## Publications

74 results found

Duan Y, North Ridao M, Eaton M,
et al., 2022, Non-intrusive semi-analytical uncertainty quantification using Bayesian quadrature with application to CFD simulations, *International Journal of Heat and Fluid Flow*, Vol: 93, Pages: 1-17, ISSN: 0142-727X

To improve the safety, reliability, and performance of complex engineering systems, it is crucial to understand and quantify uncertainties. This paper presents a framework to non-intrusively and semi-analytically quantify the parametric uncertainty within CFD simulations using Bayesian quadrature (BQ). An in-house uncertainty quantification (UQ) code based upon this mathematical framework is developed. The code is then validated by applying it to quantify the uncertainty due to a varying parameter in a simple analytical test function. The mean and variance obtained using BQ are compared with those obtained from the analytical solution and stochastic simulation using the Latin hypercube sampling (LHS) method. The validation test case shows that BQ outperforms the LHS approach in terms of computational efficiency and accuracy. The UQ code is then utilised to characterise the uncertainty (due to the unknown inlet flow profile) of CFD predicted operating parameters of an industrial scale butterfly valve, as well as the uncertainties (due to the unknown high-wavenumber damping factor ) of a SAS-SST simulated bluff-body flow. It is found that the entry flow profile presents non-ignorable effects on the valve operating parameters. Meanwhile, the variance of the valve operating parameters changes with the valve opening. For the bluff-body flow, large variances of predicted flow properties exist in the region where the separate shear layer dominates because of varying. Moreover, the effect of is more significant on the turbulence quantities, as it acts on the generation of turbulent eddies directly.

Duan Y, Ahn JS, Eaton MD,
et al., 2021, Quantification of the uncertainty within a SAS-SST simulation caused by the unknown high-wavenumber damping factor, *Nuclear Engineering and Design*, Vol: 381, Pages: 1-12, ISSN: 0029-5493

This paper aims to quantify the uncertainty in the SAS-SST simulation of a prism bluff-body flow due to varyingthe higher-wavenumber damping factor (Cs). Instead of performing the uncertainty quantification on the CFDsimulation directly, a surrogate modelling approach is adopted. The mesh sensitivity is first studied and thenumerical error due to the mesh is approximated accordingly. The Gaussian processes/Kriging method is used togenerate surrogate models for quantities of interest (QoIs). The suitability of the surrogate models is assessedusing the leave-one-out cross-validation tests (LOO-CV). The stochastic tests are then performed using the crossvalidated surrogate models to quantify the uncertainty of QoIs by varying Cs. Four prior probability densityfunctions (such as U(0, 1), N(0.5, 0.12), Beta(2, 2) and Beta(5, 1.5)) of Cs are considered.It is demonstrated in this study that the uncertainty of a predicted QoI due to varying Cs is regionallydependent. The flow statistics in the near wake of the prism body are subject to larger variance due to theuncertainty in Cs. The influence of Cs rapidly decays as the location moves downstream. The response of differentQoIs to the changing Cs varies greatly. Therefore, the calibration of Cs only using observations of one variablemay bias the results. Last but not least, it is important to consider different sources of uncertainties within thenumerical model when scrutinising a turbulence model, as ignoring the contributions to the total error may leadto biased conclusions.

Duan Y, Eaton MD, Bluck MJ, 2021, Fixed inducing points online Bayesian calibration for computer models with an application to a scale-resolving CFD simulation, *Journal of Computational Physics*, Vol: 434, Pages: 1-14, ISSN: 0021-9991

