitations of other continuous crystallizers, allowing for longer residence times and crystal growth phases. Understanding continuous crystallizer phenomena is the basis for a well-designed crystallization process, especially for small-scale applications in the pharmaceutical and fine chemical industry. Hydrodynamics and suspension behavior, for example, affect agglomeration, breakage, attrition, and ultimately crystallizer blockage. In practice, however, it is time-consuming to investigate these phenomena experimentally for each new material system. In this contribution, a flow map is developed in five steps through a combination of experiments, CFD simulations, and dimensionless numbers. Accordingly, operating parameters can be specified depending on ATC design and material system used, where suspension behavior is suitable for high-quality crystalline products. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

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 prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, 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 linearized problems 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. © The Author(s) 2021.

This paper is concerned with the application of Finite Element Methods (FEM) and NewtonMultigrid solvers to simulate thixotropic flows using quasi-Newtonian modeling. The thixotropy phenomena are introduced to yield stress material by taking into consideration the internal 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 structure 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 introducing 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 pressure 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, Univelt Inc., All rights reserved.

The numerical simulation of immiscible multiphase flow problems, particularly including drops and bubbles, is very important in many applications, and performing accurate, robust and efficient numerical computations has been the object of numerous research and simulation projects for many years. One of the main challenges for the underlying numerical methods – besides the fact that the computational simulation of the incompressible Navier–Stokes equations is challenging by itself – is that the position of the moving interface between two fluids is unknown and must be determined as part of the boundary value problem to be solved. In this contribution, we provide a compact description of state-of-the-art numerical solvers for such multiphase flow problems, namely interface tracking and interface capturing methods. It is demonstrated that corresponding discretization and solution approaches which are based on Finite Element and Discrete Projection methods for the Navier–Stokes equations, combined with corresponding numerical tools for both interface capturing, resp., tracking approaches, lead to robust, accurate, flexible, and efficient simulation tools. Moreover, we present several numerical test cases of benchmarking type which first of all shall help to evaluate the quality of the underlying flow solvers. In particular, we describe the settings for a quantitative 3D Rising Bubble benchmark which can be used for ‘simple’ validation and evaluation of multiphase CFD codes without the necessity of complex postprocessing operations. Finally, we also provide numerical reference values for a ‘Taylor bubble’ setting, and we show simulation results of a reactive Taylor bubble flow in the framework of estimating reaction parameters to match corresponding experimentally obtained results. All reference benchmark quantities can be downloaded from www.featflow.de. © 2021 Elsevier B.V.

This work describes a numerical framework developed for the efficient and accurate simulation of microfluidic applications related to two leading experiments of the DFG SPP 1740 research initiative, namely the ‘Superfocus Micromixer’ and the ‘Taylor bubble flow’. Both of these basic experiments are considered 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 (FeatFlow Homepage, www.featflow.de, version from July 2020 [5]). A demonstration of the developed simulation tool is based on already defined validation cases and on suitable examples being responsible for the determination of the related convergence properties (in the range of targeted process parameter values) of the developed numerical framework. 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, The Author(s), under exclusive license to Springer Nature Switzerland AG.

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”. © 2021, Springer Nature Switzerland AG.

To benefit from current trends in HPC hardware, such as increasing availability 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. © 2021, Springer Nature Switzerland AG.

Taylor bubbles are bubbles that are filling a capillary fixed or rising in a liquid flow. Due to their well-defined size, shape and velocity the flow field around Taylor bubbles is well controllable. Furthermore, only a small amount of gas and liquid is required for experiments, making the study of Taylor bubbles simple and safe. In this chapter, Taylor bubbles are used to study mass transfer with chemical reaction in dependency of various fluid dynamic conditions. Experimental setups are presented to study chemical reactions at Taylor bubbles and in Taylor flows. Furthermore, a Taylor bubble benchmark experiment is presented and used for the validation of numerical simulations that are performed with a front tracking technique in a FEM framework to enable a high resolution of interfaces and boundary layers. It is shown that the interplay of mass transfer, hydrodynamics and chemical reactions can be successfully determined using experimental and numerical methods presented. It is shown further how the selectivity of competitive-consecutive reactions can be influenced by the local fluid dynamics, especially in the wake of a Taylor bubble or within the liquid slug in a Taylor flow. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

The implementation of traditional sensors is a drawback when investigating mass transfer phenomena within microstructured devices, since they disturb the flow and reactor characteristics. An Arduino based slider setup is developed, which is equipped with a computer-vision system to track gas–liquid slug flow. This setup is combined with an optical analytical method allowing to compare experimental results against CFD simulations and investigate the entire lifetime of a single liquid slug with high spatial and temporal resolution. Volumetric mass transfer coefficients are measured and compared with data from literature and the mass transfer contribution of the liquid film is discussed. © 2020 The Authors. AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers.

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. We combined the open-source computational fluid dynamics (CFD) package FEATFLOW with the ray tracing software of the LAT-RUB to simulate optical trap configurations. 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 that is then compared to the results of our combined CFD-optics solver. As an extension of the standard procedure, we present a method with a time-stepping scheme that contains a macro step approach. The results show that this macro time-stepping scheme provides a significant acceleration while still maintaining good accuracy. A second configuration is analyzed that involves non-spherical geometries such as micro rotors. We proceed to compare simulation results of the final angular velocity of the micro rotor with experimental measurements. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In the Exa-Dune project we have developed, implemented and optimised numerical algorithms and software for the scalable solution of partial differential equations (PDEs) on future exascale systems exhibiting a heterogeneous massively parallel architecture. In order to cope with the increased probability of hardware failures, one aim of the project was to add flexible, application-oriented resilience capabilities into the framework. Continuous improvement of the underlying hardware-oriented numerical methods have included GPU-based sparse approximate inverses, matrix-free sum-factorisation for high-order discontinuous Galerkin discretisations as well as partially matrix-free preconditioners. On top of that, additional scalability is facilitated by exploiting massive coarse grained parallelism offered by multiscale and uncertainty quantification methods where we have focused on the adaptive choice of the coarse/fine scale and the overlap region as well as the combination of local reduced basis multiscale methods and the multilevel Monte-Carlo algorithm. Finally, some of the concepts are applied in a land-surface model including subsurface flow and surface runoff. © The Author(s) 2020.

