A new error indicator based on stresses for adaptive meshing with Hermite boundary elements

This work presents a new error indicator for adaptive meshing with Hermite boundary elements. The error indicator proposed here directly measures, in an approximate fashion, the error in the numerical solution for the stresses. The basic idea behind the error indicator is to compare, on an element by element basis, a solution for the stresses obtained with Hermite elements with a second, “reduced” solution. This “reduced” solution is obtained approximating the field variables inside each element using Lagrangian shape functions together with the nodal values for the displacements and tractions that were obtained from the analysis with Hermite elements. In this sense, it is assumed that the bigger the difference between the two solutions, the bigger the error in the numerical solution for the stresses corresponding to the model with Hermite elements. Since in the scheme presented here both numerical solutions are obtained from the same analysis, the cost associated with the computation of the error indicator is minimal. Due to its simplicity and reliability, this new error indicator is very convenient to lead adaptive processes. Furthermore, it can also be used in models with multiple subregions.     

Application of infinite elements to phase change situations on deforming meshes

In recent years progress has been made in applying moving and deforming mesh systems to phase change problems. This allows the numerical attention where it is needed, near the migrating phase change zone. In spatially unbounded problems one hopes that numerically finite outer boundaries either escape significant activity or are automatically pushed further away as activity nears. Not infrequently this approach fails. Temperature activity often spreads more rapidly than phase change, thereby reaching far boundaries; stretching of the mesh by movement of far boundaries can challenge mesh control and cause ill-conditioning. In this paper the advantages of time dependent mesh adaption are enhanced by the joining of a new formulation for infinite elements to far boundaries. This is accomplished through a co-ordinate transformation within the framework of conventional 2-D quadratic, biquadratic, and linear–quadratic elements. Standard 2 by 2 Gauss–Legendre quadrature suffices throughout and normal Galerkin finite element features are undisturbed, including strict conservation of energy. The formulation is independent of global co-ordinates, entails no restrictions on the unknown function and should be applicable to other problem types. All test cases on quadrilateral and triangular grids show very significant improvements with infinite elements relative to comparable solution systems using strictly finite grids.     

Numerical and spectral investigations of Trefftz infinite elements

The numerical and spectral performance of novel infinite elements for exterior problems of time‐harmonic acoustics are examined. The formulation is based on a functional which provides a general framework for domain‐based computation of exterior problems. Two prominent features simplify the task of discretization: the infinite elements mesh the interface only and need not match the finite elements on the interface. Various infinite element approximations for two‐dimensional configurations with circular interfaces are reviewed. Numerical results demonstrate the good performance of these schemes. A simple study points to the proper interpretation of spectral results for the formulation. The spectral properties of these infinite elements are examined with a view to the representation of physics and efficient numerical solution. Copyright © 1999 John Wiley & Sons, Ltd.     

Solution of three-dimensional, anisotropic, nonlinear problems of magnetostatics using two scalar potentials, finite and infinite multipolar elements and automatic mesh generation

The two-scalar potentials idea has been used with success for the computation of static magnetic fields in the presence of nonlinear isotropic magnetic materials by the finite element method. In this communication we formulate the two-scalar-potentials method for anisotropic materials and present a computer program and the solution of an example problem. The use of infinite multipolar elements is also discussed. Several advanced methods and ideas are employed by the program: scalar potentials, rather than vector potentials, giving only one unknown quantity; the finite element method, in which the solution is approximated by a continuous function; the Galerkin method to solve the differential equations; accurate infinite elements, which avoid the introduction of an artificial boundary for unbounded problems; automatic mesh generation, which means that the user can construct a large mesh and represent a complicated geometry with little effort; automatic elimination of nodes outside the iron, which restricts the iterations to the nonlinear anisotropic region with economy of computer time; use of sparse matrix technology, which represents a further economy in computer time when assembling the linear equations and solving them by either Gauss elimination or iterative techniques such as the conjugated gradient method, etc. The combination of these techniques is very convenient.     

