Multigrid Simulation Research Articles

Multigrid simulations of non-Newtonian fluid flow and heat transfer in a ventilated square cavity with mixed convection and baffles

The impact of baffles on a convective heat transfer of a non-Newtonian fluid is experimentally studied within a square cavity. The non-Newtonian fluid is pumped into the cavity through the inlet and subsequently departs from the cavity via the outlet. Given the inherent non-linearity of the model, a numerical technique has been selected as the method for obtaining the outcomes. Primarily, the governing equations within the two-dimensional domain have been discretized using the finite element method. For approximating velocity and pressure, we have employed the reliable P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{P}}_{2}$$\\end{document}–P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{P}}_{1}$$\\end{document} finite element pair, while for temperature, we have opted for the quadratic basis. To enhance convergence speed and accuracy, we employ the powerful multigrid approach. This study investigates how key parameters like Richardson number (Ri), Reynolds number (Re), and baffle gap bg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ ext{b}}}_{{\ ext{g}}}$$\\end{document} influence heat transfer within a cavity comprising a non-Newtonian fluid. The baffle gap (bg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${b}_{g}$$\\end{document}) has been systematically altered within the range of 0.2–0.6, and for this research, three distinct power law indices have been selected namely: 0.5, 1.0, and 1.5. The primary outcomes of the investigation are illustrated through velocity profiles, streamlines, and isotherm visualizations. Furthermore, the study includes the computation of the Nuavg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${Nu}_{avg}$$\\end{document}(average Nusselt number) across a range of parameter values. As the Richardson number (Ri) increases, Nuavg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${Nu}_{avg}$$\\end{document} also rises, indicating that an increase in Ri results in augmented average heat transfer. Making the space between the baffles wider makes heat flow more intense. This, in turn, heats up more fluid within the cavity.

Fast Multigrid Simulations of Pinning in REBCO With Highly Resistive Nanorods

A geometric multigrid method is applied to the frozen-field time-dependent Ginzburg–Landau equations for anisotropic superconductors with spatially varying material properties. This reduces the simulation time for critical current determination in large 3D systems by up to two orders of magnitude compared with a single grid. Critical current density calculations are presented for a model of REBCO tape with nanorod artificial pinning centers. The model consists of an anisotropic superconducting matrix with embedded nanorods which are treated as isotropic highly resistive cylinders. The effects of varying the density, splay, and conductivity of the nanorods are considered. Increasing the resistivity of the nanorods profoundly affects the critical current density in high fields, where current paths are percolative in the vicinity of matrix-pin interfaces and persist well above the upper critical field of the superconducting matrix.

Modeling Three-Dimensional Anisotropic Structures of Reservoir Lithofacies Using Two-Dimensional Digital Outcrops

Reservoir heterogeneity is a key geological problem that restricts oil and gas exploration and development of clastic rocks from the early to late stages. Existing reservoir heterogeneity modeling methods such as multiple-point geostatistics (MPS) can accurately model the two-dimensional anisotropic structures of reservoir lithofacies. However, three-dimensional training images are required to construct three-dimensional reservoir lithofacies anisotropic structures models, and the method to use reservoir heterogeneity model of fewer-dimensional to obtain a three-dimensional model has become a much-focused research topic. In this study, the outcrops of the second member of Qingshuihe Formation (K1q2) in the northwestern margin of the Junggar Basin, which are lower cretaceous rocks, were the research target. The three-dimensional reservoir heterogeneity model of the K1q2 outcrop was established based on the unmanned aerial vehicle (UAV) digital outcrops model and MPS techniques, and the “sequential two-dimensional conditioning data” (s2Dcd) method was modified based on a sensitivity parameter analysis. Results of the parametric sensitivity analysis revealed that the isotropic multigrid simulations demonstrate poor performance because of the lack of three-dimensional training images, conditioning data that are horizontally discrete and vertically continuous, and distribution of lithofacies that are characterized by large horizontal continuities and small thicknesses. The reservoir lithofacies anisotropic structure reconstructions performed well with anisotropic multigrids. The simulation sequence of two-dimensional surfaces for generating the three-dimensional anisotropic structure of reservoir lithofacies models should be reasonably planned according to the actual geological data and limited hard data. In additional to this, the conditional probability density function of each two-dimensional training image should be fully utilized. The simulation results using only one two-dimensional section will have several types of noises, which is not consistent with the actual geological background. The anisotropic multigrid simulations and two-dimensional training image simulation sequence, proposed in this paper as “cross mesh, refinement step by step”, effectively reduced the noise generated, made full use of the information from the two-dimensional training image, and reconstructed the three-dimensional reservoir lithofacies anisotropic structures models, thus conforming to the actual geological conditions.