In this paper, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluids is introduced based on the idea, that the extra-stress tensor in the momentum equation of the flow model can be replaced by the product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity. As potential advantage (which can be demonstrated for fully developed channel, resp., pipe flows), this approach offers the possibility to reduce the full nonlinear viscoelastic three-field model to a generalized Tensor Stokes problem, avoiding the need of considering a separate stress tensor in the solution process. Moreover, an (artificial) diffusive operator is introduced into the momentum equation in the case of “no solvent” viscoelastic fluids, which is related to the physics of the problem leading to an improved numerical behaviour w.r.t. approximation and convergence properties of the iterative solvers. After validating the Tensor Diffusion approach for fully developed channel flow configurations, it is evaluated in the context of more general two-dimensional flow settings of benchmarking character, too, taking into account a symmetrized formulation. Numerical simulations for several prototypical flow configurations illustrate the mathematical and numerical properties of this new approach and can be viewed as proof-of-concept for further development. © 2020 Elsevier B.V.

In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied 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. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.

Based on the benchmark results in Hysing et al (Int J Numer Methods Fluids 60(11):1259–1288, 2009) for a 2D rising bubble, we present the extension towards 3D providing test cases with corresponding reference results, following the suggestions in Adelsberger et al (Proceedings of the 11th world congress on computational mechanics (WCCM XI), Barcelona, 2014). 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. © Springer Nature Switzerland AG 2019.

In this article we present a Radial Basis Function (RBF)-Finite Difference (FD) level set based method for the 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. © Springer Nature Switzerland AG 2019.

In heavy oil pipelining, a major challenge is to reliably achieve high rates of pressure drop reduction in stable annular flow patterns. Sarmadi et al. (2017), introduced a novel methodology for efficient transportation of heavy oil via a triple-layer core-annular flow. The lubricating outer layer is separated from the core by a shaped skin comprising a yield stress fluid. The lubricant reduces the pressure drop, the unyielded skin eliminates interfacial instabilities and the shaped interface produces lift force in the lubrication layer to balance buoyancy of the core. Sarmadi et al. (2018), studied how to sculpt the interface in stable controlled way and how to establish the flow. Here we present three dimensional (3D) triple-layer computations which capture the buoyant motion of the core to reach its equilibrium position. The 3D computations are performed using a finite element method and adaptively aligned meshes to track dynamically the interfaces, benchmarked against axisymmetric computations from Sarmadi et al. (2018). The study shows that these flows may stably become established with control over interface shape, but development lengths (times) for the core to attain equilibrium are relatively long, meaning extensive 3D computation. We also present a simplified analytical model using the lubrication approximation and equations of motion for the lubricant and skin layers. This model allows us to quickly estimate motion to the balanced configuration for a given shape and initial conditions. © 2019 Elsevier B.V.

In the present contribution, we compare (quantitatively) different mixed least-squares finite element methods (LSFEMs) with respect to computational costs and accuracy. Various first-order systems are derived based on the residual forms of the equilibrium equation and the continuity condition. The first formulation under consideration is a div-grad first-order system resulting in a three-field formulation with total stresses, velocities, and pressure (S-V-P) as unknowns. Here, the variables are approximated in H(div) × H1 × L2 on triangles and in H1 × H1 × L2 on quadrilaterals. In addition to that a reduced stress-velocity (S-V) formulation is derived and investigated. S-V-P and S-V formulations are promising approaches when the stresses are of special interest, e.g., for non-Newtonian, multiphase or turbulent flows. The main focus of the work is drawn to performance and accuracy aspects on the one side for finite elements with different interpolation orders and on the other side on the usage of efficient solvers, for instance of Krylov-space or multigrid type. © 2018 Inderscience Enterprises Ltd.

In this note, we show the link between the classical continuous surface stress and continuous surface force approaches together with special finite element method techniques toward a fully implicit level set method. Based on a modified surface stress formulation, neither normals nor curvature has to be explicitly calculated. The method is space-dimension independent. Prototypical numerical tests of benchmarking character for a rising 2D bubble are provided for validating the accuracy of this new approach. We show additionally that the explicit redistancing can be avoided using a nonlinear PDE so that a fully implicit and even monolithic formulation of the corresponding multiphase problem gets feasible. Copyright © 2018 John Wiley & Sons, Ltd.

Continuous processing gains importance in the fine chemical and pharmaceutical industries where crystallization is an important downstream operation. Seeded cooling crystallization of the l-alanine/water system was investigated under similar conditions, i.e., temperature interval, cooling rate, and seed material, both in a stirred batch vessel and in a continuous plug flow crystallizer in the coiled flow inverter (CFI) design with horizontal helical tube coils (ID = 4 mm) and frequent 90° bends of the coils. Short-cut calculations based on characteristic time scales and the Damköhler number allow for comparing the batch and continuous crystallization processes. The experimental results reveal crystal growth and growth rate dispersion to be dominating on the product crystal size distribution (CSD). However, at low flow rates of approximately 31 g min-1, a moving sediment flow of the slurry was present in the CFI crystallizer, resulting in further size dispersion effects. Elevated flow rates of approximately 40 g min-1 resulted in a more homogeneous suspension flow and a product CSD comparable to batch quality. Simulation studies based on a population balance equation model strengthen the hypothesis of the solid phase residence time distribution (RTDS) to be more spread in the moving sediment flow regime, leading to a wider product CSD. © 2018 American Chemical Society.

Dense granular materials are universal in nature and some common examples of this kind of materials in our daily lives are sand, rice, sugar etc. Research in this area is motivated by numerous applications encountered in industrial processes, such as hopper discharge, chute flow, moving beds, sandpipe flow, etc. and also in geophysics for the description and prediction of natural hazards like landslide and rock avalanches. Here, a continuum description of these granular flows is appropriate. Granular materials in liquid-like state show strong non-Newtonian behavior, which is typically described by phenomenological constitutive laws with local rheology. Very often, fluidization and flows of granular materials are localized in thin shear bands, whereas the granular material remains in solid-like state outside these bands. In this paper we establish a benchmark problem for granular materials, namely as Couette flow which is a very common example for granular materials in industrial application. We formulate the rheology for the continuum approach in the framework of the regularized version of a Bingham fluid and use advanced numerical methods regarding discretization as well as solution aspects, so that we can provide mesh independent results with two different software packages, Featflow and Openfoam, which can be used for validation and evaluation of the different methods and approaches. The authors thank the German Research Foundation DFG for funding under the grants TU 102/44-1 and SCHW 1168/6-1 within the pilot for transnational cooperation of the Dutch Technology Foundation STW and the DFG. This work was also supported by DAAD. © 2018 Elsevier B.V.

