HP90: A general and flexible Fortran 90 hp-FE code

A general 2D-hp-adaptive Finite Element (FE) implementation in Fortran 90 is described. The implementation is based on an abstract data structure, which allows to incorporate the full hp-adaptivity of triangular and quadrilateral finite elements. The h-refinement strategies are based on h2-refinement of quadrilaterals and h4-refinement of triangles. For p-refinement we allow the approximation order to vary within any element. The mesh refinement algorithms are restricted to 1-irregular meshes. Anisotropic and geometric refinement of quadrilateral meshes is made possible by additionally allowing double constrained nodes in rectangles. The capabilities of this hp-adaptive FE package are demonstrated on various test problems. Received: 18 December 1997 / Accepted: 17 April 1998     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

Mesh refinement strategies for solving singularly perturbed reaction-diffusion problems

We consider the numerical approximation of a singularly perturbed reaction-diffusion problem over a square. Two different approaches are compared namely: adaptive isotropic mesh refinement and anisotropic mesh refinement. Thus, we compare the h-refinement and the Shishkin mesh approaches numerically with PLTMG software [1]. It is shown how isotropic elements lead to over-refinement and how anisotropic mesh refinement is much more efficient in thin boundary layers.     

栅格法三维六面体网格局部加密算法

有限元数值分析的精度和效率与网格单元的划分质量以及疏密程度密切相关,针对三维六面体网格单元之间疏密过渡必须平缓和协调的要求,提出了一套基于8分法的六面体网格加密模板,并给出了相应的数据结构和模板应用方式.为使所有加密单元都有相对应的加密模板,建立了加密信息场调整规则;对需要进行加密的区域首先补充加密单元,按照节点加密属性调整加密信息场,然后根据单元加密属性对加密单元进行分类,按照全加密单元、面加密单元、边加密单元以及过渡加密单元的顺序依次采用相应的模板进行加密,从而实现三维六面体网格的局部协调加密.实例结果表明,采用该套加密模板的六面体网格局部加密算法能够保障密集网格向稀疏网格的平缓和协调过渡,所生成的网格可满足有限元数值计算的要求.     

Mesh generation and mesh refinement procedures for the analysis of concrete shells

In this paper, a mesh generation and mesh refinement procedure for adaptive finite element (FE) analyses of real-life surface structures are proposed. For mesh generation, the advancing front method is employed. FE meshes of curved structures are generated in the respective 2D parametric space of the structure. Thereafter, the 2D mesh is mapped onto the middle surface of the structure. For mesh refinement, two different modes, namely uniform and adaptive mesh refinement, are considered. Remeshing in the context of adaptive mesh refinement is controlled by the spatial distribution of the estimated error of the FE results. Depending on this distribution, remeshing may result in a partial increase and decrease, respectively, of the element size. In contrast to adaptive mesh refinement, uniform mesh refinement is characterized by a reduction of the element size in the entire domain. The different refinement strategies are applied to ultimate load analysis of a retrofitted cooling tower. The influence of the underlying FE discretization on the numerical results is investigated.     

Local refinement of three-dimensional finite element meshes

Mesh refinement is an important tool for editing finite element meshes in order to increase the accuracy of the solution. Refinement is performed in an iterative procedure in which a solution is found, error estimates are calculated, and elements in regions of high error are refined. This process is repeated until the desired accuracy is obtained.Much research has been done on mesh refinement. Research has been focused on two-dimensional meshes and three-dimensional tetrahedral meshes ([1] Ning et al. (1993) Finite Elements in Analysis and Design, 13, 299–318; [2] Rivara, M. (1991) Journal of Computational and Applied Mathematics 36, 79–89; [3] Kallinderis; Vijayar (1993) AIAA Journal,31, 8, 1440–1447; [4] Finite Element Meshes in Analysis and Design,20, 47–70). Some research has been done on three-dimensional hexahedral meshes ([5] Schneiders; Debye (1995) Proceedings IMA Workshop on Modelling, Mesh Generation and Adaptive Numerical Methods for Partial Differential Equations). However, little if any research has been conducted on a refinement algorithm that is general enough to be used with a mesh composed of any three-dimensional element (hexahedra, wedges, pyramids, and/or retrahedra) or any combination of three-dimensional elements (for example, a mesh composed of part hexahedra and part wedges). This paper presents an algorithm for refinement of three-dimensional finite element meshes that is general enough to refine a mesh composed of any combination of the standard three-dimensional element types.     

Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves

In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L²-norm is in both cases of optimal order and proportional to O(Δt²+h^p+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Die numerische Behandlung der Poisson-Gleichung auf einem Gebiet mit einspringenden Ecken

The first derivatives of a solution of the Poisson Equation on a polygonal domain may be unbounded. Using the usual method of finite differences on an equidistant mesh (meshsizeh) we observe an error, which behaves likeh ^α?ε (ε>0 arbitrary), at a corner with interior angle π/α. With local mesh refinement it is possible to obtain uniformly (almost) quadratic convergence and to suppress the observed pollution effect.     

Local error estimation and adaptive remeshing scheme for least-squares mixed finite elements

Following the theme of our previous work on least-squares finite elements [10,28], we describe an adaptive remeshing scheme using local residuals as the error indicator. This choice of indicator is natural (and exact at the element level!) in the norm associated with the corresponding least-squares statement. The remeshing strategy applied here involves mesh enrichment by point insertion in a Delaunay scheme. Several refined grids and error plots are included for a representative model elliptic boundary-value problem.     

Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

The paper presents postprocessing techniques based on locally improved finite element (FE) solutions of the basic field variables. This opens up the possibility to control both “strain energy” terms and “kinetic energy” terms in the governing equations. The proposed postprocessing technique on field variables is essentially a least square fit of the prime variables (displacements) at superconvergent points. Its performance is compared with other well-known techniques, showing a good performance. A h-adaptive FE strategy for acoustic problems is presented where, for adaptive mesh generation and remeshing the commercial software package i-deas has been applied and for the FE analysis the commercial software package sysnoise. The paper also presents an adaptive h-version FE approach to control the discretisation error in free vibration analysis. The postprocessing technique used here is a mix of local and global updating methods. Rapid convergence of the preconditioned conjugate gradient method is enhanced by choosing the initial trial eigenmodes as the superconvergent patch recovery technique for displacements improved FE eigenmodes. Numerical examples show nice properties of the final local and global updated solution as a basis for an error estimator and the error indicator in an adaptive process.     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

Automated adaptive 3D forming simulation processes

In this study, an automated adaptive mesh control scheme, based on local mesh modifications, is developed for the finite element simulations of 3D metal-forming processes. Error indicators are used to control the mesh discretization errors, and an h-adaptive procedure is conducted. The mesh size field used in the h-adaptive procedure is processed to control the discretization and geometric approximation errors of the evolving workpiece mesh. Industrial problems are investigated to demonstrate the capabilities of the developed scheme.     

Adaptive mixed finite element method for Reissner–Mindlin plate

Mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of Reissner–Mindlin plate models. Automatic adaptive mesh-refining algorithms are an important tool to improve the approximation behavior of the finite element discretization. In this paper, a reliable and robust residual-based a posteriori error estimate is derived, which evaluates a t-depending residual norm based on results in [D. Arnold, R. Falk, R. Winther, Math. Modell. Numer. Anal. 31 (1997) 517–557]. The localized error indicators suggest an adaptive algorithm for automatic mesh refinement. Numerical examples prove that the new scheme is efficient.     

Adaptive remeshing process with quadrangular finite elements

Since the quality of FEM analysis directly depends on the quality of meshes, various mesh adaptation schemes have been researched. There are two stages on adaptive finite element analysis; to derive error measure and to control meshes based on error measure. The former has been well researched among applied mathematicians. However, the importance of the latter aspect wasn't considered enough. Even if the error measures were well estimated, the total performance of mesh adaptation might be poor with a poor mesh control. This paper proposes an effective mesh control scheme for h-adaptation, or adaptive remeshing scheme with the explicit relation between interpolation theory based on error measure and desirable mesh size. Total mesh adaptation is controlled by introducing Quality Index, or the ratio between the total error norm and the total energy norm which represents the quality of the total meshes; specifying the desirable value of Quality Index, then the adaptive remeshing process can handle it and Quality Index is almost converged to the given value. Since the full automatic feature of the mesh generator is a prerequisite for adaptive remeshing, the author also discusses the algorithm of the quadrangular mesh generator for arbitrary domains. After evaluation on a linear problem, it's confirmed that the proposed mesh control scheme and the proposed error measure-mesh size relations are acceptable. The incompatible case for mesh adaptation is also discussed in this paper.     

