Adaptive Mesh Refinement Technique Research Articles

A NEW CODE FOR NONLINEAR FORCE-FREE FIELD EXTRAPOLATION OF THE GLOBAL CORONA

Reliable measurements of the solar magnetic field are still restricted to the photosphere, and our present knowledge of the three-dimensional coronal magnetic field is largely based on extrapolation from photospheric magnetogram using physical models, e.g., the nonlinear force-free field (NLFFF) model as usually adopted. Most of the currently available NLFFF codes have been developed with computational volume like Cartesian box or spherical wedge while a global full-sphere extrapolation is still under developing. A high-performance global extrapolation code is in particular urgently needed considering that Solar Dynamics Observatory (SDO) can provide full-disk magnetogram with resolution up to $4096\times 4096$. In this work, we present a new parallelized code for global NLFFF extrapolation with the photosphere magnetogram as input. The method is based on magnetohydrodynamics relaxation approach, the CESE-MHD numerical scheme and a Yin-Yang spherical grid that is used to overcome the polar problems of the standard spherical grid. The code is validated by two full-sphere force-free solutions from Low & Lou's semi-analytic force-free field model. The code shows high accuracy and fast convergence, and can be ready for future practical application if combined with an adaptive mesh refinement technique.

Spectral WENO schemes with Adaptive Mesh Refinement for models of polydisperse sedimentation

AbstractThe sedimentation of a polydisperse suspension with particles belonging to N size classes (species) can be described by a system of N nonlinear, strongly coupled scalar first‐order conservation laws. Its solutions usually exhibit kinematic shocks separating areas of different composition. Based on the so‐called secular equation [J. Anderson, Lin. Alg. Appl. 246, 49–70 (1996)], which provides access to the spectral decomposition of the Jacobian of the flux vector for this class of models, Bürger et al. [J. Comput. Phys. 230, 2322–2344 (2011)] proposed a spectral weighted essentially non‐oscillatory (WENO) scheme for the numerical solution of the model. It is demonstrated that the efficiency of this scheme can be improved by the technique of Adaptive Mesh Refinement (AMR), which concentrates computational effort on zones of strong variation. Numerical experiments for the cases N = 4 and N = 7 are presented.

Sliding Grids and Adaptive Grid Refinement for RANS Simulation of Ship-Propeller Interaction

The paper describes a computational approach for a numerical propulsion test. Key techniques concern the computation of free-surface viscous flows around propellers using the sliding grid technique and accurate wave capturing around the hull. An advanced numerical technique resolves very small-scale flow motion such as tip vortex. Efficiency and accuracy are balanced using an adaptive mesh refinement technique. This approach is validated against the KCS test case proposed for the Tokyo 2005 and Gothenburg 2010 CFD validation workshops.

Development of MHD solver based on an adaptive mesh refinement technique

Abstract An accurate model is developed for numerical simulation of 3D magnetohydrodynamic multiphase flows at a low magnetic Reynolds number, in which a consistent and conservative scheme has been applied to calculate the Lorentz force and continuous surface force model is used to calculate the surface tension based on the volume of fluid method for capturing the interface using a PLIC (Piecewise Linear Interface Calculation) to reconstruct the interface. An adaptive mesh refinement technique is used to improve the accuracy and efficiency by automatically refining (relaxing) the meshes near (outside of) the surface, the Hartmann layers and side layers, respectively. Multi-grid method is utilized to accelerate the convergence of Poisson equations. Numerical results are compared with the theoretical or experimental results to validate and verify the numerical solver. A bubble rising in a stagnant liquid metal with/without the influence of magnetic field is simulated and compared with the experimental works.

Block-structured Adaptive Mesh Refinement - Theory, Implementation and Application

Structured adaptive mesh refinement (SAMR) techniques can enable cutting-edge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for distributed memory computers. An overview of the background of commonly used finite volume discretizations for gas dynamics is included and typical benchmarks to quantify accuracy and performance of the dynamically adaptive code are discussed. Large-scale simulations of shock-induced realistic combustion in non-Cartesian geometry and shock-driven fluid-structure interaction with fully coupled dynamic boundary motion demonstrate the applicability of the discussed techniques for complex scenarios.