This paper presents a new efficient monolithic finite element solution scheme to treat the set of PDEs governing a 2D, biphasic, saturated theory of porous media model with intrinsically coupled and incompressible solid and fluid constituents for infinitesimal and large elastic deformation. Our approach, which inherits some of its techniques from CFD, is characterized by the following aspects: (1) it only performs operator evaluation with no additional Gateaux derivatives. In particular, the computations of the time-consuming material tangent matrix are not involved here; (2) it solves the non-linear dynamic problem with no restriction on the strength of coupling; (3) it is more efficient than the linear uvp solver discussed in previous works; (4) it requires weaker derivatives, and hence, lower-order FE can be tested; and (5) the boundary conditions are reduced, solution independent and more convenient to apply than in the old uvp formulation. For the purpose of validation and comparison, prototypical simulations including analytical solutions are carried out, and at the end, an adaptive time stepping procedure is introduced to handle the rapid change in the numbers of nonlinear iterations that may occur. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Following our previous work on the application of the diffuse interface coupled lattice Boltzmann-level set (LB-LS) approach to benchmark computations for 2D rising bubble simulations, this paper investigates the performance of the coupled scheme in 3D two-phase flows. In particular, the use of different lattice stencils, e.g., D3Q15, D3Q19 and D3Q27 is studied and the results for 3D rising bubble simulations are compared with regard to isotropy and accuracy against those obtained by finite element and finite difference solutions of the Navier–Stokes equations. It is shown that the method can eventually recover the benchmark solutions, provided that the interface region is aptly refined by the underlying lattice. Following the benchmark simulations, the application of the method in solving other numerically subtle problems, e.g., binary droplet collision and droplet splashing on wet surface under high Re and We numbers is presented. Moreover, implementations on general purpose GPUs are pursued, where the computations are adaptively refined around the critical parts of the flow. © 2016 Elsevier Ltd

In this work, we provide a unified and comparative description of the most prominent phase field based two-phase flow models and present our numerical results of the application of Galerkin-based Isogeometric Analysis (IGA) to incompressible Navier–Stokes–Cahn–Hilliard (NSCH) equations in velocity–pressure–phase field-chemical potential formulation. For the approximation of the velocity and pressure fields, LBB compatible non-uniform rational B-spline spaces are used which can be regarded as smooth generalizations of Taylor–Hood pairs of finite element spaces. The one-step θ-scheme is used for the discretization in time. The static and rising bubble, in addition to the nonlinear Rayleigh–Taylor instability flow problems, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme. © 2017 Elsevier Inc.

In this paper, we develop extended one-step methods for solving optimal control problem for systems governed by delay differential equations(DDEs). The optimal control problem governed by ordinary differential equations have been considered in this paper as a case of the DDEs with the absence of delays. The proposed problem is reduced to either a constrained or unconstrained minimization problem according to the nature of the dynamic system and the given conditions. Pontryagin’s maximum (or minimum) principle is used to characterize the optimal controls. Numerical results with simulations compared with other methods are presented to show the efficiency of the methodology. © 2016, Springer-Verlag Berlin Heidelberg.

Process intensification of engineering applications in the framework of reacting flows in micromixer devices attracts the attention of engineers and scientists from various fields. With the steadily increasing available computational resources, the traditional experimentally supported investigations may be extended by computational ones. For this purpose, a simulation framework based on state-of-the-art numerical techniques extended with special grid deformation techniques has been developed. Its validation in terms of comparison with computational and experimental results in reacting as well as in non-reacting frameworks has been performed on the basis of the T-mixer and SuperFocus mixer, respectively. The computational efficiency of the developed tool is shown to be applicable for optimization tasks, such as reverse engineering purposes. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We present a unique approach for integrating research in High Performance Computing (HPC) as well as photovoltaic (PV) solar farming and battery technologies into a container-based compute center designed for a maximum of energy efficiency, performance and extensibility/scalability. We use NVIDIA Jetson TK1 boards to build a considerably dimensioned cluster of 60 low-power GPUs, attach a 7.5 kWp solar farm and a 8 kWh Lithium-Ion battery power supply and integrate everything into a single-container, standalone housing. We demonstrate the success of our system by evaluating the performance and energy efficiency for common versatile dense and sparse linear algebra kernels as well as a full CFD code. By this work we can show, that with current technology, energy consumption-induced follow-up cost of HPC can be reduced to zero. © Springer International Publishing AG 2017.

In this contribution we present advances concerning efficient parallel multiscale methods and uncertainty quantification that have been obtained in the frame of the DFG priority program 1648 Software for Exascale Computing (SPPEXA) within the funded project EXA-DUNE. This project aims at the development of flexible but nevertheless hardware-specific software components and scalable high-level algorithms for the solution of partial differential equations based on the DUNE platform. While the development of hardware-based concepts and software components is detailed in the companion paper (Bastian et al., Hardware-based efficiency advances in the EXA-DUNE project. In: Proceedings of the SPPEXA Symposium 2016, Munich, 25–27 Jan 2016), we focus here on the development of scalable multiscale methods in the context of uncertainty. © Springer International Publishing Switzerland 2016.

We analyze energy efficiency of a 3D coastal ocean simulator on Haswell and Cortex-A15 architectures and propose a simple yet effective way to model energy-to-solution on different hardware platforms. The work also demonstrates that using processors from the field of embedded/mobile computing can increase the energy efficiency by 50%. © 2016, Springer-Verlag Berlin Heidelberg.

A multiphase Lattice Boltzmann (LB) scheme coupled with a level set interface capturing model is used for the simulation of multiphase flows, and in particular, rising bubbles under moderate and high density and viscosity ratios. We make use of consistent time integration and force discretization schemes in particular for pressure forces along with using multiple relaxation time (MRT) form of the collision in the LB equation which enables us to preserve stability and accuracy for high density and critical Eo numbers. We first present the solution for the standard test of a static bubble in order to show the accuracy of the solution with respect to the Laplace law for pressure and also the spurious velocity level. We present quantitative benchmark computations and error analysis for the 2D rising bubble test cases being further validated against high precision finite element solutions in Hysing et al. (2009). Furthermore, by applying efficient multi-core and many core general purpose GPU (GPGPU) implementations outlines, we demonstrate that the desired parallel scaling characteristics of general LBM solutions are well preserved for the proposed coupled computations. The presented implementations are shown to outperform the available GPU-based phase-field LBM solvers in terms of computational time, turning the scheme into a desirable choice for massive multiphase simulations in three dimensions. © 2015 Elsevier Ltd.

We present advances concerning efficient finite element assembly and linear solvers on current and upcoming HPC architectures obtained in the frame of the EXA-DUNE project, part of the DFG priority program 1648 Software for Exascale Computing (SPPEXA). In this project, we aim at the development of both flexible and efficient hardware-aware software components for the solution of PDEs based on the DUNE platform and the FEAST library. In this contribution, we focus on node-level performance and accelerator integration, which will complement the provenMPI-level scalability of the framework. The higher-level aspects of the EXADUNE project, in particular multiscale methods and uncertainty quantification, are detailed in the companion paper (Bastian et al., Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE. In: Proceedings of the SPPEXA Symposium, 2016). © Springer International Publishing Switzerland 2016.

