An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space–time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.     

Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing

Metric tensors play a key role to control the generation of unstructured anisotropic meshes. In practice, the most well established error analysis enables to calculate a metric tensor on an element basis. In this paper, we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools. First, the unit mesh metric is defined and well justified on a node basis, by using the statistical concept of length distribution tensors. Then, the interpolation error analysis is performed on the projected approximate scalar field along the edges. The error estimate is established on each edge whatever the dimension is. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. Several examples of interpolation error are proposed as well as preliminary results of anisotropic adaptation for interface and free surface problem using a level set method.     

Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling in three steps.First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation.3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

双参数曲面的泡泡网格化方法

为将双参数曲面离散成高质量的网格,首先在参数域内利用各向异性的非均匀泡泡布点方法优化布点,然后用各向异性Delaunay三角化方法将参数域网格化,最后用映射法得到双参数曲面的离散网格.参数域中的节点由二阶黎曼度量矩阵控制,该度量矩阵由三维曲面的网格度量矩阵和曲面参数方程的梯度计算得到.数值算例表明,泡泡布点法在参数域上能生成满足度量矩阵要求的节点集,将节点连接成网格并投影回曲面,所得曲面网格具有很高的质量.     

一种基于预估-修正策略的非定常流网格自适应加密/稀疏方法

给出一种非定常流动数值模拟的网格自适应处理方法.在"求解流动方程-自适应调整网格"的流程中,引入预估-修正步.根据自适应周期内每个时间步上的流场预估解,计算单元上的事后误差估算值.建立考虑解演变的网格自适应指示器,并进行多层次单元加密-稀疏的动态网格自适应处理.在自适应网格上重新计算流场.每个自适应周期中,流动演变区域的网格获得加密;而前一个周期中的特征现象已离开区域的网格被稀疏.应用边界非协调的当地DFD(Domain-Free Discretization)方法求解流动方程.为验证网格自适应处理方法,针对静止圆柱和自推进游鱼的流动进行了数值实验.     

Minimization of the Truncation Error by Grid Adaptation

A new grid adaptation strategy, which minimizes the truncation error of a pth-order finite difference approximation, is proposed. The main idea of the method is based on the observation that the global truncation error associated with discretization on nonuniform meshes can be minimized if the interior grid points are redistributed in an optimal sequence. The method does not explicitly require the truncation error estimate, and at the same time, it allows one to increase the design order of approximation globally by one, so that the same finite difference operator reveals superconvergence properties on the optimal grid. Another very important characteristic of the method is that if the differential operator and the metric coefficients are evaluated identically by some hybrid approximation, then the single optimal grid generator can be employed in the entire computational domain independently of points where the hybrid discretization switches from one approximation to another. Generalization of the present method to multiple dimensions is presented. Numerical calculations of several one-dimensional and one two-dimensional test examples demonstrate the performance of the method and corroborate the theoretical results.     

A priori mesh quality metric error analysis applied to a high-order finite element method

Characterization of computational mesh’s quality prior to performing a numerical simulation is an important step in insuring that the result is valid. A highly distorted mesh can result in significant errors. It is therefore desirable to predict solution accuracy on a given mesh. The HiFi/SEL high-order finite element code is used to study the effects of various mesh distortions on solution quality of known analytic problems for spatial discretizations with different order of finite elements. The measured global error norms are compared to several mesh quality metrics by independently varying both the degree of the distortions and the order of the finite elements. It is found that the spatial spectral convergence rates are preserved for all considered distortion types, while the total error increases with the degree of distortion. For each distortion type, correlations between the measured solution error and the different mesh metrics are quantified, identifying the most appropriate overall mesh metric. The results show promise for future a priori computational mesh quality determination and improvement.     

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.     

A second-order 3D electromagnetics algorithm for curved interfaces between anisotropic dielectrics on a Yee mesh

A new frequency-domain electromagnetics algorithm is developed for simulating curved interfaces between anisotropic dielectrics embedded in a Yee mesh with second-order error in resonant frequencies. The algorithm is systematically derived using the finite integration formulation of Maxwell’s equations on the Yee mesh. Second-order convergence of the error in resonant frequencies is achieved by guaranteeing first-order error on dielectric boundaries and second-order error in bulk (possibly anisotropic) regions. Convergence studies, conducted for an analytically solvable problem and for a photonic crystal of ellipsoids with anisotropic dielectric constant, both show second-order convergence of frequency error; the convergence is sufficiently smooth that Richardson extrapolation yields roughly third-order convergence. The convergence of electric fields near the dielectric interface for the analytic problem is also presented.     