梯度复合材料应力强度因子计算的梯度扩展单元法

推导了一种适用于梯度复合材料断裂特性分析的梯度扩展单元, 采用细观力学方法描述材料变化的物理属性, 通过线性插值位移场给出了4节点梯度扩展元随空间位置变化的刚度矩阵, 并建立了结构的连续梯度有限元模型。通过将梯度单元的计算结果与均匀单元以及已有文献结果进行对比, 证明了梯度扩展有限元(XFEM)的优越性, 并进一步讨论了材料参数对裂纹尖端应力强度因子(SIF)的影响规律。研究结果表明: 随着网格密度的增加, 梯度单元的计算结果能够迅速收敛于准确解, 均匀单元的计算误差不会随着网格细化而消失, 且随着裂纹长度和属性梯度的增大而增大; 属性梯度和涂层基体厚度比的增大导致涂覆型梯度材料的SIF增大; 裂纹长度的增加和连接层基体厚度比的减小均导致连接型梯度材料的SIF增大。     

An hp‐adaptive finite element method for scattering problems in computational electromagnetics

An hp‐adaptive finite element (FE) approach is presented for a reliable, efficient and accurate solution of 3D electromagnetic scattering problems. The radiation condition in the far field is satisfied automatically by approximation with infinite elements (IE). Near optimal discretizations that can effectively resolve local rapid variations in the scattered field are sought adaptively by mesh refinements blended with graded polynomial enrichments. The p‐enrichments need not be spatially isotropic. The discretization error can be controlled by a self‐adaptive process, which is driven by implicit or explicit a posteriori error estimates. The error may be estimated in the energy norm or in a quantity of interest. A radar cross section (RCS) related linear functional is used in the latter case. Adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for a number of generic problems are given and compared to previously published results. Copyright © 2004 John Wiley & Sons, Ltd.     

Mapped infinite elements for 3-D vector potential magnetic problems

Numerous engineering problems, especially those in electromagnetics, often require the treatment of the unbounded continua. Mapped infinite elements have been developed for the solution of 3-D magnetic vector potential equations in infinite domain that may be used in conjunction with the standard finite elements. The electromagnetic field equations are written in terms of the magnetic vector potential for the infinite domain, and 3-D mapped infinite eiement formulation based on these equations is presented in detail. A series of magnetostatics and eddy current problems are solved to demonstrate the validity and efficiency of the procedure. These numerical results indicate that the combined finite–infinite element procedure is computationally much more economical for the solution of unbounded electromagnetic problems, especially when using the vector potential formulation, as the number of system equations decreases substantially compared to the finite element only procedure. The present procedure shows promise for the treatment of large practical industrial 3-D eddy current problems with manageable computer resources.     

Continuously deforming finite elements for the solution of parabolic problems,with and without phase change

A number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries. A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation. The method is applied to the two-phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly. Basis functions based on Hermite polynomials are used, to allow exact specification of the flux-latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.     

Mapped infinite elements in soil consolidation

The problems of consolidation often involve unbounded continua. The most common solution is then achieved by limiting the studied area through the introduction, at finite distance, of a boundary on which adequate conditions on the field variables are imposed. In this paper mapped infinite elements are used to model the far-field solution. Spatial discretization is hence performed on the basis of both finite and infinite elements. In two examples, solutions involving finite and infinite elements are compared with known analytical solutions and it is shown that an excellent agreement can be achieved by the use of mapped infinite elements.     

Isoparametric Hermite elements

Isoparametric Hermite elements are created using Bogner–Fox–Schmit rectangles on a reference domain and mapping these numerically onto the computational domain. The difficulties involved in devising explicit C¹ shape functions for isoparametric elements are thus avoided, and the resulting elements have all the benefits of full C¹ continuity, the simplicity of the Bogner–Fox–Schmit element and the geometrical flexibility expected from higher-order isoparametric elements. The numerical mapping consists in the finite element solution of a linear boundary value problem, which is inexpensive and is carried out as a preprocessing operation—the required derivatives of the mapping then being supplied to the main analysis as data. Some care is required in defining the differential boundary conditions, and guidance on this is provided. Examples are given showing the success of the mapping procedure, and the use of the resulting elements in the solution of some boundary value problems. The numerical results confirm a convergence analysis provided for the new isoparametric Hermite element.     