The aim of this contribution is to present a new Newton-type solver for yield stress fluids, for instance for viscoplastic Bingham fluids. In contrast to standard globally defined (‘outer’) damping strategies, we apply weighting strategies for the different parts inside of the resulting Jacobian matrices (after discretizing with FEM), taking into account the special properties of the partial operators which arise due to the differentiation of the corresponding nonlinear viscosity function. Moreover, we shortly discuss the corresponding extension to fluids with a pressure-dependent yield stress which are quite common for modelling granular material. From a numerical point of view, the presented method can be seen as a generalized Newton approach for non-smooth problems. © Springer International Publishing Switzerland 2016.

In this paper, we derive a class of extended one-step methods of order m for solving delay-differential equations. This class includes methods of fourth and fifth order of accuracy. Also, the class of these methods depends on two free parameters. A convergence theorem and convergence factor of these methods are given. Stability regions for such methods are determined in terms of the time-lag t . Some numerical examples are given to illustrate the effectiveness of the numerical schemes. © 2015 NSP.

Abstract In this article we present a new implicit numerical scheme for reaction-diffusion-advection equations on an evolving in time hypersurface Γ(t). The partial differential equations are solved on a stationary quadrilateral, resp., hexahedral mesh. The zero level set of the time dependent indicator function ø(t) implicitly describes the position of Γ(t). The dominating convective-like terms, which are due to the presence of chemotaxis, transport of the cell density and surface evolution may lead to the non-positiveness of a given numerical scheme and in such a way cause appearance of negative values and give rise of nonphysical oscillations in the numerical solution. The proposed finite element method is constructed to avoid this problem: implicit treatment of corresponding discrete terms in combination with the algebraic flux correction (AFC) techniques make it possible to obtain a sufficiently accurate solution for reaction-diffusion-advection PDEs on evolving surfaces. © 2015 Elsevier B.V.

A mass conserving, diffused interface, coupled Lattice Boltzmann-level set scheme is proposed and numerically studied for the simulation of high density and viscosity ratio multiphase flows. The approach is based on the pressure evolution formulation of the lattice Boltzmann equation, which is then coupled with the level set equation to capture a diffused level set function. Multiple relaxation time collision and isotropic force discretization techniques are employed to further reinforce the stability and accuracy of the LBM solver and a monolithic approach is employed to convect the level set function with minimal computations to preserve a smooth density profile. We present extensive investigations for numerical accuracy through performing benchmark simulations for rising bubble problems. It is observed that the proposed scheme is successful in producing accurate results as compared to those from high precision 2D finite element solutions in Hysing et al. (2009), offering a remarkable improvement in mass conservation and characteristic benchmark quantities upon our previous sharp interface coupled model described in Safi and Turek (2014). Moreover, parallel scalability of the coupled LB scheme is shown to be preserved through performing efficient CPU- and GPGPU-based computations. © 2015 Elsevier Ltd. All rights reserved.

The performance of two commercial simulation codes, Ansys Fluent and Comsol Multiphysics, is thoroughly examined for a recently established two-phase flow benchmark test case. In addition, the commercial codes are directly compared with the newly developed academic code, FeatFlow TP2D. The results from this study show that the commercial codes fail to converge and produce accurate results, and leave much to be desired with respect to direct numerical simulation of flows with free interfaces. The academic code on the other hand was shown to be computationally efficient, produced very accurate results, and outperformed the commercial codes by a magnitude or more. Copyright © 2015 Inderscience Enterprises Ltd.

This paper presents our numerical results of the application of Isogeometric Analysis (IGA) to the velocity-pressure formulation of the steady state as well as to the unsteady incompressible Navier-Stokes equations. For the approximation of the velocity and pressure fields, LBB compatible B-spline spaces are used which can be regarded as smooth generalizations of Taylor-Hood pairs of finite element spaces. The single-step θ-scheme is used for the discretization in time. The lid-driven cavity flow, in addition to its regularized version and flow around cylinder, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme. © 2015 Elsevier Inc. All rights reserved.

Least-squares finite element methods are motivated, beside others, by the fact that in contrast to standard mixed finite element methods, the choice of the finite element spaces is not subject to the LBB stability condition and the corresponding discrete linear system is symmetric and positive definite. We intend to benefit from these two positive attractive features, on one hand, to use different types of elements representing the physics as for instance the jump in the pressure for multiphase flow and mass conservation and, on the other hand, to show the flexibility of the geometric multigrid methods to handle efficiently the resulting linear systems. With the aim to develop a solver for non-Newtonian problems, we introduce the stress as a new variable to recast the Navier-Stokes equations into first order systems of equations. We numerically solve S-V-P, Stress-Velocity-Pressure, formulation of the incompressible Navier-Stokes equations based on the least-squares principles using different types of finite elements of low as well as higher order. For the discrete systems, we use a conjugate gradient (CG) solver accelerated with a geometric multigrid preconditioner. In addition, we employ a Krylov space smoother which allows a parameter-free smoothing. Combining this linear solver with the Newton linearization results in a robust and efficient solver. We analyze the application of this general approach, of using different types of finite elements, and the efficiency of the solver, geometric multigrid, throughout the solution of the prototypical benchmark configuration ‘flow around cylinder’. © Springer International Publishing Switzerland 2015.

We present numerical simulations of a film stretching process between two rolls of different temperature and rotational velocity. Film stretching is part of the industrial production of sheets of plastics which takes place after the extrusion process. The goal of the stretching of the sheet material is to rearrange the orientation of the polymer chains. Thus, the final products have more smooth surfaces and homogeneous properties. In numerical simulation, the plastic sheet is modelled geometrically as a membrane and rheologically as a polymer melt. The thickness of the membrane is not assumed to be constant but rather depends on the rheology of the polymer and the heat transfer. The rheology of the sheet material is governed by a viscoelastic fluid and is coupled to the flow model. An A-stable time integrator is applied to the systems in which the continuous spatial system is discretized within the FEM framework at each time step. The resulting discrete systems are solved via Newton-multigrid techniques. Moreover, a level set method is used to capture the free surface. We obtain similar results for test configurations with available results from literature and present “neck-in” as well as “dog-bone” effects. © Springer International Publishing Switzerland 2015.

