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Abstract:  Amongst the various approaches of `meshless' method, the Partition-of-unity concept married with the traditional finite-element method, namely PUFEM, has emerged to be competitive in solving the boundary-value problems. It inherits most of the advantages from both techniques except that the beauty of being `meshless' vanishes. This paper presents an alternative approach to solve singular boundary-value problems. It follows the basic PUFEM procedures. The salient feature is to enhance the quality of the influence functions, either over one single nodal cover or multi-nodal-covers. In the vicinity of the singularity, available asymptotic analytical solution is employed to enrich the influence function. The beauty of present approach is that it facilitates easy replacement of the influence functions. In other words, it favors the `influence-function refinement' procedure in a bid to search for more accurate solutions. It is analogous to the `p-version refinement' in the traditional finite-element procedures. The present approach can yield very accurate solution without adopting refined meshes. As a result, the quantities around the singularity can be evaluated directly once the nodal values are solved. No additional post-processing is needed. Firstly, the formulation of the present PUFEM approach is described. Subsequently, illustrative examples show the application to three classical singular benchmark problems having various orders of singularity. Results obtained through mesh refinements, single-nodal-cover refinements or multi-nodal-cover refinements are compared.

A new method is presented in this paper for computing the natural frequencies of a Timoshenko beam with an arbitrary number of transverse open cracks. The essence of this new method lies in the use of a kind of modified Fourier series (MFS) which is developed particularly for a Timoshenko beam having an arbitrary number of transverse open cracks. Unlike the conventional Fourier series, the modified Fourier series can approach a function with internal geometrical discontinuities. Based on the modified Fourier series, one can treat the cracked Timoshenko beam in the usual way and thus reduces the problem to a simple one. By using the present method, only standard linear eigenvalue equations, rather than non-linear algebraic equations, need to be solved. All the formulae are expressed in matrix form that renders the task of computer coding quite straightforward.

A new method is presented in this paper for computing the natural frequencies of a non-uniform beam with an arbitrary number of transverse open cracks. The essence of this new method lies in the use of a kind of modified Fourier series (MFS) which is specially developed for a beam with transverse open cracks. Unlike conventional Fourier series, modified Fourier series can approach a function with internal geometrical discontinuities. Based on the modified Fourier series, one can treat the cracked beam in the most usual way and thus reduce the problem to a simple one. The beam can be of non-uniform cross section and the number of cracks can be arbitrary. By using the present method, only standard linear eigenvalue equations, rather than non-linear algebraic equations, need to be solved. All the formulae are expressed in matrix form which renders the programming quite straightforward.

Abstract:  The stability of the coupling BEM/FEM scheme as applied in 2-D elastodynamic problems is studied further. The linear θ method, which is more stable than the standard BEM scheme (Mansur, 1983), makes the coupling BEM/FEM scheme more stable also. But due to the inter-influence of these two kinds of numerical methods – FEM and BEM, the coupling of BEM and FEM is less stable than both BEM algorithm and FEM algorithm. Unlike in FEM, any error in BEM scheme will affect all the later results, which makes the BEM procedure tends to be less stable. And even more, a procedure that is stable when only BEM is used can be unstable in the coupling BEM/FEM procedure due to the effect of the oscillations by FEM. The procedure used in this paper is to reduce such kind of oscillations caused by the FEM scheme, so that it will not cause instability problem to the BEM scheme and further maintain the stability of the coupling BEM/FEM scheme. Although little numerical damping will be introduced by the new method, the stability is greatly improved and the computer time is nearly the same.

This paper presents the free vibration analysis of arbitrary thin shell structures by using a newly developed spline finite element. The new element has three salient features in its formulation: (i) the use of B-spline shape functions for the interpolations of both in-plane and out-of-plane displacements of a general thin shell element; (ii) the use of “displacement constraints” and “parameters shifting” to construct a finite element model; (iii) that a proposed modified version of the Koiter’s thin shell theory is employed. The element is doubly curved, has nine primary nodes and eight auxiliary nodes, and has a total of 63 degrees of freedom. It is formulated through the conventional C1displacement approach, and is capable of modelling sharp corners, arbitrary shapes and multiple junctions of thin shell structures. The numerical examples discussed include spherical panels, cylindrical panels and shells of revolution, as well as single- and double-cell boxes. These examples are typical shells of negative, zero and positive Gaussian curvatures. The effects of aspect ratios, distorted meshes and junctions of shells on the performance of the element are studied. It is shown that the new spline finite element is a reliable, versatile, accurate and efficient thin shell element suitable for the analysis of arbitrary thin shell structures.