Solving the Euler and Navier–Stokes equations by the AMR–CESE method

The application of the AMR–CESE method for solving the Euler and Navier–Stokes equations is presented. The method is a combination of the space–time conservation element and solution element (CESE) method and the adaptive mesh refinement (AMR) technique. Its implementation is based on modification of the original CESE method and utilization of the framework of a parallel-AMR package PARAMESH to manage the block-AMR grid system. Furthermore, a variable time step algorithm is introduced to realize adaptivity of the method in both space and time. A test suite of standard problems for Euler and Navier–Stokes flows are calculated and the results show high resolution, high efficiency and versatility of the method for both shock capture and resolving boundary layer.

Adaptive Grid Use in Air Quality Modeling

The predictions from air quality models are subject to many sources of uncertainty; among them, grid resolution has been viewed as one that is limited by the availability of computational resources. A large grid size can lead to unacceptable errors for many pollutants formed via nonlinear chemical reactions. Further, insufficient grid resolution limits the ability to perform accurate exposure assessments. To address this issue in parallel to increasing computational power, modeling techniques that apply finer grids to areas of interest and coarser grids elsewhere have been developed. Techniques using multiple grid sizes are called nested grid or multiscale modeling techniques. These approaches are limited by uncertainty in the placement of finer grids since pertinent locations may not be known a priori, loss in solution accuracy due to grid boundary interface problems, and inability to adjust to changes in grid resolution requirements. A different approach to achieve local resolution involves using dynamic adaptive grids. Various adaptive mesh refinement techniques using structured grids as well as mesh enrichment techniques on unstructured grids have been explored in atmospheric modeling. Recently, some of these techniques have been applied to full blown air quality models. In this paper, adaptive grid methods used in air quality modeling are reviewed and categorized. The advantages and disadvantages of each adaptive grid method are discussed. Recent advances made in air quality simulation owing to the use of adaptive grids are summarized. Relevant connections to adaptive grid modeling in weather and climate modeling are also described.

A 2D Meshless Solver Based on AUSM+ and MUSCL Scheme

When solving Euler equations of fluid dynamics around complex configurations, a meshless method which doesn’t use the information of the meshes may have the advantages to simulate the flow field. In this paper, a simple and accurate meshless algorithm based on AUSM+ and MUSCL scheme has been described. To assess the capability for capturing shock position and the resolvability of the complex shock waves, a number of typical test cases are calculated with the meshless method. The results obtained are in good agreement with the exact solution or some high-resolution numerical method as well as flux-corrected transport (FCT) algorithm with adaptive mesh refinement (AMR) technique.

Magnetic Reconnection Under Solar Coronal Conditions with the 2.5D AMR Resistive MHD Model

The evolutionary process of magnetic reconnection under solar coronal conditions is investigated with our recently developed 2.5D adaptive mesh refinement (AMR) resistive magneto hydrodynamics (MHD) model. We reveal the successive fragmentation and merging of plasmoids in a long-thin current sheet with Lundquist number Rm = 5.0 × 104. It is found that several big magnetic islands are formed eventually, with many slow-mode shocks bounding around the outflow regions. The multi-scale hierarchical-like structures of the magnetic reconnection are well resolved by the model and the AMR technique of the model can capture many fine pictures (e.g., the near-singular diffusion regions) of the development and simultaneously it can save a great deal of computing resources.