Mesh generation for thin layered domains and its application to parallel multigrid simulation of groundwater flow

The generation of detailed three dimensional meshes for the simulation of groundwater flow in thin layered domains is crucial to capture important properties of the underlying domains and to reach a satisfying accuracy. At the same time, this level of detail poses high demands both on suitable hardware and numerical solver efficiency. Parallel multigrid methods have been shown to exhibit near optimal weak scalability for massively parallel computations of density driven flow. A fully automated parameterized algorithm for prism based meshing of coarse grids from height data of individual layers is presented. Special structures like pinch outs of individual layers are preserved. The resulting grid is used as a starting point for parallel mesh and hierarchy creation through interweaved projected refinement and redistribution. Efficiency and applicability of the proposed approach are demonstrated for a parallel multigrid based simulation of a realistic sample problem.

Briefing: High-performance computing for city-scale modelling and simulations

Smarter cities and infrastructure are essential if the world is to respond effectively to the critical challenges it faces. As of 2008, and for the first time in human history, more than half of the world's population now live in cities. Yet cities increasingly need to be able to do more with less, to provide for the well-being of their citizens in a truly sustainable way. City-scale simulations are becoming a possibility with advancements in the computing hardware and software technologies. Recent advancements in distributed scaling using Apache Spark is exemplified through a case-study on simulating the traffic flow behaviour of London at per-hour time resolution. The case-study on the response of ground and structures in central Tokyo shows the advancements in multigrid techniques and Finite Element simulations. This paper discusses the advancements and challenges in high-performance computing (hardware, algorithms, software, and visualisation techniques) in achieving the possibility of real-time city-scale simulations.

Improved direct radiation pattern estimation in FDTD

Application of direct radiation pattern estimation in finite difference time domain (FDTD) simulation environment as an alternative to cumbersome near-to-far field transformation is considered. With the proposed improvement, direct estimation of the radiation pattern from the field strength samples in FDTD method becomes significantly less sensitive to the antenna geometry and thus more applicable to practically relevant cases. The main idea is to use field samples taken at two different radii to account for antenna geometry. The samples are taken at radii that are as close as possible to the antenna but still not too close to spoil the correctness of the results. The introduced improvement, combined with multigrid simulation domain and higher order accuracy FDTD formulations, reduces the necessary size of the computational domain, which directly leads to reduced memory consumption and a smaller error due to numerical dispersion.

Grid refinement for entropic lattice Boltzmann models.

We propose a multidomain grid refinement technique with extensions to entropic incompressible, thermal, and compressible lattice Boltzmann models. Its validity and accuracy are assessed by comparison to available direct numerical simulation and experiment for the simulation of isothermal, thermal, and viscous supersonic flow. In particular, we investigate the advantages of grid refinement for the setups of turbulent channel flow, flow past a sphere, Rayleigh-Bénard convection, as well as the supersonic flow around an airfoil. Special attention is paid to analyzing the adaptive features of entropic lattice Boltzmann models for multigrid simulations.

