P2Q2Iso2D=2D Isoparametric FEM in Matlab

A short Matlab implementation realizes a flexible isoparametric finite element method up to quadratic order for the approximation of elliptic problems in two-dimensional domains with curved boundaries. Triangles and quadrilaterals equipped with varying quadrature rules allow for mesh refinement. Numerical examples for the Laplace equation with mixed boundary conditions indicate the flexibility of isoparametric finite elements.     

Two different approaches for the treatment of boundary singularities

The finite element treatment of boundary singularities in elliptic problems has demanded that special techniques be developed. Many of these use some form of singular element in the neighborhood of a singular point. For a homogenous second-order problem, defined in a domain with a polygonal boundary, we studied three cases with different singularity orders. Some results about the accuracy of the solution are presented. Numerical results have been obtained using Akin singular triangular elements with three and six nodes and quadrilateral elements having four and eight nodes. The behavior of the gradient in each of these elements is also analyzed. However, these elements are not completely satisfactory, so that an alternative technique using curved isoparametric elements is given here. The results obtained with the two methods are compared. Conclusions about numerical accuracy of each method, the order of integration and the simplicity of application are made.     

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method (IPFEM) is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.     

On Error Estimate of Space-Time Finite Element Methods for Parabolic Equations in a Time-Dependent Domain

We consider linear parabolic equations in a space-time domain with curved boundaries and nonhomogeneous Dirichlet boundary conditions and discuss their approximations with isoparametric space-time finite elements. A general error estimate is proved and applied to some elements of practical interest.     

Splitting extrapolation for solving second order elliptic systems with curved boundary in by using d-quadratic isoparametric finite element

Splitting extrapolation for solving second order elliptic systems with curved boundary in by using isoparametric d-quadratic element Q₂ is presented, which is a new technique for solving large scale scientific and engineering problems in parallel. By means of domain decomposition, a large scale multidimensional problem with curved boundary is turned into many discrete problems involving several grid parameters. The multivariate asymptotic expansions of isoparametric d-quadratic Q₂ finite element errors with respect to independent grid parameters are proved for second order elliptic systems. Therefore after solving smaller problems with similar sizes in parallel, a global fine grid approximation with higher accuracy is computed by the splitting extrapolation method.     

Fourteen-Node Mixed Stiffness Element and Its Computational Comparisons with Twenty-Node Isoparametric Element

In this paper, a family of 3-dimensional elements different from isoparametric serendipity is developed according to the variational principle and the convergence criteria of the mixed stiffness finite element method. For the new family, which is named mixed stiffness elements, the number of nodes on the quadratic element is not 20 but 14. Theoretical analysis and various computational comparisons have found the mixed stiffness element superior over the isoparametric serendipity element, especially a substantial improvement in computational efficiency can be achieved by replacing the 20 node-isoparametric element with the 14-node mixed stiffness element.     

The Use of Parabolic Arcs in Matching Curved Boundaries in the Finite Element Method

If isoparametric coordinates are used to deal with curved boundariesin the finite element method, the original boundary is implicitlyreplaced by a series of parabolic or cubic arcs. The equationsof these arcs involve parameters which are the coordinates ofpoints on the curved side, and a simple procedure is outlinedfor choosing these parameters in such a way that each arc isa parabola which passes through four points of the originalcurve thus ensuring a good approximation to it.     

一组适合于带纵向裂纹柱体圣维南扭转的等参数元素

本文给出一组新的适合于带纵向裂纹柱体圣维南扭转的等参数元素,它们分别是八结点等参数元素、在裂纹尖端具有r-1/2奇异性的“1/4”八结点等参数元素及八结点过渡元素.利用这些元素,对含有径向纵裂纹的圆柱体进行了圣维南扭转计算.计算结果表明,本文给出的等参数元素有较高的精度、较好的收敛性及快的收敛速度,同时还具有程序简单、省机时等优点,所以特别适合于实际的工程计算.     

一族新的三维体元——混合刚度有限元

对于三维连续介质的有限元分析,一个通用的二次有限单元体是所谓20节点等参元.尽管这个元素已富有成效地被普遍应用,但是,它需要的节点自由度太多与其达到的二次多项式的逼近精度却十分不相称,显得计算效率很低.基于混合刚度有限元法的一     

Cascadic Multigrid Method for Isoparametric Finite Element with Numerical Integration

The purpose of this paper is to study the cascadic multigrid method for the secondorder elliptic problems with curved boundary in two-dimension which are discretized by the isoparametric finite element method with numerical integration. We show that the CCG method is accurate with optimal complexity and traditional multigrid smoother (likesymmetric Gauss-Seidel, SSOR or damped Jacobi iteration) is accurate with suboptimal complexity.     