Closed form stiffness matrices and error estimators for plane hierarchic triangular elements

Closed form expressions for the stiffness matrix and a simple error estimator and error indicator are derived for plane straight sided triangular finite elements in elasticity problems. The calculation of the error estimator is performed on an element by element basis, and is found to be very accurate and efficient. In general, the solutions for benchmark problems using the error indicators for selective refinement of the regions show accelerated convergence when compared to the convergence rate of solutions using uniform mesh refinement. Evaluation of the stiffness matrices and error estimators using explicit formulations is found to be several times faster than numerical integration.     

Anisotropic mixed finite elements for elasticity

In this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal–normal continuous symmetric tensor‐valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulation of transient heat conduction using one‐dimensional mapped infinite elements

Many engineering problems exist in physical domains that can be said to be infinitely large. A common problem in the simulation of these unbounded domains is that a balance must be met between a practically sized mesh and the accuracy of the solution. In transient applications, developing an appropriate mesh size becomes increasingly difficult as time marches forward. The concept of the infinite element was introduced and implemented for elliptic and for parabolic problems using exponential decay functions. This paper presents a different methodology for modeling transient heat conduction using a simplified mesh consisting of only two‐node, one‐dimensional infinite elements for diffusion into an unbounded domain and is shown to be applicable for multi‐dimensional problems. A brief review of infinite elements applied to static and transient problems is presented. A transient infinite element is presented in which the element length is time‐dependent such that it provides the optimal solution at each time step. The element is validated against the exact solution for constant surface heat flux into an infinite half‐space and then applied to the problem of heat loss in thermal reservoirs. The methodology presented accurately models these phenomena and presents an alternative methodology for modeling heat loss in thermal reservoirs. Copyright © 2010 John Wiley & Sons, Ltd.     

Automatic adaptive refinement for plate bending problems using Reissner–Mindlin plate bending elements

The influence of the presence of singular points and boundary layers associated with the edge effects in a Reissner–Mindlin (RM) plate in the design of an optimal mesh for a finite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An effective adaptive refinement strategy for the solution of plate bending problems based on the RM plate bending model is developed. This two-stage adaptive strategy is designed to control both the total and the shear error norms of a plate in which both singular points and boundary layers are present. A series of three different order assumed strain RM plate bending elements has been used in the adaptive refinement procedure. The locations of optimal sampling points and the effect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested refinement procedure is effective and that optimally refined meshes can be generated. It is also found that all the plate bending elements used can attain their full convergence rates regardless of the presence of singular points and boundary layers inside the problem domain. Boundary layer effects are well captured in all the examples tested and the use of a second stage of refinement to control the shear error is justified. In addition, tests on the Zienkiewicz–Zhu error estimator show that their performances are satisfactory. Finally, tests of the relative effectiveness of the plate bending elements used have also been made and it is found that while the higher order cubic element is the most accurate element tested, the quadratic element tested is the most efficient one in terms of CPU time used. © 1998 John Wiley & Sons, Ltd.     

Exponential finite elements for diffusion–advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non‐isoparametric exponentially‐varying interpolation functions, based on exact, one‐ and two‐dimensional solutions of the Laplace‐transformed differential equation. Two eight‐noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight‐ and 12‐noded polynomial elements. The exponential element based on two‐dimensional analytical solutions fails basic tests of convergence. The one based on one‐dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight‐noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd.     

