Adaptive moving mesh methods for two-dimensional resistive magneto-hydrodynamic PDE models

An adaptive moving mesh technique is applied to magneto-hydrodynamics (MHD) model problem. The moving mesh strategy is based on the approach proposed in Li et al. [Li R, Tang T, Zhang P. Moving mesh methods in multiple dimensions based on harmonic maps. J Comput Phys 2001;170:562-88] to separate the mesh-moving and PDE evolution at each time step. The Magneto-hydrodynamic equations are discretized by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation with the directional splitting monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations

This paper is devoted to the numerical simulation of the Navier–Stokes–Korteweg equations, a phase-field model for water/water-vapor two-phase flows. We develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher-order partial–differential operator that represents capillarity. We introduce a new refinement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions. Finally, we present several numerical examples in two and three dimensions that illustrate the effectiveness and robustness of our approach.     

A solution of one-dimensional moving boundaries problems by the finite-element method

A finite-element formulation for the solution of one-dimensional problems with moving boundaries is considered. The movement of the boundaries may not be known a priori. The term “conformal mesh” is defined, and a mesh undergoing conformal deformation is shown to have some numerical advantages. The formulation is applied to linear shape functions. An auxiliary method is suggested, which replaces the repetitive solution of a system of algebraic equations with a one-time solution of an eigenvalue problem. The examples used as models for this method are one- and two-phase Stefan problems and the problem of thermal displacement and stresses in a wall undergoing ablation.     

Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment.     

一种用于分布参数系统移动边界的自适应剖分数值方法

本文提出通过对具有移动边界分布参数系统中的移动边界的一步预报,自适应生成剖分网格,然后通过系统的焓方程应用有限元方法求解,得到具有移动边界的分布参数系统的数值解.结果表明,这种方法较好地解决了用有限元方法求解该类系统的数值解时遇到的移动边界附近数值解精度与网格剖分过细所导致的计算量过大的矛盾.为具有移动边界的分布参数系统的建模和仿真提供了一种有效的数值计算方法,同时也为研究系统的控制、估计、辨识等问题的数值方法打下了基础.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.     

A study of moving mesh methods applied to a thin flame propagating in a detonator delay element

We study the application of moving mesh methods to a one-dimensional (time dependent) detonator delay element problem. We consider moving mesh methods based on the equidistribution principle derived by Huang et al. [1]. Adaptive mesh methods have been widely used recently to solve time dependent partial differential equations having large solution gradients. Significant improvements in accuracy and efficiency are achieved by adapting the nodes (mesh points) so that they are concentrated about areas of large solution variations. Each system of equations for the moving mesh methods is solved in conjunction with the detonator problem. In this paper, the system of ordinary differential equations that results (after discretising in space) is solved using the double precision version of the stiff ordinary differential equation solver DASSL. The numerical results clearly demonstrate that the moving mesh methods are capable of tracking the deflagration wave as it travels down the detonator delay element more accurately and more efficiently than a fixed mesh method.     

A comprehensive generalized mesh system for CFD applications

The numerical approaches used for the solution of governing equations of fluid flow are dictated highly by the topology of the domain discretization. Two of the most commonly used discretization approaches are the structured and unstructured topologies. This paper describes the discretization of the domain using generalized elements with an arbitrary number of nodes to combine the advantages of both the structured and unstructured methodologies. Numerical algorithms for the solution of the governing equations for generalized mesh, an approach for handling mesh movement applicable to rotating machineries, and the application of this framework for overset meshes to handle moving body problems are discussed. A library-based approach has been adopted for the implementation of overset capability for the framework. The results from the application of this framework for various applications are presented.     

A moving mesh approach to an ice sheet model

A moving mesh approach to the numerical modelling of problems governed by nonlinear time-dependent partial differential equations (PDEs) is applied to the numerical modelling of glaciers driven by ice diffusion and accumulation/ablation. The primary focus of the paper is to demonstrate the numerics of the moving mesh approach applied to a standard parabolic PDE model in reproducing the main features of glacier flow, including tracking the moving boundary (snout). A secondary aim is to investigate waiting time conditions under which the snout moves.     

Parallel computational methods for 3D simulation of a parafoil with prescribed shape changes

In this paper we describe parallel computational methods for 3D simulation of the dynamics and fluid dynamics of a parafoil with prescribed, time-dependent shape changes. The mathematical model is based on the time-dependent, 3D Navier-Stokes equations governing the incompressible flow around the parafoil and Newton's law of motion governing the dynamics of the parafoil, with the aerodynamic forces acting on the parafoil calculated from the flow field. The computational methods developed for these 3D simulations include a stabilized space-time finite element formulation to accommodate for the shape changes, special mesh generation and mesh moving strategies developed for this purpose, iterative solution techniques for the large, coupled nonlinear equation systems involved, and parallel implementation of all these methods on scalable computing systems such as the Thinking Machines CM-5. As an example, we report 3D simulation of a flare maneuver in which the parafoil velocity is reduced by pulling down the flaps. This simulation requires solution of over 3.6 million coupled, nonlinear equations at every time step of the simulation.     