Error assessment and mesh adaptivity for regularized continuous failure models

This paper deals with the adaptive finite element analysis of structural failure. A gradient-enhanced damage model has been chosen to simulate material degradation. Since this model is regularized in the post-peak regime, the finite element solution does not suffer from pathological mesh dependence and thus converges to an objective solution upon mesh refinement. However, the error analyses have shown that the error in the nonlocal equivalent strain field becomes dominant during the post-peak loading stages. The accuracy of the nonlocal equivalent strain field (and the corresponding damage quantity) also greatly influences the accuracy of the quantity of interest. Two error measures have been proposed. The goal-oriented error estimates have provided similar error distributions, although some small differences have been found in the softening regime. Objective error estimates, together with adaptive criteria, have been used to perform automated h-adaptivity during computation.     

Poisson-Boltzmann方程的并行有限元求解及网格自适应生成

说明如何利用并行自适应有限元软件平台PHG 求解生物分子溶液体系的非线性Poisson-Boltzmann方程,并介绍一种解决这类问题的方法,它将网格生成与自适应计算过程结合在一起,可自动产生合适的网格,避免复杂的曲面网格生成步骤.之前的网格生成工作有：（1） TMSmesh生成高斯曲面的三角网格; （2） TransforMesh删除自相交的三角网格; （3） ISO2Mesh提高表面网格质量3个步骤.而基于PHG的自适应加密模块可以在逐次调整网格的同时保持动态负载平衡,高效地得到计算网格用于近似求解非线性Poisson-Boltzmann方程.计算了小球模型和AChE系统,分别从误差指示子下降阶和溶剂化能收敛的角度验证了方法的有效性,并且还将网格生成算法成功地应用于gA离子通道.     

A posteriori error estimates for the primary and dual variables for the div first-order least-squares finite element method

We propose a posteriori error estimators for first-order div least-squares (LS) finite element method for linear elasticity, Stokes equations and general second-order scalar elliptic problems. Our main interest is obtaining a posteriori error estimators for the dual variables (fluxes, strains, stress, etc.) which are main quantity of interest in many applications. We also provide a posteriori error estimators for the primary variable. These estimators are obtained from the local least-squares functional by assigning weight coefficients scaling the respective residuals. The weight coefficients are given in terms of local meshsize h_K. We establish the global upper bounds and local lower bounds for the estimators. The estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution (the indicators and estimators are identical). Numerical experiments show a superior performance of our a posteriori estimators for user-specific norm over the standard LS functional.     

Development of an object-oriented finite element program with adaptive mesh refinement for multi-physics applications

In this paper, an object-oriented framework for numerical analysis of multi-physics applications is presented. The framework is divided into several basic sets of classes that enable the code segments to be built according to the type of problem to be solved. Fortran 2003 was used in the development of this finite element program due to its advantages for scientific and engineering programming and its new object-oriented features. The program was developed with h-type adaptive mesh refinement, and it was tested for several classical cases involving heat transfer, fluid mechanics and structural mechanics. The test cases show that the adaptive mesh is refined only in the localization region where the feature gradient is relatively high. The overall mesh refinement and the h-adaptive mesh refinement were justified with respect to the computational accuracy and the CPU time cost. Both methods can improve the computational accuracy with the refinement of mesh. The overall mesh refinement causes the CPU time cost to greatly increase as the mesh is refined. However, the CPU time cost does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved by up to 90%, especially for the simulated system with a large number of elements and nodes.     

Direct (Re)Meshing for Efficient Surface Processing

We propose a novel surface remeshing algorithm. While many remeshing algorithms are based on global parametrization or local mesh optimization, our algorithm is closely related to surface reconstruction techniques and it requires no explicit parameterization. Our approach is based on the advancing‐front paradigm, and it can be used to both incrementally remesh the complete surface, or simply to remesh a portion of it with a high‐quality mesh. It is accurate, fast, robust, and suitable for use with interactive mesh processing applications that require local remeshing. We show a number of applications, including matching the resolution of meshes when doing Boolean operations such as unions and intersections. We also show how to adapt the algorithm to blend and merge mixed‐mode objects — for example, to compute the union of a point‐set surface and a triangle mesh.