J. Billingham
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View article: The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 2. The initial-boundary value problem on a finite domain
The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 2. The initial-boundary value problem on a finite domain Open
In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely the nonlocal Fisher-KPP equation in one spatial dimension, \begin{equation*} u_t = D u_{xx} + u(1-\phi *u),…
View article: A Shape-Newton Method for Free-Boundary Problems Subject to the Bernoulli Boundary Condition
A Shape-Newton Method for Free-Boundary Problems Subject to the Bernoulli Boundary Condition Open
We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the nonlinear Bernoulli equation. The method is a Newton-like scheme that employs shape derivatives of the…
View article: The 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 3. The effect of perturbations in the kernel
The 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 3. The effect of perturbations in the kernel Open
In the third part of this series of papers, we address the same Cauchy problem that was considered in part 1, namely the nonlocal Fisher-KPP equation in one spatial dimension, $u_t = D u_{xx} + u(1-ϕ_T*u)$, where $ϕ_T*u$ is a spatial convo…
View article: The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line
The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line Open
We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, \begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*} where $\phi *u$ is a spatial convolution with the top hat kernel, …
View article: Lubrication flow in grinding
Lubrication flow in grinding Open
In the machining process known as grinding, fluid is applied to regulate the temperature of the workpiece and reduce the risk of expensive thermal damage. The factors that influence the transport of this grinding fluid are not well underst…
View article: The inverse heat placement problem in metal additive manufacturing: Why a rational approach is needed for powder bed fusion
The inverse heat placement problem in metal additive manufacturing: Why a rational approach is needed for powder bed fusion Open
Powder bed fusion (PBF) uses an energy beam to scan a powder bed surface, heat it locally, and consolidate the material to form a part. The choice of energy beam paths is typically based on user experience. Simple beam path strategies (e.g…
View article: Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth
Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth Open
We employ the multiphase, moving boundary model of Byrne et al. (2003, Appl. Math. Lett., 16, 567–573) that describes the evolution of a motile, viscous tumour cell phase and an inviscid extracellular liquid phase. This model comprises two…
View article: Efficient and Scalable Inverse Kinematics for Continuum Robots
Efficient and Scalable Inverse Kinematics for Continuum Robots Open
With their flexible nature, continuum robots offer hyper-redundancy regarding their workspace; their backbone can take many shapes upon a single tip position and orientation. Deciphering which backbone shape to use under certain conditions…
View article: A Shape-Newton Method for Free-boundary Problems Subject to The Bernoulli Boundary Condition
A Shape-Newton Method for Free-boundary Problems Subject to The Bernoulli Boundary Condition Open
We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the Bernoulli equation. The Newton-like scheme is developed by employing shape derivatives in the weak for…
View article: A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem
A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem Open
High demand for components with complex geometries at macro and micro levels drives the development of additive manufacturing (AM). However, the scientific basis for designing energy beam scanning strategies (e.g. beam scanning speed, beam…
View article: The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics
The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics Open
In this paper, we consider the classical Riemann problem for a generalized Burgers equation, with a spatially dependent, nonlinear sound speed, with , which decays algebraically with increasing distance from a fixed spatial origin. When , …
View article: Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth
Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth Open
We analyse a multiphase, moving boundary model that describes solid tumour growth. We consider the evolution of a motile, viscous cell phase and an inviscid extracellular liquid phase. The model comprises two partial differential equations…
View article: Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near local limit
Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near local limit Open
We study non-negative travelling wave solutions, u ≡ U ( x − ct ) with constant wavespeed c > 0, of the cubic nonlocal Fisher-KPP equation in one spatial dimension, namely, …
View article: On modelling coolant penetration into the microchannels at the tool-workpiece interface
On modelling coolant penetration into the microchannels at the tool-workpiece interface Open
A network of microchannels is formed at the interface between the cutting tool and the workpiece during machining due to their rough surface structures. The penetration of coolant into these microchannels has a great effect on the machined…
View article: Slugging in horizontal channel flow
Slugging in horizontal channel flow Open
The stratified flow of gas–liquid mixtures is important in many industries, in particular oil and gas, where the formation of liquid slugs is usually undesirable. In this paper, we analyse the hydraulic model for turbulent, two-layer, gas–…
View article: Non-local effects on travelling waves arising in a moving-boundary reaction–diffusion model
Non-local effects on travelling waves arising in a moving-boundary reaction–diffusion model Open
We examine travelling wave solutions of the partial differential equation u t = u xx + u (1 − u ∗ ϕ ) on a moving domain x ⩽ L ( t ), where u ∗ ϕ is the spatial convolution of the population density with a kernel ϕ ( y ). We provide asympt…
View article: Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near locl limit
Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near locl limit Open
Figures for paper
View article: Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near locl limit
Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near locl limit Open
Figures for paper
View article: A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem
A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem Open
Codes for the paper RSPA-2022-0386.R1 entitled "A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem." Codes include raster and spiral beam path cases, considering beam power, speed, and p…
View article: A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem
A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem Open
Codes for the paper RSPA-2022-0386.R1 entitled "A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem." Codes include raster and spiral beam path cases, considering beam power, speed, and p…
View article: A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem
A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem Open
Codes for the paper RSPA-2022-0386.R1 entitled "A rational approach to beam path planning in additive manufacturing: the inverse heat placement problem." Codes include raster and spiral beam path cases, considering beam power, speed, and p…
View article: Slow travelling wave solutions of the nonlocal Fisher-KPP equation
Slow travelling wave solutions of the nonlocal Fisher-KPP equation Open
© 2020 IOP Publishing Ltd & London Mathematical Society. We study travelling wave solutions, u = U(x - ct), of the nonlocal Fisher- KPP equation in one spatial dimension, dimension, (Display equation presented), with D = 1 and c = 1, where…
View article: A dam break driven by a moving source: a simple model for a powder snow avalanche
A dam break driven by a moving source: a simple model for a powder snow avalanche Open
We study the two-dimensional, irrotational flow of an inviscid, incompressible fluid injected from a line source moving at constant speed along a horizontal boundary, into a second, immiscible, inviscid fluid of lower density. A semi-infin…
View article: Time-dependent manufacturing processes lead to a new class of inverse problems
Time-dependent manufacturing processes lead to a new class of inverse problems Open
The control of time-dependent, energy beam manufacturing processes has been achieved in the past through trial-and-error approaches. We identify key research gaps and generic challenges related to inverse problems for these processes that …
View article: The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow
The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow Open
We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressibl…
View article: The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow
The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow Open
The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration $\unicode[STIX]{x1D70E}$ ) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to t…
View article: Equilibrium solutions of three player Kuhn poker with $N>3$ cards: A new numerical method using regularization and arc-length continuation
Equilibrium solutions of three player Kuhn poker with $N>3$ cards: A new numerical method using regularization and arc-length continuation Open
We study the equilibrium solutions of three player Kuhn poker with $N>3$ cards. We compute these solutions as a function of the initial pot size, $P$, using a novel method based on regularizing the system of polynomial equations and inequa…