This paper proposes a novel fixed inducing points online Bayesian calibration (FIPO-BC) algorithm to efficiently learn the model parameters using a benchmark database. The standard Bayesian calibration (STD-BC) algorithm provides a statistical method to calibrate the parameters of computationally expensive models. However, the STD-BC algorithm does not scale well with regard to the number of data points and also it lacks an online learning capability. The proposed FIPO-BC algorithm greatly improves the computational efficiency of the algorithm and, in addition, enables online calibration to be performed by executing the calibration on a set of predefined inducing points.To demonstrate the procedure of the FIPO-BC algorithm, two tests are performed, finding the optimal value and exploring the posterior distribution of 1) the parameter in a simple function, and 2) the high-wave number damping factor in a scale-resolving turbulence model (scale adaptive simulation shear-stress transport model/SAS-SST). The results (such as the calibrated model parameter and its posterior distribution) of FIPO-BC with different inducing points are compared to those of STD-BC. It is found that FIPO-BC and STD-BC can provide very similar results, once the predefined set of inducing points in FIPO-BC is sufficiently fine. Given that fewer datapoints are needed in the proposed FIPO-BC algorithm, compared to the STD-BC algorithm, it will be a more computational efficient algorithm. In our demonstration test cases, the proposed FIPO-BC algorithm is at least ten times faster than the STD-BC algorithm. Meanwhile, the online feature of the FIPO-BC allows continuous updating of the calibration outputs and potentially reduces the workload on generating the database.

Giustini G, Kim H, I Issa R,
et al., 2020, Modelling microlayer formation in boiling sodium, *Fluids*, Vol: 5, Pages: 1-19, ISSN: 2311-5521

During boiling at a solid surface, it is often the case that a liquid layer of a few microns of thickness (’microlayer’) is formed beneath a bubble growing on the heated surface. Microlayers have been observed forming beneath bubbles in various transparent fluids, such as water and refrigerants, subsequently depleting due to evaporation, thus contributing significantly to bubble growth and possibly generating the majority of vapor in a bubble. On the other hand, boiling of opaque fluids, such as liquid metals, is not amenable to optical observations, and microlayers have not yet been observed in liquid metals. Among that class of fluids is sodium, suitable as a coolant for nuclear reactors and as the working fluid in phase-change solar power receivers. In order to support these applications, it is necessary to understand the boiling behavior of sodium and identify the parameters that might influence microlayer formation during boiling of this important fluid. This paper presents simulations of the hydrodynamics of sodium vapor bubble growth at a surface. An interface capturing flow solver has been implemented in the OpenFOAM code and used to predict the behavior of a sodium vapor bubble near a solid surface in typical boiling conditions. The methodology has been validated using recently reported direct experimental observations of microlayer formation in water and then applied to sodium boiling cases. Simulations indicate that microlayers are formed in sodium in a similar fashion to water. Comparison of simulation results with an extant algebraic model of microlayer formation showed good agreement, which increases confidence in the current predictions of microlayer formation. Typical values of microlayer thickness thus computed indicate that the microlayer is likely to play an important role during bubble growth in sodium.

Georgaka S, Stabile G, Star K,
et al., 2020, A hybrid reduced order method for modelling turbulent heat transfer problems, *COMPUTERS & FLUIDS*, Vol: 208, ISSN: 0045-7930

Ahn JS, Bluck M, 2020, Isogeometric analysis of the time-dependent incompressible MHD equations, *International Journal of Computational Fluid Dynamics*, Vol: 34, Pages: 226-248, ISSN: 1029-0257

This paper presents an isogeometric (IGA) solver for time-dependent, incompressible magnetohydrodynamics (MHD). In this paper, a combination of inf-sup stable mixed discretisations is considered to discretise the hydrodynamic pair (i.e. velocity and pressure) and magnetic pair (i.e. magnetic field and magnetic pressure). The one step θ-method is used for the temporal discretisation. Manufactured and analytical solutions are considered to assess convergence, accuracy and performance of the IGA solver. Moreover, benchmark problems involving a MHD flow over an obstacle and MHD flow under non-uniform magnetic field are considered to examine the effect of the magnetic field on the flow behaviour.

Georgaka S, Stabile G, Rozza G,
et al., 2020, Parametric POD-Galerkin model order reduction for unsteady-state heat transfer problems, *Communications in computational physics*, Vol: 27, Pages: 1-32, ISSN: 1815-2406

A parametric reduced order model based on proper orthogonal decomposition with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power reactor cooling systems. Thermal mixing of different temperature coolants in T-junction pipes leads to temperature fluctuations and this could potentially cause thermal fatigue in the pipe walls. The novelty of this paper is the development of a parametric ROM considering the three dimensional, incompressible, unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume regime. Two different parametric cases are presented in this paper: parametrization of the inlet temperatures and parametrization of the kinematic viscosity. Different training spaces are considered and the results are compared against the full order model. The first test case results to a computational speed-up factor of 374 while the second test case to one of 211.

