An evaluation of lattice Boltzmann schemes for porous medium flow simulation

We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar–Gross–Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate several fluid–solid boundary conditions including: (1) the standard bounce-back (SBB) scheme, (2) the linearly interpolated bounce-back (LIBB) scheme, (3) the quadratically interpolated bounce-back (QIBB) scheme, and (4) the multi-reflection (MR) scheme. Three-dimensional flow through two porous media—a body-centered cubic (BCC) array of spheres and a random-sized sphere-pack—are examined in this study. For flow past a BCC array of spheres, we validate the linear LBE model by comparing its results with the nonlinear LBE model. We investigate systematically the viscosity-dependence of the computed permeability, the discretization error, and effects due to the choice of relaxation parameters with the MRT and BGK schemes. Our results show unequivocally that the MRT–LBE model is superior to the BGK–LBE model, and interpolation significantly improves the accuracy of the fluid–solid boundary conditions.     

On the solution of linear boundary value problems by extended invariant imbedding

Unstable linear boundary value problems can be solved by the method of Invariant Imbedding in a stable manner. Instead of integration of the system equations this method requires the integration of a matrix Riccati equation, which depends on the boundary values of the problem. The dimension of the Riccati equation is determined by a suitable decoupling of the system equations. Invariant Imbedding now fails, if this decoupling does not correspond with the boundary condition. In addition, the Riccati equation has to be solved once more for each new boundary condition. An extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. This “extended” boundary value is solved by a modified Invariant Imbedding. The resulting “Extended (Dual) Invariant Imbedding” is always applicable and does not depend on the boundary conditions. The corresponding “extended” Riccati equation has to be integrated only once “offline”. If the boundary condition is changed, only systems of linear equations have to be solved “online”.     

Improved coupling of time integration and hydrodynamic interaction in particle suspensions using the lattice Boltzmann and discrete element methods

This paper introduces improvements to the simulation of particle suspensions using the lattice Boltzmann method (LBM) and the discrete element method (DEM). First, the benefit of using a two-relaxation-time (TRT) collision operator, instead of the popular Bhatnagar–Gross–Krook (BGK) collision operator, is demonstrated. Second, a modified solid weighting function for the partially saturated method (PSM) for fluid–solid interaction is defined and tested. Results are presented for a range of flow configurations, including sphere packs, duct flows, and settling spheres, with good accuracy and convergence observed. Past research has shown that the drag, and consequently permeability, predictions of the LBM exhibit viscosity-dependence when used with certain boundary conditions such as bounce-back or interpolated bounce-back, and this is most pronounced when the BGK collision operator is employed. The improvements presented here result in a range of computational viscosities, and therefore relaxation parameters, within which drag and permeability predictions remain invariant. This allows for greater flexibility in using the relaxation parameter to adjust the LBM timestep, which can subsequently improve synchronisation with the time integration of the DEM. This has significant implications for the simulation of large-scale suspension phenomena, where the limits of computational hardware persistently constrain the resolution of the LBM lattice.     

Abstract interpretation of logic programs using magic transformations

In dataflow analysis of logic programs, information must be propagated according to the control strategy of the language under consideration. However, for languages with top-down control flow, naive top-down dataflow analyses may have problems guaranteeing completeness and/or termination. What is required in the dataflow analysis is a bottom-up fixpoint computation, guided by the (possibly top-down) control strategy of the language. This paper describes the application of the magic templates algorithm, originally devised as a technique for efficient bottom-up computation of logic programs, to dataflow analysis of logic programs. It turns out that a direct application of the magic templates algorithm can result in an undesirable loss in precision, because connections between “calling patterns” and the corresponding “success patterns” may be lost. We show how the original magic templates algorithm can be modified to avoid this problem, and prove that the resulting analysis algorithm is at least as precise as any other abstract interpretation that uses the same abstract domain and abstract operations.     

