# Mark A. KelmansonUniversity of Leeds · Department of Applied Mathematics

Mark A. Kelmanson

## About

82

Publications

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Citations since 2017

## Publications

Publications (82)

Two novel visualisation tools on extreme flooding are considered and their appeal discussed. First, an overview of the Wetropolis Flood Demonstrator is given. This is a portable set-up that showcases what a return period of an extreme flood event is: it has been received well by the general Second, a novel graphical cost-effectiveness diagnostic ai...

Impact Case Study 2021 link: https://results2021.ref.ac.uk/impact/0ad7c1be-8e91-4aac-ab57-6c1e873cd3f1?page=1
Summary:
Researchers in the School of Mathematics have developed the “ Wetropolis” flood-outreach demonstrator to enable visualisation of the so-called return periods of extreme rainfall and flooding events. Wetropolis, initially designed...

A new, cost-effective and widely applicable tool is developed for simulating three-dimensional (3D) water-wave motion in the context of the maritime-engineering sector, with specific focus on the formation and analysis of extreme waves generated within in-house experimental wave tanks. The resulting ``numerical wave tank'' is able to emulate realis...

O. Bokhove, --- 2020: Evidence entitled "A new tool for communicating cost-effectiveness of flood-mitigation schemes":
https://committees.parliament.uk/writtenevidence/9641/pdf/
For the UK Government Department of Environment, Food and Rural Affairs Committee inquiry into flooding:
https://committees.parliament.uk/work/107/flooding
See for "full" f...

Inspired by the Boxing Day 2015 flood of the River Aire in Leeds, UK, and subsequent attempts to mitigate adverse consequences of flooding, the goals considered are: (i) to revisit the concept of flood-excess volume (FEV) as a complementary diagnostic for classifying flood events; (ii) to establish a new roadmap/protocol for assessing flood-mitigat...

As interest mounts in nature‐based solutions (NBS) for flood mitigation as complementary options to civil‐engineering measures, possible flood‐protection strategies have become more diverse and hence complicated to assess. We offer a straightforward and educational protocol targeted for effectiveness analysis and decision making involving stakehold...

A novel procedure is proposed for the a priori computation of error bounds for the ubiquitous Nyström solver applied to one-dimensional Fredholm integro-differential equations. The distinctive feature of the new approach is that the bounds are computed not only to spectral accuracy, but also explicitly, and in terms of only the numerical solution i...

This paper offers a protocol for conducting a quantified assessment of the relative merits of both existing and proposed methods of Natural Flood Management (NFM). Assessment is based on the rarely used concept of flood-excess volume (FEV), which approximately quantifies the volume of water one wishes to eliminate via flood-mitigation schemes, and...

The goals of this paper are threefold, namely to:(i) define the rarely used concept of flood-excess volume (FEV) as the flood volume above a chosen river-level threshold of flooding; (ii) show how to estimate FEV for the Boxing Day Flood of 2015 of the River Aire in the UK; and,(iii) analyse the use of FEV in evaluating a hypothetical flood-allevia...

The questions we address in the present article are the following:
(i) whether (extreme) river floods can be prevented or seriously mitigated by the introduction of beavers in the wild, and (ii) for which river catchments does flood mitigation by beaver activity (not) work? By using the concept of flood-excess volume (FEV) for four rivers in the UK...

Spectrally accurate a priori error estimates for Nyström-method approximate solutions of one-dimensional Fredholm integro-differential equations (FIDEs) are obtained indirectly by transforming the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE), which is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and...

We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them....

We present the theory underlying and computational implementation of analytical predictions of error bounds for the approximate solution of one-dimensional Fredholm integral equations of the second kind. Through asymptotic estimates of near-supremal operator norms, readily implementable formulae for the error bounds are computed explicitly using on...

Extending the authors’ recent work [15] on the explicit computation of error bounds for Nystrom solvers applied to one-dimensional Fredholm integro-differential equations (FIDEs), presented herein is a study of the errors incurred by first transforming (as in, e.g., [21]) the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE). The VFIE i...

