Interpolation functions in control volume finite element method

 The main contribution of this paper is the study of interpolation functions in control volume finite element method used in equal order and applied to an incompressible two-dimensional fluid flow. Especially, the exponential interpolation function expressed in the elemental local coordinate system is compared to the classic linear interpolation function expressed in the global coordinate system. A quantitative comparison is achieved by the application of these two schemes to four flows that we know the analytical solutions. These flows are classified in two groups: flows with privileged direction and flows without. The two interpolation functions are applied to a triangular element of the domain then; a direct comparison of the results given by each interpolation function to the exact value is easily realized. The two functions are also compared when used to solve the discretized equations over the entire domain. Stability of the numerical process and accuracy of solutions are compared. Received: 20 October 2002 / Accepted: 2 December 2002     

The treatment of natural boundary conditions in the finite element and finite difference methods

To isolate the relative accuracy in the numerical approximation of the natural boundary conditions, a plane- stress example is presented for which the field equations using finite element and finite difference methods are identical. For this example, and by implication a wider class of problem, it is demonstrated that by using conventional practice in both methods the implicit representation of the natural boundary conditions in the finite element gives rise to a lower numerical accuracy than that in the less convenient explicit satisfaction of these boundary conditions in finite difference method.     

Generalized finite element analysis with T-complete boundary solution functions

The use of a complete and nonsingular set of Trefftz functions in the solution of quasi-harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary-type approximation. Procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated.     

Comparison of boundary element and finite element methods in the inelastic torsion of prismatic shafts

The accuracy and computational efficiency of the boundary element and the finite element methods are compared in this paper for problems of time-dependent inelastic torsion of prismatic shafts. Several cross-sections and two types of twisting history are considered in the numerical examples. The shaft material is assumed to obey an elastic-time hardening creep constitutive model.     

Coupling finite and boundary element methods for two-dimensional potential problems

A finite-element-boundary-element (FE-BE) coupling method based on a weighted residual variational method is presented for potential problems, governed by either the Laplace or the Poisson equations. In this method, a portion of the domain of interest is modelled by finite elements (FE) and the remainder of the region by boundary elements (BE). Because the BE fundamental solutions are valid for infinite domains, a procedure that limits the effect of the BE fundamental solution to a small region adjacent to the FE region, called the transition region (TR), is developed. This procedure involves a judicious choice of functions called the transition (T) functions that have unit values on the BE-TR interface and zero values on the FE-TR interface. The present FE-BE coupling algorithm is shown to be independent of the extent of the transition region and the choice of the transition functions. Therefore, transition regions that extend to only one layer of elements between FE and BE regions and the use of simple linear transition functions work well.     

Semi-implicit formulation of the immersed finite element method

The immersed finite element method (IFEM) is a novel numerical approach to solve fluid–structure interaction types of problems that utilizes non-conforming meshing concept. The fluid and the solid domains are represented independently. The original algorithm of the IFEM follows the interpolation process as illustrated in the original immersed boundary method where the fluid velocity and the interaction force are explicitly coupled. However, the original approach presents many numerical difficulties when the fluid and solid physical properties have large mismatches, such as when the density difference is large and when the solid is a very stiff material. Both situations will lead to divergent or unstable solutions if not handled properly. In this paper, we develop a semi-implicit formulation of the IFEM algorithm so that several terms of the interfacial forces are implicitly evaluated without going through the force distribution process. Based on the 2-D and 3-D examples that we study in this paper, we show that the semi-implicit approach is robust and is capable of handling these highly discontinuous physical properties quite well without any numerical difficulties.     

A posteriori estimates of the solution error caused by discretization in the finite element,finite difference and boundary element methods

A theory is described which guarantees an upper and lower bound estimate of the discretization error in numerical solutions of elliptic boundary value problems. This method gives bounded global estimates of the error in the energy norm. Pointwise estimates of the error in the solution variable or its derivatives can then be obtained if the numerical solution is exhibiting pointwise monotonic convergence. The versatility of this method is illustrated by its application to numerical solutions from finite element, finite difference and boundary element methods.     

Dual-mixed hp finite element methods using first-order stress functions and rotations

 A two-field dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satisfied a priori by introducing the tensor of first-order stress functions. It is pointed out that the use of six properly chosen first-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacement-pressure formulation of elasticity and the velocity-pressure formulation of Stokes flow. Selection of stable mixed hp finite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p finite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp finite elements is presented and compared to displacement-based hp finite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h- and p-extensions. Received 22 October 1999     

Elastic-plastic analysis of crack in adhesive joint by combination of boundary element and finite element methods

This paper concerns with a combination method of the boundary element method and the finite element method for elastic-plastic analyses. The combination method proposed here is based on the substructure method using the conjugate gradient method. This combination method has the advantage of saving CPU time and memory storage size over the finite element method. The combination method is applied to a J-integral analysis of a crack in an adhesive joint. The effect of bond thickness on the J-integral is discussed.     