Duan Y, Cooling C, Ahn JS,
et al., 2019, Using a Gaussian process regression inspired method to measure agreement between the experiment and CFD simulations, *International Journal of Heat and Fluid Flow*, Vol: 80, ISSN: 0142-727X

This paper presents a Gaussian process regression inspired method to measure the agreement between experiment and computational fluid dynamics (CFD) simulation. Because of misalignments between experimental and numerical outputs in spatial or parameter space, experimental data are not always suitable for quantitative assessing the numerical models. In this proposed method, the cross-validated Gaussian process regression (GPR) model, trained based on experimental measurements, is used to mimic the measurements at positions where there are no experimental data. The agreement between an experiment and the simulation is mimicked by the agreement between the simulation and GPR models. The statistically weighted square error is used to provide tangible information for the local agreement. The standardised Euclidean distance is used for assessing the overall agreement.The method is then used to assess the performance of four scale-resolving CFD methods, such as URANS k-ω-SST, SAS-SST, SAS-KE, and IDDES-SST, in simulating a prism bluff-body flow. The local statistically weighted square error together with standardised Euclidean distance provide additional insight, over and above the qualitative graphical comparisons. In this example scenario, the SAS-SST model marginally outperformed the IDDES-SST and better than the other two other, according to the distance to the validated GPR models.

Ahn J, Bluck M, 2019, Isogeometric analysis of the steady-state incompressible mhd equations, *SIAM Journal on Scientific Computing*, Vol: 41, Pages: B396-B424, ISSN: 1064-8275

This paper presents an isogeometric (IGA) solver for steady-state incompressiblemagnetohydrodynamics (MHD). MHD is the study of the behavior of electrically conducting fluidsand can be viewed mathematically as a coupled system: the Navier--Stokes equations (for the fluid)and a reduced form of Maxwell's equations (for the electromagnetic field). A key feature of MHDflow is the potential development of very strong shear, usually in proximity to walls. This resultsin two correlated demands on numerical simulation: the need to represent the geometry and thenear-wall shear accurately. In addition, for both the Navier--Stokes and the Maxwell's equations,appropriate discretizations are required for the problem to be well-posed. IGA analysis is a variant ofthe conventional finite element (FE) method, but utilizing the underlying approximations commonlyused in computer-aided design (CAD) to represent geometry, basis functions, and test functions.As a result, IGA can represent curved shapes such as circles and conic sections exactly using B-splines and nonuniform rational B-splines (NURBS). Furthermore, IGA can obtain much improvedaccuracy in the computed solution, for a given number of degrees of freedom, due to its inherentsmoothness and higher continuity of basis functions when compared with standard FE and finitevolume (FV) methods. To address the issue of well-posedness, a set of stable IGA discretizationsfor the Navier--Stokes and Maxwell's equations is developed and incorporated into the IGA solver.A detailed convergence study is carried out to verify the convergence, accuracy, and performance ofthe IGA scheme for certain benchmark cases.

Bluck M, 2019, Analytical solutions to non-uniform surface heat transfer with volumetric sources in magnetohydrodynamic duct flow, *Journal of Heat Transfer-Transactions of the ASME*, Vol: 141, ISSN: 0022-1481

A detailed understanding of the flow of a liquid metal in a rectangular duct subject to a strong transverse magnetic field is vitalin a number of engineering applications, notably for proposed blanket technologies for fusion reactors. Fusion reactors offer thepotential for clean base-load energy and their development is now entering an engineering phase where the practical means bywhich the energy released can be converted into useful heat must be addressed. To such ends, this article considers the convectiveheat transfer processes for fully developed laminar magnetohydrodynamic (MHD) flows in rectangular ducts of the kind proposedin some blanket designs. Analytical solutions which incorporate the non-uniformity of peripheral temperature and heat flux and theeffect of volumetric heating, are developed as functions of magnetic field strength and duct aspect ratio. A distinct feature of theseMHD problems, not yet addressed in the literature, is that unlike the conventional characterisation of heat transfer by a Nusseltnumber, it is necessary to generalise the concept to vectors and matrices of Nusselt coefficients, due to the extreme anisotropy ofboth the flow and heating. The new analytical results presented here capture more complex heat transfer behaviour than non-MHDflows and in particular chracterize the importance of aspect ratio. The importance of these new results lie not only in the improvedunderstanding of this complex process, but also in the provision of characterisations of convective heat transfer which underpinprogress toward systems scale simulations of fusion blanket technology which will be vital for the realisation of practical fusionreactors.