We introduce a new model of nonlinear dispersive waves generated by wavemakers in deep water, coupled to a shallowwater model with wave breaking, modelled as hydraulics bores, in the shore zone. Coupling of deep- and shallow-water models requires the formulation of an advanced space-time technology able to stably capture the free-surface dynamics....

We present a novel approach to fluid-structure interactions (FSI) that preserves energy and phase-space structure owing to the variational and Hamiltonian techniques used. We posit a variational principle (VP), for nonlinear potential-flow wave dynamics coupled to a nonlinear hyperelastic mast, and derive its linearization. Both linear and nonlinea...

We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with (linear) elasticity, and the coupling between them....

An MoD-funded research programme based in Applied Mathematics at Leeds University has resulted in demonstrable long-term and ongoing benefits on diverse fronts for beneficiaries in a range of public and private sectors. First, by guaranteeing robustness and reliability of bespoke numerical methods for the MoD, the joint research led to substantial...

New expressions for computable error bounds are derived for Nyström method approximate solutions of one-dimensional second-kind Fredholm integral equations. The bounds are computed using only the numerical solution, and so require no a priori knowledge of the exact solution. The analysis is implemented on test problems with both well-behaved and “R...

New mechanisms are discovered regarding the effects of inertia in the transient Moffatt–Pukhnachov problem (J. Méc., vol. 187, 1977, pp. 651–673) on the evolution of the free surface of a viscous film coating the exterior of a rotating horizontal cylinder. Assuming two-dimensional evolution of the film thickness (i.e. neglecting variation in the ax...

A flexible, fully automated, computer-algebra algorithm is developed for solving a class of non-linear partial-differential evolution equations arising frequently in the modeling of two-dimensional transient free-surface viscous thin-film flows. The method, which is formulated for solving spatially periodic problems, is based upon an explicit multi...

A method is presented for improving the accuracy of the widely used Gauss-Legendre Nyström method for determining approximate solutions of Fredholm integral equations of the second kind on finite intervals. The authors' recent continuous-kernel approach is generalised in order to accommodate kernels that are either singular or of limited continuous...

New stability results for the widely studied paradigm ``rotating cylinder coating flow'' problem are found using a novel multiple-timescale asymptotic approach that is not only fully automated within an algebraic-manipulator platform, but also more widely applicable to diverse evolution equations, particularly those arising in thin-film flow on spa...

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical...

A novel pseudo-three-timescale asymptotic procedure is developed and implemented for obtaining accurate approximations to solutions of an evolution equation arising in thin-film free-surface viscous flow. The new procedure, which employs strained fast and slow timescales, requires considerably fewer calculations than its standard three-timescale co...

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh s...

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.

A rarely noted, economically unrealistic feature of Goodwins (1967; 1972) celebrated growth-cycle model is that its state variables, the wage share of output and the employment proportion, can exceed unity. We propose a novel extension of the two-variable dynamical system which ensures that its solutions remain within the economically feasible regi...

We present a simple reaction kinetics model to describe the polymer synthesis used by Lusignan et al. [Phys. Rev. E 60, 5657 (1999)] to produce randomly branched polymers in the vulcanization class. Numerical solution of the rate equations gives probabilities for different connections in the final product, which we use to generate a numerical ensem...

Synopsis We present a general algorithm for predicting the linear rheology of branched polymers. While the method draws heavily on existing theoretical understanding of the relaxation processes in entangled polymer melts, a number of new concepts are developed to handle diverse polymer architectures including branch-on-branch structures. We validat...

An arbitrary Lagrangian--Eulerian finite element method is described for the solution of time-dependent, three-dimensional, free-surface flow problems. Many flows of practical significance involve contact lines, where the free surface meets a solid boundary. This contact line may be pinned to a particular part of the solid but is more typically fre...

