Special Shape Functions Research Articles

Analytical integrations and SIFs computation in 2D fracture mechanics

AbstractAnalytical integrations, in the framework of linear elastic problems modelled by means of boundary integral equations, have been considered in a previous publication (Int. J. Numer. Methods Eng. 2002; 53(7):1695–1719): the present note aims at extending the subject to linear elastic fracture mechanics. In such a context, special shape functions have been recently proposed (SIAM J. Appl. Math. 1998; 58: 428–455) in order to increase accuracy in stress intensity factors approximation: the closed form solution for ‘integrals’ that arise from the boundary element method is a goal of the present work. Exploiting the analytical integrations, asymptotical analysis around the crack tip are made possible, with the purpose of formulating a coherent and accurate correlation between approximated stress intensity factors and crack opening displacements over the crack tip straight special elements. Copyright © 2006 John Wiley & Sons, Ltd.

On the Analysis of an EFG Method Under Large Deformations and Volumetric Locking

This work investigates a modified element-free Galerkin (MEFG) method when applied to large deformation processes. The proposed EFG method enables the direct imposition of the essential boundary conditions, as a result of the kronecker delta property of the special shape functions, constructed in the neighborhood of the essential boundary. The plasticity model assumes a multiplicative decomposition of the deformation gradient into an elastic and a plastic part and considers a J 2 elasto-plastic constitutive relation that accounts for a nonlinear isotropic hardening. The constitutive model is written in terms of the rotated Kirchhoff stress and of the conjugate logarithmic strain measure. A total Lagrangian formulation is considered in order to improve the computational performance of the proposed algorithm. Here, aspects related to the volumetric locking are numerically investigated and an F-bar approach is considered. Some numerical results are presented, under axisymmetric and plane strain assumption, in order to attest the performance of the proposed method.

Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method

This paper presents an efficient meshless method for analyzing cracked piezoelectric structures subjected to mechanical and electrical loading. In this method, an element free Galerkin (EFG) formulation, an enriched basic function and some special shape functions that contain discontinuous derivatives are employed. Based on the moving least squares (MLS) interpolation approach, the EFG method is one of the promising methods for dealing with problems involving progressive crack growth. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing procedure required in the conventional finite element method (FEM) is avoided. The numerical results show that the proposed method can yield an accurate near-tip stress field in an infinite piezoelectric plate containing an interior hole. In another example studying a ceramic multilayer actuator, the proposed model was found to be accurate in the simulation of stress and electric field concentrations arround the abrupt end of an internal electrode.

Near-wall velocity profile with adaptive shape functions for turbulent forced convection

This article applies Reichardt's velocity profile to turbulent convection analysis. In contrast to a conventional law-of-the-wall formulation with three regions, a single profile represents the entire region from the viscous sublayer to the fully turbulent region. Special shape functions are developed for this profile in a hybrid finite element/volume formulation, together with dissipation rate boundary conditions, which are consistent with the velocity profile modeling. Applications to turbulent channel flow are successfully predicted with a k− ɛ turbulence model.

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

A Petrov–Galerkin transformation that eliminates spurious oscillations for heterogeneous diffusion–reaction equations

In this paper a transformation technique is developed for diffusion–reaction equations of autocatalytic type, which possess spatially variable coefficients, that removes the possibility of spurious oscillations when using a backward Euler time discretization with large time steps. The approach can be considered as a construction of “special shape functions” yielding a stable Petrov–Galerkin type approximation.

Fully discrete hp-finite elements: fast quadrature

A fully discrete hp-finite element method (FEM) is presented. It combines the features of the standard hp-FEM (conforming Galerkin formulation, variable order quadrature schemes, geometric meshes, static condensation) and of the spectral element method (special shape functions and spectral quadrature techniques). The speed-up (relative to standard hp elements) is analyzed in detail both theoretically and computationally.

On the solution of three-dimensional thermoelastic mixed-mode edge crack problems by the dual boundary element method

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems. The DBEM incorporates two pairs of independent boundary integral equations: namely, the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. Special shape functions are implemented to model the r behaviour of crack tip fields. Stress intensity factors were calculated with the crack opening displacement formulae and by using standard and special elements at the crack front. To demonstrate the accuracy and efficiency of the proposed method, several examples were solved and the results compared, where possible, with existing solutions.

A variable order infinite element for multi-domain boundary element modelling of acoustic radiation and scattering

A direct collocation multi-domain boundary element method for acoustic radiation and scattering problems in unbounded domains is introduced. For effective modelling of the acoustic field variables along the infinite interfaces of adjacent subdomains a new infinite boundary element is formulated. The element is based on a finite to infinite geometry mapping and a special shape function of variable order combining an appropriate amplitude decay and wavelike variation. The direct collocation scheme is of particular interest when modelling different acoustic media in adjacent subdomains. Other benefits of this multi-domain boundary element method include banded system matrices for better computational efficiency and the fact that no singularities are present due to the non-uniqueness problem, common for conventional boundary integral methods. Both two-dimensional and axisymmetric modelling of acoustic radiation and scattering problems are presented.

A wormhole with a special shape function

This paper presents a wormhole solution using a special shape function. The solution suggests that, at least mathematically, the region requiring exotic matter can be made arbitrarily small.

EFG approximation with discontinuous derivatives

A technique for incorporating discontinuities in derivatives into meshless methods is presented. The technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives. The special shape functions have compact support which results in banded matrix equations. The technique is described in element-free Galerkin context, but is easily applicable to other meshless methods and projections. Numerical results for problems in one and two dimensions are reported. © 1998 John Wiley & Sons, Ltd.

