基于非结构自适应网格技术的高超声速流动数值模拟

发展了一种基于非结构网格的自适应方法,对高超声速无粘流场进行了数值模拟。根据流场参数的变化梯度确定加密边,由加密准则进行自适应网格剖分后得到分布合理的较密网格。通过预先生成的初始极密表面网格将边界的加密点投影到边界上,使得边界保持初始外形。通过求解三维Euler 方程,对三维双椭球高超声速绕流问题进行了数值模拟,计算结果和实验数据相吻合,表明了该文所建立方法的正确性和可靠性。     

On the discrete core of quadrilateral mesh refinement

We present a new approach to quadrilateral mesh refinement, which reduces the problem to its structural core. The resulting problem formulation belongs to a class of discrete problems, network‐flow problems, which has been thoroughly investigated and is well understood. The network‐flow model is flexible enough to allow the simultaneous incorporation of various aspects such as the control of angles and aspect ratios, local density control, and templates (meshing primitives) for the internal refinement of mesh elements. We show that many different variants of the general quadrilateral mesh‐refinement problem are covered. In particular, we present a novel strategy, which provably finds a conformal refinement unless there is none. Copyright © 1999 John Wiley & Sons, Ltd.     

An adaptive mesh refinement technique for two-dimensional shear band problems

We have developed an adaptive mesh refinement technique that rezones the given domain for a fixed number of quadrilateral elements such that fine elements are generated within the severely deformed region and coarse elements elsewhere. Loosely speaking, the area of an element is inversely proportional to the value of the deformation measure at its centroid. Here we use the temperature rise at a material point to gauge its deformations which is reasonable for the shear band problem since the material within the shear band is deformed intensely and is heated up significantly. It is shown that the proposed mesh refinement technique is independent of the initial starting mesh, and that the use of an adaptively refined mesh gives thinner shear bands, and shaper temperature rise and the growth of the second invariant of the plastic strain-rate within the band as compared to that for a fixed mesh having the same number of nodes. The method works well even when the deformation localizes into more than one narrow region.     

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub?domain smoothed Galerkin method is proposed to integrate the advantages of mesh?free Galerkin method and FEM. Arbitrarily shaped sub?domains are predefined in problems domain with mesh?free nodes. In each sub?domain, based on mesh?free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high?order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub?domain by dividing the sub?domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub?domains. The mesh?free properties of Galerkin method are retained in each sub?domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub?domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.     

A feed-back approach to error control in finite element methods: application to linear elasticity

Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.     

Smart computational fracture of materials and structures

The use of the finite element method for complex engineering problems is now common. To ease the burden on the engineer the development of smart or adaptive computational methods is now required to model complex problems. In this paper we investigate the development of an adaptive finite element method for fracture-related problems. The adaptive method involves various stages which include the finite element analysis, error estimation/indication, mesh refinement and fracture/failure analysis in a loop. Some simple error estimators, based on stress projection, are used to investigate the adaptive finite element process. Element refinement is based on three schemes; the first and second are a simple and hierarchical refinement scheme with transitioning which avoids the need for constraint equations between element boundaries. Another scheme based on constraint equations between elements is also examined. The energy norm is used to estimate the element error. The software has the ability to introduce a discrete fracture in the structure according to standard fracture analysis practice. Crack tip parameters are calculated using a least-squares fit of the displacements into the asymptotic crack tip displacement field. Some simple examples are used to investigate the adaptive process, its behavior and some of the practical problems encountered. The convergence and equilibrium of the adaptive process, in terms of global error in the energy norm, are investigated. In the example the same problem is analyzed using both a fine computational grid and a coarse one. The coarse mesh is then adapted using the three different procedures available. The estimated error in the solution and the stress intensity are shown against the number of elements and number of iterations. Some further areas of research in adaptive finite element analysis are discussed.     