Multi-grid simulation of counter flow pedestrian dynamics with emotion propagation

Much insight into pedestrian flow dynamics has been achieved by conducting simulations and experiments. However, the combination of pedestrian movement and psychological influence in discrete models has not been well investigated. Here we employ a multi-grid model to understand pedestrian dynamics in counter flow, coupled with the effect of emotion propagation in a crowd stampede induced by panic. The time evolution of average speed under conditions with and without emotion propagation is obtained. Though emotion propagation increases the desired speed, it easily leads to congestion because of an increase in competition in counter flow. According to the speed–density relationship, a function of the crowd density and percentage of pedestrians in counter flow is employed to identify the flowability of pedestrians in a long channel. Thus, three regimes, i.e., lane formation, the transition stage and clogging, are presented. This is useful in the organization of safe mass events with respect to crowd density. The force distribution and cumulative distribution of the largest force that pedestrians bear are further analyzed to understand a tragic incident related to crushing such as the Cambodian stampede. These results provide an estimation of the number of injuries from the perspective of force. It is hoped that this research will be helpful in crowd management.

Bulge formation in disk galaxies with MOND

The formation of galaxies and their various components can be stringent tests of dark matter models and of gravity theories. In the standard cold dark matter (CDM) model, spheroids are formed through mergers in a strongly hierarchical scenario, and also in the early universe through dynamical friction in clumpy galaxies. More secularly, pseudo-bulges are formed by the inner vertical resonance with bars. The high efficiency of bulge formation is in tension with observations in the local universe of a large amount of bulge-less spiral galaxies. In the present work, the formation of bulges in very gas-rich galaxies, as those in the early universe, is studied in the Milgrom's MOdified Newtonian Dynamics (MOND), through multi-grid simulations of the non-linear gravity, including the gas dissipation, star formation and feedback. Clumpy disks are rapidly formed, as in their Newtonian equivalent systems. However, the dynamical friction is not as efficient, in the absence of dark matter halos, and the clumps have no time to coalesce into the center to form bulges, before they are eroded by stellar feedback and shear forces. Previous work has established that mergers are less frequent in MOND, and classical bulges are expected less massive. It is now shown that gas-rich clumpy galaxies in the early universe do not form bulges. Since pseudo-bulges are formed with a similar rate as in the Newtonian equivalent systems, it can be expected that the contribution of pseudo-bulges is significantly higher in MOND.

Geomodelling of carbonate mounds using two-point and multipoint statistics

Abstract The Late Carboniferous strata exposed in the Sacramento Mountains in Southern New Mexico, USA, have long been considered classic exposures documenting reciprocal high-frequency mixed carbonate–siliciclastic cyclicity and shelf-edge algal-mound growth. The growth style and internal architecture of these phylloid algae mounds depend on their position on the shelf and are controlled by potential accommodation space, depth of the photic zone and hydrodynamic energy. The combination of these parameters results in a laterally variable amount of reworked phylloid algae debris and in-situ mound core facies along the depositional profile. This variable architecture can be observed on the outcrop and results in a complex distribution of these two lithofacies in three dimensions that is challenging to reproduce in a 3D geocellular model. Two geostatistical estimation algorithms are used to stochastically model carbonate buildups: surface-based and multipoint statistics (MPS)-based. The surface-based model uses two-point statistics and is built by first recreating the overall geometry of the mound and then reproducing the internal architecture using indicator Gaussian simulation, but requires strong secondary trend data to reproduce the correct facies architecture. The MPS model successfully recreates both the geometry and internal architecture of the mound, but requires a complicated training image and complex multigrid simulation that would be hard to implement in subsurface. This comparison demonstrates that modelling carbonate buildup geometry and internal architecture is not trivial and requires complex workflow with secondary trends. These secondary trends require a significant amount of prior knowledge that is easily extracted from outcrop observations, but would be difficult to assess in subsurface data.