Lattice Boltzmann simulation of turbulent flow laden with finite-size particles

The paper describes a particle-resolved simulation method for turbulent flow laden with finite size particles. The method is based on the multiple-relaxation-time lattice Boltzmann equation. The no-slip boundary condition on the moving particle boundaries is handled by a second-order interpolated bounce-back scheme. The populations at a newly converted fluid lattice node are constructed by the equilibrium distribution with non-equilibrium corrections. MPI implementation details are described and the resulting code is found to be computationally efficient with a good scalability. The method is first validated using unsteady sedimentation of a single particle and sedimentation of a random suspension. It is then applied to a decaying isotropic turbulence laden with particles of Kolmogorov to Taylor microscale sizes. At a given particle volume fraction, the dynamics of the particle-laden flow is found to depend mainly on the effective particle surface area and particle Stokes number. The presence of finite-size inertial particles enhances dissipation at small scales while reducing kinetic energy at large scales. This is in accordance with related studies. The normalized pivot wavenumber is found to not only depend on the particle size, but also on the ratio of particle size to flow scales and particle-to-fluid density ratio.     

A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows

A continuous adjoint formulation for inverse design problems in external aerodynamics and turbomachinery is presented. The advantage of the proposed formulation is that the objective function gradient does not depend upon the variation of field geometrical quantities, such as metrics variations in the case of structured grids. The final expression for the objective function gradient includes only boundary integrals which can readily be calculated in both structured and unstructured grids; this is feasible in design problems where the objective function is either a boundary integral (pressure deviation along the solid walls) or a field integral (the entropy generation over the flow domain). The formulation governs inviscid and viscous flows; it takes into account the streamtube thickness variation terms in quasi-3D cascade designs or rotational terms in rotating blade design problems. The application of the method is illustrated through a number of design problems concerning isolated airfoils, a 3D duct, 2D, quasi-3D and 3D, stationary and rotating turbomachinery blades.     

Optimal parameters of a Perfectly-Matched Layer for nanophotonics problems

In the solution of evolutionary problems in electrodynamics and in the numerical Finite Difference Time Domain (FDTD) method with Perfectly Matched Layer (PML) boundary conditions, reflections of electromagnetic waves from the boundaries inevitably arise. The nature and magnitude of these reflections depend on many parameters. We have demonstrated the method of calculating the coefficient of reflection for the one-dimensional (1D) case. A comparison with real 3D numerical experiments shows the high accuracy of the calculated theoretical reflection. On this basis, a method for selecting the optimal parameters of a perfectly matched layer is proposed.     

Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization

We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of solid boundary conditions. We show that a porosity model is ideally suited for topology optimization purposes and models no-slip boundary conditions with sufficient accuracy when compared to interpolation bounce-back conditions. Augmenting the porous boundary condition with a shaping factor, we define a generalized geometry optimization formulation and derive the corresponding sensitivity analysis for the single relaxation LBM for both topology and shape optimization applications. Using numerical examples, we verify the accuracy of the analytical sensitivity analysis through a comparison with finite differences. In addition, we show that for fluidic topology optimization a scaled volume constraint should be used to obtain the desired “0-1” optimal solutions.     

非均匀分布颗粒群中的曳力分布

本文采用格子Boltzmann方法(LBM)在图形处理器(GPU)上计算了由静止圆柱阵列组成的团聚物周期单元内的不可压缩流体流动,流固交界面处采用直接反弹以实现无滑移边界,每个圆柱上的曳力通过统计动量交换直接求得。根据LBM求得的流体速度,对于团聚物中的单圆柱按能量最小多尺度(EMMS)模型计算平均曳力系数,并考察了将聚团近似为均匀悬浮的临界条件。对颗粒雷诺数Re_p在0～10之间的80种固相份额的模拟结果表明,密相空隙率可以表征这种临界条件。当固相份额恒定时,该临界空隙率随着Re_p的增加而降低;当Re_p恒定时,该临界空隙率随着固相份额的增加而降低。     

