Dr. Matthias Kirchhart - Vortex Methods for Incompressible Flows

AICES Graduate School, RWTH Aachen University

Abstract

Vortex methods are numerical schemes for solving the incompressible
Navier–Stokes equations. These equations accurately describe the motion
of both gases and liquids as we encounter them in everyday life, i.e.,
at velocities far below the speed of sound and not subject to extreme
temperatures or pressures. It is hard to overestimate their importance
in engineering applications, where they can for example be used to
minimise air-resistance and thereby fuel consumption of cars. However,
current numerical schemes for these equations face severe problems when
applied to turbulent flows: stringent time-step constraints,
instabilities, or the introduction of significant amounts of artificial,
spurious viscosity make their application infeasible or render the
results unusable.

Vortex methods, on the other hand, are particle methods that are based
on the vorticity formulation of the Navier–Stokes equations. This
formulation comes with two main benefits: the pressure variable is
eliminated and the system consists of separate dynamic and kinematic
parts, which can be treated independently with semi-analytical schemes.
The dynamic part is discretised using particles, which are then
convected with the flow. This natural treatment of convection renders
the method virtually free of artificial viscosity. The kinematic part of
the equations is solved using a solver based on the Biot–Savart law,
which guarantees incompressibility in the strong, point-wise sense. In
addition, in the two-dimensional case, the resulting schemes can be
shown to also conserve circulation, linear momentum, angular momentum,
and energy. These properties make vortex methods an interesting
alternative to current, mesh-based alternatives.

In this talk we will first describe a simple vortex method in the
unbounded, two-dimensional setting to illustrate the intuition of
vortex methods. We then present basic results from their analysis,
before moving on to discuss some of the problems vortex methods are
facing in three-dimensional bounded domains. We present recent research
results on one of these problems and conclude with an outlook to further
research opportunities.

References
[1] G.-H. Cottet and P. D. Koumoutsakos. Vortex Methods. Theory and
Practice. Cambridge University Press, 2000. ISBN: 0521621860.

[2] A. J. Majda and A. L. Bertozzi. Vorticity and Incompressible Flow.
Cambridge University Press, Nov. 2001. ISBN: 0521630576.

[3] M. Kirchhart and S. Obi. ‘A Smooth Partition of Unity Finite Element
Method for Vortex Particle Regularization’. In: SIAM Journal on
Scientific Computing 39.5 (Oct. 2017), pp. A2345–A2364. ISSN: 1064–8275.
DOI: 10.1137/17M1116258.

Prof. Duncan Lockerby, Ph.D. - Simulating Low-Speed Micro-Scale Gas Flows: A Method of Fundamental Solutions

School of Engineering, University of Warwick, UK

Abstract

Creeping flow (also known as Stokes flow) is flow at vanishingly small Reynolds number. This classic topic has many applications, particularly in micro- and nano-flow technology, owing to both the small scale and the low speeds encountered. In these conditions the inertia of the fluid can be neglected, and (when local-equilibirum assumptions are valid) the momentum and continuity equation reduce to the Stokes equations. A common mathematical tool for the analysis of creeping flows is a fundamental singularity solution to the Stokes equations, known as the Stokeslet (first derived by Lorentz in 1897). This fundamental solution – in essence a Green’s function for the Stokes equations – is the flow response to a Dirac delta forcing term applied to the momentum equation.

In the most straightforward use, a flow field is represented by a superposition of Stokeslets (positioned outside of the domain) that are given a combination of strengths chosen to satisfy the same number of conditions at nodes on the boundary. This approach is known as the method of fundamental solutions (MFS) or the superposition method.  The approach has the advantage of having a flow domain that is meshless, and a dimensionality that is reduced in order by one (the boundary is discretized rather than the volume). The Stokeslet is also in itself of interest, as a fundamental solution to the Stokes equations, and can be used to conveniently derive certain analytical results.

In dilute gas flows departed far from local thermodynamic equilibrium (i.e. at non negligible Knudsen numbers, such as in rarefied conditions or at the micro/nano scale) the Stokes constitutive law becomes invalid/inaccurate. As such, the Stokeslet has limited applicability in the analysis of creeping gas flows at the micro/nano scale. However, constitutive closures exist that extend the applicability of the continuum treatment to higher Knudsen numbers. Notably, Grad’s family of moment equations(and particularly their ‘regularised’ counterparts; e.g. the R13 equations due to Struchtrup and Torrilhon) have attracted significant attention in recent years.

In this talk we introduce fundamental solutions to the the linearised R13 equations for very low Reynolds number flows; equivalent to the Stokeslet, but applicable to higher Knudsen numbers. A simple numerical implementation of the method of fundamental solutions (MFS) is presented for some three-dimensional creeping flows. Incorporation of these new fundamental solutions into the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere, capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated.

Prof. Karen Willcox, Ph.D. - Data-Driven Operator Inference for Learning Physics-Based Low-Dimensional Models

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, USA

Abstract

This talk presents a non-intrusive data-driven approach for learning low-dimensional models for systems governed by time-dependent partial differential equations. Projection-based model reduction constructs the operators of a reduced model by projecting the equations of a full high-fidelity model onto a reduced space. Traditionally, this projection is intrusive, which means that the full-model operators are required explicitly in assembled form or implicitly through a routine that returns the action of the operators on a given vector; however, in many situations these full-model operators may be inaccessible. Our non-intrusive operator inference approach solves an optimization problem to infer approximations of the reduced operators directly from input and state data, without requiring the full model itself. The inferred operators are the solution of a least-squares problem and converge, with sufficient state trajectory data, in the Frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full-model operators. Thus, while the approach is data-driven, it also embeds the physical constraints associated with the underlying system governing equations. Joint work with Benjamin Peherstorfer (U. Wisconsin Madison).

Prof. Nicholas Higham, Ph.D. - Multiprecision Algorithms

Abstract

• IEEE standard arithmetic and availability in hardware and software. Motivation for low precision from applications, including machine learning. Applications requiring high precision. Simulating low precision for testing purposes. Software for high precision. Challenges of implementing algorithms in low precision.
• Basics of rounding error analysis. Examples of error analyses of algorithms, focusing on issues relating to low precision.
• Solving linear systems using mixed precision: iterative renement, hybrid direct-iterative methods. Multiprecision algorithms for matrix functions, focusing on the matrix logarithm.

Prof. Denis Noble, Ph.D. - From Pacing the Heart to the Pace of Evolution

Department of Physiology, Anatomy and Genetics, University of Oxford, England

Abstract

Multi-mechanism interpretations of cardiac pacemaker function reveal the extent to which many physiological functions are buffered against genomic change. Contrary to Schrodinger's claim in What is Life? (1944) which led to the Central Dogma of Molecular Biology (Crick 1970), biological functions at higher levels harness stochasticity at lower levels. This harnessing of stochasticity is a prerequisite for the processes by which the pace of evolution can be accelerated through guided control of mutation rates and of buffering by regulatory networks in organisms.

Schrodinger E. 1944 What is life? Cambridge, UK: Cambridge University Press.
Crick FHC. 1970 Central dogma of molecular biology. Nature 227, 561 – 563. (doi:10.1038/227561a0)
Noble D. Dance to the Tune of Life. Biological Relativity. Cambridge University Press 2016
Noble D. Evolution viewed from physics, physiology and medicine. Interface Focus 2017, 7, 20160159.
Noble R & Noble D. Was the Watchmaker Blind? Or was she One-eyed? Biology, 2017, 6, 47.