AC 2 finite element and interpolation

In this paper, we define so-called QT triangulation of a given partition with quadrilateral, establish aC ² finite element and interpolation of the space of piecewise bivariate polynomial of total degree 6, and in the process obtain a local bases for the space.     

Recent developments in the Trefftz method for finite element and boundary element applications

In 1926 E. Trefftz published a paper about a variational formulation which utilizes boundary integrals. Almost half a century later researchers became interested again in the ideas of Trefftz when the potential advantage of the Trefftz-method for an efficient use in numerical application on a computer was recognized. The concept of Trefftz can be used both for finite element and boundary element applications. A crucial ingredient of the Trefftz- method is a set of linearly independent trial functions which a priori satisfy the governing differential equations under consideration. In this paper an overview of some recent developments to construct trial functions for the Trefftz-method in a systematic manner is given. Using different types of approximation functions (singular or non-singular) we can obtain very accurate finite element and boundary element algorithms.     

Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements

Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallel-epipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.     

Anisotropic finite element modeling for patient-specific mandible

This paper presents an ad hoc modular software tool to quasi-automatically generate patient-specific three-dimensional (3D) finite element (FE) model of the human mandible. The main task is taking into account the complex geometry of the individual mandible, as well as the inherent highly anisotropic material law. At first, by computed tomography data (CT), the individual geometry of the complete range of mandible was well reproduced, also the separation between cortical and cancellous bone. Then, taking advantage of the inherent shape nature as ‘curve’ long bone, the algorithm employed a pair of B-spline curves running along the entire upper and lower mandible borders as auxiliary baselines, whose directions are also compatible with that of the trajectory of maximum material stiffness throughout the cortical bone of the mandible. And under the guidance of this pair of auxiliary baselines, a sequence of B-spline surfaces were interpolated adaptively as curve cross-sections to cut the original geometry. Following, based on the produced curve contours and the corresponding curve cross-section surfaces, quite well structured FE volume meshes were constructed, as well as the inherent trajectory vector fields of the anisotropic material (orthotropic for cortical bone and transversely isotropic for cancellous bone). Finally, a sensitivity analysis comprising various 3D FE simulations was carried out to reveal the relevance of elastic anisotropy for the load carrying behavior of the mandible.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

Anisotropic mesh adaptation for finite element solution of Anisotropic Porous Medium Equation

Anisotropic Porous Medium Equation (APME) is developed as an extension of the Porous Medium Equation (PME) for anisotropic porous media. A special analytical solution is derived for APME for time-independent diffusion. Anisotropic mesh adaptation for linear finite element solution of APME is discussed and numerical results for two dimensional examples are presented. The solution errors using anisotropic adaptive meshes show second order convergence.     

Application of the finite element method to machinery

The finite element method has given the designer of machinery components and structures a new tool for analysis of stresses, temperatures and dynamic vibrations.In the following some applications of the tool to different problems within this field are shown (by some examples), using the SESAM-69 complex.Three-dimensional solids as well as three-dimensional plate and bar structures are treated, and the results are compared to analytical or experimental solutions where such are known.These examples should verify the FEM as very attractive to the modem machinery-designer.     

Some results of finite element applications in finite elasticity

This paper examines a number of problems connected with the finite element analysis of finite elastic deformations. A brief review of formulation of equations governing finite deformations of highly elastic elements is given. The convergence of finite element approximations for static problems in elasticity is studied. Incremental stiffness equations are derived in general form and various types of incremental loading techniques are examined. A number of representative solved problems in finite elasticity are given.     

Application of the finite element method to off-shore structures

The merits of finite element methods in connection with the design of offshore structures are discussed using the Ekofisk Tank as an example. Various design approaches and procedures are evaluated on the same basis and some of the experiences gained are mentioned. The conclusion is drawn that the finite element method is a very powerful tool in the evaluation of new designs, but to be successful, it requires good planning and experienced personnel.The procedures presently applied for determination of wave loads do not represent a very realistic modelling of the actual storm conditions. To show what improvements can be made in this field, an outline of how the irregular wave system may be formulated and how the calculated loads may be applied in the stress analysis is given. The procedure is described with reference to a semisubmersible drilling platform but the principle may be applied to any offshore structure.     

Fundamentals of the finite element method

The present paper gives a brief review of the finite element method. After a historical review, the organization of the finite element analysis in two steps, an element analysis and a system analysis, is described for a simple frame problem.The generalization of this idea to two-and three-dimensional problems is explained and the application of simple types of elements is discussed. The extension to more complex elements is outlined, and finally the capability of the method in solving non-linear and dynamic problems is sketched.     

Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes

The present paper deals with an anisotropic mesh adaptation (AMA) of triangulation which can be employed for the numerical solution various problems of physics. AMA tries to construct an optimal triangulation of the domain of computation in the sense that an “error” of the solution of the problem considered is uniformly distributed over the whole triangulation. First, we describe the main idea of AMA. We define an optimal triangle and an optimal triangulation. Then we describe the process of optimization of the triangulation and the complete multilevel computational process. We apply AMA to a problem of CFD, namely to inviscid compressible flow. The computational results for a channel flow are presented. Received: 11 December 1997 / Accepted: 16 February 1998     

Gauss-Jacobi quadrature rules for n-simplex with applications to finite element methods

For numerical integrations over the n-simplex, the Gauss-Jacobi quadrature operators are presented definitively. It is demonstrated in finite-element applications that the Gauss-Jacobi quadratures could be highly effective under the isoparametric mapping as well as under the semi-radial singularity mapping.     

A finite element method to model progressive fracturing

In this paper, a theoretical basis is established for a finite element iterative algorithm for load-displacement analysis of elasticity problems in which the elastic body fractures under load. Location of fractures is determined by testing a Coulomb friction criterion along inter-element edges. Convergence of the algorithm to a physically-meaningful solution is proved, and is illustrated by results for a cantilever-type problem.     

A flow-condition-based interpolation finite element procedure for incompressible fluid flows

A new finite element procedure for the solution of the incompressible Navier–Stokes equations is presented. In the Petrov–Galerkin formulation employed, the velocities are interpolated using the flow conditions over the elements and the pressure is interpolated to satisfy the inf–sup condition for incompressible analysis. Element control volumes are employed to satisfy local mass and momentum conservation (as in finite volume methods), which corresponds to using step functions as weight functions in the finite element method. An important achievement of the discretization scheme is that no artificial parameters are set in the scheme to reach stability for low and high Reynolds (and Péclet) number flows. The solutions of nontrivial test problems are presented to demonstrate the capability and potential of the scheme.     

Stabilized low-order finite elements for frictional contact with the extended finite element method

Contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity, in which the normal contact pressure exhibits spurious oscillation. This oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined. Using a Lagrange multipliers formulation we trace this problem to non-satisfaction of the LBB condition associated with equal-order interpolation of slip and normal component of traction. In this paper, we employ a stabilized finite element formulation based on the polynomial pressure projection (PPP) technique, which was used successfully for Stokes equation and for coupled solid-deformation–fluid-diffusion using low-order mixed finite elements. For the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method. We use low-order linear triangular elements (tetrahedral elements for 3D) for both slip and normal pressure degrees of freedom, and show the efficacy of the stabilized formulation on a variety of plane strain, plane stress, and three-dimensional problems.     

Numerical integration in the finite element method

The theoretical predictions of [1] concerning the needed accuracy in the numerical integration of curvilinear (isoparametric) finite elements is confirmed experimentally. The theoretical arguments and numerical results arrived at here suggest a way to lump the mass matrix with no accuracy loss. In the finite element analysis of almost inextentional shells and nearly incompressible materials where the lack of zero energy modes severely impedes convergence, numerical integration has the beneficial effect of reducing the rank of some stiffness matrices assuring thereby the presence of discrete zero modes.     

Application of the finite fourier series to the boundary element method

Interpolation functions using finite Fourier series are formulated and their applications to the boundary element method are proposed. Interpolation functions proposed here are derived by calculating the finite Fourier series of relative displacement and traction vectors between two adjacent nodes on the boundary. These interpolation functions may vary in higher order, including linear variation. Numerical examples are examined to evaluate the efficiency of the proposed method.     

Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering

In this study, we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right-hand side data coming from forcing functions or boundary conditions. We show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage, and we demonstrate that block iterative methods for algebraic systems with multiple right-hand sides can be used to advantage to compute this solution. We also show how to generate a variety of statistical quantities from the computed solution. Finally, we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering.     

On the reformulation of the finite element method

The finite element method has seen widespread application in various fields of engineering sciences since its conception as an aid to structural analysis and design.Work by the author[1–5] as well as others [6–12] has resulted in the reexamination of the finite element method as it is presently applied. In essence, this work focuses on the need to incorporate consideration of the idealization geometry as a dependent variable in the formulation. A technique utilized by the author to incorporate this variable is based on considering the Lagrangian a function of generalized displacements and idealization geometry [5]. It is shown that this reformulation process yields auxiliary equation in addition to the equations normally satisfied by the finite element method.In this paper the author briefly summarizes the salient feature of the reformulation process. The correct formulation for the elastostatic, elastodynamic and buckling problem are indicated. The merits of this approach are verified numerically. In the elastostatic problem this approach is shown to yield improved values for stresses and displacements. In the eigenvalue problem improved values for buckling loads and/or natural frequencies are obtained.A practical test for convergence is shown to arise naturally. An alternative strategy developed by the author [3–5] to avoid the necessity of global optimization is summarized.     

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A _h u _h ,u _h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h ²) error estimate is given for these conditions.