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.     

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.     

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.     

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.