적응적 메쉬세분화기법과 분할격자기법을 이용한 극한 도시홍수 실험 모의

적응적 메쉬세분화기법과 분할격자기법을 적용한 2차원 천수방정식모형을 활용하여 구조물을 고려한 극한 홍수 실험을 모의하였다. 본 연구에 사용된모형은 두 격자생성 기법을함께 사용함으로서 복잡한 경계를보다 적은 격자로 효율적으로 표현하는 것이 가능하며, 동적 적응 메쉬세분화기법을 사용하여 흐름이 빠르게 변하는 영역에서 정확도를 유지하면서도 효율적으로 계산하는 것이 가능하다. HLLC 리만근사해법과 MUSCL 기법을 적용하여 시공간상에서 2차정도를 유지하며, 댐붕괴파와 같은 불연속적인 흐름을 정확하게 모의할 수 있다. 모형의 검증을 위해 IMPACT 프로젝트에서 수행한 도시지역 극한홍수실험을 모의하였다. 실험결과와 모의결과가 양호하게 일치하는 것을 확인하였으며, 천이류 현상과 함께 구조물에 의한 홍수파 전달 양상이 의미있는 수준으로 모의된 것을 확인하였다. 또한 분할격자기법의 사용으로 모델의 격자 민감도가 향상되었다. 본 모델은 댐붕괴와 같이 내수침수현상이 지배적이지 않은 도시범람을 모의하는데 활용할 수 있을 것으로 기대된다. Two-dimensional shallow water model based on the cut cell and the adaptive mesh refinement techniques is presented in this paper. These two mesh generation methods are combined to facilitate modeling of complex geometries. By using dynamically adaptive mesh, the model can achieve high resolution efficiently at the interface where flow changes rapidly. The HLLC Reimann solver and the MUSCL method are employed to calculate advection fluxes with numerical stability and precision. The model was applied to simulate the extreme urban flooding experiments performed by the IMPACT (Investigation of Extreme Flood Processes and Uncertainty) project. Simulation results were in good agreement with observed data, and transient flows as well as the impact of building structures on flood waves were calculated with accuracy. The cut cell method eased the model sensitivity to refinement. It can be concluded that the model is applicable to the urban flood simulation in case the effects of sewer and stormwater drainage system on flooding are relatively small like the dam brake.

Implementation and application of adaptive mesh refinement for thermochemical mantle convection studies

Numerical modeling of mantle convection is challenging. Owing to the multiscale nature of mantle dynamics, high resolution is often required in localized regions, with coarser resolution being sufficient elsewhere. When investigating thermochemical mantle convection, high resolution is required to resolve sharp and often discontinuous boundaries between distinct chemical components. In this paper, we present a 2‐D finite element code with adaptive mesh refinement techniques for simulating compressible thermochemical mantle convection. By comparing model predictions with a range of analytical and previously published benchmark solutions, we demonstrate the accuracy of our code. By refining and coarsening the mesh according to certain criteria and dynamically adjusting the number of particles in each element, our code can simulate such problems efficiently, dramatically reducing the computational requirements (in terms of memory and CPU time) when compared to a fixed, uniform mesh simulation. The resolving capabilities of the technique are further highlighted by examining plume‐induced entrainment in a thermochemical mantle convection simulation.

Numerical simulation of current sheet formation in a quasiseparatrix layer using adaptive mesh refinement

The formation of a thin current sheet in a magnetic quasiseparatrix layer (QSL) is investigated by means of numerical simulation using a simplified ideal, low-β, MHD model. The initial configuration and driving boundary conditions are relevant to phenomena observed in the solar corona and were studied earlier by Aulanier et al. [Astron. Astrophys. 444, 961 (2005)]. In extension to that work, we use the technique of adaptive mesh refinement (AMR) to significantly enhance the local spatial resolution of the current sheet during its formation, which enables us to follow the evolution into a later stage. Our simulations are in good agreement with the results of Aulanier et al. up to the calculated time in that work. In a later phase, we observe a basically unarrested collapse of the sheet to length scales that are more than one order of magnitude smaller than those reported earlier. The current density attains correspondingly larger maximum values within the sheet. During this thinning process, which is finally limited by lack of resolution even in the AMR studies, the current sheet moves upward, following a global expansion of the magnetic structure during the quasistatic evolution. The sheet is locally one-dimensional and the plasma flow in its vicinity, when transformed into a comoving frame, qualitatively resembles a stagnation point flow. In conclusion, our simulations support the idea that extremely high current densities are generated in the vicinities of QSLs as a response to external perturbations, with no sign of saturation.