A polygonal element approach to random heterogeneous media with rigid ellipses or elliptical voids

A hybrid polygonal element method is proposed for the micromechanical analysis of heterogeneous materials with randomly distributed rigid ellipses or elliptical voids. An arbitrarily-shaped n-sided polygonal mesh is used to discretize a random heterogeneous medium. The conformal mapping technique is employed to construct special shape functions for a polygonal super-element containing an elliptical inclusion. The performance of the developed polygonal super-element is verified against analytical solutions. As in the case of circular inclusion, one polygonal super-element containing an elliptical inclusion replaces many traditional displacement-based elements without sacrificing numerical accuracy. The versatility of the polygonal element method is demonstrated by applying the method to a random heterogeneous cantilever beam problem.

Geometrically Nonlinear Formulations of Beams in Flexible Multibody Dynamics

In this paper, the equations of motion of flexible multibody systems are derived using a nonlinear formulation which retains the second-order terms in the strain-displacement relationship. The strain energy function used in this investigation leads to the definition of three stiffness matrices and a vector of nonlinear elastic forces. The first matrix is the constant conventional stiffness matrix; the second one is the first-order geometric stiffness matrix; and the third is a second-order stiffness matrix. It is demonstrated in this investigation that accurate representation of the axial displacement due to the foreshortening effect requires the use of large number or special axial shape functions if the nonlinear stiffness matrices are used. An alternative solution to this problem, however, is to write the equations of motion in terms of the axial coordinate along the deformed (instead of undeformed) axis. The use of this representation yields a constant stiffness matrix even if higher order terms are retained in the strain energy expression. The numerical results presented in this paper demonstrate that the proposed new approach is nearly as computationally efficient as the linear formulation. Furthermore, the proposed formulation takes into consideration the effect of all the geometric elastic nonlinearities on the bending displacement without the need to include high frequency axial modes of vibration.

A thermo-elastic-plastic model with special elements in a cutting process with tool flank wear

A coupled model of the thermo-elastic-plastic material under large deformation for orthogonal cutting with a pre-honed land simulating tool flank wear is constructed. The stress concentration of workpiece at tool tip is especially considered. The singular elements are utilized in the workpiece around the tip of the tool to simulate the stress concentration. The elastic-plastic and the geometric stiffness matrices were derived based on the special shape function. The cutting tool is incrementally advanced forward in a step-by-step process from an incipient stage of tool-workpiece engagement to a steady state of chip formation. A numerical technique gradually reducing displacement was developed to deal with a new generative node to contact with the tool flank.

Special Finite Element Methods for a Class of Second Order Elliptic Problems with Rough Coefficients

In this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered. Problems of the type considered arise in the analysis of unidirectional composites, where the coefficients represent the properties of the material. Several methods for this class of problems are presented, and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients. The methods are referred to as special finite element methods because they are of finite element type but employ special shape functions, chosen to accurately model the unknown solution.

Special finite element shape functions for axisymmetric magnetic problems

New infinite elements which can be used for a solution of axisymmetric magnetic problems with open boundaries are presented. It is supposed that the new elements are being developed for the vector potential A(r,z) and the modified vector potential rA formulation. The elements can represent any type of decay towards infinity, and the element matrix when the decay function is 1/ rho /sup n/, n>or=1, rho /sup 2/=r/sup 2/+z/sup 2/ is given explicity. Standard finite elements for the rA formulation yield very inaccurate results near r=0. The introduction, close to the symmetry axis, of so-called axial elements with special shape functions is proposed. The formulation is so simple that closed-form expressions for the element matrix are obtained. Numerical results are presented for a problem having an analytical solution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transition Plate‐Bending Elements for Compatible Mesh Gradation

In this paper, a new approach to the development of the transition plate‐bending elements with variable midside nodes is presented. These elements will facilitate abrupt changes in mesh refinement between domains consisting of quadrilateral elements. For compatible mesh gradation, special shape functions with slope discontinuity at midside nodes are used. The transition elements are formulated based on the Mindlin‐Reissner plate theory to take shear deformation into account. To evaluate the proper element stiffness pertinent to shear, the substitute shear strain polynomials in Cartesian coordinates are constructed. A modified Gaussian integration is adopted to eliminate the slope discontinuity problem in the element domain. Some numerical examples are presented that illustrate the validity and effectiveness of these transition elements for the efficient analysis of plate‐bending problems.

BEM with mixed special shape functions for 2D eddy current problems

A boundary element formulation with mixed special shape functions, which are linear and hyperbolic functions, is derived. The hyperbolic shape function is formulated together with the skin depth; thus, it is directly related to the actual physical characteristics of the eddy current field. Numerical examples show that this formulation has a higher precision in a wide range of the skin depths than the one with only a linear shape function and is relatively independent of the skin depth. >

Boundary‐element analysis of wave scattering from cracks

AbstractA multi region boundary‐element method is employed to study the scattering of time‐harmonic waves by cracks in a three‐dimensional elastic solid. Special shape functions are employed to model the near‐crack singularity. The multi region formulation enables removal of singularities from all boundary integral equations. The matter of fictitious eigenfrequencies is discussed and resolved. Values of dynamic stress intensity factors for three problems involving penny‐shaped cracks are compared with results of other investigators.

Special finite element shape functions for 2-D eddy current problems

In this paper a new concept of finite element shape functions for 2-D eddy current problems is described. The skin depth dependent shape functions are formulated for triangular elements. The formulation is so simple that closed — form expressions for the element matrix are obtained. In order to test the new technique, a simple two-dimensional test problem, for which an analytical solution exists, is analysed.