Duan Y, Jackson C, Eaton M,
et al., 2019, An assessment of eddy viscosity models on predicting performance parameters of valves, *Nuclear Engineering and Design*, Vol: 342, Pages: 60-77, ISSN: 0029-5493

The major objective of the present study is to evaluate the performance of a range of turbulent eddy viscosity models in the prediction of macro-parameters (flow coefficient (CQ) and force coefficient (CF)), for certain types of valve, including the conic valve, the disk valves, and the compensated valve. This has been achieved by comparison of numerical predictions with experimental measurements available in the literature. The examined turbulence models include most of the available turbulent eddy viscosity models in STAR-CCM+ 12.04. They are the standard k-ε model, realizable k-ε model, k-ω-sst model, V2F model, EB k-ε model and the Lag EB k-ε models.The low-Re turbulence models (k-ω-sst, V2F, EB k-ε and Lag EB k-ε) perform worse than the high-Re models (standard k-ε and realizable k-ε). For the conic valve, the performance of different turbulent models varies little; the standard k-ε model shows a marginal advantage over the others. The performance of the turbulence models changed greatly, however, for prediction of CQ and CF of the disk and compensated valves. In general, the realizable k-ε model is demonstrated to be a robust choice for both valve types. Although the EB k-ε may marginally outperform it in the prediction of CF at large disk valve opening.The effects of the unknown entry flow and initialization conditions are also studied. The predictions are more sensitive to the entry flow condition when the valve opening is large. Additionally, the uncertainties caused by unknown entry conditions are comparable to overall modelling errors in some cases. For flow systems with multiple stable flow-states coexisting in the flow domain, the output of the numerical models can also be affected by the initialization conditions.When the streamline curvature and secondary flow is modest like conical valve flow, the nonlinear modification of the standard k-ε mode

Duan Y, Eaton MD, Bluck MJ, et al., 2018, ASSESSMENTS OF DIFFERENT TURBULENCE MODELS IN PREDICTING THE PERFORMANCE OF A BUTTERFLY VALVE, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

Ahn JS, Bluck M, Eaton M, et al., 2018, A VALIDATION OF RELAP ON PREDICTING NUCLEAR POWER PLANT PHENOMENA, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

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Duan Y, Eaton MD, Bluck MJ, et al., 2018, A VALIDATION OF CFD METHODS ON PREDICTING VALVE PERFORMANCE PARAMETERS, ASME Power Conference 2018 (POWER2018), Publisher: AMER SOC MECHANICAL ENGINEERS

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Pearson RJ, Bluck MJ, Murphy ST, 2017, A symbiotic approach to compact fission and fusion reactors, 2017 ANS Meeting and Nuclear Technology Expo, Pages: 378-381, ISSN: 0003-018X

Bluck MJ, Wolfendale MJ, 2017, An analytical solution to the heat transfer problem in thick-walled hunt flow, *International Journal of Heat and Fluid Flow*, Vol: 64, Pages: 103-111, ISSN: 0142-727X

The flow of a liquid metal in a rectangular duct, subject to a strong transverse magnetic field is of interest in a number of applications. An important application of such flows is in the context of coolants in fusion reactors, where heat is transferred to a lead-lithium eutectic. It is vital, therefore, that the heat transfer mechanisms are understood. Forced convection heat transfer is strongly dependent on the flow profile. In the hydrodynamic case, Nusselt numbers and the like, have long been well characterised in duct geometries. In the case of liquid metals in strong magnetic fields (magnetohydrodynamics), the flow profiles are very different and one can expect a concomitant effect on convective heat transfer. For fully developed laminar flows, the magnetohydrodynamic problem can be characterised in terms of two coupled partial differential equations. The problem of heat transfer for perfectly electrically insulating boundaries (Shercliff case) has been studied previously (Bluck et al., 2015). In this paper, we demonstrate corresponding analytical solutions for the case of conducting hartmann walls of arbitrary thickness. The flow is very different from the Shercliff case, exhibiting jets near the side walls and core flow suppression which have profound effects on heat transfer.

