## Publications

304 results found

Gong Y, Young DJ, Atkinson C,
et al., 2020, Modelling of the degradation of martensitic stainless steels by the Boudouard reaction, *CORROSION SCIENCE*, Vol: 173, ISSN: 0010-938X

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- Citations: 5

Atkinson C, 2019, Some axially symmetric contact problems, *Journal of Engineering Mathematics*, Vol: 116, Pages: 23-28, ISSN: 0022-0833

Axisymmetric stress analysis problems are considered where mixed boundary conditions apply over the interior of a cylinder. A brief indication of some generalisations is also given.

Dunn IH, Daou R, Atkinson C, 2019, A straightforward 2 omega technique for the measurement of the Thomson effect, *Review of Scientific Instruments*, Vol: 90, ISSN: 0034-6748

We present a simplified, rapid, and accurate method for the measurement of the thermoelectric Thomson coefficient by the dynamical heating of a suspended wire by an alternating current. By applying a temperature gradient across the wire, we find that the response at the second harmonic of the excitation frequency is directly proportional to the Thomson coefficient. The absolute thermoelectric coefficient of a single material can therefore be extracted with high precision by using a phase sensitive detector. We test our method on platinum and nickel wires and develop both analytical and numerical models to determine the leading sources of error.

Akbay T, Kilner JA, Ishihara T,
et al., 2019, Explicit solution to extract self-diffusion and surface exchange coefficients from isotope back-exchange experiments, *The Journal of Physical Chemistry C*, Vol: 123, Pages: 258-264, ISSN: 1932-7447

Multistep 18O isotope exchange procedures and subsequent analytical techniques can be used to elucidate the effect of ambient gas atmospheres on the transport properties of oxide ion-conducting materials utilized in high-temperature solid oxide devices for electrochemical energy conversion. In this contribution, we provide an explicit solution to the one-dimensional transient diffusion equation to estimate oxygen self-diffusion and surface exchange coefficients of oxide ion conducting materials exposed to multistep 18O exchange procedures. Although an analytical solution exists for representing the diffusion profiles of labeled species obtained from a single-step isotope exchange procedure, it is not applicable to the diffusion profiles resulted from consecutive procedures with dynamically altered initial and surface boundary conditions. Hence, a new analytical solution is found for the diffusion problem representing the isotope back-exchange procedure in a semi-infinite spatial domain. The explicit solution is then used to determine the self-diffusion and surface exchange coefficients as fitting parameters for tracer gas diffusion profiles obtained from multistep isotope exchange experiments conducted in different oxidizing gas atmospheres. It is demonstrated that the explicit solution provides a great flexibility in analyzing the effects of oxidizing gas atmospheres on transport properties of oxide ion conducting materials.

Atkinson C, Mokkhavesa S, Ingpochai P, 2019, Strategic Asset Allocation for Life Insurers with Stochastic Liability, *EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS*, Vol: 12, Pages: 1315-1336, ISSN: 1307-5543

Barba D, Alabort E, Pedrazzini S,
et al., 2017, On the microtwinning mechanism in a single crystal superalloy, *Acta Materialia*, Vol: 135, Pages: 314-329, ISSN: 1359-6454

The contribution of a microtwinning mechanism to the creep deformation behaviour of single crystal superalloy MD2 is studied. Microtwinning is prevalent for uniaxial loading along at for the stress range 625 to MPa and for 625 MPa. Using quantitative stereology, the twin fraction and twin thickness are estimated; this allows the accumulated creep strain to be recovered, in turn supporting the role of the microtwinning mode in conferring deformation. Atom probe tomography confirms the segregation of Cr and Co at the twin/parent interface, consistent with the lowering of the stacking fault energy needed to support twin lengthening and thickening. A model for diffusion-controlled growth of twins is proposed and it is used to recover the measured creep strain rate. The work provides the basis for a thermo-mechanical constitutive model of deformation consistent with the microtwinning mechanism.

