High-order adaptive methods for parabolic systems

We consider the adaptive solution of parabolic partial differential systems in one and two space dimensions by finite element procedures that automatically refine and coarsen computational meshes, vary the degree of the piecewise polynomial basis and, in one dimension, move the computational mesh. Two-dimensional meshes of triangular, quadrilateral, or a mixture of triangular and quadrilateral elements are generated using a finite quadtree procedure that is also used for data management. A posteriori estimates, used to control adaptive enrichment, are generated from the hierarchical polynomial basis. Temporal integration, within a method-of-lines framework, uses either backward difference methods or a variant of the singly implicit Runge-Kutta (SIRK) methods. A high-level user interface facilitates use of the adaptive software.     

Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method  and  for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.     

二维矢量边界推进非结构网格生成方法

提出二维矢量边界推进生成非结构三角形网格方法并证明其可行性.根据流场边界尺度布置边界节点并运用矢量边界推进方法生成背景网格,运用符号面积函数和概率筛选方法布置初始点阵,提出Spring-Laplace方法优化节点位置,同时利用边交换技术优化网格结构.该方法可包含任意点源、线源和内嵌边界,可自由进行局部自适应加密或稀疏,实现任意平面域内与尺度要求一致的高效光滑三角网格剖分.     

反应流体的移动网格动理学格式

在移动网格上构造一种反应流的动理学格式.首先利用BGK模型推导含化学反应的流体力学方程组,并利用其积分形式构造移动网格上离散格式,再利用自适应移动网格方法得到网格速度,最后利用时间精确的动理学数值方法构造数值通量,得到移动网格单元上新的物理量.一维与二维的数值实验表明这种格式同时具有高精度、高分辨率的特点.     

A parallel solution – adaptive method for three-dimensional turbulent non-premixed combusting flows

A parallel adaptive mesh refinement (AMR) algorithm is proposed and applied to the prediction of steady turbulent non-premixed compressible combusting flows in three space dimensions. The parallel solution-adaptive algorithm solves the system of partial-differential equations governing turbulent compressible flows of reactive thermally perfect gaseous mixtures using a fully coupled finite-volume formulation on body-fitted multi-block hexahedral meshes. The compressible formulation adopted herein can readily accommodate large density variations and thermo-acoustic phenomena. A flexible block-based hierarchical data structure is used to maintain the connectivity of the solution blocks in the multi-block mesh and to facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. For calculations of near-wall turbulence, an automatic near-wall treatment readily accommodates situations during adaptive mesh refinement where the mesh resolution may not be sufficient for directly calculating near-wall turbulence using the low-Reynolds-number formulation. Numerical results for turbulent diffusion flames, including cold- and hot-flow predictions for a bluff-body burner, are described and compared to available experimental data. The numerical results demonstrate the validity and potential of the parallel AMR approach for predicting fine-scale features of complex turbulent non-premixed flames.     

Adaptive mesh refinement for stochastic reaction–diffusion processes

We present an algorithm for adaptive mesh refinement applied to mesoscopic stochastic simulations of spatially evolving reaction–diffusion processes. The transition rates for the diffusion process are derived on adaptive, locally refined structured meshes. Convergence of the diffusion process is presented and the fluctuations of the stochastic process are verified. Furthermore, a refinement criterion is proposed for the evolution of the adaptive mesh. The method is validated in simulations of reaction–diffusion processes as described by the Fisher–Kolmogorov and Gray–Scott equations.     

A particle–particle hybrid method for kinetic and continuum equations

We present a coupling procedure for two different types of particle methods for the Boltzmann and the Navier–Stokes equations. A variant of the DSMC method is applied to simulate the Boltzmann equation, whereas a meshfree Lagrangian particle method, similar to the SPH method, is used for simulations of the Navier–Stokes equations. An automatic domain decomposition approach is used with the help of a continuum breakdown criterion. We apply adaptive spatial and time meshes. The classical Sod’s 1D shock tube problem is solved for a large range of Knudsen numbers. Results from Boltzmann, Navier–Stokes and hybrid solvers are compared. The CPU time for the hybrid solver is 3–4 times faster than for the Boltzmann solver.     

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauß–Seidel method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.     

基于任意拉格朗日—欧拉型移动网格的反应流广义黎曼问题方法

提出一种在自由重映移动网格下的广义黎曼问题方法模拟反应流.该方法基于显式的自由重映移动网格广义黎曼问题的解.为保证在时间和空间上的高精度，应用广义黎曼问题方法构造数值通量.为保证反应区的高分辨率，采用变分法生成自适应移动网格.该方法不仅能够保证网格质量，而且能有效地避免任意拉格朗日—欧拉方法中由于显式重映过程而带来的数值误差.包括CJ爆轰及不稳定爆轰的数值实验说明该格式的精确性和鲁棒性，证明这种移动网格下的二阶广义黎曼问题方法可以较好地捕捉反应流的间断与光滑结构.     

二维非线性抛物方程广义差分法/有限体积法的自适应计算

给出三角网上二维非线性抛物方程广义差分法(有限体积法)的一种基于残量估计的后验误差估计,并在此基础上设计了自适应计算方案,以适应物理解在时空的大梯度变化.提出了适合发展方程自适应计算的三角网数据结构(不是树状结构)和灵活的局部粗化算法.     

The adaptive immersed interface finite element method for elliptic and Maxwell interface problems

We propose self-adaptive finite element methods with error control for solving elliptic and electromagnetic problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. New error indicators are introduced to control the error due to non-body-fitted meshes. Flexible h-adaptive strategies are developed, which can be systematically extended to a large class of interface problems. Extensive numerical experiments are performed to support the theoretical results and to show the competitive behavior of the adaptive algorithm even for interfaces involving corner or tip singularities.     