An Adaptive Multimoment Global Model on a Cubed Sphere

Abstract An adaptive global shallow-water model is proposed on cubed-sphere grid using the multimoment finite volume scheme and the Berger–Oliger adaptive mesh refinement (AMR) algorithm that was originally designed for a Cartesian grid. On each patch of the cubed-sphere grid, the curvilinear coordinates are constructed in a way that the Berger–Oliger algorithm can be applied directly. Moreover, an algorithm to transfer data across neighboring patches is proposed to establish a practical integrated framework for global AMR computation on the cubed-sphere grid. The multimoment finite volume scheme is adopted as the fluid solver and is essentially beneficial to the implementation of AMR on the cubed-sphere grid. The multimoment interpolation based on both volume-integrated average (VIA) and point value (PV) provides the compact reconstruction that makes the present scheme very attractive not only in dealing with the artificial boundaries between different patches but also in the coarse–fine interpolations required in the AMR computations. The single-cell-based reconstruction avoids involving more than two nesting levels during interpolations. The reconstruction profile of constrained interpolation profile–conservative semi-Lagrangian scheme with third-order polynomial function (CIP-CSL3) is adopted where the slope parameter provides a flexible and convenient switching to get the desired numerical properties, such as high-order (fourth order) accuracy, monotonicity, and positive definiteness. Numerical experiments with typical benchmark tests for both advection equation and shallow-water equations are carried out to evaluate the proposed model. The numerical errors and the corresponding CPU times of numerical experiments on uniform and adaptive meshes verify the performance of the proposed model. Compared to the uniformly refined grid, the AMR technique is able to achieve the similar numerical accuracy with less computational cost.

PROTOSTELLAR COLLAPSE OF MAGNETO-TURBULENT CLOUD CORES: SHAPE DURING COLLAPSE AND OUTFLOW FORMATION

We investigate protostellar collapse of molecular cloud cores by numerical simulations, taking into account turbulence and magnetic fields. By using the adaptive mesh refinement technique, the collapse is followed over a wide dynamic range from the scale of a turbulent cloud core to that of the first core. The cloud core is lumpy in the low density region owing to the turbulence, while it has a smooth density distribution in the dense region produced by the collapse. The shape of the dense region depends mainly on the mass of the cloud core; a massive cloud core tends to be prolate while a less massive cloud core tends to be oblate. In both cases, anisotropy of the dense region increases during the isothermal collapse. The minor axis of the dense region is always oriented parallel to the local magnetic field. All the models eventually yield spherical first cores supported mainly by the thermal pressure. Most of turbulent cloud cores exhibit protostellar outflows around the first cores. These outflows are classified into two types, bipolar and spiral flows, according to the morphology of the associated magnetic field. Bipolar flow often appears in the less massive cloud core. The rotation axis of the first core is oriented parallel to the local magnetic field for bipolar flow, while the orientation of the rotation axis from the global magnetic field depends on the magnetic field strength. In spiral flow, the rotation axis is not aligned with the local magnetic field.

Full Particle-in-Cell Simulation on a Small-Scale Magnetosphere Using Uniform and Nested Grid Systems

Solar wind interaction with an artificial magnetosphere is investigated by means of a full particle-in-cell simulation. The resultant momentum transfer of solar wind plasmas may provide the propulsive force for a magnetic sail, which is a potential next-generation interplanetary flight system. These simulations are performed using two different simulation codes. One is a traditional code employing a uniform grid system, and the other is a newly developed code with an adaptive mesh refinement (AMR) technique. Even in a small magnetosphere having a scale smaller than the ion inertia length, ions are scattered at the front of the magnetosphere. In this region, an electron-scale current structure is observed, and the electromagnetic interaction with the coil current density, which createsthe magnetosphere, causes a propulsive force. The current density structure observed in the AMR simulation is in good agreement with that resulting from the traditional code. The AMR code is expected to be a powerful tool to demonstrate this solar wind interaction under realistic conditions at a reasonable numerical cost. c

A Multi-Scale Electromagnetic Particle Code with Adaptive Mesh Refinement and Its Parallelization

Abstract Space plasma phenomena occur in multi-scale processes from the electron scale to the magnetohydrodynamic scale. In order to investigate such multi-scale phenomena including plasma kinetic effects, we started to develop a new electromagnetic Particle-In-Cell (PIC) code with Adaptive Mesh Refinement (AMR) technique. AMR can realize high-resolution calculation saving computer resources by generating and removing hierarchical cells dynamically. In the parallelization, we adopt domain decomposition method and for good locality preserving and dynamical load balancing, we will use the Morton ordered curve. In the PIC method, particle calculation occupies most of the total calculation time. In our AMR-PIC code, time step intervals are also refined. To realize the load balancing between processes in the domain decomposition scheme, it is the most essential to consider the number of particle calculation loops for each cell among all hierarchical levels as a work weight for each processor. Therefore, we calculate the work weights based on the cost of particle calculation and hierarchical levels of each cell. Then we decompose the domain according to the Morton curve and the work weight, so that each processor has approximately the same amount of work. By performing a simple one-dimensional simulation, we confirmed that the dynamic load balancing is achieved and the computation time is reduced by introducing the dynamic domain decomposition scheme.