In this paper, we discuss solution techniques of Newton-multigrid type for the resulting nonlinear saddle-point block-systems if higher order continuous Galerkin-Petrov (cGP(k)) and discontinuous Galerkin (dG(k)) time discretizations are applied to the nonstationary incompressible Navier-Stokes equations. In particular for the cGP(2) method with quadratic ansatz functions in time, which lead to 3rd order accuracy in the L2-norm and even to 4th order superconvergence in the endpoints of the time intervals, together with the finite element pair Q2/P1disc for the spatial approximation of velocity and pressure leading to a globally 3rd order scheme, we explain the algorithmic details as well as implementation aspects. All presented solvers are analyzed with respect to their numerical costs for two prototypical flow configurations. © 2014 IMACS.

We discuss extended one-step methods of order three for the numerical solution of delay-differential equations. A convergence theorem and the numerical studies regarding the convergence factor of these methods are given. Also, we investigate the stability properties of these methods. The results of the theoretical studies are illustrated by numerical examples. © 2014 NSP Natural Sciences Publishing Cor.

In this paper, the previously described monolithic approach [6] for the stationary discrete Boltzmann equation is extended to time-dependent problems. In general, both collision and advection operators are discretized on nonuniform grids as opposed to the standard Lattice Boltzmann method. Implicit time-stepping schemes are applied for an accurate and robust numerical treatment of the nonstationary flow problems. The resulting coupled system of equations is treated using special numerical methods for PDE's. As in the steady case, we apply a full Newton method for the nonlinear problems, but we also discuss possible variants of semi-implicit schemes all of which lead to nonsymmetric linear systems. The preconditioning of the used Krylov-space methods, resp., the construction of corresponding smoothing operators in the applied multigrid approaches is closely connected to the underlying short characteristic-upwinding discretization, yielding the exact inverse of the transport operators even for unstructured meshes due to a special numbering technique. Numerical results are given for the proposed solvers analysing the efficiency depending on the Mach number, time step and mesh size, while accuracy and stability of the complete space-time discretization are demonstrated for prototypical flow configurations at various timesteps. © de Gruyter 2014.

We solve the V-V-P, vorticity-velocity-pressure, formulation of the stationary incompressible Navier-Stokes equations based on the least-squares finite element method. For the discrete systems, we use a conjugate gradient (CG) solver accelerated with a geometric multigrid preconditioner for the complete system. In addition, we employ a Krylov space smoother inside of the multigrid which allows a parameter-free smoothing. Combining this linear solver with the Newton linearization, we construct a very robust and efficient solver. We use biquadratic finite elements to enhance the mass conservation of the least-squares method for the inflow-outflow problems and to obtain highly accurate results. We demonstrate the advantages of using the higher order finite elements and the grid independent solver behavior through the solution of three stationary laminar flow problems of benchmarking character. The comparisons show excellent agreement between our results and those of the Galerkin mixed finite element method as well as available reference solutions. © 2013 Elsevier Inc.

Biological microorganisms swim with flagella and cilia that execute nonreciprocal motions for low Reynolds number (Re) propulsion in viscous fluids. This symmetry requirement is a consequence of Purcell' s scallop theorem, which complicates the actuation scheme needed by microswimmers. However, most biomedically important fluids are non-Newtonian where the scallop theorem no longer holds. It should therefore be possible to realize a microswimmer that moves with reciprocal periodic body-shape changes in non-Newtonian fluids. Here we report a symmetric 'micro-scallop', a single-hinge microswimmer that can propel in shear thickening and shear thinning (non-Newtonian) fluids by reciprocal motion at low Re. Excellent agreement between our measurements and both numerical and analytical theoretical predictions indicates that the net propulsion is caused by modulation of the fluid viscosity upon varying the shear rate. This reciprocal swimming mechanism opens new possibilities in designing biomedical microdevices that can propel by a simple actuation scheme in non-Newtonian biological fluids.

This study concerns the determination of the significant factors for an innovative deburring process: low pressure abrasive water-jet blasting. The abrasive medium aluminum oxide (Al2O3) is classified according to the individual characteristics of different grain sizes. Then, the particle behavior in the air jet is analyzed with an optical measuring method, Particle Image Velocimetry (PIV); the velocity profile and the particle distribution of the dispersed system are obtained. Computational Fluid Dynamics (CFD) simulations were verified by comparing the experimental and numerical results, and the velocity range for the abrasive particles has been specified. Copyright © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We present an implicit finite element method for a class of chemotaxis models in three spatial dimensions. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density. To enforce the discrete maximum principle, the standard Galerkin discretization is constrained using a local extremum diminishing flux limiter. To demonstrate the efficiency and robustness of this approach, we solve blow-up problems in a 3D chemostat domain. To give a flavor of more complex and realistic chemotactic applications, we investigate the pattern dynamics and aggregating behavior of the bacteria Escherichia coli and Salmonella typhimurium. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature. © 2012 Elsevier B.V. All rights reserved.

In this paper, we present fully implicit continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time-stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank-Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well-known LBB-stable finite element pair Q2/P1disc of third-order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka-like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t.CPU and numerical costs. As a first test problem, we consider a classical 'flow around cylinder' benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary 'flow through a Venturi pipe'. The objective of this simulation is to control the instantaneous and mean flux through this device. © 2013 John Wiley & Sons, Ltd.

We present special Newton-multigrid techniques for stationary incompressible nonlinear flow models discretized by the high order LBB-stable Q2P1 element pair. We treat the resulting nonlinear and the corresponding linear discrete systems by a fully coupled monolithic approach to maintain high accuracy and robustness, particularly with respect to different rheological behaviors and also regarding different problem sizes and types of nonlinearity. Here, local pressure Schur complement techniques are presented as a generalization of the classical Vanka smoother. The discussed methodology is implemented for the well-known flow around cylinder benchmark configuration for generalized Newtonian as well as non-Newtonian flows including non-isothermal, shear/pressure dependent and viscoelastic effects. © 2012 John Wiley & Sons, Ltd.

In this paper we present an implicit finite element method for a class of chemotaxis models where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.