Atkinson C, quek G, 2017, Portfolio selection in discrete time with transaction costs and power utility function: a perturbation analysis, *Applied Mathematical Finance*, Vol: 24, Pages: 77-111, ISSN: 1350-486X

In this article, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.

Safarov N, Atkinson C, 2017, Natural gas-fired power plants valuation and optimization under levy copulas and regime switching, *International Journal of Theoretical and Applied Finance*, Vol: 20, ISSN: 0219-0249

In this work, we analyze a stochastic control problem for the valuation of a natural gas power station while taking into account operating characteristics. Both electricity and gas spot price processes exhibit mean-reverting spikes and Markov regime-switches. The Lévy regime-switching model incorporates the effects of demand-supply fluctuations in energy markets and abrupt economic disruptions or business cycles. We make use of skewed Lévy copulas to model the dependence risk of electricity and gas jumps. The corresponding coupled Hamilton–Jacobi–Bellman (HJB) equations are solved by an explicit finite difference method. The numerical approach gives us both the value of the plant and its optimal operating strategy depending on the gas and electricity prices, current temperature of the boiler and time. The surfaces of control strategies and contract values are obtained by implementing the numerical method for a particular example.

Atkinson C, safarov N, 2016, Natural gas-fired power plants valuation and optimisation under Levy copulas and regime-switching, Publisher: SSRN

In this work we analyse a stochastic control problem for the valuation of a natural gas power station while taking into account operating characteristics. Both electricity and gas spot price processes exhibit mean-reverting spikes and Markov regime-switches. The Levy regime-switching model incorporates the effects of demand-supply fluctuations in energy markets and abrupt economic disruptions or business cycles. We make use of skewed Levy copulas to model the dependence risk of electricity and gas jumps. The corresponding HJB equation is the non-linear PIDE which is solved by an explicit finite difference method. The numerical approach gives us both the value of the plant and its optimal operating strategy depending on the gas and electricity prices, current temperature of the boiler and time. The surfaces of control strategies and contract values are obtained by implementing the numerical method for a particular example.

Atkinson C, Sherwood JD, 2016, Added mass of a pair of discs at small separation, *European Journal of Applied Mathematics*, Vol: 28, Pages: 687-706, ISSN: 1469-4425

Inviscid irrotational flow around a pair of coaxial disks is considered in the limit in which thedistance2hbetween the disks is small compared to their radiusa. The disks have zero thickness andaccelerate away from one another along their common axis. The added massMof each acceleratingdisk is increased by the presence of the other disk. Analyticpredictions are obtained whenh/a≪1,withM∼πa/(8h)−ln(h/a)/2 + 0.77875 +· · ·. The termO(a/h)can be obtained by meansof an inviscid analysis of approximately unidirectional flow within the gap between the disks, butthe correction terms have not been reported previously. Theirrotational flow problem satisfies Neu-mann boundary conditions on the surface of the disks, but is otherwise analogous to the Dirichletproblem of the capacitance of a pair of charged disks, which has been the subject of much study andcontroversy.

Atkinson C, 2016, A simple model of dynamic cleavage producing constant speed crack propagation in a sample of finite width, *SIAM Journal on Applied Mathematics*, Vol: 76, Pages: 1344-1354, ISSN: 0036-1399

A simple model of dynamic cleavage under longitudinal shear deformation is analyzed as an attempt to understand how ancient lapidaries might have produced smooth cleaved surfaces. Attention is concentrated on the time signature of the load applied to the specimen in order to maintain a constant stress intensity factor (or energy release rate) at the crack tip as it propagates away from the location of the applied load at a constant speed. The analysis is also applicable to situations in which fracture requires a time dependent energy release rate. An example of the cleavage of diamond is considered.

