Block-based Adaptive Mesh Refinement Research Articles

Wake and air entrainment properties of transom stern over a wide range of Froude numbers

In the present study, high-fidelity simulations of the wake behind a transom stern are performed with a block-based adaptive mesh refinement technology. By transom stern, we mean a square-ended stern of a ship, which is a favorable design for the high-speed ship. The sharp volume of fluid method is adopted to capture the gas–liquid interface, and the immersed boundary method is applied to simulate the boundaries of ship hull. Simulation results show that the V-like diverging wave along with air entrainment constitute the main characteristics of the wake. Air cavity of various scales is captured and tracked by the cavity-detection algorithm. Thus, the spatial and temporal distribution of the number and volume of air cavity is obtained in the simulation. Different draft Froude numbers are considered to analyze their influence on the wake. The wave profile, distribution of air cavity, turbulence kinetic energy, and the air entrainment features of the wakes behind dry and wetted stern are compared quantitatively. Numerical results demonstrate the present solver is capable of reproducing the main characteristics of wake behind a high-speed transom stern.

Vortexlet formation in Schardin's problem

The present study focuses on the shock diffraction problem over a triangle wedge for Mach numbers of M=1.3, 1.5, 1.7, and 2.0 by using a two-dimensional, high-order, in-house Euler solver. The solver is based on a family of advection upstream splitting method in combination with a central essentially non-oscillatory scheme and benefits a block-based adaptive mesh refinement algorithm to resolve the regions that contain discontinuities. High accuracies in time and space, and adaptive mesh refinement capabilities of the solver allow us to investigate vortexlet formation mechanism in detail. Our results reveal that there are two different types of vortexlet formation mechanisms. While the first type of formation is observed at all Mach numbers considered here, the second type arises when the Mach number is greater than 1.3. This difference results from their driving mechanisms, which are the upward moving accelerated shock and embedded shock in the primary vortex. In addition to their driving mechanisms, two types are also different in terms of their locations.

Enhanced anisotropic block-based adaptive mesh refinement for three-dimensional inviscid and viscous compressible flows

A parallel anisotropic block-based adaptive mesh refinement (AMR) algorithm is proposed for the prediction of physically complex flow problems having disparate spatial and temporal scales and exhibiting highly anisotropic features on three-dimensional multi-block body-fitted hexahedral meshes with non-uniform grid blocks. The proposed AMR scheme makes use of a binary tree hierarchical data structure to permit anisotropic refinement of grid blocks in a preferred coordinate direction as dictated by appropriately selected physics-based refinement criteria. The anisotropic coarsening of the grid blocks in a manner that is independent of the refinement history allows the mesh to rapidly re-adapt for evolving unsteady flow applications. Moreover, the proposed anisotropic AMR scheme adopts a non-uniform representation of the cells within each block by directly using the neighboring cells as the ghost cells, even at a grid resolution change. This affords a number of computational advantages especially related to evaluating the solution in the ghost cells as well as to ensuring flux conservation at block interfaces. A modified upwind-based finite-volume spatial discretization scheme is applied in conjunction with the AMR scheme to the solution of Euler and Navier-Stokes equations for inviscid and viscous compressible gaseous flows. Steady-state and time-varying flow problems are considered on anisotropic adapted meshes. The potential flexibility and efficiency of the proposed enhanced anisotropic AMR scheme are demonstrated for the simulation of a number of representative flows of varying complexity.

Gmunu: paralleled, grid-adaptive, general-relativistic magnetohydrodynamics in curvilinear geometries in dynamical space–times

ABSTRACT We present an update on the General-relativistic multigrid numerical (Gmunu) code, a parallelized, multidimensional curvilinear, general relativistic magnetohydrodynamics code with an efficient non-linear cell-centred multigrid elliptic solver, which is fully coupled with an efficient block-based adaptive mesh refinement module. To date, as described in this paper, Gmunu is able to solve the elliptic metric equations in the conformally flat condition approximation with the multigrid approach and the equations of ideal general-relativistic magnetohydrodynamics by means of high-resolution shock-capturing finite-volume method with reference metric formularised multidimensionally in Cartesian, cylindrical, or spherical geometries. To guarantee the absence of magnetic monopoles during the evolution, we have developed an elliptical divergence cleaning method by using the multigrid solver. In this paper, we present the methodology, full evolution equations and implementation details of Gmunu and its properties and performance in some benchmarking and challenging relativistic magnetohydrodynamics problems.