Numerical solvers based on population balance equations (PBE) coupled with flow equations are a promising approach to simulate liquid/gas-liquid dispersed flows, which are very commonly observed in nature and in industrial processes. The challenges for the numerical solution of the coupled equation systems are discussed and detailed numerical recipes are presented whose main ingredients are the method of classes, positivity-preserving linearization and the high-order FEM-AFC schemes, additional to the FeatFlow in-house flow solver package. Liquid-liquid flows through static mixers and dispersed phase systems in a flat bubble column are studied with the accordingly developed computational tool. The suggested recipes were validated by comparing the numerical results against experimental data. The state of the art on the numerical simulation of liquid/gas-liquid dispersed flows with population balance equations coupled to computational fluid dynamics, and the related numerical challenges are explained. A novel approach is presented to numerically simulate bubbly flows at moderate gas holdups. The suggested recipes are validated with experimental data. Copyright © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Numerical simulation of incompressible multiphase flows with immiscible fluids is still a challenging field, particularly for 3D configurations undergoing complex topological changes. In this paper, we discuss a 3D FEM approach with high-order Stokes elements (Q2/P1) for velocity and pressure on general hexahedral meshes. A discontinuous Galerkin approach with piecewise linear polynomials (dG(1)) is used to treat the Level Set function. The developed methodology allows the application of special redistancing algorithms which do not change the position of the interface. We explain the corresponding FEM techniques for treating the advection steps and surface tension effects, and validate the corresponding 3D code with respect to both numerical test cases and experimental data. The corresponding applications describe the classical rising bubble problem for various parameters and the generation of droplets from a viscous liquid jet in a coflowing surrounding fluid. Both of these applications can be used for rigorous benchmarking of 3D multiphase flow simulations. © 2013 Springer Science+Business Media Dordrecht.

The occlusional performance of sole endoluminal stenting of intracranial aneurysms is controversially discussed in the literature. Simulation of blood flow has been studied to shed light on possible causal attributions. The outcome, however, largely depends on the numerical method and various free parameters. The present study is therefore conducted to find ways to define parameters and efficiently explore the huge parameter space with finite element methods (FEMs) and lattice Boltzmann methods (LBMs). The goal is to identify both the impact of different parameters on the results of computational fluid dynamics (CFD) and their advantages and disadvantages. CFD is applied to assess flow and aneurysmal vorticity in 2D and 3D models. To assess and compare initial simulation results, simplified 2D and 3D models based on key features of real geometries and medical expert knowledge were used. A result obtained from this analysis indicates that a combined use of the different numerical methods, LBM for fast exploration and FEM for a more in-depth look, may result in a better understanding of blood flow and may also lead to more accurate information about factors that influence conditions for stenting of intracranial aneurysms. © 2013 Frank Weichert et al.

We describe our FE-gMG solver, a finite element geometric multigrid approach for problems relying on unstructured grids. We augment our GPU- and multicore-oriented implementation technique based on cascades of sparse matrix-vector multiplication by applying strong smoothers. In particular, we employ Sparse Approximate Inverse (SPAI) and Stabilised Approximate Inverse (SAINV) techniques. We focus on presenting the numerical efficiency of our smoothers in combination with low- and high-order finite element spaces as well as the hardware efficiency of the FE-gMG. For a representative problem and computational grids in 2D and 3D, we achieve a speedup of an average of 5 on a single GPU over a multithreaded CPU code in our benchmarks. In addition, our strong smoothers can deliver a speedup of 3.5 depending on the element space, compared to simple Jacobi smoothing. This can even be enhanced to a factor of 7 when combining the usage of approximate inverse-based smoothers with clever sorting of the degrees of freedom. In total the FE-gMG solver can outperform a simple (multicore-) CPU-based multigrid by a total factor of over 40. © 2012 Elsevier Ltd.

The characteristic-based split (CBS) method has been widely used in the finite element community to facilitate the numerical solution of Navier-Stokes (NS) equations. However, this computational algorithm has rarely been employed in the finite volume context and the stabilization of the numerical solution procedure has traditionally been addressed differently in volume-based numerical schemes. In this article, the CBS-based finite volume algorithm is employed to formulate and solve a number of laminar incompressible flow and convective heat transfer problems. Both explicit and implicit versions of the algorithm are first explained and validated in the context of the solution of a lid-driven cavity problem and a backward facing step (BFS) flow problem. The modified algorithm, capable of modelling the coupling between the momentum and energy balance equations, is then introduced and used to solve a buoyancy-driven cavity flow problem. Computational results show that the CBS finite volume algorithm can be reliably used in the solution of laminar incompressible heat and fluid flow problems. © 2012 Copyright Taylor and Francis Group, LLC.

Numerically challenging, comprehensive benchmark cases are of great importance for researchers in the field of CFD. Numerical benchmark cases offer researchers frameworks to quantitatively explore limits of the computational tools and to validate them. Therefore, we focus on simulations of challenging benchmark tests, laminar and transient 3D flows around a cylinder, and aim to establish a new comprehensive benchmark case by doing direct numerical simulations with three distinct CFD software packages, OpenFOAM, Ansys-CFX and FeatFlow which employ different numerical approaches to the discretisation of the incompressible Navier-Stokes equations. All the software tools successfully pass the benchmark tests and show a good agreement such that the benchmark result was precisely determined. As a main result, the CFD software package with high order finite element approximation has been found to be computationally more efficient and accurate than the ones adopting low order space discretisation methods. Copyright © 2012 Inderscience Enterprises Ltd.

In the framework of finite element discretizations, we introduce a fully nonlinear Newton-like method and a linearized second order approach in time applied to certain partial differential equations for chemotactic processes incorporating two entities, a chemical agent and the reacting population of certain biological organisms/species. We investigate the benefit of a corresponding monolithic approach and the decoupled variant. In particular, we analyze accuracy, efficiency and stability of different methods and their dependences on certain parameters in order to identify a well suited finite element solver for chemotaxis problems. © 2011 Elsevier Ltd. All rights reserved.

We present special finite element and multigrid techniques for solving prototypical cerebral aneurysm hemodynamics problems numerically. An arbitrary Lagrangian-Eulerian (ALE) formulation is employed for this fluid-structure interaction (FSI) application. We utilize the well-known high order finite element pair Q2P1 for discretization in space to gain high accuracy and robustness and perform as time-stepping a fully implicit second order accurate time integrator. The resulting nonlinear discretized algebraic system is solved by an iterative Newton solver which approximates the Jacobian matrix by the divided difference approach, and the resulting linear system is solved by means of Krylov type and geometrical multigrid solvers with a Vanka-like smoother. The aim of this paper is to study the interaction of the elastic walls of an aneurysm with the geometrical shape of an implanted stent structure for prototypical 2D configurations. Preliminary results for the stent-assisted occlusion of a cerebral aneurysm and a qualitative analysis of the behavior of the elasticity of the walls vs. the geometrical details of the stent for prototypical flow situation are presented. Additionally, our approach is designed in such a way that complicated realistic constitutive relations for biomechanics applications for blood vessel simulations can be easily integrated. © 2011 IMACS.

In this paper we discuss numerical simulation techniques using a finite element approach in combination with the fictitious boundary method (FBM) for rigid particulate flow configurations in 3D. The flow is computed with a multigrid finite element solver (FEATFLOW), the solid particles are allowed to move freely through the computational mesh which can be static or adaptively aligned by a grid deformation method allowing structured as well as unstructured meshes. We explain the details of how we can use the FBM to simulate flows with complex geometries that are hard to describe analytically. Stationary and time-dependent numerical examples, demonstrating the use of such geometries are provided. Our numerical results include well-known benchmark configurations showing that the method can accurately and efficiently handle prototypical particulate flow situations in 3D with particles of different shape and size. © 2011 John Wiley & Sons, Ltd.

