p-version hierarchical three dimensional curved shell element for elastostatics

This paper presents a hierarchical three dimensional curved shell finite element formulation based on the p-approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders p_ξ, p_η and p_ξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders p_ξ, p_η and p_ξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders p_ξ, p_η and p_ξ are a subset of those corresponding to the orders (p_ξ + 1), (p_η +1) and (p_ξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries. The geometry of the element is described by the co-ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = ?1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.     

Locking-free curved beam element based on curvature

The formulation of a curved beam element with 3 nodes for curvature to eliminate the shear/membrane locking phenomenon is presented. The element is based on curvature so that it may represent the bending energy fully, and the shear/membrane strain energy is incorporated into the formulation by the equilibrium equations. To deal with general boundary conditions, a transformation matrix between nodal curvature and nodal displacement vector is introduced. Several examples are presented in order to verify the element formulation and its analytical capability. The solutions obtained reveal that the element describes the curved beam behaviour quite correctly and efficiently, showing no locking phenomena, and that it is also applicable to the analysis of both thin and thick curved beams.     

p-Version hierarchical three dimensional curved shell element for heat conduction

This paper presents a finite element formulation for a three dimensional nine node p-version hierarchical curved shell element for heat conduction where the element temperature approximation can be of arbitrary order p , p , and p in the , and directions. This is accomplished by first, constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators for each of the three directions , and using Lagrange interpolating polynomials and then taking their products (sometimes also called tensor products). The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the equivalent heat vectors are hierarchical also i.e. the element properties corresponding to polynomial orders p , p , and p are a subset of those corresponding to (p +1), (p +1), and (p +1). The element formulation ensures C ⁰ continuity. The curved shell geometry is constructed in the usual way by taking the coordinates of the nodes lying on the middle surface of the element (=0) and the nodal thickness vectors. The element properties i.e. element matrices and the equivalent heat vectors are derived using weak formulation (or quadratic functional) of the three dimensional F ourier heat conduction equation and the hierarchical element temperature approximation. The element formulation is equally effective for very thin as well as extremely thick shells. Numerical examples are presented to demonstrate the accuracy, efficiency, modeling convenience, faster rate of convergence and over all superiority of the present formulation. The h-approximation results are presented for comparison purposes.     

Geometrically non-linear formulation for three dimensional curved beam elements with large rotations

This paper presents a geometrically non-linear formulation (GNL) for the three dimensional curved beam elements using the total Lagrangian approach. The element geometry is constructed using co-ordinates of the nodes on the centroidal or reference axis and the orthogonal nodal vectors representing the principal bending directions. The element displacement field is described using three translations at the element nodes and three rotations about the local axes

1 The element displacement field has also been described in the literature using Euler parameters, Milenkovic parameters, or Rodriges parameters representing the effects of large rotations.

. The GNL three dimensional beam element formulations based on these element approximations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining non-linear nodal terms in the definition of the element displacement field, and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper non-linear functions representing the effects of nodal rotations. The details of the element properties are presented and discussed. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements. A comparison of the results obtained from the present formulation with those available in the literature using a linearized element approximation clearly demonstrate the superiority of the formulation in terms of large load steps, large rotations between two load steps and extremely good convergence characteristics during equilibrium iterations. The displacement approximation of these elements is fully compatible with the isoparametric curved shell elements (with large rotations), and since the elements possess offset capability, these elements can also serve as stiffeners for the curved shells.     

An isoparametric quadratic thick curved beam element

A three-noded curved beam element with transverse shear deformation, based on independent isoparametric quadratic interpolations, is designed from field-consistency principles. It is shown that a quadratic element that is field-inconsistent in membrane strain suffers from ‘membrane locking’—i.e. an error of the second kind propagates indefinitely as the element length to thickness ratio and/or the element length to radius of curvature ratio increase, in nearly inextensional bending. However, field-inconsistency in shear strain does not lead to ‘shear locking’ but degrades its performance to exactly that of a field-consistent linear element. It is also seen that field-inconsistency leads to severe axial force and shear force oscillations. Error estimates for locking are derived, wherever possible, and confirmed by numerical experiments. The field-consistent element offered here is the most efficient quadratic curved beam element possible.     