Palazzi A, Bluck MJ, Lo S, et al., 2016, Coupling RELAP5-3D and STAR-CCM+ for simulations of steady and transient single phase flows, 2016 International Congress on Advances in Nuclear Power Plants (ICAPP 2016), Publisher: Curran Associates, Inc., Pages: 2202-2211

A coupling scheme was developed to couple the CFD software STAR-CCM+ v10.04 and the system code RELAP5-3D v4.2.1. In this paper the structure of the scheme is presented, together with validations for single phase flow in smooth pipes in both transient and steady state cases. Attention was also given to the problem of reconstructing the profile at the inlet of the CFD model under the hypothesis of fully developed flow. This problem arises when flow data has to be passed from the one-dimensional system code to the three-dimensional CFD software.

Wolfendale MJ, Bluck MJ, 2015, A coupled systems code-CFD MHD solver for fusion blanket design, *FUSION ENGINEERING AND DESIGN*, Vol: 98-99, Pages: 1902-1906, ISSN: 0920-3796

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Bluck MJ, Wolfendale MJ, Marquis AJ, 2015, An analytical solution to the heat transfer problem in Shercliff flow, *International Journal of Heat and Mass Transfer*, Vol: 86, Pages: 542-549, ISSN: 0017-9310

The study of flow in a rectangular duct, subject to a strong transverse magnetic field is of interest in a number of applications. An important application of such flows is in the context of coolants, where the principle issue of interest is convective heat transfer. For fully developed laminar flows, the problem can be characterised in terms of two coupled partial differential equations. In the case of perfectly electrically insulating boundaries, there is a well known analytical solution due to Shercliff, which provides the velocity and induced magnetic field profiles. In this paper, we demonstrate analytical solutions to and heat transfer problems for the Shercliff case in rectangular ducts and obtain temperature profiles and corresponding Nusselt numbers as functions of aspect ratio and Hartmann number.

Bluck MJ, Wolfendale MJ, 2015, An analytical solution to electromagnetically coupled duct flow in MHD, *Journal of Fluid Mechanics*, Vol: 771, Pages: 595-623, ISSN: 0022-1120

The flow of an electrically conducting fluid in an array of square ducts, separated by arbitrary thickness conducting walls, subject to an applied magnetic field is studied. The analytical solution presented here is valid for thick walls and is based on the homogeneous solution obtained by Shercliff (Math. Proc. Camb. Phil. Soc., vol. 49 (01), 1953, pp. 136-144). Arrangements of ducts arise in a number of applications, most notably in fusion blankets, where liquid metal is used both as coolant and for tritium generation purposes. Analytical solutions, such as those presented here, provide insight into the physics and important benchmarking and validation data for computational magnetohydrodynamics (MHD), as well as providing approximate flow parameters for 1D systems codes. It is well known that arrays of such ducts with conducting walls exhibit varying degrees of coupling, significantly affecting the flow. An important practical example is the so-called Madarame problem (Madarame et al., Fusion Technol., vol. 8, 1985, pp. 264-269). In this work analytical results are derived for the relevant hydrodynamic and magnetic parameters for a single duct with thick walls analogous to the Hunt II case. These results are then extended to an array of such ducts stacked in the direction of the applied magnetic field. It is seen that there is a significant coupling affect, resulting in modifications to pressure drop and velocity profile. In certain circumstances, counter-current flow can occur as a result of the MHD effects, even to the point where the mean flow is reversed. Such phenomena are likely to have significant detrimental effects on both heat and mass transfer in fusion applications. The dependence of this coupling on parameters such as conductivities, wall thickness and Hartmann number is studied.