Core zone scatterplots: A new approach to feature extraction for visual displays

This paper presents a new scheme for mapping high dimensional data onto two-dimensional viewing spaces. The mapping procedure is carried out in two stages. In the first stage, the fuzzy c-means (FCM) algorithm is applied to the N-dimensional data to find membership functions of clusters in the data. Core subsets are then selected from the original data based upon threshold values applied to the membership functions found by FCM. In the second stage feature vectors in the selected “core” subsets are submitted to various feature extraction mappings, which yield scatterplots of the image points in 2D space. The proposed approach has two significant advantages over many previous schemes. First, changes in the core structure imposed on the original data under feature extraction can be used to gauge the relative quality of competing extraction techniques. And second, the cores provide a way to generalize almost any known method, resulting in new extraction algorithms. We also discuss various ways to color the selected data that enhance the 2D display. Our approach incorporates a means for assessing the “quality” of the 2D display via parameters which provide an evaluation of (i) the validity of clusters in the original data set and (ii) the relative ability of various extraction mappings to preserve certain well-defined structural properties of the original data. The feasibility of our approach is illustrated using two sets of data: the well known Iris data; and a set of flow cytometric data. Color displays are used to visually assess scatterplot configurations in 2-space.     

Extension and application of an algorithm for systematic identification of weak coupling and partitions in dynamic system models

This paper reviews and extends a technique to detect weak coupling (one-way coupling or complete decoupling) among elements of a dynamic system model, and to partition and reduce models in which weak coupling is found. The ability to partition a model increases the potential for physical-domain model reduction, and allows parallel simulation of smaller individual submodels that can reduce computation time. Negligible constraint equation terms are identified and eliminated in a bond graph by converting inactive power bonds to modulated sources. If separate bond graphs result, between which all modulating signals move from a “driving” subgraph to a “driven” one, then one-way coupling exists in the model and it can be separated into driving and driven partitions. Information flow between the subgraphs is one-way.In this paper the algorithm is extended to models in which two-way information flow from modulating signals precludes complete partitioning. It is shown for several classes of modulating signals that, under certain conditions the signal is “weak” and therefore can be eliminated. Removal of weak signals allows partitioning of the longitudinal and pitch dynamics of a medium-duty truck model. The intensity of dynamic coupling and the potential for model reduction are shown to depend on the magnitude of system parameters and the severity of inputs such as road roughness.     

Febie—a combined finite element-boundary integral equation method

A method of combining finite element and boundary integral equations is described wherein the boundary integral equation forms a “natural boundary condition” in the variational procedure upon which the finite element method is based. The technique is illustrated by a problem of wave resonance in a harbor.     

Virtual viewpoints reconstruction via Fourier slice transformation

We propose a new method of calculating the arbitrary viewpoints for auto‐stereoscopic display. The three‐dimensional (3D) object is first virtually reconstructed in 3D spaces by mapping each pixel with a depth according to the depth mapping. We then calculate the Fourier spectrum of the 3D object by the fast Fourier transformation. The arbitrary viewpoints are reconstructed by “slicing” the 3D Fourier spectrum. To repair “black hole” artifacts, the regions in the background are calculated by advanced boundary in‐painting. Experimental results show the effectiveness and accuracy of the proposed algorithm to calculate viewpoints with arbitrary viewing angles. A comparison is also presented, which indicates that the proposed algorithm is more accurate than conventional method, and the advanced boundary in‐painting can save three quarters of time than the conventional in‐painting method.     