A new higher-order hybrid-mixed curved beam element

The purpose of this work is to show the successful use of nodeless degrees of freedom in developing a highly accurate, locking free hybrid-mixed C⁰ curved beam element. In the performance evaluation process of the present field-consistent higher-order element, the effect of field consistency and the role of higher-order interpolation on both displacement-type and hybrid-mixed-type elements are carefully examined. Several benchmark tests confirm the superior behaviour of the present element. © 1998 John Wiley & Sons, Ltd.     

A three dimensional beam profile monitor based on residual and trace gas ionization

A three dimensional beam profile monitor based on tracking the ionization of the residual gas molecules in the evacuated beam pipe is described. Tracking in position and time of the ions and electrons produced in the ionization enables simultaneous position sampling in three dimensions. Special features which make it possible to sample very low beam currents were employed. The characteristics of this detector make it particularly suitable for sampling beams produced at radioactive beam facilities, provided an auxiliary gas feed can be utilized.     

The curved beam/deep arch/finite ring element revisited

In this paper, an attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformations. It is shown that with elements of finite size (i.e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the ‘second kind’ and a physical phenomenon called ‘membrane locking’. The findings here emerge from recent research on the effect of reduced integration on shallow curved beam elements and on the use of coupled displacement fields in finite rings. The failures which have occurred in earlier attempts to use independent polynomial displacement fields for curved elements may not have been due to neglect of rigid body motions or failure to achieve constant strain states, but because of locking due to spurious constraints. These emerge in the penalty limits of extreme thinness (an inextensional regime), when exact integration of the energy functional of an element based on low order independent interpolations for the in-plane and normal displacements is used. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. The much maligned ‘cubic in w–lincar in u’ curved beam element is now reworked to show its excellent behaviour in all situations. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables. In this manner, we can restore an old element to respectability and thereby indicate clearly the underlying principles. These are: the importance of ‘field consistency’ so that arch and shell problems can be modelled consistently by independent polynomial displacement fields, and the role that reduced integration or some equivalent construction can play to achieve this.     

Boundary element method applied to three dimensional thermoelastic contact

In this paper we present a boundary element method to analyze and solve three dimensional frictionless thermoelastic contact problems. Although many problems in engineering can be solved with one-dimensional or two-dimensional models, those simplifications there are not possible in many others, such as the design of microelectronics packages. We calculate the stresses, movements, temperatures and thermal gradients on 3D solids. A thermal resistance at the contact zone depends on the local pressure is considered. The problem is solved by a double iterative method, so that in the final solution do not appear tensions in the contact zone or penetrations between the two solids. The solutions are compared with other works, where possible, to validate the method.     

p-version least-squares finite element formulation of Burgers' equation

A p-version least-squares finite element formulation for non-linear problems is presented and applied to the steady-state, one-dimensional Burgers' equation. The second-order equation is recast as a set of first-order equations which permit the use of C⁰ elements. The primary and auxiliary variables are approximated using equal-order p-version hierarchical approximation functions. The system of non-linear simultaneous algebraic equations resulting from the least-squares process is solved using Newton's method with a line search. The use of ‘exact’ and ‘reduced’ quadrature rules is investigated and the results are compared. The formulation is found to produce excellent results when the ‘exact’ integration rule is used. The combination of least-squares finite element formulation and p-version works extremely well for Burgers' equation and appears to have great potential in fluid dynamics problems.     

p-Version two-dimensional curved beam element using piecewise hierarchical approximation for laminated composites

