## Publications

38 results found

Wu W, Feng X, Xu H, 2022, Improved Deep Neural Networks with Domain Decomposition in Solving Partial Differential Equations, *Journal of Scientific Computing*, Vol: 93, ISSN: 0885-7474

An improved neural networks method based on domain decomposition is proposed to solve partial differential equations, which is an extension of the physics informed neural networks (PINNs). Although recent research has shown that PINNs perform effectively in solving partial differential equations, they still have difficulties in solving large-scale complex problems, due to using a single neural network and gradient pathology. In this paper, the proposed approach aims at implementing calculations on sub-domains and improving the expressiveness of neural networks to mitigate gradient pathology. By investigations, it is shown that, although the neural networks structure and the loss function are complicated, the proposed method outperforms the classical PINNs with respect to training effectiveness, computational accuracy, and computational cost.

Xing B, Li D, Zhang B,
et al., 2021, Vortex structures evolution in supersonic mixing layers with different inlet Reynolds numbers based on the Lagrangian method, *AIP ADVANCES*, Vol: 11

Lai P, Wang R, Zhang W,
et al., 2021, Parameter optimization of open-loop control of a circular cylinder by simplified reinforcement learning, *PHYSICS OF FLUIDS*, Vol: 33, ISSN: 1070-6631

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- Citations: 4

Xu H, Zhang W, Wang Y, 2021, Explore missing flow dynamics by physics-informed deep learning: The parameterized governing systems, *PHYSICS OF FLUIDS*, Vol: 33, ISSN: 1070-6631

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- Citations: 15

Qiu S, Cheng Z, Xu H,
et al., 2021, On the characteristics and mechanism of perturbation modes with asymptotic growth in trailing vortices, *JOURNAL OF FLUID MECHANICS*, Vol: 918, ISSN: 0022-1120

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- Citations: 2

Xiong C, Qi X, Gao A,
et al., 2020, The bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime, *JOURNAL OF FLUID MECHANICS*, Vol: 896, ISSN: 0022-1120

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- Citations: 8

Xu H, Cantwell C, Monteserin C,
et al., 2018, Spectral/hp element methods: Recent developments, applications, and perspectives, *Journal of Hydrodynamics*, Vol: 30, Pages: 1-22, ISSN: 1001-6058

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

Xu H, Mughal SM, Gowree E,
et al., 2017, Destabilisation and modification of Tollmien-Schlichting disturbances by athree dimensional surface indentation, *Journal of Fluid Mechanics*, Vol: 819, Pages: 592-620, ISSN: 1469-7645

We consider the influence of a smooth three-dimensional (3-D) indentation on the instability of an incompressible boundary layer by linear and nonlinear analyses. The numerical work was complemented by an experimental study to investigate indentations of approximately 11δ99 and 22δ99 width at depths of 45 %, 52 % and 60 % of δ99 , where δ99 indicates 99% boundary layer thickness. For these indentations a separation bubble confined within the indentation arises. Upstream of the indentation, spanwise-uniform Tollmien–Schlichting (TS) waves are assumed to exist, with the objective to investigate how the 3-D surface indentation modifies the 2-D TS disturbance. Numerical corroboration against experimental data reveals good quantitative agreement. Comparing the structure of the 3-D separation bubble to that created by a purely 2-D indentation, there are a number of topological changes particularly in the case of the widest indentation; more rapid amplification and modification of the upstream TS waves along the symmetry plane of the indentation is observed. For the shortest indentations, beyond a certain depth there are then no distinct topological changes of the separation bubbles and hence on flow instability. The destabilising mechanism is found to be due to the confined separation bubble and is attributed to the inflectional instability of the separated shear layer. Finally for the widest width indentation investigated ( 22δ99 ), results of the linear analysis are compared with direct numerical simulations. A comparison with the traditional criteria of using N -factors to assess instability of properly 3-D disturbances reveals that a general indication of flow destabilisation and development of strongly nonlinear behaviour is indicated as N=6 values are attained. However N -factors, based on linear models, can only be used to provide indications and severity of the destabilisation, since the process of disturbance breakdown to turbu

Xu H, Lombard J, Sherwin S, 2017, Influence of localised smooth steps on the instability of a boundary layer, *Journal of Fluid Mechanics*, Vol: 817, Pages: 138-170, ISSN: 1469-7645