非矩形区域上半导体问题的交替方向有限元方法

Transient behavior of semiconductor with heat-conduction on nonrectangular is studied using isoparametric elements and an approximation to the Jacobian of the isoparametric map.Concentration and heat-conduction equations are solved by alternating-direction methods and electric potential equation is approximated by finite element method.Optimal order error estimates in L^2 are demonstrated using the theory and technique of a prior estimate of differential equation.     

Elliptic Problems in Curved Domains Using Cubature Points Based Triangular Spectral Elements and Isoparametric Mappings

Using the cubature points based triangular spectral element method and isoparametric mappings, we provide accuracy results for elliptic problems in non polygonal domains. Two regimes of convergence, associated to the bulk and to the boundary of the computational domain are clearly discerned and an efficient way to define the isoparametric mapping is proposed.     

Hamilton等参元与智能叠层板的半解析法

根据压电材料修正后的Hellinger-Reissner(H-R)变分原理,建立了各向异性压电材料4节点Hamilton等参元的一般形式．为智能叠层板自由振动问题和带有压电块的叠层悬臂梁的瞬态响应等问题提出了一种新的半解析法．数学模型的基本步骤:将压电层和主体层看成独立的三维体,在平面内离散各层,分别建立各层的方程;根据主体层和压电层在连接界面上广义应力和广义位移的连续条件,联立主体层和压电层的方程得到全结构的控制方程．等参元不限制智能板侧面的几何边界形状、板的厚度和层数,有广泛的应用领域．     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.     

Numerical modelling of subcritical open channel flow using the K-ε turbulence model and the penalty function finite element technique

A numerical model has been developed that employs the penalty function finite element technique to solve the vertically averaged hydrodynamic and turbulence model equations for a water body using isoparametric elements. The full elliptic forms of the equations are solved, thereby allowing recirculating flows to be calculated. Alternative momentum dispersion and turbulence closure models are proposed and evaluated by comparing model predictions with experimental data for strongly curved subcritical open channel flow. The results of these simulations indicate that the depth-averaged two-equation k-ε turbulence model yields excellent agreement with experimental observations. In addition, it appears that neither the streamline curvature modification of the depth-averaged k-ε model, nor the momentum dispersion models based on the assumption of helicoidal flow in a curved channel, yield significant improvement in the present model predictions. Overall model predictions are found to be as good as those of a more complex and restricted three-dimensional model.     

The invertibility of the isoparametric mapping for pyramidal and prismatic finite elements

Summary. We consider the isoparametric transformation, which maps a given reference element onto a global element given by its vertices, for multi-linear finite elements on pyramids and prisms. We present easily computable conditions on the position of the vertices, which ensure that the isoparametric transformation is bijective. Received May 7, 1999 / Revised version received April 28, 2000 / Published online December 19, 2000     

Isoparametric finite-element approximation of a Steklov eigenvalue problem

We study the isoparametric variant of the finite-element method(FEM) for an approximation of Steklov eigenvalue problems forsecond-order, selfadjoint, elliptic differential operators.Error estimates for eigenfunctions and eigenvalues are derived.We prove the same estimate for eigenvalues as that obtainedin the case of conforming finite elements provided that theboundary of the domain is well approximated. Some algorithmicaspects arising from the FE isoparametric discretization ofthe Steklov problems are analysed. We finish this paper withnumerical results confirming the considered theory.     

Coons-patch macroelements in two-dimensional parabolic problems

Having recently obtained encouraging results in elliptic and hyperbolic problems, this paper summarizes previous work and further investigates the performance of large isoparametric finite elements based on the Coons–Gordon interpolation formula in the analysis of two-dimensional parabolic potential problems. The latter formula allows the global interpolation of the potential within the whole problem domain and leads to the so-called Coons-patch-macroelements (CPM), where the degrees of freedom appear primarily at the element boundaries but in the general case it is also possible to use any desirable number of internal nodes. Mathematical and numerical aspects such as the relationship between boundary-only Coons-patch macroelements and Serendipity type elements, the systematic and straightforward way of adding internal nodes, the procedure of merging dissimilar domains and, finally, efficient numerical integration schemes are discussed. Numerical results on typical static (Laplace) and time-dependent thermal problems sustain the proposed method, which is successfully compared with conventional bilinear finite elements and exact analytical solutions.     

An implementation of stress discontinuity in the boundary element method and application to gear teeth

The boundary element method as it applies to three-dimensional elasto-static problems is implemented using isoparametric quadratic elements. A scheme for resolving the problems of traction vector discontinuity is presented and the required additional equations are derived. Example problems considered, including the stress analysis of a three-dimensional gear tooth, demonstrate that high accuracy may be achieved using a relatively small number of elements if continuity of displacements and discontinuity of tractions are properly implemented.