Stefan Turek

ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

Fluid dynamical problems are often conceptualized in unbounded domains. However, most methods of numerical simulation then require a truncation of the conceptual domain to a bounded one, thereby introducing artificial boundaries. Here we analyse out experience in choosing artificial boundary conditions implicitly through the choice of variational formulations. We deal particularly with a class of problems that involve the prescription of pressure drops and/or net flux conditions.

Benchmark Computations of Laminar Flow Around a Cylinder

An overview of benchmark computations for 2D and 3D laminar flows around a cylinder is given, which have been defined for a comparison of different solution approaches for the incompressible Navier-Stokes equations developed within the Priority Research Programme. The exact definitions of the benchmarks are recapitulated and the numerical schemes and computers employed by the various participating groups are summarized. A detailed evaluation of the results provided is given, also including a comparison with a reference experiment. The principal purpose of the benchmarks is discussed and some general conclusions which can be drawn from the results are formulated.

Efficient Solvers for Incompressible Flow Problems

Preface.- Notation.- Motivation for current research.- Derivation of Navier-Stokes solvers.- Other mathematical components.- Numerical comparisons of Navier-Stokes solvers.- Conclusions and outlook.- The enclosed CD-ROM.The complete table of contents can be found on the Internet: http://www.springer.de

Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow

We describe new benchmark settings for the rigorous evaluation of different methods for fluid-structure interaction problems. The configurations consist of laminar incompressible channel flow around an elastic object which results in self-induced oscillations of the structure. Moreover, characteristic flow quantities and corresponding plots are provided for a quantitative comparison.

Performance and accuracy of hardware-oriented native-, emulated-and mixed-precision solvers in FEM simulations

In this survey paper, we compare native double precision solvers with emulated- and mixed-precision solvers of linear systems of equations as they typically arise in finite element discretisations. The emulation utilises two single float numbers to achieve higher precision, while the mixed precision iterative refinement computes residuals and updates the solution vector in double precision but solves the residual systems in single precision. Both techniques have been known since the 1960s, but little attention has been devoted to their performance aspects. Motivated by changing paradigms in processor technology and the emergence of highly-parallel devices with outstanding single float performance, we adapt the emulation and mixed precision techniques to coupled hardware configurations, where the parallel devices serve as scientific co-processors. The performance advantages are examined with respect to speedups over a native double precision implementation (time aspect) and reduced area requirements for a chip (space aspect). The paper begins with an overview of the theoretical background, algorithmic approaches and suitable hardware architectures. We then employ several conjugate gradient (CG) and multigrid solvers and study their behaviour for different parameter settings of the iterative refinement technique. Concrete speedup factors are evaluated on the coupled hardware configuration of a general-purpose CPU and a graphics processor. The dual performance aspect of potential area savings is assessed on a field programmable gate array (FPGA). In the last part, we test the applicability of the proposed mixed precision schemes with ill-conditioned matrices. We conclude that the mixed precision approach works very well with the parallel co-processors gaining speedup factors of four to five, and area savings of three to four, while maintaining the same accuracy as a reference solver executing everything in double precision.