A semi-implicit finite element scheme and a Newton-like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low-order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi-implicit scheme is derived via a time-lagged linearization of the Jacobian operator, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. © 2011 John Wiley & Sons, Ltd.

We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier-Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems. © 2012 IMACS. Published by Elsevier B.V. All rights reserved.

Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. © 2011 Elsevier Ltd.

We discuss numerical properties of continuous Galerkin-Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs. © 2011 de Gruyter.

The derivation of macroscopic models for particle-laden gas flows is reviewed. Semi-implicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust. © 2011 Elsevier B.V. All rights reserved.

In this paper, we present numerical techniques for one-way coupling of CFD and Population Balance Equations (PBE) based on the incompressible flow solver FeatFlow which is extended with Chien's Low-Reynolds number k-e{open} turbulence model, and breakage and coalescence closures. The presented implementation ensures strictly conservative treatment of sink and source terms which is enforced even for geometric discretization of the internal coordinate. The validation of our implementation which covers wide range of computational and experimental problems enables us to proceed into three-dimensional applications as, turbulent flows in a pipe and through a static mixer. The aim of this paper is to highlight the influence of different formulations of the novel theoretical breakage and coalescence models on the equilibrium distribution of population, and to propose an implementation strategy for three-dimensional one-way coupled CFD-PBE model. © 2011 Elsevier Ltd.

In this work, we discuss special numerical techniques for viscoelastic flow problems given in log-conformation reformulation (LCR). In particular, we consider Oldroyd-B and Giesekus-type fluids. We utilize a fully coupled monolithic finite element approach that treats all the numerical variables simultaneously. Thus, it is possible to do a direct steady approach and to avoid pseudo-timeNewton method handles the discrete nonlinear system, which results from the FEM discr stepping with correspondingly small time step sizes in the case of a nonsteady approach. The etization with consistent edge-oriented FEM stabilization techniques. In each nonlinear step, a direct sparse solver or a geometrical multigrid solver with special Vanka smoother deals with the resulting linear subproblems. Moreover, local grid refinement helps to reduce the computational efforts and to increase the accuracy of functional values. The merit of the presented methodology, for the well-known 'flow around cylinder' benchmark problem, is that we can obtain the discrete approximations by using a direct steady approach. Thus, the numerical effort can be rather independent of the examined We numbers. Furthermore, the 'black box' techniques can deal with any given viscoelastic models easily, hereby showing the same advantageous numerical convergence behaviour of the above mentioned fluids. © 2010 Elsevier B.V.

This is an experimental and numerical study of dry, frictional powder flows in the quasi-static and intermediate regimes using the geometry of the Couette device. We measure normal and shear stresses on the shearing surface and extract from the data, constitutive equations valid in the slow frictional, quasi-static and the intermediate (dense), collisional regimes of flow. This constitutive equation is then used in a new, specially developed FEM solver (FeatFlow-Ouazzi et al., 2005 [18]) to obtain solutions of the continuum equations of motion as well as stress and velocity distributions in the powder. While the measurements to obtain the constitutive equation are performed in a concentric Couette device, the numerical scheme is used to predict the torque and stresses in two additional geometries. These geometries are an eccentric Couette where the inner, rotating cylinder is placed off-center with different eccentricities and a more complicated geometry where a cylindrical body is introduced in the middle between the rotating and stationary cylinders and obstructs part of the shearing gap. The purpose of these calculations is to show the versatility of the numerical solution. © 2009 Elsevier B.V. All rights reserved.

In this paper, we present special discretization and solution techniques for the numerical simulation of the Lattice Boltzmann equation (LBE). In Hübner and Turek (Computing, 81:281-296, 2007), the concept of the generalized mean intensity had been proposed for radiative transfer equations which we adapt here to the LBE, treating it as an analogous (semi-discretized) integro-differential equation with constant characteristics. Thus, we combine an efficient finite difference-like discretization based on short-characteristic upwinding techniques on unstructured, locally adapted grids with fast iterative solvers. The fully implicit treatment of the LBE leads to nonlinear systems which can be efficiently solved with the Newton method, even for a direct solution of the stationary LBE. With special exact preconditioning by the transport part due to the short-characteristic upwinding, we obtain an efficient linear solver for transport dominated configurations (macroscopic Stokes regime), while collision dominated cases (Navier-Stokes regime for larger Re numbers) are treated with a special block-diagonal preconditioning. Due to the new generalized equilibrium formulation (GEF) we can combine the advantages of both preconditioners, i.e. independence of the number of unknowns for convection-dominated cases with robustness for stiff configurations. We further improve the GEF approach by using hierarchical multigrid algorithms to obtain grid-independent convergence rates for a wide range of problem parameters, and provide representative results for various benchmark problems. Finally, we present quantitative comparisons between a highly optimized CFD-solver based on the Navier-Stokes equation (FeatFlow) and our new LBE solver (FeatLBE). © 2009 Springer-Verlag.

In this paper a projection method for the Navier-Stokes equations with Coriolis force is considered. This time-stepping algorithm takes into account the Coriolis terms both on prediction and correction steps. We study the accuracy of its semi-discretized form and show that the velocity is weakly first-order approximation and the pressure is weakly order 1/2 approximation. © 2009 Birkhäuser Verlag, Basel.

FEAST (Finite Element Analysis and Solutions Tools) is a Finite Element-based solver toolkit for the simulation of PDE problems on parallel HPC systems, which implements the concept of 'hardware-oriented numerics', a holistic approach aiming at optimal performance for modern numerics. In this paper, we describe this concept and the modular design that enables applications built on top of FEAST to execute efficiently, without any code modifications, on commodity-based clusters, the NEC SX 8 and GPU-accelerated clusters. We demonstrate good performance and weak and strong scalability for the prototypical Poisson problem and more challenging applications from solid mechanics and fluid dynamics. Copyright (C) 2010 John Wiley & Sons, Ltd.

A macroscopic two-fluid model of compressible particle-laden gas flows is considered. The governing equations are discretized by a high-resolution finite element method based on algebraic flux correction. A multidimensional limiter of TVD type is employed to constrain the local characteristic variables for the continuous gas phase and conservative fluxes for a suspension of solid particles. Special emphasis is laid on the efficient computation of steady state solutions at arbitrary Mach numbers. To avoid stability restrictions and convergence problems, the characteristic boundary conditions are imposed weakly and treated in a fully implicit manner. A two-way coupling via the interphase drag force is implemented using operator splitting. The Douglas-Rachford scheme is found to provide a robust treatment of the interphase exchange terms within the framework of a fractional-step solution strategy. Two-dimensional simulation results are presented for a moving shock wave and for a steady nozzle flow. © 2009 Elsevier B.V. All rights reserved.

