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.     

Managing complex data and geometry in parallel structured AMR applications

Adaptive mesh refinement (AMR) is an increasingly important simulation methodology for many science and engineering problems. AMR has the potential to generate highly resolved simulations efficiently by dynamically refining the computational mesh near key numerical solution features. AMR requires more complex numerical algorithms and programming than uniform fixed mesh approaches. Software libraries that provide general AMR functionality can ease these burdens significantly. A major challenge for library developers is to achieve adequate flexibility to meet diverse and evolving application requirements. In this paper, we describe the design of software abstractions for general AMR data management and parallel communication operations in SAMRAI, an object-oriented C++ structured AMR (SAMR) library developed at Lawrence Livermore National Laboratory (LLNL). The SAMRAI infrastructure provides the foundation for a variety of diverse application codes at LLNL and elsewhere. We illustrate SAMRAI functionality by describing how its unique features are used in these codes which employ complex data structures and geometry. We highlight capabilities for moving and deforming meshes, coupling multiple SAMR mesh hierarchies, and immersed and embedded boundary methods for modeling complex geometrical features. We also describe how irregular data structures, such as particles and internal mesh boundaries, may be implemented using SAMRAI tools without excessive application programmer effort. This work was performed under the auspices of the US Department of Energy by University of California Lawrence Livermore National Laboratory under contract number W-7405-Eng-48 and is released under UCRL-JRNL-214559.     

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.     

Flow field computation for the high voltage gas blast circuit breaker with the moving boundary

We studied the gas dynamics for the ideal gas in the simplified high voltage (HV) gas blast circuit breaker with the moving boundary. The piston and the electric contact are moving. Since the boundary is moving, it is difficult for the ordinary finite difference (FD) method or the finite element (FE) method to compute the solution. For the purpose of numerical simplicity and efficiency, we introduced an upwind meshfree scheme which is an excellent scheme for the time varying domain. Despite the low coding and computational cost, the numerical simulation is successfully conducted. Our method is even more efficient when considering a three-dimensional computation with a moving boundary.     

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.     

Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection–diffusion equations

In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.     

Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids

When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentric pressures. Second, meshes must align with potentially complex moving surfaces and boundaries, necessitating continuous remeshing. Unfortunately, sacrificing mesh quality in favour of speed yields inaccurate velocities and simulation artifacts. We describe how to eliminate the boundary‐matching constraint by adapting recent embedded boundary techniques to tetrahedra, so that neither air nor solid boundaries need to align with mesh geometry. This enables the use of high quality, arbitrarily graded, non‐conforming Delaunay meshes, which are simpler and faster to generate. Temporal coherence can also be exploited by reusing meshes over adjacent timesteps to further reduce meshing costs. Lastly, our free surface boundary condition eliminates the spurious currents that previous methods exhibited for slow or static scenarios. We provide several examples demonstrating that our efficient tetrahedral embedded boundary method can substantially increase the flexibility and accuracy of adaptive Eulerian fluid simulation.     

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.     

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.     

Numerical Studies of Adaptive Finite Element Methods for Two Dimensional Convection-Dominated Problems

In this paper, we study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction problem in the two-dimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids), we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical approximation is sensitive to the symmetry of the grid in the region where the solution is smooth. On the basis of these two observations, we develop a multilevel-homotopic-adaptive finite element method (MHAFEM) by combining streamline diffusion finite element method, anisotropic mesh adaptation, and the homotopy of the diffusion coefficient. We use numerical experiments to demonstrate that MHAFEM can efficiently capture boundary or interior layers and produce accurate solutions.     

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.     

Modelling and simulation of moving contact line problems with wetting effects

This paper presents a numerical scheme for computing moving contact line flows with wetting effects. The numerical scheme is based on Arbitrary Lagrangian Eulerian (ALE) finite elements on moving meshes. In the computations, the wetting effects are taken into account through a weak enforcement of the prescribed equilibrium contact angle into the model equations. The equilibrium contact angle is included in the variational form of the model by replacing the curvature with Laplace Beltrami operator and integration by parts. This weak implementation allows that the contact angle determined by the numerical scheme differs from the equilibrium value and develops a certain dynamics. The Laplace Beltrami operator technique with an interface/boundary resolved mesh is well-suited for describing the dynamic contact angle observed in experiments. We consider the spreading and the pendant liquid droplets to investigate this implementation of the contact angle. It is shown that the dynamic contact angle tends to the prescribed equilibrium contact angle when time goes to infinity. However, the dynamics of the contact angle is influenced by the slip at the moving contact line. This work has been partially supported by the German Research Foundation (DFG) through the grant To143/9.     

Development of adaptive mesh refinement scheme for engine spray simulations

Spray modeling is a critical component to engine combustion and emissions simulations. Accurate spray modeling often requires a fine computational mesh for better numerical resolutions. However, computations with a fine mesh will require extensive computer time. This study developed a methodology that uses a locally refined mesh in the spray region. The fine mesh virtually moves with the liquid spray. Such adaptive mesh refinement can enable greater resolution of the liquid-gas interaction while incurring only a small increase in the total number of computational cells. The present study uses an h-refinement adaptive method. A face-based approach is used for the inter-level boundary condition. The prolongation and restriction procedure preserves conservation of properties in performing grid refinement/coarsening. The refinement criterion is based on the total mass of liquid drops and fuel vapor in each cell. The efficiency and accuracy of the present adaptive mesh refinement scheme is described in the paper. Results show that the present scheme can achieve the same level of accuracy in modeling sprays with significantly lower computational cost as compared to a uniformly fine mesh.     

Array-based,parallel hierarchical mesh refinement algorithms for unstructured meshes

In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE’s using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in-memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework “Mesh Oriented dAtaBase” (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. We also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.     

Moving Mesh Method with Error-Estimator-Based Monitor and Its Applications to Static Obstacle Problem

The main objective of this work is to demonstrate that sharp a posteriori error estimators can be employed as appropriate monitor functions for moving mesh methods. We illustrate the main ideas by considering elliptic obstacle problems. Some important issues such as how to derive the sharp estimators and how to smooth the monitor functions are addressed. The numerical schemes are applied to a number of test problems in two dimensions. It is shown that the moving mesh methods with the proposed monitor functions can effectively capture the free boundaries of the elliptic obstacle problems and reduce the numerical errors arising from the free boundaries.     

低雷诺数湍流模型的网格特征

低雷诺数模型目前主要应用于二维简单流动的数值仿真中,为研究该湍流模型在三维复杂流动计算中的网格特征,选取不同系列的车身面网格尺寸、车身壁面第一层边界层与壁面法向高度以及边界层层数等3组网格参数,利用ANSYS对阶背式MIRA模型外流场进行数值仿真.数值仿真结果与风洞试验的结果对比表明：数值计算得到的车身表面平均y+值随面网格尺寸增加而呈现减小趋势;网格方案对气动力因数和车身表面压力因数分布影响显著,气动阻力因数仿真值与试验值误差的变化区间为0.83%~7.93%,气动升力因数误差变化区间为10%~104%;气动阻力因数和气动升力因数均随着边界层层数的增加而增大,边界层层数为5时可以得到兼顾气动力因数精度和车身表面压力因数精度的较优仿真结果.     

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.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.