Combined stiffening and in-plane boundary conditions effects on the buckling of circular cylindrical stiffened-shells

The influence of in-plane boundary conditions on the critical loads of axially compressed simply supported stiffened cylindrical shells, stiffened by stringers and by combinations of rings and stringers is studied. It is observed that the axial restraint, u = 0, at the edges and the dimensions of either the stringers or the rings characterize the type of influence experienced. In shells stiffened by “medium” and “heavy” stringers the axial restraint is a predominant factor and the “weak in shear”, N_xφ = 0, B.Cs. have only a slight secondary effect. For such shells a “stiffening” effect is observed for SS2 and SS4 B.Cs. As the stringers become “weaker” the influence of the axial restraint diminishes and the isotropic or ring-stiffened like type of behavior, “sensitivity” to the vanishing of the circumferential restraint, overcomes the “stiffening” effect due to u = 0.In shells stiffened by combinations of rings and stringers the influence of the in-plane boundary conditions is governed by the relative magnitudes of the rings and stringers under consideration. Combinations of “heavy” stringers and “weak” rings behave like stringer-stiffened shells, exhibiting the “stiffening” effect due to u = 0 whereas shells stiffened by “heavy” rings and “light” stringers tend to behave like ring-stiffened shells, revealing their “sensitivity” to the “weak in shear” boundary conditions, N_xφ = 0.     

An artificial neural network-based multiscale method for hybrid atomistic-continuum simulations

This paper presents an artificial neural network-based multiscale method for coupling continuum and molecular simulations. Molecular dynamics modelling is employed as a local “high resolution” refinement of computational data required by the continuum computational fluid dynamics solver. The coupling between atomistic and continuum simulations is obtained by an artificial neural network (ANN) methodology. The ANN aims to optimise the transfer of information through minimisation of (1) the computational cost by avoiding repetitive atomistic simulations of nearly identical states, and (2) the fluctuation strength of the atomistic outputs that are fed back to the continuum solver. Results are presented for prototype flows such as the isothermal Couette flow with slip boundary conditions and the slip Couette flow with heat transfer.     

Optical flow 3D segmentation and interpretation: a variational method with active curve evolution and level sets

This study investigates a variational, active curve evolution method for dense three-dimensional (3D) segmentation and interpretation of optical flow in an image sequence of a scene containing moving rigid objects viewed by a possibly moving camera. This method jointly performs 3D motion segmentation, 3D interpretation (recovery of 3D structure and motion), and optical flow estimation. The objective functional contains two data terms for each segmentation region, one based on the motion-only equation which relates the essential parameters of 3D rigid body motion to optical flow, and the other on the Horn and Schunck optical flow constraint. It also contains two regularization terms for each region, one for optical flow, the other for the region boundary. The necessary conditions for a minimum of the functional result in concurrent 3D-motion segmentation, by active curve evolution via level sets, and linear estimation of each region essential parameters and optical flow. Subsequently, the screw of 3D motion and regularized relative depth are recovered analytically for each region from the estimated essential parameters and optical flow. Examples are provided which verify the method and its implementation     

Analysis of the stochastic “unilateral contact” problem

In the present paper the stochastic analysis of structures with “unilateral contact”-boundary conditions is considered. Initially the problem of “unilateral contact” is formulated in a variational inequality form. Then the theorems of minimum potential and complementary energy, which are derived to account for this type of boundary conditions permit us to formulate the analysis as a nonlinear programming problem. To embed stochastic parameters in the deterministic model, an analytic Monte Carlo procedure is used. Numerical examples on foundation structures illustrate the developed analysis.     

A high-performance parallel implementation of the certified reduced basis method

The certified reduced basis method (herein RB method) is a popular approach for model reduction of parametrized partial differential equations. In this paper we introduce new techniques that are required to efficiently implement the Offline “Construction stage” of the RB method on high-performance parallel supercomputers. This enables us to generate certified RB models for large-scale three-dimensional problems that can be evaluated on standard workstations and other “thin” computing resources with speedup of many orders of magnitude compared to the corresponding full order model. We use our implementation to perform detailed numerical studies for two computationally expensive model problems: a natural convection fluid flow problem and a “many parameter” heat transfer problem. In the heat transfer problem, we exploit the computational efficiency of the RB method to perform a detailed study of “snapshot” selection in the Greedy algorithm, and we also examine statistics of the output sensitivity derivatives to obtain a “global” view of the relative importance of the parameters.