We present an efficient method for the simulation of laminar fluid flows with free surfaces including their interaction with moving rigid bodies, based on the two-dimensional shallow water equations and the Lattice-Boltzmann method. Our implementation targets multiple fundamentally different architectures such as commodity multicore CPUs with SSE, GPUs, the Cell BE and clusters. We show that our code scales well on an MPI-based cluster; that an eightfold speedup can be achieved using modern GPUs in contrast to multithreaded CPU code and, finally, that it is possible to solve fluid-structure interaction scenarios with high resolution at interactive rates. © 2010 Springer-Verlag.

We calculate numerically the solutions of the stationary Navier-Stokes equations in two dimensions, for a square domain with particular choices of boundary data. The data are chosen to test whether bounded disturbances on the boundary can be expected to spread into the interior of the domain. The results indicate that such behavior indeed can occur, but suggest an estimate of general form for the magnitudes of the solution and of its derivatives, analogous to classical bounds for harmonic functions. The qualitative behavior of the solutions we found displayed some striking and unexpected features. As a corollary of the study, we obtain two new examples of non-uniqueness for stationary solutions at large Reynolds numbers. © 2010 Springer-Verlag Berlin Heidelberg.

Recently, we introduced and mathematically analysed a new method for grid deformation (Grajewski et al., 2009) [15] we call basic deformation method (BDM) here. It generalises the method proposed by Liao et al. (Bochev et al., 1996; Cai et al., 2004; Liao and Anderson, 1992) [4,6,20]. In this article, we employ the BDM as core of a new multilevel deformation method (MDM) which leads to vast improvements regarding robustness, accuracy and speed. We achieve this by splitting up the deformation process in a sequence of easier subproblems and by exploiting grid hierarchy. Being of optimal asymptotic complexity, we experience speed-ups up to a factor of 15 in our test cases compared to the BDM. This gives our MDM the potential for tackling large grids and time-dependent problems, where possibly the grid must be dynamically deformed once per time step according to the user's needs. Moreover, we elaborate on implementational aspects, in particular efficient grid searching, which is a key ingredient of the BDM. © 2010 IMACS.

Comparative benchmark results for different solution methods for fluid-structure interaction problems are given which have been developed as collaborative project in the DFG Research Unit 493. The configuration consists of a laminar incompressible channel flow around an elastic object. Based on this benchmark configuration the numerical behavior of different approaches is analyzed exemplarily. The methods considered range from decoupled approaches which combine Lattice Boltzmann methods with hp-FEM techniques, up to strongly coupled and even fully monolithic approaches which treat the fluid and structure simultaneously. © 2011 Springer.

An Arbitrary Lagrangian-Eulerian (ALE) formulation is applied in a fully coupled monolithic way, considering the fluid-structure interaction (FSI) problem as one continuum. The mathematical description and the numerical schemes are designed in such a way that general constitutive relations (which are realistic for biomechanics applications) for the fluid as well as for the structural part can be easily incorporated. We utilize the LBB-stable finite element pairs Q 2 P 1 and P 2 + P 1 for discretization in space to gain high accuracy and perform as time-stepping the 2nd order Crank-Nicholson, respectively, a new modified Fractional-Step-θ-scheme for both solid and fluid parts. The resulting discretized nonlinear algebraic system is solved by a Newton method which approximates the Jacobian matrices by a divided differences approach, and the resulting linear systems are solved by direct or iterative solvers, preferably of Krylov-multigrid type. For validation and evaluation of the accuracy and performance of the proposed methodology, we present corresponding results for a new set of FSI benchmark configurations which describe the self-induced elastic deformation of a beam attached to a cylinder in laminar channel flow, allowing stationary as well as periodically oscillating deformations. Then, as an example of FSI in biomedical problems, the influence of endovascular stent implantation on cerebral aneurysm hemodynamics is numerically investigated. The aim is to study the interaction of the elastic walls of the aneurysm with the geometrical shape of the implanted stent structure for prototypical 2D configurations. This study can be seen as a basic step towards the understanding of the resulting complex flow phenomena so that in future aneurysm rupture shall be suppressed by an optimal setting of the implanted stent geometry. © 2011 Springer.

We present numerical techniques for solving the problem of fluid structure interaction with a compressible elastic material in a laminar incompressible viscous flow via fully coupled monolithic Arbitrary Lagrangian-Eulerian (ALE) formulation. The mathematical description and the numerical schemes are designed in such a way that more complicated constitutive relations can be easily incorporated. The whole domain of interest is treated as one continuum and we utilize the well known Q 2 P 1 finite element pair for discretization in space to gain high accuracy. We perform numerical comparisons for different time stepping schemes, including variants of the Fractional-Step-θ-scheme, Backward Euler and Crank-Nicholson scheme for both solid and fluid parts. The resulting nonlinear discretized algebraic system is solved by a quasi-Newton method which approximates the Jacobian matrices by the divided differences approach and the resulting linear systems are solved by a geometric multigrid approach. In the numerical examples, a cylinder with attached flexible beam is allowed to freely rotate around its axis which requires a special numerical treatment. By identifying the center of the cylinder with one grid point of the computational mesh we prescribe a Dirichlet type boundary condition for the velocity and the displacement of the structure at this point, which allows free rotation around this point. We present numerical studies for different problem parameters on various mesh types and compare the results with experimental values from a corresponding benchmarking experiment. © 2010 Springer-Verlag Berlin Heidelberg.

We present different kernels based on Lattice-Boltzmann methods for the solution of the two-dimensional Shallow Water and Navier-Stokes equations on fully structured lattices. The functionality ranges from simple scenarios like open-channel flows with planar beds to simulations with complex scene geometries like solid obstacles and non-planar bed topography with dry-states and even interaction of the fluid with floating objects. The kernels are integrated into a hardware-oriented collection of libraries targeting multiple fundamentally different parallel hardware architectures like commodity multicore CPUs, the Cell BE, NVIDIA GPUs and clusters. We provide an algorithmic study which compares the different solvers in terms of performance and numerical accuracy in view of their capabilities and their specific implementation and optimisation on the different architectures. We show that an eightfold speedup over optimised multithreaded CPU code can be obtained with the GPU using basic methods and that even very complex flow phenomena can be simulated with significant speedups without loss of accuracy.