We consider a smooth, spanwise-uniform forward facing step de ned by the Gauss error function of height 4-30% and four times the width of the local boundary layer thickness δ_99. The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the two-dimensional Tollmien-Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward facing smooth steps is conducted by linear analysis. The results indicate thatfor a TS wave with a frequency F 2 [140; 160] (F=! =U21 106 where ! and U1 denote the perturbation angle frequency and freestream velocity magnitude, respectively), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a at plate boundary layer. Furthermore, it is observed thattwo smooth forward facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20% of the local boundary layer thickness for a xed width parameter, the TS wave is amplified and thereby a destabilisation e ect is introduced. Therefore, stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward facingsmooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favorably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter F = 150 and in the absence of other externalperturbations, two forward facing smooth steps of height 5% and 12% of the boundary lay

Xu H, Mughal MS, Gowree ER, et al., 2016, Effect of a 3d indentation on boundary layer instability, ICAS 2016, 30th Congress of the International Council of the Aeronautical Sciences, Publisher: ICAS

Xu H, Mughal MS, Gowree ER, et al., 2016, Effect of a 3d surface indentation on boundary layer stability, 24th International Congress of Theoretical and Applied Mechanics ICTAM 2016, Publisher: ICAS

We are concerned about effect of a 3D surfaceindentation on instability and laminar-turbulenttransition in a boundary layer. For naturaltransition in a boundary layer, the transitiononset is dominated by growth of the Tollmien-Schlichting (TS) wave and its subsequentsecondary instability. In the paper, both linearanalysis and nonlinear calculations are carriedout to address the 3D surface indentation effecton amplifying TS waves’ amplitudes andprompting transition onset. By the linearanalysis, we address sudden amplification of theTS modes by a separation bubble in a surfaceindentation region. The nonlinear calculationsare implemented to validate the traditionaltransition criteria predicted by the linear theorywhen a 3D indentation is present. Finally,applicability of the traditional transitioncriteria is assessed.

He Y, Xu H, Chen Z, 2016, A second-order decoupled implicit/explicit method of the 3D primitive equations of ocean II: finite element spatial discretization, *International Journal for Numerical Methods in Engineering*, ISSN: 0029-5981

A fully discrete second-order decoupled implicit/explicit method is proposed for solving 3D primitive equations of ocean in the case of Dirichlet boundary conditions on the side, where a second-order decoupled implicit/explicit scheme is used for time discretization, and a finite element method based on the P1(P1) − P1−P1(P1) elements for velocity, pressure and density is used for spatial discretization of these primitive equations. Optimal H1−L2−H1 error estimates for numerical solution inline image and an optimal L2 error estimate for inline image are established under the convergence condition of 0 < h≤β1,0 < τ≤β2, and τ≤β3h for some positive constants β1,β2, and β3. Furthermore, numerical computations show that the H1−L2−H1 convergence rate for numerical solution inline image is of O(h + τ2) and an L2 convergence rate for inline image is O(h2+τ2) with the assumed convergence condition, where h is a mesh size and τ is a time step size. More practical calculations are performed as a further validation of the numerical method.

Xu H, Sherwin S, Hall P,
et al., 2016, The behaviour of Tollmien-Schlichting waves undergoing small-scale localised distortions, *Journal of Fluid Mechanics*, Vol: 792, Pages: 499-525, ISSN: 1469-7645