A new 3-D multi-fluid model with the application in bubble dynamics using the adaptive mesh refinement

Violent pulsating bubbles behave diversely in different circumstances. It is a multi-scale problem in both space and time. In 3-D problems, the numerical simulation is usually too expensive to implement in practice with a fixed grid. In this paper, a 3-D multi-fluid model is established based on the Eulerian finite element method and the adaptive mesh refinement technique to investigate the bubble evolution and its toroidal motion near a solid vertical wall. The mixture formula for compressible multi-fluid flow is adopted to ensure conservativeness. By means of the block-based adaptive mesh refinement, the accuracy and the efficiency of the simulation are well balanced. The present model is validated by comparing the results with an underwater explosion experiment and the existing numerical results. The results agree well and a fast convergence is observed. Then, several cases with different buoyancy parameters are simulated, and the toroidal bubble motion and their pressure load on the solid wall are analyzed. The bubble’s motion exhibits complex physics, such as the formation of the crescent-shaped bubble, the air cushion effect during the jet penetration, and the nonlinear relationship between the jet impact pressure and the angle between the jet and the opposite bubble surface.

Effect of the solar wind density on the evolution of normal and inverse coronal mass ejections

Context. The evolution of magnetised coronal mass ejections (CMEs) and their interaction with the background solar wind leading to deflection, deformation, and erosion is still largely unclear as there is very little observational data available. Even so, this evolution is very important for the geo-effectiveness of CMEs. Aims. We investigate the evolution of both normal and inverse CMEs ejected at different initial velocities, and observe the effect of the background wind density and their magnetic polarity on their evolution up to 1 AU. Methods. We performed 2.5D (axisymmetric) simulations by solving the magnetohydrodynamic equations on a radially stretched grid, employing a block-based adaptive mesh refinement scheme based on a density threshold to achieve high resolution following the evolution of the magnetic clouds and the leading bow shocks. All the simulations discussed in the present paper were performed using the same initial grid and numerical methods. Results. The polarity of the internal magnetic field of the CME has a substantial effect on its propagation velocity and on its deformation and erosion during its evolution towards Earth. We quantified the effects of the polarity of the internal magnetic field of the CMEs and of the density of the background solar wind on the arrival times of the shock front and the magnetic cloud. We determined the positions and propagation velocities of the magnetic clouds and thus also the stand-off distance of the leading shock fronts (i.e. the thickness of the magnetic sheath region) and the deformation and erosion of the magnetic clouds during their evolution from the Sun to the Earth. Inverse CMEs were found to be faster than normal CMEs ejected in the same initial conditions, but with smaller stand-off distances. They also have a higher magnetic cloud length, opening angle, and mass. Synthetic satellite time series showed that the shock magnitude is not affected by the polarity of the CME. However, the density peak of the magnetic cloud is dependent on the polarity and, in case of inverse CMEs, also on the background wind density. The magnitude of the z-component of the magnetic field was not influenced by either the polarity or the wind density.

High-Order Finite-Volume Method with Block-Based AMR for Magnetohydrodynamics Flows

A high-order central essentially non-oscillatory (CENO) finite volume scheme combined with a block-based adaptive mesh refinement (AMR) algorithm is proposed for the solution of the ideal magnetohydrodynamics equations. The high-order CENO finite-volume scheme is implemented with fourth-order spatial accuracy within a flexible multi-block, body-fitted, hexahedral grid framework. An important feature of the high-order adaptive approach is that it allows for anisotropic refinement, which can lead to large computational savings when anisotropic flow features such as isolated propagating fronts and/or waves, shocks, shear surfaces, and current sheets are present in the flow. This approach is designed to handle complex multi-block grid configurations, including cubed-sphere grids, where some grid blocks may have degenerate edges or corners characterized by missing neighboring blocks. A procedure for building valid high-order reconstruction stencils, even at these degenerate block edges and corners, is proposed, taking into account anisotropic resolution changes in a systematic and general way. Furthermore, a non-uniform or heterogeneous block structure is used where the ghost cells of a block containing the solution content of neighboring blocks are stored directly at the resolution of the neighbors. A generalized Lagrange multiplier divergence correction technique is applied to achieve numerically divergence-free magnetic fields while preserving high-order accuracy on the anisotropic AMR grids. Parallel implementation and local grid adaptivity are achieved by using a hierarchical block-based domain partitioning strategy in which the connectivity and refinement history of grid blocks are tracked using a flexible binary tree data structure. Physics-based refinement criteria as well as the CENO smoothness indicator are both used for directing the mesh refinement. Numerical results, including solution-driven anisotropic refinement of cubed-sphere grids, are presented to demonstrate the accuracy and efficiency of the approach.