A Parallel Implementation of ALE Moving Mesh Technique for FSI Problems using OpenMP

This paper investigates a high performance implementation of an Arbitrary Lagrangian Eulerian moving mesh technique on shared memory systems using OpenMP environment. Moving mesh techniques are considered an integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and Fluid Structure Interaction (FSI). The moving mesh technique adopted in this work is based on the notion of nodes relocation, subjected to a certain evolution as well as constraint conditions. A conjugate gradient method augmented with preconditioning is employed for solution of the resulting system of equations. The proposed algorithm, initially, reorders the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh scheme. Numerical simulations are conducted on the multicore AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D problems. The quality of generated meshes is checked by comparing the element Jacobians in the reference and current meshes, and by keeping track of the change in the interior angles in triangles and tetrahedrons. Overall, 51 and 72% efficiencies in terms of speedup are achieved for both the parallel mesh reordering and ALE moving mesh algorithms, respectively.     

An arbitrary Lagrangian Eulerian formulation for residual distribution schemes on moving grids

The arbitrary Lagrangian Eulerian formulation is derived for the residual distribution method on moving meshes. The system of Euler equations is discretized on moving meshes and in case of deforming meshes a geometrical source term has to be taken into account. A conservative linearization guarantees the conservation property of the discretized equations.From the geometric conservation law we obtain the appropriate integration points in time for the cell fluctuation and a guideline for how to distribute the geometrical source term.Testcases include the flow around a transonic oscillating airfoil and a convected vortex. In the first case a rigidly moving mesh is employed, while in the other testcase a deforming mesh is used to investigate the influence of the geometrical source term on the solution.     

On Resistive MHD Models with Adaptive Moving Meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics.     

Direct Discretization Method for the Cahn–Hilliard Equation on an Evolving Surface

We propose a simple and efficient direct discretization scheme for solving the Cahn–Hilliard (CH) equation on an evolving surface. By using a conservation law and transport formulae, we derive the CH equation on evolving surfaces. An evolving surface is discretized using an unstructured triangular mesh. The discrete CH equation is defined on the surface mesh and its dual surface polygonal tessellation. The evolving triangular surfaces are then realized by moving the surface nodes according to a given velocity field. The proposed scheme is based on the Crank–Nicolson scheme and a linearly stabilized splitting scheme. The scheme is second-order accurate, with respect to both space and time. The resulting system of discrete equations is easy to implement, and is solved by using an efficient biconjugate gradient stabilized method. Several numerical experiments are presented to demonstrate the performance and effectiveness of the proposed numerical scheme.     

Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws

In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method.     

基于物理模型的烟雾模拟

在分析研究流体物理特性算法基础上,提出一种新的烟雾模拟实现方法。该方法基于物理模型的求解简化方程模拟烟雾的动态变化过程。模型中采用了非粘性欧拉方程,通常它比其他用粘性Navier-Stoke方程建模的更适合用来对气体进行建模并且减少计算量。实验验证该模型还可以正确处理烟雾与移动的物体之间的相互作用。     

基于非结构滑移网格技术的旋翼型无人机空气动力学并行数值模拟方法

在现代飞行器设计中,数值模拟方法以低成本、高效率和高灵活性等优点成为研究飞行器空气动力学的重要方法.在旋翼型无人机流场模拟中,由于旋翼与机身存在相互作用,为获得精确模拟结果需要对整个无人机的流场进行模拟,因此,有效地模拟旋翼与机身的相对运动是实现成功模拟的关键步骤,这使得此类模拟问题极具挑战性.文章设计了一套求解旋翼型无人机空气动力学数值模拟问题的基于非结构滑移网格技术的高可扩展并行计算方法.该方法对控制方程的离散,在空间方向采用非结构移动网格有限元方法,时间推进采用全隐式二阶向后差分格式,最后采用一种并行Newton-Krylov-Schwarz方法求解离散后的非线性方程组.作为应用,文章对一个真实旋翼型无人机模型在悬停状态下的外流场进行了数值模拟,获得了一些非常详细的流场信息.数值结果显示,算法在天河2号上使用4 096个处理器核时仍具有接近线性的并行加速比,这为下一步开展旋翼型无人机的高保真度快速模拟奠定了良好的基础.     

An immersed boundary technique for simulating complex flows with rigid boundary

A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.