This paper is concerned with the behaviour of Tollmien-Schlichting (T-S) waves experiencingsmall localised distortions within an incompressible boundary layer developingover a flat-plate. In particular, the distortion is produced by an isolated roughness elementlocated at Rexc = 440 000. We considered the amplification of an incoming T-S wavegoverned by the two-dimensional linearised Navier-Stokes equations, where the base flowis obtained from the two-dimensional non-linear Navier-Stokes equations. We comparethese solutions with asymptotic analyses which assume a linearised triple-deck theory forthe base flow and determine the validity of this theory in terms of the height of the smallscalehumps/indentations taken into account. The height of the humps/indentations isdenoted by h which is considered to be less than or equal to xcRe−5/8xc(correspondingto h/δ99 < 6% for our choice of Rexc). The rescaled width ˆd(≡ d/(xcRe−3/8xc)) ofthe distortion is of the order O(1) and the width d is shorter than the T-S wavelength(λT S = 11.3 δ99).We observe that for distortions which are smaller than 0.1 of the inner deck height(h/δ99 < 0.4%) the numerical simulations confirm the asymptotic theory in the vicinityof the distortion. For larger distortions which are still within the inner deck (0.4% <h/δ99 < 5.5%) and where the flow is still attached the numerical solutions show thatboth humps and indentations are destabilising and deviate from the linear theory evenin the vicinity of the distortion.We numerically determine the transmission coefficient which provides the relative amplificationof the T-S wave over the distortion as compared to the flat-plate. We observethat for small distortions, h/δ99 < 5.5%, where the width of the distortion is of orderof the boundary layer a maximum amplification of only 2% is achieved. This amplificationcan however be increased as the width of the distortion is increased or

He Y, Zhang Y, Xu H,
et al., 2016, First-order decoupled finite element method of the three-dimensional primitive equations of the ocean, *SIAM Journal on Scientific Computing*, Vol: 38, Pages: A273-A301, ISSN: 1095-7197

This paper is concerned with a first-order fully discrete decoupled method for solving the three-dimensional (3D) primitive equations of the ocean with the Dirichlet boundary conditions on the side, where a decoupled semi-implicit scheme is used for the time discretization, and the $P_1(P_1)-P_1-P_1(P_1)$ finite element for velocity, pressure, and density is used for the spatial discretization of these equations. The $H^1-L^2-H^1$ optimal error estimates for the numerical solution $(u_h^n,p_h^n,\theta_h^n)$ and the $L^2$ optimal error estimate for $(u^n_h,\theta_h^n)$ are established under the restriction of $0<h\le \beta_1$ and $0<\tau\le \beta_2$ for some positive constants $\beta_1$ and $\beta_2$. Moreover, numerical investigations are provided to show that the first-order decoupled method is of almost unconditional convergence with accuracy $\mathcal{O}(h+\tau)$ in the $H^1$-norm and $\mathcal{O}(h^2+\tau)$ in the $L^2$-norm for solving the 3D primitive equations of the ocean. Numerical results are given to verify the theoretical analysis.

Xu H, Mughal MS, Sherwin S, 2015, Effect of a 3D surface depression on boundary layer transition, 68th Annual Meeting of the APS Division of Fluid Dynamics

The influence of a three-dimensional surface depression on the transitional boundary layer is investigated numerically. In the boundary layer transition, the primary mode is a Tollmien-Schlichting (TS) wave which is a viscous instability. These modes are receptive to surface roughness interacting with free stream disturbances and/or surface vibrations. In this paper, numerical calculations are carried out to investigate the effect of the depression on instability of the boundary layer. In order to implement linear analysis, two/three (2D/3D)-dimensional nonlinear Navier-Stokes equations are solved by spectral element method to generate base flows in a sufficient large domain. The linear analyses are done by the parabolic stability equations (PSE). Finally, a DNS calculation is done to simulate the boundary layer transition.

Cantwell CD, Moxey D, Comerford A,
et al., 2015, Nektar++: an open-source spectral/hp element framework, *Computer Physics Communications*, Vol: 192, Pages: 205-219, ISSN: 0010-4655

Nektar++ is an open-source software framework designed to support the development of high-performance scalable solvers for partial differential equations using the spectral/hphp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hphp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains.

Zhang Y, Xu H, He Y, 2015, On two-level Oseen iterative methods for the 2D/3D steady Navier Stokes equations, *COMPUTERS & FLUIDS*, Vol: 107, Pages: 89-99, ISSN: 0045-7930

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- Citations: 3

Xu H, Hall P, sherwin S, 2014, Effect of curvature modulation on Gortler vortices in boundary layers, 67th Annual Meeting of the APS Division of Fluid Dynamics

Xu H, Sherwin S, Hall P, 2014, Transmission coefficient of Tollmien-Schlichting waves undergoing small indentation/hump distortion, The 29th Congress of the International Council of the Aeronautical Sciences