A contribution to error estimation and mapping algorithms for a hierarchical adaptive FE-strategy in finite elastoplasticity

The aim of this contribution is the presentation of an adaptive finite element procedure for the solution of geometrically and physically non-linear problems in structural mechanics. Within this context, the attention is mainly directed on the error estimation and hierarchical strategies for mesh refinement and coarsening in the case of finite elasto-plastic deformations. An important but sensitive aspect of adaptation approaches of the space discretization is the calculation of mechanical field variables for the modified mesh. Procedures of mesh refinement and coarsening imply the determination of strains, stresses and internal variables at the nodes and the Gauss points of new elements based on the transfer of the required data from the former mesh. In order to improve the efficiency as well as the convergence behaviour of the adaptive FE process an approach of data transfer primarily related to nodal values is presented. It is characterized by solving the initial value problem not only at the Gauss points but additionally at the nodes of the elements.     

Announcements

A new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities. In it, the singularity is eliminated form the computational domain in an exact fashion. This is in contrast to other common methods, such as those which use a refined mesh in the singularity region, or those which use special singular finite elements. In them, the singularity is treated as a part of the numerical scheme. The new method is illustrated on an elliptic differential equation in a domain containing a re-entrant corner. Numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region. Both methods require about the same computing time.     

自适应有限元线法在二维无穷域问题中的应用

无穷域问题广泛存在于实际工程中,半解析、半离散的数值计算方法—有限元线法（Finite ElementMethod of Lines,简称FEMOL）对其具有较好的适应性。在已有的映射型FEMOL无穷单元理论的基础上,基于单元能量投影（Element Energy Projection,简称EEP）法的自适应FEMOL被应用于二维无穷域问题的求解。用户只需输入稀疏的初始网格和误差限,算法即自动生成优化的FEMOL网格,该网格上常规单元和无穷单元的FEMOL解均按最大模度量满足给定误差限。文中首先介绍二维FEMOL的原理策略、无穷单元的构建,然后概述基于EEP法的自适应FEMOL算法,并讨论其对无穷域问题的适用性,之后对圆柱绕流的Poisson方程问题、带孔无穷大板单向拉伸的弹性力学平面问题、受圆形均布荷载半空间体的三维轴对称问题进行了自适应分析,最终不仅给出了满足误差限的函数（位移）解,也给出了具有优良性态的导数（应力）解,从而为无穷域问题的求解提供了一种高效可靠的新途径。     

Automated computer simulation of two-dimensional elastostatic problems by the finite element method

A software system for the automated computer simulation of two-dimensional elastostatic problems by the finite element method is described. This system consists of two parts, automated mesh generation and automated stress analysis. The mesh generation is based on a method in which equilateral triangles are generated successively in the unmeshed region. Automated mesh refinement is carried out in the latter part of the simulation process. The stress analysis is based on the assumed stress hybrid method and the successive over relaxation method. The computer program developed for this paper can generate a succession of increasingly refined triangular meshes until a certain mesh convergence criterion is achieved. The mesh convergence criterion is based on a comparison of nodal stresses in successive analyses until all the stress differences are within a specified tolerance.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

An evaluation of equivalent-single-layer and layerwise theories of composite laminates

A review of equivalent-single-layer and layerwise laminate theories is presented and their computational models are discussed. The layerwise theory advanced by the author is reviewed and a variable displacement finite element model and the mesh superposition techniques are described. The variable displacement finite elements contain several different types of assumed displacement fields. By choosing appropriate terms from the multiple displacement field, an entire array of elements with different orders of kinematic refinement can be formed. The variable kinematic finite elements can be conveniently connected together in a single domain for global-local analyses, where the local regions are modeled with refined kinematic elements. In the finite element mesh superposition technique an independent overlay mesh is superimposed on a global mesh to provide localized refinement for regions of interest regardless of the original global mesh topology. Integration of these two ideas yields a very robust and economical computational tool for global-local analysis to determine three-dimensional effects (e.g. stresses) within localized regions of interest in practical laminated composite structures.     

3D mesh refinement in compliance with a specified node spacing function

A simple and efficient refinement procedure for the three-dimensional tetrahedral element mesh based on successive bisection of edges is proposed. The quality of the elements generated can be guaranteed if the subdivision is performed in the sequence according to the length of the line segments to be divided. Such an order of priority can be determined by a simple sorting process on all the line segments for which refinement is needed. This list of ordered line segments has to be updated from time to time to take into account of the new line segments generated during the subdivision process. From the examples studied, the CPU time for mesh refinement bears a linear relationship with the number of elements generated, with a refinement rate of more than 50 000 elements per second on a IBM Power Station 3BT. Shape optimization procedures can be applied to the refined mesh to further improve the quality of the elements. The refinement scheme is useful as part of a general three-dimensional mesh generation package, or as the mesh refinement module in an adaptive finite element analysis.     