Atkinson C, papanastasiou P, papamichos E, 2016, Hydraulic fracturing in CO2 geological storage, ARMA 50th US Rock Mechanics/Geomechanics Symposium, Publisher: American Rock Mechanics Association

Among the outstanding technical issues in CO2 geological storage is the risk of hydraulic fracturing and migration of CO2 in upper formations and to the atmosphere, the wellbore CO2 injectivity and the storage capacity of the formation. We present a contribution on these issues based on conclus ions of the modelling work of hydraulic fracturing in weak formations. The work was initially carried out with finite element analysis of a fully coupled elastoplastic hydraulic fracturing model and l ater was extended to analytical work based on a Mohr-Coulomb dislocation model where the complete slip process that is distributed around the crack tip was replaced by superdislocations that are placed in the effective centers of plastic deformation. Scaling of the FEM and analytical results enables the identification of a dominant parameter which defines the regimes of brittle to ductile propagation and the limit at which a mode-1 fracture cannot advance. We found that a hydraulically induced vertical fracture from CO2 injection is more likely to propagate horizontally than vertically, remaining contained in the storage zone. The horizontal fracturepropagation will have a positive effect on the injectivity and storage capacity of the formation.

Atkinson C, Huang R, 2016, A theoretical model for capacitance measurement of liquid films in an annular flow, *Mathematics-in-Industry Case Studies*, Vol: 7, ISSN: 1913-4967

This paper develops mathematical models for a proposed production logging tool, which uses capacitance sensors to measure liquid annulus thickness and liquid type for a primarily gas-carrying wellbore. A semi-analytical method is devised which uses Fourier analysis to account for the complicated system geometry and reduces the problem to singular integral equations over a sensor electrode. Approximate solutions of these integral equations give expressions for capacitance, which is calculated for different liquids and varying liquid annuli thicknesses and electrode positions. These results are compared with capacitance values found using COMSOL Multiphysics, a finite element package. Good agreement is obtained between these two methods, with a discrepancy range of 1–8 %. Capacitance values obtained range from 32 to 93 pF/m, and sensitivity is estimated to range from 0.8 to 60 %, depending mainly on the proximity of the electrode to the liquid and also on the permittivity and thickness of the liquid layer. Practical implications of the study are also discussed. Realistically useful measurements, where ~0.2 mm of liquid is detectable, can potentially be made using an electrode 4–6 mm away from the inner wellbore wall.

Safarov N, Atkinson C, 2016, Natural gas storage valuation and optimization under time-inhomogeneous exponential L´evy processes., *International Journal of Computer Mathematics*, Vol: 94, Pages: 2147-2165, ISSN: 1029-0265

We analyse a stochastic control problem for the valuation of a natural gas storage facility while takinginto account operating characteristics. The underlying natural gas spot price dynamics is assumedto follow a time-inhomogeneous exponential L´evy process [6]. This underlying incorporates commonfeatures of gas spot prices such as seasonality, mean-reversion and price spikes with seasonal jumpintensities. The corresponding HJB equation is the non-linear PIDE which is solved by an explicitfinite difference method. The numerical approach gives us both the value of the storage contract andoptimal operating strategy depending on the gas price, current level of gas in the storage and time.The effects of seasonality and seasonal jump intensity are investigated by observing the surfaces ofcontrol strategies and contract values.

Papanastasiou P, Papamichos E, Atkinson C, 2016, On the risk of hydraulic fracturing in CO2 geological storage, *International Journal for Numerical and Analytical Methods in Geomechanics*, Vol: 40, Pages: 1472-1484, ISSN: 1096-9853

We present a contribution on the risk of hydraulic fracturing in CO2 geological storage using an analytical model of hydraulic fracturing in weak formations. The work is based on a Mohr–Coulomb dislocation model that is extended to account for material with fracture toughness. The complete slip process that is distributed around the crack tip is replaced by superdislocations that are placed in the effective centers. The analytical model enables the identification of a dominant parameter, which defines the regimes of brittle to ductile propagation and the limit at which a mode-1 fracture cannot advance. We examine also how the corrosive effect of CO2 on rock strength may affect hydraulic fracture propagation. We found that a hydraulically induced vertical fracture from CO2 injection is more likely to propagate horizontally than vertically, remaining contained in the storage zone. The horizontal fracture propagation will have a positive effect on the injectivity and storage capacity of the formation. The containment in the vertical direction will mitigate the risk of fracturing and migration of CO2 to upper layers and back to the atmosphere. Although the corrosive effect of CO2 is expected to decrease the rock toughness and the resistance to fracturing, the overall decrease of rock strength promotes ductile behavior with the energy dissipated in plastic deformation and hence mitigates the mode-1 fracture propagation.