Urban growth and aerosol effects on convection over Houston. Part II: Dependence of aerosol effects on instability

Our studies of the urban growth and aerosol effects on convection over Houston were performed in two phases. The first phase (Part I) used a sea-breeze-induced convective case (August 24 2000) as a benchmark for a series of sensitivity experiments varying the city size and the intensity of aerosol sources linked to sub-grid urban area fractions. The use of National Land Cover Dataset (NLCD) for three different years allowed an objective way to design the land-use sensitivity experiments as well as to consider the spatial distribution and differential intensities of the urban aerosol sources. This paper extends Part I to explore how the relative intensification of the cells downwind of the city (due to urban aerosol sources) can change for environments with more or less instability. This second series of numerical experiments consists of a fairly large number of multi-grid simulations (almost 100) that varied not only the intensity of the urban sources but also the value of CAPE. Our results were in agreement with previous studies; higher cloud condensation nuclei (CCN) concentrations reduced the sizes of cloud droplets and increased their probability of reaching supercooled levels and thus reducing coalescence. Therefore, downwind convective cells were intensified by additional release of latent heat of freezing. However, our results show a non-monotonic behavior on surface precipitation. This interesting response was linked to the riming efficiency reduction of ice particles when aerosol concentrations are greatly enhanced. Therefore, a greater fraction of the ice-phase condensed water mass is transported out of the storm as pristine ice crystals instead of being transferred to precipitating water species. Moreover, the precipitation efficiency of cells downwind of the city exhibited a similar behavior, increasing when CCN concentrations are initially increased and then decreasing when aerosol concentrations are further enhanced. The CCN concentration needed to reach maximum precipitation efficiency is higher for more unstable environments.

Multi-grid simulation of pedestrian counter flow with topological interaction

We investigate the dynamics of pedestrian counter flow by using a multi-grid topological pedestrian counter flow model. In the model, each pedestrian occupies multi- rather than only one grid, and interacts with others in the form of topological interaction, which means that a moving pedestrian interacts with a fixed number of those nearest neighbours coming from the opposite direction to determine his/her own moving direction. Thus the discretization of space and time are much finer, the decision making process of the pedestrian is more reliable, which all together makes the moving behaviour and boundary conditions much more realistic. When compared with field observations, it can be found that the modified model is able to reproduce well fitted pedestrian collective behaviour such as dynamical variation of lane formation, clustering of pedestrians in the same direction, etc. The fundamental diagram produced by the model fits also well with field data in the free flow region. Further analyses indicate that with the increase of the size of pedestrian counter flow system, it becomes harder for the system to transit into a jamming state, while the increase of interaction range does not change the transition point from free flow to jamming flow in the multi-grid topological counter flow model. It is also found that the asymmetry of the injection rate of pedestrians on the boundaries has direct influence on the process of transition from free flow to jamming flow, i.e., a symmetric injection makes it easier for the system to transit into jamming flow.

Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations

We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the $L_{s}^{\infty}(0,T;H_{h}^{1})$ norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the $L_{s}^{2}(0,T;H_{h}^{2})$ norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.

Parallel unstructured multigrid simulation of 3D unsteady flows and fluid–structure interaction in mechanical heart valve using immersed membrane method

In this work, a second-order accurate immersed membrane method (IMM) is adopted to simulate the fluid–structure interaction phenomena in the mechanical heart valves (MHVs). The leaflets of the MHV are immersed in the fluid flows and move on top of the fixed fluid mesh. The blood flow is computed by a 3D parallel unstructured multigrid implicit finite-volume Navier–Stokes solver for incompressible flows. The opening and closing phases of a St. Jude 29 mm MHV are computed under pulsatile inflow to investigate the blood–leaflet interactions. The results show that the moment generated by the fluid pressure is the major cause for the valve motions, while the moment due to the fluid shear stresses is almost negligible. It is also observed that near the end of the opening phase the valve opening speed decelerates, so the valve leaflets have a cushioning effect and avoid a sudden impact on the hinges. For closing phase, jet flows are formed in the central channel and squeeze flows occur in the side channels near the fully closed positions.