Material characterization and interfacial boundary identification by solving an inverse elasto statics boundary elements problem

An inverse geometry problem of identifying simultaneously two irregular interfacial boundaries along with the mechanical properties of the interface domain located between the components of multiple (three) connected regions is investigated. A discrete number of displacement measurements obtained from a uniaxial tension test are used as extra information to solve this inverse problem. A unique combination of global and local optimization method is used, that is, the imperialist competitive algorithm (ICA) to find the best initial guesses of the unknown parameters to be used by the local optimization methods, that is, the conjugate gradient method (CGM) and the simplex method (SM). The CGM and SM are used in series. The performance of these local optimization methods is dependents on the initial guesses of the unknown boundaries and the mechanical properties, that is, Poisson’s ratio and Young’s modulus, so ICA provides the best initial guesses. The boundary elements method is employed to solve the direct two-dimensional (2D) elastostatics problem. A fitness function, which is the summation of squared differences between measured and computed displacements at identical locations on the exterior boundary, is minimized. Several example problems are solved and the accuracy of the obtained results is discussed. The influence of the value of the material properties of the subregions and the effect of measurement errors on the estimation process are also addressed.     

A general methodology for structural shape optimization problems using automatic adaptive remeshing

The proposed methodology is based on the use of the adaptive mesh refinement (AMR ) techniques in the context of 2D shape optimization problems analysed by the finite element method. A suitable and very general technique for the parametrization of the optimization problem, using B-splines to define the boundary, is first presented. Then mesh generation, using the advancing frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines. finite element mesh, structural behaviour and error estimator) is studied in detail. The sensitivities of the finite element mesh and error estimator permit their projection from one design to the next one leading to an a priori knowledge of the finite element error distribution on the new design without the necessity of any additional structural analysis. With this information the mesh refinement criterion permits one to build up a finite element mesh on the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked by means of several examples.     

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

This paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution ( u , p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Mesh refinement and iterative solution methods for finite element computations

Global and element residuals are introduced to determine a posteriori, computable, error bounds for finite element computations on a given mesh. The element residuals provide a criterion for determining where a finite element mesh requires refinement. This indicator is implemented in an algorithm in a finite element research program. There it is utilized to automatically refine the mesh for sample two-point problems exhibiting boundary layer and interior layer solutions. Results for both linear and nonlinear problems are presented. An important aspect of this investigation concerns the use of adaptive refinement in conjunction with iterative methods for system solution. As the mesh is being enriched through the refinement process, the solution on a given mesh provides an accurate starting iterate for the next mesh, and so on. A wide range of iterative methods are examined in a feasibility study and strategies for interweaving refinement and iteration are compared.     

Tetrahedral mesh improvement using swapping and smoothing

Automatic mesh generation and adaptive refinement methods for complex three-dimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face- and edge-swapping techniques, which change local connectivity, and optimization-based mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimization-based smoothing is highly effective in eliminating very small and very large angles. High-quality meshes are obtained in a computationally efficient manner by using optimization-based smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes. © 1997 John Wiley & Sons, Ltd.     

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

二维泊松方程问题Lagrange型有限元p型超收敛算法

该文针对二维泊松方程问题的Lagrange型有限元法提出了一种p型超收敛算法。该法受有限元线法对二维问题降维思想的启发,基于网格结点位移的天然超收敛性,通过从网格中取出一行对边相邻的单元作一子域,将子域内各单元另一对边解答取为原有限元解答,在子域上建立真解近似满足的局部偏微分方程边值问题,对该局部边值问题,沿对边方向单向提高单元阶次进行有限元求解获得单元对边上的超收敛解。单元另一对边上的超收敛解可通过另一方向的单元行类似获得。在单元边超收敛解的基础上,依次取出各个单元,以单元边位移超收敛解为Dirichlet边界条件,双向提高单元阶次对原泊松方程问题进行有限元求解即可获得全域超收敛解。数值算例表明,通过简单的后处理计算本法可显著提高解答的精度和收敛阶。