Cinosi N, Walker SP, Bluck MJ,
et al., 2014, CFD simulation of turbulent flow in a rod bundle with spacer grids (MATIS-H) using STAR-CCM+, *Nuclear Engineering and Design: an international journal devoted to the thermal, mechanical, materials, and structural aspects of nuclear fission energy*, Vol: 279, Pages: 37-49, ISSN: 0029-5493

This paper presents the CFD simulation of the turbulent flow generated by a model PWR spacer grid within a rod bundle. The investigation was part of the MATIS-H benchmark exercise, organized by the OECD-NEA, with measurements performed at the KAERI facilities in Daejeon, Korea. The study employed the CD-Adapco code Star-CCM+. An initial sensitivity study was conducted to attempt to assess the importance to the overall flow of components such as the outlet plenum and the end support grid; these were shown to be able to be safely neglected, but the tapered end portion of the rods was found to be significant, and this was incorporated in the model analyzed. A RANS model using any of K-epsilon, K-omega and Reynolds-stress turbulence models was found to be adequate for the prediction of mean velocity profiles, but they all three underestimate the time-averaged turbulent velocity components. Vorticity seems to be better predicted, although the measured values of vorticity are only presented via colored contour plots, making quantitative comparison rather difficult. Circulation, calculated via an integral for each channel, seems to be well predicted by all three models.

Palazzi A, Bluck MJ, Lo S, 2014, A coupled RELAP5-3D/STAR-CCM+ simulation for the calculation of friction factor in pipes, Pages: 1323-1329

The thermal hydraulic analysis of nuclear reactors is largely performed by what are known as "System Codes". These codes predict the flows in the complex network of pipes, pumps, vessels and heat exchangers that together form the thermal hydraulic systems of a nuclear reactor. These codes have been used for many decades, and are now very well established, and given this long process of refinement, they are able to produce remarkably accurate predictions of plant behaviour under both steady and transient conditions. Modern CFD is able to produce high-quality predictions of flows in complex geometries, but only with the use of large computing resources. It would be impractical to build a CFD model of, for example, the entire primary circuit of a PWR. However, much of the primary circuit may well be able to be modelled with adequate fidelity using a cheaper one-dimensional system code, and it may only be in a limited part of the circuit that full three-dimensional effects are important. In this paper, the coupling tool present in the STAR-CCM+ v8.02 software was used, together with RELAP5-3D v4.0.3, to perform a coupled analysis of single phase flow in a circular pipe in order to evaluate Moody's friction factor. Three different cases were studied, following two different models. The first case consisted in using smooth walls for the pipe, whereas, in the second and the third ones, the roughness of the wall was set to 20 μm and 50 μm respectively. The first model consisted in using RELAP5-3D for the upstream part of the pipe and STAR-CCM+ for the downstream part. In the second model, instead, STAR-CCM+ was used for the upstream part of the pipe and RELAP5-3D for the downstream part. The results obtained with these two models have been compared with each other to check for possible incongruences. Furthermore, all the results have been compared with standalone simulations performed with RELAP5-3D and STAR-CCM+ and with experimental data (Moody's diagram).

Cinosi N, Walker SP, Bluck M, et al., 2014, CFD ANALYSIS OF THE FLOW IN CRUD-COATED NUCLEAR ROD BUNDLES, 22nd International Conference on Nuclear Engineering (ICONE22), Publisher: AMER SOC MECHANICAL ENGINEERS

Bluck MJ, Cinosi N, Walker SP, 2014, A multilevel hierarchical preconditioner for multiscale EM scattering

Bluck M, Cinosi N, Walker S, 2014, A multilevel hierarchical preconditioning method for multiscale scattering bodies

This paper considers the solution of the electric field integral equation (EFIE). Hierarchical conforming bases are developed which are subsequently used in the construction of a multilevel Schwarz type preconditioner. The effectiveness of this approach is demonstrated by the computation of scattering from sub-wavelength scattering bodies. The resulting schemes are shown to be faster than conventional schemes up to an order of magnitude.

Bluck M, Cinosi N, Walker S, 2013, A multilevel hierarchical preconditioner technique for multiscale and complex em scattering bodies, Pages: 1519-1522

This paper considers the solution of the combined field integral equation (CFIE) for lossy/dielectric problems. Hierarchical conforming bases are developed which are subsequently used in the construction of a multilevel Schwarz type preconditioner. The effectiveness of this approach is demonstrated by the computation of scattering in free space from a range of spheres with different permittivity (non PEC material). The resulting schemes are shown to be faster than conventional schemes. © 2013 IEEE.