A three node two-dimensional laminated composite curved beam finite element formulation for linear static analysis is presented where the displacement approximation for the laminate is piecewise hierarchical and is derived based on p-version. The displacement approximation for the element is developed by establishing a hierarchical displacement approximation for each lamina of the laminate and then by imposing interlamina continuity conditions of displacement at the interfaces between the laminas. The approximation functions and the nodal variables for each lamina are derived directly from the Lagrange family of interpolation functions of order p and p . This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the and directions for the three and one node equivalent configurations that correspond to p +1 and p +1 equally spaced nodes in the and directions and then taking their products. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the interlamina continuity conditions of displacements. The element formulation ensures C ⁰ continuity of displacement across the interelement as well as interlamina boundaries.The individual lamina stiffness matrices and the equivalent nodal force vectors are derived using principle of virtual work and the hierarchical displacement approximation for the laminas. Interlamina continuity conditions are used to construct the transformation matrices for the laminas. These matrices permit transformation of the lamina degrees of freedom to the laminate degrees of freedom. Using these transformation matrices, individual lamina stiffness matrices and the equivalent load vectors are transformed and then summed to obtain the laminate element stiffness matrix and equivalent load vectors. There is no restriction on the number of laminas and their lay up pattern. Each layer can be generally orthotropic. The material directions and the layer thickness may vary from point to point within each lamina. The geometry of the curved beam element is defined by the nodes located at the middle surface of the element and the lamina thicknesses.Numerical examples are presented to demonstrate the accuracy, efficiency, convergence characteristics and the advantages of the present formulation. The results obtained from the present formulation are compared with the analytical solution and with those reported in the literature.     

p-version least-squares finite element formulation for convection-diffusion problems

This paper presents a p-version least-squares finite element formulation of the convection-diffusion equation. The second-order differential equation describing convection-diffusion is reduced to a series of equivalent first-order differential equations for which the least-squares formulation is constructed using the same order of approximation for each of the dependent variables. The hierarchical approximation functions and the nodal variable operators are established by first constructing the one-dimensional hierarchical approximation functions of orders pξ and pη and the corresponding nodal variable operators in ξ and η-direction and then taking their products. Numerical results are presented and compared with analytical and numerical solutions for a two-dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation. The Gaussian quadrature rule used to calculate the numerical values of the element matrices, vectors as well as the error functional I(E), is established based on the highest degree of the polynomial in the integrands. It is demonstrated that this quadrature rule with the present p-version formulation produces excellent results for very low as well as extremely high Peclet numbers (10-10⁶) and, furthermore, the error functional I (sum of the integrals of E²) is a monotonically decreasing function of the number of degrees of freedom as the p-levels are increased for a fixed mesh. It is shown that exact integration with the h-version (linear and parabolic elements) produces inaccurate solutions at high Peclet numbers. Results are also presented using reduced integration. It is shown that the reduced integration with p-version produces accurate values of the primary variable even for relatively low p-levels but the error functional I (when calculated using the proper integration rule) has a much higher value (due to errors in the derivatives of the primary variable) and is a non-monotonic function of the degrees of freedom as p-levels are increased for a fixed mesh. Similar behaviour of the error functional I is also observed for the h-models using linear elements when reduced integration is used. Although the h-models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased, the error values are inferior to the p-version results using exact integration.     

Improvement of stress singular element for crack problems in three dimensional boundary element method

In the paper, a new kind of stress singular element is introduced for crack problems. This kind of element is more simple and widely used than those presented before. In the paper, a cube with embedded circular crack and a first kind Benchmark problem are studied. The study shows that using quarter-point element and the stress singular element can obviously improve the accuracy. The influences of methods estimating stress intensity factor on accuracy are also studied.     

The p-version of finite element method for shell analysis

A new quadrature scheme and a family of hierarchical assumed strain elements have been developed to enhance the performance of the displacement-based hierarchical shell elements. Various linear iterative procedures have been examined for their suitability to solve system of equations resulting from hierarchic shell formulations.     

A variable thickness,curved beam and shell stiffening element with shear deformations

A family of 2-D and 3-D superparametric curved beam elements are introduced as a special form of general 2-D and 3-D isoparametric continuum elements. Each beam ‘subelement’ is rectangular incross section, however several elements may be offset and overlayed for the analysis of structural members of quite general section properties. A number of examples are presented in order to illustrate the validity and range of application of the element.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

Complex wavenumber Fourier analysis of the p-version finite element method

High-order finite element discretizations of the reduced wave equation have frequency bands where the solutions are harmonic decaying waves. In these so called stopping bands, the solutions are not purely propagating (real wavenumbers) but are attenuated (complex wavenumbers). In this paper we extend the standard dispersion analysis technique to include complex wavenumbers. We then use this complex Fourier analysis technique to examine the dispersion and attenuation characteristics of the p-version finite element method. Practical guidelines are reported for phase and amplitude accuracy in terms of the spectral order and the number of elements per wavelength.