# Events

# I³MS - Lockerby Seminar

## 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.

# Willcox Seminar

## 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).

# EU Regional School - Higham Seminar

## 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.

# CHARLEMAGNE DISTINGUISHED LECTURE SERIES - Noble Seminar

## 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.

# I³MS - Kiendl Seminar

## Dr. Josef Kiendl - Isogeometric Methods in Structural Analysis

Department of Marine Technology, Norwegian University of Science and Technology, Norway

### Abstract

Isogeometric analysis is a novel method of computational analysis where functions used to describe geometries in Computer Aided Design (CAD) are adopted as basis for analysis. Due to this unified geometric representation, the model transfer from design to analysis, called mesh generation, is omitted providing a better integration of design and analysis. NURBS are the most widespread technology in today’s CAD modeling tools and therefore are adopted as basis functions for analysis. Apart from the geometrical advantages, NURBS-based isogeometric analysis has proven superior approximation properties compared to standard finite element analysis for many different applications. Furthermore, the high continuity between elements also allows the discretization of higher order PDEs, which is especially useful in structural mechanics, where the classical plate and shell theories, based on Kirchhoff’s kinematic assumption, can be implemented in a straightforward way.

We show an isogeometric shell analysis framework with formulations ranging from linear, geometrically nonlinear, and fully nonlinear shell models. All formulations are based on the Kirchhoff-Love shell theory and are rotation-free, i.e., using only displacement degrees of freedom. These formulations are then employed for the simulation of various problems of structural mechanics, including large deformations, buckling, elastoplasticity, and brittle fracture as well as for fluid-structure-interaction problems including high-fidelity FSI simulations of offshore wind turbine blades and bioprosthetic heart valves.

A further development of IGA are isogeometric collocation methods (IGA-C), where the partial differential equations are solved in the strong form. This avoids the need of computing integrals by numerical quadrature and, thus, reduces the computational costs by several orders. We show isogeometric collocation formulations for different problems in structural mechanics, like spatial beams, plates, and shells.