Economy and convergence of notch stress analysis using boundary and finite element methods

The BEM is shown to be superior to FEM with respect to economy (man hour and data processing) and convergence (with increasing number of boundary nodes) in the elasic notch stress analysis of compact components.     

A comparative study of the boundary and finite element methods for the Helmholtz equation in two dimensions

The performance of the boundary and finite element methods for the Helmholtz equation in two dimensions is investigated. To facilitate the comparison, the system of linear equations arising from the finite element formulation is reduced to a smaller system involving the boundary values of the unknown function and its normal derivative alone. The difference between the boundary and finite element solutions is then expressed in terms of a difference matrix operating on the boundary data. Numerical investigations show that the boundary element method is generally more accurate than the finite element method when the size of the finite elements is comparable to that of the boundary elements, especially for the Dirichlet problem where the boundary values of the solution are specified. Exceptions occur in the neighborhood of isolated points of the Helmholtz constant where eigenfunctions of the boundary integral equation arise and the boundary element method fails to produce a unique solution.     

Comparison of boundary and finite element methods for moving-boundary problems governed by a potential

Three formulations of the boundary element method (BEM) and one of the Galerkin finite element method (FEM) are compared according to accuracy and efficiency for the spatial discretization of two-dimensional, moving-boundary problems based on Laplace's equation. The same Euler-predictor, trapezoid-corrector scheme for time integration is used for all four methods. The model problems are on either a bounded or a semi-infinite strip and are formulated so that closed-form solutions are known. Infinite elements are used with both the BEM and FEM techniques for the unbounded domain. For problems with the bounded region, the BEM using the free-space Green's function and piecewise quadratic interpolating functions (QBEM) is more accurate and efficient than the BEM with linear interpolation. However, the FEM with biquadratic basis functions is more efficient for a given accuracy requirement than the QBEM, except when very high accuracy is demanded. For the unbounded domain, the preferred method is the BEM based on a Green's function that satisfies the lateral symmetry conditions and which leads to discretization of the potential only along the moving surface. This last formulation is the only one that reliably satisfies the far-field boundary condition.     

On the relationship between the finite element and finite difference methods

The integral equations which arise from application of the Galerkin-finite element scheme to the convective diffusion equation are examined to illustrate how this method represents differential equations. The formulae obtained are effectively spatial averages of standard finite difference equations written at a node. The truncation error in the finite element solution at a node is obtained for various nodal configurations.     

Multiple scale finite element methods

New temporal and spatial discretization methods are developed for multiple scale structural dynamic problems. The concept of fast and slow time scales is introduced for the temporal discretization. The required time step is shown to be dependent only on the slow time scale, and therefore, large time steps can be used for high frequency problems. To satisfy the spatial counterpart of the requirement on time step constraint, finite-spectral elements and finite wave elements are developed. Finite-spectral element methods combine the usual finite elements with the fast convergent spectral functions to obtain a faster convergence rate; whereas, finite wave elements are developed in parallel to the temporal shifting technique. Therefore, the spatial resolution is increased substantially. These methods are especially applicable to structural acoustics and linear space structures. Numerical examples are presented to illustrate the effectiveness of these methods.     

Imposing Dirichlet boundary conditions in the extended finite element method

This paper is devoted to the imposition of Dirichlet‐type conditions within the extended finite element method (X‐FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X‐FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase‐transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd.     

On the efficiency and accuracy of the boundary element method and the finite element method

The efficiency and computational accuracy of the boundary element and finite element methods are compared in this paper. This comparison is carried out by employing different degrees of mesh refinement to solve a specific illustrative problem by the two methods.     

Coupled boundary element and finite element model for fluid-filled membrane in gravity waves

This paper describes a three-dimensional, coupled boundary element and finite element model for dynamic analysis of a fluid-filled membrane in gravity waves. The model consists of three components, describing respectively, the membrane deflection and the motions of fluids inside and outside the membrane. Small amplitude assumptions of the surface waves and membrane deflection lead to linearization of the mathematical problem and an efficient solution in the frequency domain. A finite element model, based on the membrane theory of shells, relates the membrane deflection to the internal and external fluid pressure. Two boundary element models, which describe the potential flows inside and outside the membrane, are coupled to the finite element model through the kinematic and dynamic boundary conditions on the membrane. As a demonstration, the resulting model is applied to evaluate the dynamic response of a bottom-mounted fluid-filled membrane in a wave flume. Previous two-dimensional numerical model results and three-dimensional laboratory data verify and validate the present three-dimensional model. Analysis of the computed membrane response and surface wave pattern reveals intricate resonance characteristics that explain the discrepancies between the numerical model results and the laboratory data.