An arbitrary Lagrangian-Eulerian (ALE) finite element method is described for the solution of three-dimensional free-surface flow problems. The focus of this work is on extending the algorithm to include a dynamic contact line model allowing the fluid free surface, in the steady case, to form a prespecified static contact angle with a solid boundar...

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approxima...

Continuing from the work of Hinch & Kelmanson (2003 Proc. R. Soc. Lond. A459, 1193-1213), the lubrication approximation is used to investigate the drift and decay of free-surface perturbations in the viscous flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. Non-dimensional parameters correspo...

The biharmonic streamfunction is naturally employed in the complex-variable formulation of free-boundary transient problems in two-dimensional Stokes flow. In the event that analytical solutions are not obtainable, biharmonic boundary-integral methods (BBIMs) are frequently used. By using the well-known analytical solution of Hopper (J.
Fluid
Mech....

The use of boundary-conforming finite element methods is considered for the solution of surface-tension-dominated free-surface flow problems in three dimensions. This class of method is based upon the use of a moving mesh whose velocity is driven by the motion of the free surface, which is in turn determined via a kinematic boundary condition for t...

Free–surface viscous flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitationa field is considered. When the mean film thickness is small compared with the cylinder radius a, numerical simulations of the full Stokes equations reveal a surface–amplitude decay rate so slow that computational expens precludes i...

Using an adaptive finite-element (FE) scheme developed recently by the authors, we shed new light on the long-standing fundamental problem of the unsteady free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. For supportable loads, we observe that the steady-state is more read...

Presented herein is a zonal boundary element method (ZBEM) for the rapid and efficient solution of a wide class of polyelliptic boundary value problems which can be recast in integral-equation form, in domains with high aspect ratio (L⪢ 1). In contrast to the dense-matrix solution procedure of the classical BEM (CBEM), the ZBEM employs a sparse, bl...

A new method is described for the iterative solution of two-dimensional free-surface problems, with arbitrary initial geometries, in which the interior of the domain is represented by an unstructured, triangular Eulerian mesh and the free surface is represented directly by the piecewise-quadratic edges of the isoparametric quadratic-velocity, linea...

A knowledge of the complex roots λ of the transcendental eigenvalue equationsinλα=±λsinαis essential in the analysis of the slow viscous fluid flow in the neighbourhood of a sharp corner which subtends an angle α ϵ (0, 2π] to the fluid. Existing methods for finding all roots λ essentially require an a priori knowledge of the solution structure; giv...

We present a new approach to the solution of time--dependent two--dimensional free surface problems with arbitrary geometries, and describe a method for the fully automatic update of the boundary and interior meshes. For validation we consider a benchmark free--surface problem --- the Stokes--flow coalescence of two infinite parallel cylinders of e...

Presented herein is an adaptive-mesh computational method for the efficient solution of the continuum equations of compressible flow for high-velocity impact dynamics. The integral forms of the governing equations are used to derive a stable form of energy equation, using internal rather than total energy, after which the corresponding differential...

A new boundary-element method is presented for the rapid and accurate solution of viscous-flow boundary-value problems in which the inherent geometry has a high aspect ratio, R much greater than 1. As such, the method is particularly suited to the investigation of steady flow within thin-gap bearings of arbitrary geometry, in which the spatial dime...

We present a zonal boundary element method (ZBEM) for the rapid, efficient, and accurate solution of the temperature distribution within extended surfaces whose aspect ratio may be asymptotically large. Our ZBEM employs a block-tridiagonal-matrix solution technique which decouples information which is ‘far apart’ with respect to the smallest dimens...

A detailed error analysis of a regular-mcsh. 2-D Eulerian code for hyperbolic PDEs is presented herein. The code is suited to the investigation of problems in solid mechanics; in particular. of high-speed impact dynamics Explicit formulae for truncation errors are calculated for each of the provisional (pressure, stress and transport) stages of the...

Using an integral-equation method which employs the authors' new and widely applicable technique of singularity annihilation , we investigate the eddy genesis in the benchmark problem of steady viscous flow in a lid-driven cavity, two of whose opposite edges move with different velocities. The method enables us to investigate the eddy structure in...