He Y, Zhang Y, Xu H, 2013, Two-Level Newton's Method for Nonlinear Elliptic PDEs, *JOURNAL OF SCIENTIFIC COMPUTING*, Vol: 57, Pages: 124-145, ISSN: 0885-7474

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- Citations: 11

Xu H, Sagaut P, 2013, Analysis of the absorbing layers for the weakly-compressible lattice Boltzmann methods, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 245, Pages: 14-42, ISSN: 0021-9991

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- Citations: 23

Xu H, He Y, 2013, Some iterative finite element methods for steady Navier-Stokes equations with different viscosities, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 232, Pages: 136-152, ISSN: 0021-9991

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- Citations: 44

Luan HB, Xu H, Chen L,
et al., 2012, Coupling of finite volume method and thermal lattice Boltzmann method and its application to natural convection, *INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS*, Vol: 70, Pages: 200-221, ISSN: 0271-2091

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- Citations: 24

He Y, Zhang Y, Shang Y,
et al., 2012, Two-level Newton iterative method for the 2D/3D steady Navier-Stokes equations, *NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS*, Vol: 28, Pages: 1620-1642, ISSN: 0749-159X

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- Citations: 35

Xu H, Malaspinas O, Sagaut P, 2012, Sensitivity analysis and determination of free relaxation parameters for the weakly-compressible MRT-LBM schemes, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 231, Pages: 7335-7367, ISSN: 0021-9991

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- Citations: 27

Wei YK, Qian Y, Xu H, 2012, Lattice boltzmann simulations of single bubble deformation and breakup in a shear flow, *Journal of Computational Multiphase Flows*, Vol: 4, Pages: 111-118, ISSN: 1757-482X

Lattice Boltzmann method (LBM) is used to simulate the deformation and breakup of single bubble in a shear flow. Numerical simulations of single bubble deformation are qualitatively compared with experimental results in a shear flow. Respectively the rotation angle θ is quantitatively compared with experimental results according to different capillary numbers (Ca), which shows numerical simulations are in agreement with the experimental results and theoretical results. Finally, the breakup process of single bubble in a shear flow is simulated straightforwardly.

Xu H, Luan H, He Y,
et al., 2012, A lifting relation from macroscopic variables to mesoscopic variables in lattice Boltzmann method: Derivation, numerical assessments and coupling computations validation, *COMPUTERS & FLUIDS*, Vol: 54, Pages: 92-104, ISSN: 0045-7930

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- Citations: 21

Xu H, Sagaut P, 2011, Optimal low-dispersion low-dissipation LBM schemes for computational aeroacoustics, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 230, Pages: 5353-5382, ISSN: 0021-9991

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- Citations: 56

Luan H, Xu H, Chen L,
et al., 2011, Coupling of FVM and LBM for natural convection in a square cavity, *Hsi-An Chiao Tung Ta Hsueh/Journal of Xi'an Jiaotong University*, Vol: 45, Pages: 78-83, ISSN: 0253-987X

On the basis of the existing density distribution function reconstruction operator, the temperature distribution function reconstruction operator was derived to calculate heat transfer by coupling of the lattice Boltzmann method (LBM) and the finite volume method (FVM). The present coupling method was validated by the 2D natural convection flow in a square cavity with various Rayleigh numbers (Ra) from 103 to 106. The results from the coupling method agree well with those by commercial software FLUENT, and all the physical quantities cross the coupled interface smoothly. According to residual history curves it is likely that the numerical stability of the present method are better than those of the pure LBM at fine grid numbers and high Ra.

Luan HB, Xu H, Chen L,
et al., 2011, Coupling between FVM and LBM for fluid flow around a circular cylinder, *Jisuan Lixue Xuebao/Chinese Journal of Computational Mechanics*, Vol: 28, Pages: 97-101, ISSN: 1007-4708

A coupling between FVM and LBM (CFVLBM) model is proposed to resolve the flow around a circular cylinder with far-field boundary. LBM with multi-block grid is adopted near the cylinder, while FVM is used far away. Careful comparisons are done between the results of CFVLBM and that of previous publications as well as multi-block LBM, and body-fitted grid LBM. The results show that the CFVLBM are accurate and reliable. Meanwhile, the CFVLBM can greatly improve the computational efficiency.

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