A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics

We investigate a new method of solving the problem of fluid-structure interaction of an incompressible elastic object in laminar incompressible viscous flow. Our proposed method is based on a fully implicit, monolithic formulation of the problem in the arbitrary Lagrangian-Eulerian framework. High order FEM is used to obtain the discrete approximation of the problem. In order to solve the resulting systems a quasi-Newton method is applied with the linearized systems being approximated by the divided differences approach. The linear problems of saddle-point type are solved by a standard geometric multigrid with local multilevel pressure Schur complement smoothers. 1 Overview We consider the problem of viscous fluid flow interacting with an elastic body which is being deformed by the fluid action. Such a problem is encountered in many real life applications of great importance. Typical examples of this type of problem are the areas of aero-elasticity, biomechanics or material processing. For example, a good mathematical model for biological tissue could be used in such areas as early recognition or prediction of heart muscle failure, advanced design of new treatments and operative procedures, and the understanding of atherosclerosis and associated problems. Other possible applications include the development of virtual reality programs for training new surgeons or designing new operative procedures (see [12]). 1.1 Fluid structure models in biomechanics There have been several different approaches to the problem of fluid-structure interaction. Most notably these include the work of [13, 14, 15, 16] where an immersed boundary method was developed and applied to a three-dimensional model of the heart. In this model they consider a set of one-dimensional elastic fibers immersed in a three-dimensional fluid region and using a parallel supercomputer they were able to model the pulse of the heart ventricle. ⋆ This work has been supported by German Reasearch Association (DFG), Reasearch unit 493. A fluid-structure model with the wall modelled as a thin shell was used to model the left heart ventricle in [3, 4] and [18, 17]. In [7, 8] a similar approach was used to model the flow in a collapsible tube. In these models the wall is modelled by two-dimensional thin shells which can be modified to capture the anisotropy of the muscle. In reality the thickness of the wall can be significant and very important. For example in arteries the wall thickness can be up to 30% of the diameter and its local thickening can lead to the creation of an aneurysm. In the case of heart ventricle the thickness of the wall is also significant and also the direction of the muscle fibers changes through the wall. 1.2 Theoretical results The theoretical investigation of fluid structure interaction problems is complicated by the need of mixed description. While for the solid part the natural view is the material (Lagrangian) description, for the fluid it is the spatial (Eulerian) description. In the case of their combination some kind of mixed description (usually referred to as the arbitrary Lagrangian-Eulerian description or ALE) has to be used which brings additional nonlinearity into the resulting equations. In [10] a time dependent, linearized model of interaction between a viscous fluid and an elastic shell in small displacement approximation and its discretization is analyzed. The problem is further simplified by neglecting all changes in the geometry configuration. Under these simplifications by using energy estimates they are able to show that the proposed formulation is well posed and a global weak solution exists. Further they show that an independent discretization by standard mixed finite elements for the fluid and by nonconforming discrete Kirchhoff triangle finite elements for the shell together with backward or central difference approximation of the time derivatives converges to the solution of the continuous problem. In [19] a steady problem of equilibrium of an elastic fixed obstacle surrounded by a viscous fluid is studied. Existence of an equilibrium state is shown with the displacement and velocity in C and pressure in C under assumption of small data in C and domain boundaries of class C. A numerical solution of the resulting equations of the fluid structure interaction problem poses a great challenge since it includes the features of nonlinear elasticity, fluid mechanics and their coupling. The easiest solution strategy, mostly used in the available software packages, is to decouple the problem into the fluid part and solid part, for each of those parts to use some well established method of solution then the interaction is introduced as external boundary conditions in each of the subproblems. This has an advantage that there are many well tested finite element based numerical methods for separate problems of fluid flow and elastic deformation, on the other hand the treatment of the interface and the interaction is problematic. The approach presented here treats the problem as a single continuum with the coupling automatically taken care of as internal interface, which in our formulation does not require any special treatment. 2 Continuum description Let Ω ⊂ R be a reference configuration of a given body. Let Ωt ⊂ R 3 be a configuration of this body at time t. Then a one-to-one, sufficiently smooth mapping χΩ of the reference configuration Ω to the current configuration χΩ : Ω × [0, T ] 7→ R , (1) describes the motion of the body, see figure 1. The mapping χΩ depends on the choice of the reference configuration Ω which can be fixed in a various ways. Here we think of Ω to be the initial (stress-free) configuration Ω0. Thus, if not emphasized, we mean by χ exactly χΩ = χΩ0 .

Exploring weak scalability for FEM calculations on a GPU-enhanced cluster

The first part of this paper surveys co-processor approaches for commodity based clusters in general, not only with respect to raw performance, but also in view of their system integration and power consumption. We then extend previous work on a small GPU cluster by exploring the heterogeneous hardware approach for a large-scale system with up to 160 nodes. Starting with a conventional commodity based cluster we leverage the high bandwidth of graphics processing units (GPUs) to increase the overall system bandwidth that is the decisive performance factor in this scenario. Thus, even the addition of low-end, out of date GPUs leads to improvements in both performance- and power-related metrics.

A comparative study of time-stepping techniques for the incompressible Navier-Stokes equations : From fully implicit non-linear schemes to semi-implicit projection methods

We present a numerical comparison of some time-stepping schemes for the discretization and solution of the non-stationary incompressible Navier-Stokes equations. The spatial discretization is by non-conforming quadrilateral finite elements which satisfy the LBB condition. The major focus is on the differences in accuracy and efficiency between the backward Euler, Crank-Nicolson and fractional-step θ schemes used in discretizing the momentum equations. Further, the differences between fully coupled solvers and operator-splitting techniques (projection methods) and the influence of the treatment of the nonlinear advection term are considered. The combination of both discrete projection schemes and non-conforming finite elements allows the comparison of schemes which are representative for many methods used in practice. On Cartesian grids this approach encompasses some well-known staggered grid finite difference discretizations too. The results which are obtained for several typical flow problems are thought to be representative and should be helpful for a fair rating of solution schemes, particularly in long-time simulations.

Swimming by reciprocal motion at low Reynolds number

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.

Using GPUs to improve multigrid solver performance on a cluster

This paper explores the coupling of coarse and fine-grained parallelism for Finite Element (FE) simulations based on efficient parallel multigrid solvers. The focus lies on both system performance and a minimally invasive integration of hardware acceleration into an existing software package, requiring no changes to application code. Because of their excellent price performance ratio, we demonstrate the viability of our approach by using commodity Graphics Processing Units (GPUs), addressing the issue of limited precision on GPUs by applying a mixed precision, iterative refinement technique. Our results show that we do not compromise any software functionality and gain speedups of two and more for large problems.