Thin-film theory is used to derive an implicit criterion for the existence of a viscous, free-surface flow on the outer surface of a rotating circular cylinder. We estimate both the maximum fluid load supportable at a given Stokes number, and the maximum Stokes number permitting the support of a prescribed fluid load. The results of our theory are...

A numerical integral-equation method is presented for the accurate solution of viscous flow problems in which there are inherent boundary singularities. Where applicable, this method has the advantage over existing singularity subtraction and singularity incorporation methods since natural boundary conditions can be employed without any special for...

The commonly observed phenomenon of steady, viscous, free-surface flow on the outer surface of a rotating cylinder is investigated by means of an iterative, integral-equation formulation applied to the Stokes approximation of the Navier-Stokes equations. The method of solution places no restriction on the thickness of the fluid layer residing on th...

The commonly-observed phenomenon of steady, viscous, free-surface flow on the outer surface of a rotating cylinder is investigated herein. An existence criterion, based on thin-film theory, is presented and an expression derived for the maximum fluid load supportable at a given rotation rate. Results of the theory are shown to be in excellent agree...

The driven-cavity problem, a renowned bench-mark problem of computational, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive the flow: this follows from G.I. Taylor's analysis of the so-called sca...

Under the TLTP initiative, the Mathematics Departments at Imperial College and Leeds University are jointly developing a CAL method directed at supplementing the level of mathematics of students entering science and engineering courses from diverse A-level (or equivalent) backgrounds. The aim of the joint project is to maintain – even increase - th...

Under the TLTP initiative, the Mathematics Departments at Imperial College and Leeds University are jointly developing a CAL method directed at supplementing the level of mathematics of students entering science and engineering courses from diverse A‐level (or equivalent) backgrounds. The aim of the joint project is to maintain — even increase ‐ th...

A scheme is presented for the automatic generation of unstructured rectangular meshes suitable for the integration of dynamically adaptive, time dependent Eulerian codes. Our approach generates meshes quite different from those in the existing literature, which are primarily based upon block structured or grid embedding algorithms, and therefore ha...

A specially-modified boundary integral equation (BIE) method is used to investigate the viability of the singular boundary conditions of the well known driven-cavity Stokes flow problem, a bench-mark problem of computational fluid dynamics. We introduce small ‘leaks’ to replace the singularities, thus creating a perturbed, physically realizable pro...

When solving the biharmonic equation for the classical two-dimensional Stokes flow in a wedge of angle α z.ε (0, 2 π], the no-slip condition on the wedge yields the well-known and much-used transcendental eigenvalue equation where λ is a real or complex constant whose value is intimately interwoven in the construction of the solutions of the aforem...

A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of m...

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary-value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous bou...

A method is presented for assessing the nature of the error incurred in the boundary integral equation (BIE) solution of both harmonic and biharmonic boundary value problems (BVPs). It is shown to what order of accuracy the governing partial differential equation is actually represented by the approximating numerical scheme, and how raising the ord...

A numerical method is presented for the computationally efficient and accurate solution of heat transfer boundary-value problems (BVPs) within long, slender bodies (e.g., fin assemblies). The method is based on the well-known boundary integral equation (BIE) technique for the solution of steady-state two-dimensional Laplacian BVPs. Whereas the clas...

A simple blunt body sampler is considered which consists of a circular cylinder with a finite size orifice. Using this model a linear boundary integral equation (LBIE) method is employed in order to determine the potential flow past the sampler with the orifice facing the oncoming fluid. It is demonstrated that the size of the orifice is very impor...

A boundary integral equation (BIE) formulation is presented for the numerical solution of certain two dimensional nonlinear elliptic equations subject to nonlinear boundary conditions. By applying the Kirchoff transformation, all nonlinear aspects are first transferred to the boundary of the solution domain. Then the accurate solution of problems i...