Block-based adaptive mesh refinement for fluid–structure interactions in incompressible flows

In this study, an immersed boundary (IB) approach on the basis of moving least squares (MLS) interpolation is proposed for analyzing the dynamic response of a rigid body immersed in incompressible flows. An improved mapping strategy is proposed for a quick update of the signed distance field. A CIP-CSL (constraint interpolation profile - semi-Lagrangian) scheme with a compact stencil is adopted for the convective term in momentum equation. Fluid–structure interaction problems can be solved by either the weak or the strong coupling schemes according to the density ratio of the solid and fluid. This research is based on our previous studies on block-structured adaptive mesh refinement (AMR) method for incompressible flows (Liu & Hu, 2018). Present AMR-FSI solver is proved to be accurate and robust in predicting dynamics of VIV (vortex induced vibration) problems. The efficiency of the adaptive method is demonstrated by the 2D simulation of a freely falling plate with the comparison to other numerical methods. Finally, the freely falling and rising 3D sphere are computed and compared with corresponding experimental measurement.

Numerical investigation of BB-AMR scheme using entropy production as refinement criterion

ABSTRACTIn this work, a parallel finite volume scheme on unstructured meshes is applied to fluid flow for multidimensional hyperbolic system of conservation laws. It is based on a block-based adaptive mesh refinement strategy which allows quick meshing and easy parallelisation. As a continuation and as an extension of a previous work, the useful numerical density of entropy production is used as mesh refinement criterion combined with a local time-stepping method to preserve the computational time. Then, we numerically investigate its efficiency through several test cases with a confrontation with exact solution or experimental data.

Parallel implicit anisotropic block-based adaptive mesh refinement finite-volume scheme for the study of fully resolved oblique shock wave reflections

A new parallel, fully implicit, anisotropic block-based adaptive mesh refinement (AMR) finite-volume scheme is proposed, described and demonstrated for the prediction of laminar, compressible, viscous flows associated with unsteady oblique shock reflection processes. The proposed finite-volume method provides numerical solutions to the Navier–Stokes equations governing the flow of polytropic gases in an accurate and efficient manner on two-dimensional, body-fitted, multi-block meshes consisting of quadrilateral computational cells. The combination of the anisotropic AMR and parallel implicit time-marching techniques adopted is shown to readily facilitate the simulation of challenging and complex shock interaction problems, as represented by the time-accurate predictions of unsteady oblique shock reflection configurations with fully resolved internal shock structures.

Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows

In this work, we present a fast and parallel finite volume scheme on unstructured meshes applied to complex fluid flow. The mathematical model is based on a three-dimensional compressible low Mach two-phase flows model, combined with a linearised ‘artificial pressure’ law. This hyperbolic system of conservation laws allows an explicit scheme, improved by a block-based adaptive mesh refinement scheme. Following a previous one-dimensional work, the useful numerical density of entropy production is used as mesh refinement criterion. Moreover, the computational time is preserved using a local time-stepping method. Finally, we show through several test cases the efficiency of the present scheme on two- and three-dimensional dam-break problems over an obstacle.

Efficient GPU-Implementation of Adaptive Mesh Refinement for the Shallow-Water Equations

The shallow-water equations model hydrostatic flow below a free surface for cases in which the ratio between the vertical and horizontal length scales is small and are used to describe waves in lakes, rivers, oceans, and the atmosphere. The equations admit discontinuous solutions, and numerical solutions are typically computed using high-resolution schemes. For many practical problems, there is a need to increase the grid resolution locally to capture complicated structures or steep gradients in the solution. An efficient method to this end is adaptive mesh refinement (AMR), which recursively refines the grid in parts of the domain and adaptively updates the refinement as the simulation progresses. Several authors have demonstrated that the explicit stencil computations of high-resolution schemes map particularly well to many-core architectures seen in hardware accelerators such as graphics processing units (GPUs). Herein, we present the first full GPU-implementation of a block-based AMR method for the second-order Kurganov---Petrova central scheme. We discuss implementation details, potential pitfalls, and key insights, and present a series of performance and accuracy tests. Although it is only presented for a particular case herein, we believe our approach to GPU-implementation of AMR is transferable to other hyperbolic conservation laws, numerical schemes, and architectures similar to the GPU.

High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows

A high-order, central, essentially non-oscillatory (CENO), finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of the Navier-Sto...