Bluck MJ, Cinosi NN, Walker SP, 2013, A p-MUS preconditioner for the EFIE, *IEEE Transactions on Antennas and Propagation*, Vol: n/a, ISSN: 0018-926X

This paper considers the solution of the electric field integral equation (EFIE) in electromagnetics. As with associated finite element methods, their solution relies upon the construction of conforming bases. Whilst lowest order (RWG) spaces are near ubiquitous, their extension to higher order offers, potentially, a number of benefits in terms of accuracy and efficiency, which has been well documented in both finite elements and integral equation formulations. A further evolution of higher order conforming bases is the hierarchical basis. These have demonstrated considerable gains in efficiency in finite element applications. Such bases allow for the development of effective acceleration schemes, for instance, the multilevel Schwarz type preconditioner (p-MUS). An obvious question arises as to the applicability of such hierarchical bases and their associated acceleration schemes to integral equations. It is seen that the conclusions as to their efficacy depend strongly on the scattering regime. In particular, high frequency problems (those where the wavelength is the principal determinant of mesh size) are shown to benefit little from hierarchical functions. On the other hand, for {\lq low frequency\rq} problems (where geometry is the main determinant of mesh size), there are significant improvements in performance over corresponding interpolatory schemes.

Bluck MJ, Cinosi N, Walker SP, 2013, A multilevel hierarchical preconditioner for multiscale EM scattering

Integral equation schemes, both EFIE and MFIE, have become a powerful tool, particularly with the development of accelerated schemes such as the fast multipole method (FMM). Key to most such treatments is the requirement to solve matrix equations iteratively, which at their core involve matrixvector multiplications. Much of the cost of such solutions then depends on the number of iterations and the cost per iteration. The vast majority of integral equations implementations employ the simplest Rao-WiltonGlisson (RWG) basis functions on triangles. High order interpolatory bases have been developed which (in principle) offer improved accuracy for a given cost. Within the current developments, the next natural step beyond high order interpolatory methods is to arrange these bases hierarchically, as already experienced within the finite element community. Of themselves hierarchical bases offer little more than their high order interpolatory counterparts. However, as has been demonstrated in finite elements it is possible to employ this hierarchical structure to great effect in the reduction of the computational cost of the underlying iterative scheme via a multilevel Schwarz type preconditioner. In this paper we attempt to apply such hierarchical bases and their associated acceleration schemes to integral equations. The results suggest that their efficacy depend strongly on the scattering regime. In particular, high frequency problems (those where the wavelength is the principal determinant of mesh size) are shown to benefit little from hierarchical functions. Unlike their finite element counterparts, equivalent p-MUS integral equation schemes appear to offer little gain, if any, over nonhierarchical schemes. On the other hand, for ̀low frequency' problems, such as scattering of objects with sub-wavelength features (where geometry is the main determinant of mesh size), there are significant improvements in performance over corresponding interpolatory schemes. Copyright &

Bluck MJ, 2012, Conforming Hierarchical Basis Functions, *Communications in Computational Physics*, Vol: 12, Pages: 1215-1256, ISSN: 1815-2406

A unified process for the construction of hierarchical conforming bases on a range of element types is proposed based on an ab initio preservation of the under- lying cohomology. This process supports not only the most common simplicial ele- ment types, as are now well known, but is generalized to squares, hexahedra, prisms and importantly pyramids. Whilst these latter cases have received (to varying de- grees) attention in the literature, their foundation is less well developed than for the simplicial case. The generalization discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair [16–18] and Arnold et al. [4] for simplices. The process of forming hierar- chical bases involves a recursive orthogonalization and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or bet- ter growth in condition number.

Cinosi N, Bluck M, Walker S, 2012, Application of Time-Domain BIE to sub-wavelength Scattering Bodies with Finite Conductivity, International Conference on Electromagnetics in Advanced Applications (ICEAA) / IEEE-APS Topical Conference on Antennas and Propagation in Wireless Communications (IEEE APWC) / URSI Electromagnetic Environment and Interference Symposium (EEIS), Publisher: IEEE, Pages: 1177-1179

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