Unstructured Multigrid Simulations of Turbulent Launch Vehicle Flows Including a Propulsive Jet

A study of the flowfield around axisymmetric launch vehicles in different flight conditions is performed. In particular, the first Brazilian Satellite Launcher second-stage configuration is analyzed considering the case with and without a propulsive jet in the vehicle base. The study is performed using a cell-centered finite volume formulation on unstructured grids. Different spatial discretization schemes are compared, including a centered and an upwind scheme. The upwind scheme is a second-order version of the Liou flux vector-splitting scheme. Turbulence effects are accounted for using two one-equation turbulence closure models, namely, the Baldwin and Barth and the Spalart and Allmaras models. An agglomeration multigrid algorithm is used to accelerate the converge to steady state of the numerical solutions. The numerical results obtained are compared with experimental data, as well as with previous structured grid simulations.

Efficient and accurate time adaptive multigrid simulations of droplet spreading

AbstractAn efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non‐linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time‐stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three‐dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics. Copyright © 2004 John Wiley & Sons, Ltd.

Multigrid simulations of detached shock waves

AbstractThe use of multigrid convergence acceleration techniques is investigated for supersonic and hypersonic flows over blunt bodies. In these cases detached shock waves occur that usually complicate convergence for multigrid methods. It is shown that a simple damping of the restricted residual error enables convergence without the application of expensive upwind restriction or prolongation operators. The achieved reduction in CPU time increases with increasing free stream Mach number. A similar technique may be used for reactive flows where the restricted residual error is damped in regions of high chemical activity. A spherical nose projectile moving at Mach 6.46 in a stochiometric hydrogen–air mixture serves as a test case to demonstrate the efficiency of this approach in case of combustion. Copyright© 2004 John Wiley & Sons, Ltd.

Multigrid Simulation for High-Frequency Solutions of the Helmholtz Problem in Heterogeneous Media

The Helmholtz problem is hard to solve in heterogeneous media, in particular, when the wave number is real and large. The problem is neither coercive nor Hermitian symmetric. This article concerns the V-cycle multigrid (MG) method for high-frequency solutions of the Helmholtz problem. Since we need to choose at least 10--12 grid points per wavelength for stability, the coarse grid problem is still large. To solve the coarse grid problem efficiently, a nonoverlapping domain decomposition method is adopted without introducing another coarser subspace correction. Various numerical experiments have shown that the convergence rate of the resulting MG method is independent on the grid size and the wave number, provided that the coarse grid problem is fine enough for the solution to capture characteristics of the physical problem.

Multigrid Simulation of Electromagnetic Actuators

AbstractA recently developed multigrid simulation scheme for coupled magneto‐mechanical systems is presented. This scheme allows the efficient calculation of 3D dynamic motions of elastic ferromagnetic materials in a magnetic field. The 3D mechanical problem is discretized with nodal elements and the 3D magnetic problem with edge elements. Both meshes may be chosen independently and therefore, they can be adapted to the physical requirements of the mechanical and magnetic field. By applying fast multigrid solution techniques, very short solution times can be achieved even for large scale 3D problems. Using the developed scheme the transient behavior of a solenoid valve is calculated.

Automated Simulation of Electrode Processes: Quantitative Mechanistic Analysis via Working Surface Interpolation

An efficient automated simulation of the transport-limited electrolysis in a channel flow cell, based on the multigrid method, is developed and is used to compute the working curve for the transport-limited current within the Lévêque approximation. An accurate universal two-dimensional working surface (making neither the Lévêque approximation nor ignoring axial diffusion) is presented for the steady-state transport-limited current from which the simulated response may be interpolated. The multigrid simulation is extended to include homogenous ECE kinetics and a working surface is presented for analysis of this mechanism within the Lévêque approximation. This is then modified using a Gauss−Newton method to solve the nonlinear EC2E system for which an analogous working surface is generated. The effect of axial diffusion on the measurement of rate constants for these mechanisms is discussed. The merits of bilinear, cubic convolution, bicubic spline, and artificial neural network methods are assessed as interpolation methods for working surfaces. Working surface interpolation is shown to be a viable cost-effective alternative to repeated simulation methods, especially for problems of more than one dimension. A free data-processing service using the working surface data together with a suitable interpolation method is available via the World Wide Web (http://physchem.ox.ac.uk:8000/wwwda).