This paper investigates the steady slow flow of an incompressible viscous fluid in the region between an inner circular cylinder rotating with constant angular velocity and an outer stationary cylinder of arbitrary cross section. The numerical solution technique known as the boundary integral equation method is employed in which the governing parti...

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the...

The main objective of this book has been to extend the range of application of the boundary integral equation (BIE) method and, whenever possible, to compare results with those obtained by using the more traditional “space discretisation” techniques such as finite difference (FD) and finite element (FE). The philosophy has been to gradually build u...

A biharmonic boundary integral equation (BBIE) method is used to solve a two dimensional contained viscous flow problem. In order to achieve a greater accuracy than is usually possible in this type of method analytic expressions are used for the piecewise integration of all the kernel functions rather than the more time-consuming method of Gaussian...

The mathematical formulation of many problems in physics and engineering involving rates of change with respect to two or more independent variables, leads either to a partial differential equation or to a set of such equations. These equations are supplemented by a set of prescribed boundary conditions to constitute a boundary value problem (BVP),...

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations.
The classical BBIE gives poo...

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's Theorem is used to reformulate the differential equation as a pair of coupled integral equations. The classical BBIE gives poo...

solutions of the biharmonic equation governing steady two-dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the...

A biharmonic boundary integral equation method (BBIE) is used to solve a two-dimensional contained viscous flow problem. In order to achieve a greater accuracy than is usually possible in this type of method, analytic expressions are used for the piecewise integration of all of the kernel functions rather than the more time-consuming method of Gaus...

A steady two-dimensional viscous flow through a bifurcated channel formed by two infinite parallel stationary planes separated along the channel centerline by a third semiinfinite parallel stationary plane is studied. Accurate solutions to this problem were obtained using a direct boundary integral approach, incorporating the analytic form of the s...

Presented herein is a novel and widely-applicable iterative integral equation method for the solution of 6th-order partial differential equations (PDEs); one such example arising in the theory of rotating viscous fluids is presented by way of demonstration of the method’s accuracy. The nature of the formulation permits its extension to a variety of...

This paper investigates the steady slow flow of an incompressible viscous fluid in the region between an inner circular cylinder rotating with constant angular velocity and an outer stationary cylinder of arbitrary cross section, however complex. The numerical solution technique known as the boundary integral equation method is employed in which th...

## Projects

Project (1)

A little-known fact is that, every week, two ships weighing over 100 tonnes sink in oceans, sometimes with tragic consequences. This alarming observation suggests that maritime structures may be struck by stronger waves than those they were designed to withstand. These are the legendary rogue (or freak) waves, i.e., suddenly appearing huge waves that have traumatised mariners for centuries and currently remain an unavoidable threat to ships, and to their crews and passengers. Thus motivated, an EU-funded collaboration between the Department of Applied Mathematics (Leeds University) and the Maritime Research Institute Netherlands (MARIN) supported this project, in which the ultimate goal, of importance to the international maritime sector, is to develop reliable damage-prediction tools, leading to beneficial impact in terms of both safety and costs. To understand the behaviour of rogue waves, cost-effective water-wave models are derived in both deep and shallow water. Novel mathematical and numerical strategies are introduced to capture the dynamic air-water interface and to ensure conservation of important properties. Specifically, advanced variational Galerkin finite-element methods are used to provide stable simulations of potential-flow water waves in a basin with wavemakers and seabed topography, which allows reliable simulations of rogue waves in a target area. For optimised computational speed, wave absorption is considered with a beach on which waves break and dissipate energy. Robust integrators are therefore introduced to couple the potential-flow model to shallow-water wave dynamics at the beach. Experimental validation of the numerical tank is conducted at Delft University of Technology to ensure accuracy of the simulations from the wavemaker to the beach. The numerical tank is designed for subsequent use by MARIN to investigate the damage caused by rogue waves on structures in order to update maritime design practice and to ensure safety of ships, therefore leading to a competitive commercial advantage across Europe.