Atkinson C, Sarris E, Gravanis E,
et al., 2016, On certain singular integral equations arising in the analysis of wellbore recharge in anisotropic formations, *Applied Mathematical Modelling*, Vol: 40, Pages: 343-362, ISSN: 1872-8480

The problem of determining the steady state pressure field for single and multi-well configurations with non-trivial wellbore boundary conditions is considered in this work as an integral equation problem. The aquifer, where the well configuration resides, is assumed to have different vertical and horizontal intrinsic permeabilities and it is bounded above and below by impermeable geological settings. The solutions of the integral equation, known as density functions, are studied from two points of view. First, the singular behavior of the density function is investigated by studying the singular part of the kernel of the integral equation; on this basis the density function is suitably expressed in terms of a non-singular counterpart, the reduced density function, for which a polynomial approximation is formulated and constructed numerically. The convergence of the approximation is studied with respect to the order of the polynomial and shown to be adequately fast. Second, the density functions for large depth to radius ratio and/or large horizontal to vertical permeability ratio exhibit similarity. The density functions depend on the parameters of the single-well problem through a single similarity parameter c. For large values of c , corresponding to the physical limits just mentioned, the density functions are reduced to essentially a single function, modulo a factor View the MathML source. This property simplifies considerably the analysis of all the large c cases. Considering the case of two wells, as an illustrative example, we also show that the properties of the single well case are also exhibited by the multi-well density functions.

Atkinson C, 2015, A remark on non-local theories of elasticity, piezoelectric materials etc, *International Journal of Engineering Science*, Vol: 97, Pages: 95-97, ISSN: 0090-6913

A reminder is made of difficulties with applications of non local theories (which represent stress as a weighted integral of strain) to generating finite stresses at line crack tips.

Papanastsiou P, Atkinson C, 2015, The brittleness index in hydraulic fracturing, 49th US Rock Mechanics / Geomechanics Symposium, Publisher: ARMA

We present a new definition of a brittleness index which is used as a criterion for candidate selection of rockintervals for hydraulic fracturing. The new index is a combination of material strength parameters and insitu stresses. It was derivedfrom an analytical model of hydraulic fracturing in weak formations of varying ductility. The model is based on Mohr-Coulombdislocations that are placed in the effective centres of the complete slip process that is distributed around the crack tip. The newbrittleness index varies between 0 and 1 with the one limit to correspond to brittle propagation and the other limit to a fracture thatrequires infinite energy release per unit advance. The values between 0 and 1 correspond to fracture propagation of increasingductility from brittle to small scale and finally to large scale yielding. The results are particularly interesting for predicting thepropagation of axial fractures in the horizontal direction and their confinement in the vertical direction.

Atkinson C, Coman CD, Aldazabal J, 2015, Couple stresses and the fracture of rock, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 373, ISSN: 1364-503X

An assessment is made here of the role played by the micropolar continuum theory on the cracked Brazilian disc test used for determining rock fracture toughness. By analytically solving the corresponding mixed boundary-value problems and employing singular-perturbation arguments, we provide closed-form expressions for the energy release rate and the corresponding stress-intensity factors for both mode I and mode II loading. These theoretical results are augmented by a set of fracture toughness experiments on both sandstone and marble rocks. It is further shown that the morphology of the fracturing process in our centrally pre-cracked circular samples correlates very well with discrete element simulations.