High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows

A high-order, central, essentially non-oscillatory (CENO), finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of the Navier–Stokes equations on body-fitted multi-block mesh. In contrast to other ENO schemes which require reconstruction on multiple stencils, the proposed CENO method uses a hybrid reconstruction approach based on a fixed central stencil. This feature is crucial to avoiding the complexities associated with multiple stencils of ENO schemes, providing high-order accuracy at relatively lower computational cost as well as being very well suited for extension to unstructured meshes. The spatial discretization of the inviscid (hyperbolic) fluxes combines an unlimited high-order k-exact least-squares reconstruction technique following from the optimal central stencil with a monotonicity-preserving, limited, linear, reconstruction algorithm. This hybrid reconstruction procedure retains the unlimited high-order k-exact reconstruction for cells in which the solution is fully resolved and reverts to the limited lower-order counterpart for cells with under-resolved/discontinuous solution content. Switching in the hybrid procedure is determined by a smoothness indicator. The high-order viscous (elliptic) fluxes are computed to the same order of accuracy as the hyperbolic fluxes based on a k-order accurate cell interface gradient derived from the unlimited, cell-centred, reconstruction. A somewhat novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady mesh adaptation. The proposed numerical procedure is thoroughly analyzed for advection–diffusion problems characterized by the full range of Péclet numbers, and its predictive capabilities are also demonstrated for several inviscid and laminar flows. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown. Moreover, the ability to perform mesh refinement in regions of under-resolved and/or non-smooth solution content is also demonstrated.

Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method

The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769–795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air–water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier–Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model’s capability in predicting the interaction dynamics of rising bubbles.

Numerical solution scheme for inert, disperse, and dilute gas-particle flows

A multi-velocity formulation is proposed for the solution of an Eulerian representation of an inert, disperse, and dilute particle-phase of a gas-particle flow. Single-velocity formulations are capable of predicting regions of zero particle concentration but are problematic with crossing particle trajectories or compression waves. The multi-velocity formulation described here can account for crossing particle trajectories by splitting the particle-phase into distinct velocity families which are transported separately in the flow. Switching of the particle families at solid boundaries and due to momentum transfer with the gas-phase is conducted in a manner that enforces conservation of mass, momentum, and energy. This numerical method is combined with a parallel block-based adaptive mesh refinement algorithm that is very effective in treating problems with disparate length scales. The block-based data structure lends itself naturally to domain decomposition and thereby enables efficient and scalable implementations of the algorithm on distributed-memory multi-processor architectures. Numerical results are described to demonstrate the capabilities of the approach for predicting gas-particle flows.

A Diffusion Synthetic Acceleration Method for Block Adaptive Mesh Refinement

A multidimensional block-based adaptive mesh refinement (BAMR) method for the neutral particle transport equation with diamond and linear discontinuous spatial differencing was developed several years ago. This method was implemented in the PARallel TIme-dependent SN (PARTISN) deterministic transport code. However, the only source acceleration method available with BAMR was transport synthetic acceleration. Although the block-based adaptive mesh is orthogonal, the individual mesh cells may not be simply connected. Because of this lack of simple connectivity, development of a fully consistent diffusion synthetic acceleration (DSA) method has not been possible. This paper describes the development of a DSA method based upon an additive correction to the scalar flux iterate after a transport sweep. This DSA equation is differenced using a vertex-centered diffusion discretization that is diamond-like and may be characterized as “partially” consistent. It does not appear algebraically possible to derive a diffusion discretization that is fully consistent with diamond transport differencing on AMR meshes. The diffusion matrix is symmetric positive definite, and the DSA method is effective for most applications. This BAMR-DSA solver has been implemented and tested in two dimensions for rectangular (X-Y) and cylindrical (R-Z) geometries. As expected, results confirm that a partially consistent BAMR-DSA method will introduce instabilities for extreme cases (e.g., scattering ratios approaching 1.0 with optically thick cells), but for most realistic problems, e.g., the iron-water shielding problem, the BAMR-DSA method provides an effective acceleration method.

A Block Adaptive Mesh Refinement Algorithm for the Neutral Particle Transport Equation

We describe the development and implementation of a block-based adaptive mesh refinement (AMR) algorithm for solving the discrete ordinates neutral particle transport equation. AMR algorithms allow mesh refinement in areas of interest without requiring the extension of this refinement throughout the entire problem geometry, minimizing the number of computational cells required for calculations. The block-based AMR algorithm described here is a hybrid between traditional cell or patch-based approaches and is designed to allow an efficient parallel solution of the transport equation while still reducing the cell count.This paper discusses the data structure implementation and CPU/memory efficiency for our Block AMR method, the equations and procedures used in mapping edge fluxes between blocks of different refinement levels for both diamond and linear discontinuous spatial discretizations, effects of AMR on mesh convergence, and our approach to parallelization. Comparisons between our Block AMR method and a traditional single-level mesh are presented for a sample brachytherapy problem. The Block AMR results are shown to be significantly faster for this problem (on at least a few processors), while still returning an accurate solution.