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.     

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.     

求解二维三温辐射热传导方程组的一种网格自适应方法

在求解二维三温辐射热传导方程组的过程中,设计了一类新的基于Hessian矩阵的网格自适应算法.数值实验结果表明,与现在流行的基于一阶导数或通量的网格自适应技术相比,该算法能够大幅改善系统的能量守恒误差,并具有较高的整体计算效率.     

Adaptive mesh redistribution method for domains with complex boundaries

An adaptive structured mesh redistribution method (ASMRM) that permits smooth transition from non-uniformly distributed boundary points to solution-adaptive interior points and enables the resolution of complex flow in the complex boundary region as well as away from the boundary is proposed. It is a variant of the traditional variational technique. It involves a combination of static and dynamic monitor functions, the former for mesh distribution in the vicinity of a complex boundary and the latter for mesh adaption with the evolving solution elsewhere. Its effectiveness is demonstrated on some example problems, and it is then applied to a chevron nozzle. The proposed method is shown to be capable of generating a mesh with a good balance of orthogonality and smoothness in the entire domain.     

Output error estimation for summation-by-parts finite-difference schemes

The paper develops a posteriori error estimates of integral output functionals for summation-by-parts finite-difference methods. The error estimates are based on the adjoint-weighted residual method and take advantage of a variational interpretation of summation-by-parts discretizations. The estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators. For smooth boundary-value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined. The theory is verified using linear boundary-value problems and the Euler equations.     

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.     

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.     

A Goal-Oriented Adaptive Moreau-Yosida Algorithm for Control- and State-Constrained Elliptic Control Problems

In this work, we develop an adaptive algorithm for solving elliptic optimal control problems with simultaneously appearing state and control constraints. The algorithm combines a Moreau-Yosida technique for handling state constraints with a semi-smooth Newton method for solving the optimality systems of the regularized sub-problems. The state and adjoint variables are discretized using continuous piecewise linear finite elements while a variational discretization concept is applied for the control. To perform the adaptive mesh refinements cycle we derive local error estimators which extend the goal-oriented error approach to our setting. The performance of the overall adaptive solver is assessed by numerical examples.     

A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows

A stencil adaptive lattice Boltzmann method (LBM) is developed in this paper. It incorporates the stencil adaptive algorithm developed by Ding and Shu [26] for the solution of Navier–Stokes (N–S) equations into the LBM calculation. Based on the uniform mesh, the stencil adaptive algorithm refines the mesh by two types of 5-points symmetric stencils, which are used in an alternating sequence for increased refinement levels. The two types of symmetric stencils can be easily combined to form a 9-points symmetric structure. Using the one-dimensional second-order interpolation recently developed by Wu and Shu [27] along the straight line and the D2Q9 model, the adaptive LBM calculation can be effectively carried out. Note that the interpolation coefficients are only related to the lattice velocity and stencil size. Hence, the simplicity of LBM is not broken down and the accuracy is maintained. Due to the use of adaptive technique, much less mesh points are required in the simulation as compared to the standard LBM. As a consequence, the computational efficiency is greatly enhanced. The numerical simulation of two dimensional lid-driven cavity flows is carried out. Accurate results and improved efficiency are reached. In addition, the steady and unsteady flows over a circular cylinder are simulated to demonstrate the capability of proposed method for handling problems with curved boundaries. The obtained results compare well with data in the literature.     

基于变分原理的二维热传导方程差分格式

研究二维热传导方程的差分数值模拟.用变分原理在不规则结构网格上建立热流通量形式的差分格式.将热流通量作为未知函数求泛函极值,并与温度函数联立求解.克服通常九点格式用插值方法计算网格边界上的热传导系数和网格结点上的温度所引入的误差.     

Voigt-function evaluation using a two-dimensional interpolation scheme

A method of evaluating the Voigt function for the shape of spectral lines is described. The method relies on interpolation from a two-dimensional table using simply-transformed coordinates. High-level mathematical functions are avoided wherever possible, and no series summations are made. The relative error on a 400 by 100 table is less than 3 × 10^?3 over the region where the interpolation method is faster than other expansions. Computing speed is comparable to that of built-in scientific functions.     

An efficient moving mesh spectral method for the phase-field model of two-phase flows

We develop in this paper a moving mesh spectral method for the phase-field model of two-phase flows with non-periodic boundary conditions. The method is based on a variational moving mesh PDE for the phase function, coupled with efficient semi-implicit treatments for advancing the mesh function, the phase function and the velocity and pressure in a decoupled manner. Ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method.     

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.     

一类基于RBF插值的守恒重映算法

为保证重映过程的高守恒精度和单调性,并且在间断处具有极高的分辨率,基于径向基函数(RBF)插值方法构造了一类适用于任意网格的RBF守恒重映算法,通过计算守恒误差测试重映算法的守恒精度。将该方法用于光滑函数和含有间断的函数,并与其它守恒重映方法比较,表明该方法数值结果较好。     

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.     

耦合径向基函数与多项式基函数的无网格方法

耦合径向基函数和多项式基函数,形成一种新的近似函数.该近似函数对散乱分布的离散数据点进行逼近时,只需节点信息,不需要划分网格.详细描述了耦合近似函数的建立、属性、插值行为及其形函数和形函数导数的性质.最后引入修正变分原理和单位分解积分技术求解边值问题,并给出了计算实例,表明耦合径向基函数和多项式基函数是一种非常有效的方法.     

Output-based space–time mesh adaptation for the compressible Navier–Stokes equations

This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space–time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space–time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier–Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space–time meshes are shown to be relatively insensitive to the starting mesh.     

中厚板弯曲问题的Kriging插值无网格法

基于Reissner-Mindlin板弯曲理论,将Kriging插值无网格法应用于中厚板弯曲问题,推导相应的离散方程.该方法可以只依赖于一组离散的节点建立试函数,有效地避免了复杂的网格划分和网格畸变的影响.相对于无网格法中常用的移动最小二乘近似而言,滑动Kriging插值法的形函数满足Kronecker delta函数性质,可以直接施加本质边界条件.算例分析表明,用Kriging插值无网格法分析中厚板弯曲问题,具有效率高,精度高和易于实现等优点.     

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.     

Reduced surface point selection options for efficient mesh deformation using radial basis functions

Previous work by the authors has developed an efficient method for using radial basis functions (RBFs) to achieve high quality mesh deformation for large meshes. For volume mesh deformation driven by surface motion, the RBF system can become impractical for large meshes due to the large number of surface (control) points, and so a particularly effective data reduction scheme has been developed to vastly reduce the number of surface points used. The method uses a chosen error function on the surface mesh to select a reduced subset of the surface points; this subset contains a sufficiently small number of points so as to make the volume deformation fast, and a correction function is used to correct those surface points not included. Hence, the scheme is split such that both parts are working on appropriate problems. RBFs are an excellent way of finding smooth orthogonality preserving global deformations, but are less suitable for enforcing an exact geometry for a large number of points, while a simpler approach is ideal for diffusing small changes evenly but has quality (and possibly expense) drawbacks if used for the entire volume. However, alternatives exist for the error function used to select the reduced data set, so here a comparison is made between three different options: the surface error function, the unit function and the power function. Tests run on structured and unstructured meshes show that the surface error function gives the lowest errors, but this also requires a deformed surface shape to be known in advance of the simulation. The unit and power functions both avoid the need for a deformed surface, and the unit function is shown to be superior.