This paper presents a spline finite element for bending analysis of thin‐plate structures. A new set of B‐spline shape functions is developed for the interpolation of displacements. The element has nine nodes, in the shape of an arbitrary quadrilateral with biquadratic Lagrangian shape functions for geometric interpolation. The classical Kirchhoffs plate theory is used, and the element is formulated through standard displacement approach. Although in recent years, the use of Reissner‐Mindlin plate theory is more popular among researchers, it is believed that for thin‐plate problems, elements based on Kirchhoff's plate theory is more efficient and reliable. The comparison of numerical results of present element with some highly successful Mindlin‐plate elements have favored the argument. The accuracy and validity of the element have also been investigated through the analysis of a representative set of test problems. It is shown that the use of B‐spline shape functions in general two‐dimensional finite element analysis is promising, and the potentialities of C1 element with this approach certainly deserve greater attention.

This paper presents a mixed time finite element method for vibration response analysis. The underlying variational formulation is developed from the principle of virtual work. Conventional spatial discretization techniques are adopted. Time discretization is implemented by using mixed time finite elements. The proposed formulation not only forms a unified variational basis for spatial and temporal discretization; it also derives many other robust time step integration algorithms. Unconditionally stable and high order accurate algorithms with variable numerical dissipation can be constructed systematically. Moreover, the proposed formulations can be used to solve linear as well as non-linear problems. The accuracy and stability of the derived algorithms are presented.

A new time finite-element method based on the extrapolation technique and the Galerkin time finite-element method is presented. In this method, the second-order governing differential equations of motion for dynamic problems are rewritten as a set of first order differential equations in state space. The standard Galerkin method is then employed for the temporal discretization. The algorithm is first-order accurate only. Based on the first-order Galerkin time finite-element formulation, the extrapolation technique is introduced to improve the order of accuracy. It is achieved by expressing the numerical amplification matrix of higher-order algorithm as a linear combination of the basic amplification matrices evaluated at selected instances of time. The matrices are combined with different weighting factors. The pairs of the selected instance of time and the corresponding weighting factors are free parameters. Unconditionally stable higher-order accurate formulations can be derived by properly choosing the free parameters. Algorithms up to fourth-order accurate are presented in this paper. Detailed analyses on stability, numerical dissipation and numerical dispersion are also given. Comparisons of the present algorithms with some well-known time-integration methods are presented to demonstrate the versatility of the present method, in particular its accuracy in the higher-order formulations.

A new axisymmetric element suitable for the static analysis of shells of revolution is developed in this paper. The element, which is based on classical thin shell theory, employs a set of B‐spline shape functions for the interpolation of the geometry as well as the displacements. These shape functions, which are C2 continuous, can be obtained easily by choosing a suitable linear combination of cubic splines. By adopting the standard finite element procedure, the merits of spline interpolation and the versatility of the finite element method are incorporated. In order to show the accuracy and efficiency of the proposed element, numerical examples are presented. It can be observed that in most cases, accurate results can be obtained with only a small number of elements. In the current stage of development, only linear analysis is carried out. However, with further research, the spline finite element has a bright potential of being extended to nonlinear and dynamic analysis of various shells of revolution.

The free vibration analysis of general plates by a newly developed nine-node spline plate element is presented. The formulation is based on the classical Kirchhoff thin plate theory. It employs B-spline shape functions to form the two-dimensional displacement function and biquadratic Lagrangian functions for geometric interpolation. The nine-node element has 21 degrees of freedom and is of arbitrary quadrilateral shape with curved edges. Numerical examples are presented to demonstrate the accuracy and efficiency of the element for free vibration analysis. In particular, the effects of element geometry such as aspect ratio, skewness, shape of distortion, curved boundary and irregularity of mesh are illustrated. It is shown that the present element is rather insensitive to these irregularities and that, very often, a coarse mesh analysis can yield very accurate results. Furthermore, it is shown that the element has the merit of high accuracy and efficiency without any sacrifice of general applicability. Copyright 1993, 1999 Academic Press

Abstract: In part I of this paper, we presented a consistent mathematical perspective to the formulations of Hamilton's law and unified the formulations by parametrized form with global approximation. In part II of this paper, we extend the formulations to a proper form to develop high-performance time finite elements for numerical solutions of dynamic problems. The two-field mixed formulations are emphasized and the particular features of using lower order interpolation functions are discussed.

Abstract: Many dynamic problems can be solved numerically by using Hamilton's law. The solution is expressed as a series in the time domain with undetermined coefficients. The unknown coefficients are determined by satisfying the Hamilton's law when the solution is allowed to have certain types of variations. The advantage of the method is that it can directly generate a set of algebraic equations without considering the dynamic equilibrium or the governing differential equations. In this paper, the essential features of the Hamilton's law and its variations are re-examined from the numerical perspectives. A general version of variation is proposed and the parametrized formulations are presented. The parametrized formulations unify conventional formulations and also yield many new ones. Illustrative numerical examples in this paper demonstrate that the conventional formulations may not be optimal although they may be rational.