From formal solutions to computational methods avoiding passages to the limit

Before the advent of digital computers, the so-called formal solutions were the only available solutions to differential equations. Formal solutions can be closed solutions, or solutions involving infinite algorithms. The latter involve an infinite number of algebraic operations. Truncation becomes thus necessary, and the concepts of truncation error and convergence become vital. Once digital computers became available, other kinds of computational methods could be used and it became convenient to distinguish between computational methods like finite difference and finite element methods, in which numerical analysis starts before integration, and those like classical integral methods and boundary element methods, in which numerical analysis starts after integration. The classical finite difference method, in which a mesh is required, is a particular case of the generalised difference methods, characterised by a local interpolation around each node together with the collocation technique. The generalised difference method may be regarded as a modality of the meshless techniques. The finite element method differs of the finite difference method in that the approximate solution is generated respectively by variational and by collocation techniques. Hybrid and block elements are dual generalisations of the finite element method in which compatibility and equilibrium are respectively allowed within each element. Also in what concerns the methods in which numerical analysis starts after integration, bold steps have been given toward their generalisation, like those avoiding passages to the limit.     

Finite elements for the dynamic analysis of fluid-solid systems

Several new finite elements are presented for the idealization of two- and three-dimensional coupled fluid-solid systems subjected to static and dynamic loading. The elements are based on a displacement formulation in terms of the displacement degrees-of-freedom at the nodes of the element. The formulation includes the effects of compressible wave propagation and surface sloshing motion. The use of reduced integration techniques and the introduction of rotational constraints in the formulation of the element stiffness eliminates all unnecessary zero-energy modes. A simple method is given which allows the stability of a finite element mesh of fluid elements to be investigated prior to analysis. Hence, the previously encountered problems of ‘element locking’ and ‘hour glass’ modes have been eliminated and a condition of optimum constraint is obtained. Numerical examples are presented which illustrate the accuracy of the element. It is shown that the element behaves very well for non-rectangular geometry. The optimum constraint condition is clearly illustrated by the static solution of a rigid block floating on a mesh of fluid elements.     

Diffraction and refraction of surface waves using finite and infinite elements

The wave problem is introduced and a derivation of Berkhoff's surface wave theory is outlined. Appropriate boundary conditions are described, for finite and infinite boundaries. These equations are then presented in a variational form, which is used as a basis for finite and infinite elements. The elements are used to solve a wide range of unbounded surface wave problems. Comparisons are given with other methods. It is concluded that infinite elements are a competitive method for the solution of such problems.     

Eigenproblems associated with the discrete LBB condition for incompressible finite elements

The basic stability condition for mixed/Lagrange multiplier variational principles in incompressible media problems is the LBB condition. It plays an essential role in determining whether or not the problem is well-posed and governs the choice of finite elements in the discretization. This paper examines the discrete eigenstructure of a well-known Lagrange multiplier formulation for linear elasticity or Stokes-flow. It shows how the weak incompressibility constraint is reflected in the elementary divisor structure of the eigenproblem whose solution determines the finite element approximation to the natural modes. In this context the discrete LBB condition can be seen to be a condition determining the limiting disposition, as the mesh parameter decreases, of a matrix pencil with infinite eigenvalues. Pure pressure modes and the load vectors required to transmit them are paired in cyclic subspaces of the infinite eigenspace. This pairing can be related to a well-known heuristic interpretation of the LBB condition. The relationship between the natural mode eigenproblem and the eigenproblems which determine the norm of the inverse of the discrete operator of a static or steady-flow problem is described. Finally, because of the equivalence between classes of mixed and penalty formulations, it is shown that these results apply to penalty/reduced integration finite element methods.     

Automatic conversion of triangular finite element meshes to quadrilateral elements

A method is presented for the fully automatic conversion of a general finite element mesh containing triangular elements into a mesh composed of exclusively quadrilateral elements. The initial mesh may be constructed of entirely triangular elements or may consist of a mixture of triangular and quadrilateral elements. The technique used employs heuristic procedures and criteria to selectively combine adjacent triangular elements into quadrilaterals based on preestablished criteria for element quality. Additional procedures are included to eliminate isolated triangles. The methods operates completely without user intervention once the nodal co-ordinates and element connectivity of the original mesh are supplied.