Atkinson C, Martineau PM, Khan RUA,
et al., 2015, Cleaving the Halqeh-ye-nur diamonds: a dynamic fracture analysis, *PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 373, ISSN: 1364-503X

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- Citations: 2

Pereira PJS, Atkinson C, 2014, Invariant integrals applied to nematic liquid crystals with small Ericksen number and topological defects, *INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE*, Vol: 76, Pages: 12-26, ISSN: 0020-7225

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- Citations: 1

Hill J, Atkinson C, 2013, The Acoustic Analysis of a Partially Filled Pipe, International Conference on Industrial Engineering and Management Science (ICIEMS), Publisher: DESTECH PUBLICATIONS, INC, Pages: 338-344

Atkinson C, Martinez-Esnaola JM, Elizalde MR, 2012, Contact mechanics: a review and some applications, *MATERIALS SCIENCE AND TECHNOLOGY*, Vol: 28, Pages: 1079-1091, ISSN: 0267-0836

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- Citations: 13

Atkinson C, 2012, Moving boundary problems for time fractional and composition dependent diffusion, *FRACTIONAL CALCULUS AND APPLIED ANALYSIS*, Vol: 15, Pages: 207-221, ISSN: 1311-0454

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- Citations: 17

Atkinson C, Quek G, 2012, Dynamic Portfolio Optimization in Discrete-Time with Transaction Costs, *Applied Mathematical Finance*, Vol: 19, Pages: 265-298, ISSN: 1350-486X

Abstract: A discrete-time model of portfolio optimization is studied under the effects of proportional transaction costs. A general class of underlying probability distributions is assumed for the returns of the asset prices. An investor with an exponential utility function seeks to maximize the utility of terminal wealth by determining the optimal investment strategy at the start of each time step. Dynamic programming is used to derive an algorithm for computing the optimal value function and optimal boundaries of the no-transaction region at each time step. In the limit of small transaction costs, perturbation analysis is applied to obtain the optimal value function and optimal boundaries at any time step in the rebalancing of the portfolio. Abstract: A discrete-time model of portfolio optimization is studied under the effects of proportional transaction costs. A general class of underlying probability distributions is assumed for the returns of the asset prices. An investor with an exponential utility function seeks to maximize the utility of terminal wealth by determining the optimal investment strategy at the start of each time step. Dynamic programming is used to derive an algorithm for computing the optimal value function and optimal boundaries of the no-transaction region at each time step. In the limit of small transaction costs, perturbation analysis is applied to obtain the optimal value function and optimal boundaries at any time step in the rebalancing of the portfolio.

Zhang L, Atkinson C, 2012, On some integral equations arising in the pressure analysis of inclined wells in anisotropic formations, *INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE*, Vol: 50, Pages: 198-215, ISSN: 0020-7225

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- Citations: 2

atkinson C, Ingpochai P, 2011, The Effect of Correlation andTransaction Costs on the Pricing ofBasket Options, *Applied Mathematical Finance*

ABSTRACT In this article, we examine the problem of evaluating the option price of a Europeancall option written on N underlying assets when there are proportional transaction costs in the market.Since the portfolio under consideration consists of multiple risky assets, which makes numericalmethods formidable, we use perturbation analyses. The article extends the model for option pricingon uncorrelated assets, which was proposed by Atkinson and Alexandropoulos (2006; Pricing aEuropean basket option in the presence of proportional transaction cost, Applied MathematicalFinance, 13(3), pp. 191–214). We determine optimal hedging strategies as well as option priceson both correlated and uncorrelated assets. The option valuation problem is obtained by comparingthe maximized utility of wealth with and without option liability. The two stochastic controlproblems, which arise from the transaction costs, are transformed to free boundary and partial differentialequation problems. Once the problems have been formulated, we establish optimal tradingstrategies for each of the portfolios. In addition, the optimal hedging strategies can be found bycomparing the trading strategies of the two portfolios. We provide a general procedure for solvingN risky assets, which shows that for ‘small’ correlations the N asset problem can be replacedby N (N − 1)/2 two-dimensional problems and give numerical examples for the two risky assetsportfolios.

atkinson C, 2011, developing a flow control system for a well, us 8025072 b2

To develop a flow control system for use in a well ............

Atkinson C, Osseiran A, 2011, DISCRETE-SPACE TIME-FRACTIONAL PROCESSES, *Fract. Calc. Appl. Anal.*, Vol: Vol. 14, No 2 (2011), Pages: 201-232

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