Adaptive variational curve smoothing based on level set method

This paper presents an adaptive method for variational curve smoothing based on level set implementation. A suitable cost functional is minimized via solving the derived Euler–Lagrangian equation, of which the discretization is conducted on unstructured triangular meshes by employing a simple and effective finite volume scheme. Through adaptive refinement of the mesh, the geometry features of the given curve can be well resolved in a cost-effective way. Various numerical experiments demonstrate the effectiveness and efficiency of the proposed approach.     

自适应间断有限元方法求解双曲守恒律方程

研究自适应Runge-Kutta间断Galerkin (RKDG)方法求解双曲守恒律方程组,并提出两种生成相容三角形网格的自适应算法.第一种算法适用于规则网格,实现简单、计算速度快.第二种算法基于非结构网格,设计一类基于间断界面的自适应网格加密策略,方法灵活高效.两种方法都具有令人满意的计算效果,而且降低了RKDG的计算量.     

A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties

This work proposes a domain adaptive stochastic collocation approach for uncertainty quantification, suitable for effective handling of discontinuities or sharp variations in the random domain. The basic idea of the proposed methodology is to adaptively decompose the random domain into subdomains. Within each subdomain, a sparse grid interpolant is constructed using the classical Smolyak construction [S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963) 240–243], to approximate the stochastic solution locally. The adaptive strategy is governed by the hierarchical surpluses, which are computed as part of the interpolation procedure. These hierarchical surpluses then serve as an error indicator for each subdomain, and lead to subdivision whenever it becomes greater than a threshold value. The hierarchical surpluses also provide information about the more important dimensions, and accordingly the random elements can be split along those dimensions. The proposed adaptive approach is employed to quantify the effect of uncertainty in input parameters on the performance of micro-electromechanical systems (MEMS). Specifically, we study the effect of uncertain material properties and geometrical parameters on the pull-in behavior and actuation properties of a MEMS switch. Using the adaptive approach, we resolve the pull-in instability in MEMS switches. The results from the proposed approach are verified using Monte Carlo simulations and it is demonstrated that it computes the required statistics effectively.     

Numerical simulation of laminar flames at low Mach number by adaptive finite elements

We present an adaptive finite-element method for the simulation of reactive flows in general domains including reliable error control for quantities of physical interest. On the basis of computable a posteriori error bounds the mesh is locally adjusted by a hierarchical feedback process yielding economical mesh-size distributions for a prescribed maximum number of cells. The key feature is the computational evaluation of the sensitivity of the error with respect to the local residuals by solving an associated global dual problem. This general approach is applied for computing three examples of flames in two dimensions with an increasing degree of complexity, namely a simple diffusion flame described by the flame-sheet model, an ozone decomposition flame and a methane flame in a complex burner geometry using a detailed reaction mechanism.     

Comparison of multimesh hp-FEM to interpolation and projection methods for spatial coupling of thermal and neutron diffusion calculations

Multiphysics solution challenges are legion within the field of nuclear reactor design and analysis. One major issue concerns the coupling between heat and neutron flow (neutronics) within the reactor assembly. These phenomena are usually very tightly interdependent, as large amounts of heat are quickly produced with an increase in fission events within the fuel, which raises the temperature that affects the neutron cross section of the fuel. Furthermore, there typically is a large diversity of time and spatial scales between mathematical models of heat and neutronics. Indeed, the different spatial resolution requirements often lead to the use of very different meshes for the two phenomena. As the equations are coupled, one must take care in exchanging solution data between them, or significant error can be introduced into the coupled problem. We propose a novel approach to the discretization of the coupled problem on different meshes based on an adaptive multimesh higher-order finite element method (hp-FEM), and compare it to popular interpolation and projection methods. We show that the multimesh hp-FEM method is significantly more accurate than the interpolation and projection approaches considered in this study.     

自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

BGK方法在三维非结构网格上的初步应用

将BGK计算方法从二维拓展到三维并且应用于三维非结构网格,具有重要的理论价值和实用价值.采用旋转局部座标的方法,发展了一种针对三维非结构网格的BGK计算方法.在计算过程中,将最小二乘法应用于三维非结构网格的导数计算.对三维激波管和三维欠膨胀垂直冲击射流等两个算例进行了细致分析.这两个算例的计算结果表明,该方法在三维非结构网格上的初步应用是成功的 关键词： 气动BGK方法 三维 非结构网格     

任意网格重映的样条逼近算法

在大变形流体力学问题的数值模拟中,任何方法都必须考虑网格重分或网格自适应,只要改动网格就涉及重分,或自适应后从旧的、扭曲的网格到新网格的守恒量重映,包括质量、动量和能量.在研究样条函数逼近的基础上,给出一种物理量重映的对结构网格和非结构网格均适应的算法,并给出了数值结果.     

九点格式中网格边上切向流的计算——节点自适应加权格式

针对网格扭曲的不同情形,直接考虑网格边上切向流的离散.基于扩散方程法向流连续的条件,给出离散法向流的构造,导出扭曲网格上九点计算格式中网格边上离散切向流的表达式,从而推导出加权系数的计算公式,适应于各种扭曲的网格.数值结果表明,与九点格式中节点量简单加权的方法相比,基于网格边离散切向流的节点自适应加权九点格式的精度有明显改进,迭代求解次数减少,计算效率明显提高.