A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose–Einstein condensates

Numerical computations of stationary states of fast-rotating Bose–Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method with mesh adaptivity by metric control, as an alternative approach to the commonly used high-order (finite difference or spectral) approximation methods. The mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states: an imaginary time propagation method and a Sobolev gradient descent method. We first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful. Mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases. Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state. Considerable computational time saving is obtained compared to uniform mesh computations. The new method is applied to compute difficult cases relevant for physical experiments (large nonlinear interaction constant and high rotation rates).     

Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which does not satisfy the inf-sup condition. The two-scale method consists of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure is obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation $h =\mathcal{O}(H^2)$. Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.     

Transient adaptivity applied to two-phase incompressible flows

An anisotropic adaptation process is applied to a three-dimensional incompressible two-phase flow solver. The solver uses a level set/finite element method on unstructured tetrahedral meshes. We show how the level set function can be used to build an anisotropic mesh with good properties. Some computations with a strong transient character and large densities ratios (1/1000) are presented. We show that the efficiency of the computations can be deeply enhanced by mesh adaptations.     

An alternative solution of the scattering from an anisotropic cylindrical dielectric shell

A method based on the approximate wave functions for anisotropic media and the mode-matching approach is developed to solve the problem of the electromagnetic scattering from an anisotropic cylindrical dielectric shell. The cylindrical shell is assumed to be infinite in length, and it is illuminated by a plane wave or a cylindrical wave from a line source. The problem is two-dimensional and the solutions to both types of polarization (TE and TM) are presented. The validity of this solution is verified by comparing the numerical results with those in literatures and the previous calculations based on the exact wave functions for anisotropic media. Numerical results show the higher computational efficiency of the present method for bounded anisotropic media.     

流场自适应叉树网格数值模拟三维复杂超声速流场

建立了基于分区结构网格的三维贴体叉树形网格的数据结构,并阐述了在此基础上进行自适应分裂/合并判别方法.为节省网格量和保证流场结构捕捉质量,提出对自适应程度进行区域性控制,以及对流场结构进行"保护"性预加密的优化方式.通过应用该网格对三维复杂超声速流场算例的计算,证明该方法对网格加密控制方便,对流场结构分辨率高.     

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.     

Coupling multimodelling with local mesh refinement for the numerical computation of laminar flames

In this work, we propose a twofold adaptive method for the simulation of steady reactive flows. On the one hand, locally refined meshes are used. On the other hand, two types of diffusion models are applied: a simple Fick law and a more accurate multicomponent diffusion model. The diffusion model is changed locally throughout the computational domain. An analytically derived a posteriori error estimator provides reliable information on where to refine the mesh and where to choose the appropriate diffusion model. During the adaptation process, discretization and modelling errors are equilibrated. Numerical results are presented for ozone and hydrogen laminar flames.     

Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping

The dynamic modification of the computational grid due to element displacement, deformation and edge swapping is described here in terms of suitably-defined continuous (in time) alterations of the geometry of the elements of the dual mesh. This new interpretation allows one to describe all mesh modifications within the arbitrary Lagrangian Eulerian framework, thus removing the need to interpolate the solution across computational meshes with different connectivity. The resulting scheme is by construction conservative and it is applied here to the solution of the Euler equations for compressible flows in two spatial dimensions. Preliminary two dimensional numerical simulations are presented to demonstrate the soundness of the approach. Numerical experiments show that this method allows for large time steps without causing element invalidation or tangling and at the same time guarantees high quality of the mesh elements without resorting to global re-meshing techniques, resulting in a very efficient solver for the analysis of e.g. fluid-structure interaction problems, even for those cases that require large mesh deformations or changes in the domain topology.     

湍流燃烧自适应求解技术在低排放燃烧室计算中的应用

基于各向异性非结构网格生成技术, 开发了面向复杂几何和复杂湍流燃烧问题的自适应求解算法, 并进行了程序代码的可靠性验证工作, 展示了各向异性网格自适应算法在降低问题求解规模、提高火焰面和流场计算精度等方面的优势.应用该自适应求解技术准确捕捉到了一维预混层流火焰、二维对冲火焰和三维本生灯湍流火焰的流场信息, 火焰面附近的温度、速度、组分等物理量与实验值吻合很好.对一款富油-快速混合-贫油(rich-burn, quick-mix, lean-burn, RQL)低排放发动机燃烧室进行了计算分析, 发现了燃烧室内的热声不稳定现象.