Numerical study about the nonlinear instability of the Sweet-Parker thin current sheet with shearing flows

以Harris Sheet作为初始条件, 使用数值模拟的方法, 研究了二级磁岛不稳定重联的一些性质. 在模拟中随着初始扰动的加入, Harris Sheet将演化到非线性阶段, 形成更薄的有剪切速度的电流片, 并伴有一级磁岛产生. 当Lundquist数大于或等于10⁵时, 非均匀剪切速度的Sweet-Parker电流片开始不稳定, 并有二级磁岛出现. 不稳定Sweet-Parker电流片对应的临界长宽比为65. Lundquist数越大, 演化形成的Sweet-Parker电流片越薄, 更多的二级磁岛将出现, 且沿电流片两边向外喷出, 随时间变大, 相互合并. 二级磁岛的出现使重联率增大, 并与Lundquist数之间不再满足S^-1/2的关系, 而似乎对它的依赖关系不明显.

Three-dimensional Visualization of Cosmological and Galaxy Formation Simulations

AbstractOur understanding of the structuring of the Universe from large-scale cosmological structures down to the formation of galaxies now largely benefits from numerical simulations. The RAMSES code, relying on the Adaptive Mesh Refinement technique, is used to perform massively parallel simulations at multiple scales. The interactive, immersive, three-dimensional visualization of such complex simulations is a challenge that is addressed using the SDvision software package. Several rendering techniques are available, including ray-casting and isosurface reconstruction, to explore the simulated volumes at various resolution levels and construct temporal sequences. These techniques are illustrated in the context of different classes of simulations. We first report on the visualization of the HORIZON Galaxy Formation Simulation at MareNostrum, a cosmological simulation with detailed physics at work in the galaxy formation process. We then carry on in the context of an intermediate zoom simulation leading to the formation of a Milky-Way like galaxy. Finally, we present a variety of simulations of interacting galaxies, including a case-study of the Antennae Galaxies interaction.

Grounding line migration in an adaptive mesh ice sheet model

Grounding line migration is a key process affecting the stability of marine ice sheets such as the West Antarctic ice sheet. Recent studies have shown that ice sheet models employing a fixed spatial grid (such as are commonly used for whole ice sheet simulations) cannot be used to solve this problem in a robust manner. We have developed a one‐dimensional (vertically integrated) “shelfy stream” ice sheet model that employs the adaptive mesh refinement (AMR) technique to bring higher resolution to spatially and temporally evolving subregions of the model domain. A higher‐order solver, the piecewise parabolic method (PPM), is used to compute the thickness evolution. Both AMR and PPM extend readily to greater than one dimension and could be used in full ice sheet simulations. We demonstrate that this approach can bring improvements in terms of accuracy and consistency in both grounded ice sheet and ice stream/ice shelf simulations, given the appropriate choice of refinement criteria. In particular, we demonstrate that AMR, in conjunction with a parameterization for subgrid scale grounding line position, can produce predictions of grounding line migration.

Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes

Standard and goal-oriented adaptive mesh refinement (AMR) techniques are presented for the linear Boltzmann transport equation. A posteriori error estimates are employed to drive the AMR process and are based on angular-moment information rather than on directional information, leading to direction-independent adapted meshes. An error estimate based on a two-mesh approach and a jump-based error indicator are compared for various test problems. In addition to the standard AMR approach, where the global error in the solution is diminished, a goal-oriented AMR procedure is devised and aims at reducing the error in user-specified quantities of interest. The quantities of interest are functionals of the solution and may include, for instance, point-wise flux values or average reaction rates in a subdomain. A high-order (up to order 4) Discontinuous Galerkin technique with standard upwinding is employed for the spatial discretization; the discrete ordinates method is used to treat the angular variable.