Quadratic Interpolation Functions Research Articles

A recursive algorithm for the finite-element method in scattering theory

A simple recursive form of the finite-elemental method is presented for scattering problems. Two alternative recursions are given for linear quadratic interpolation functions. Comparisons of the finite-element method are made to the logarithmic-derivative method for the case of ArKr scattering.

Nonlinear contact problems—A finite element approach implemented in ADINA

In the analysis of structures or materials which consist of two or more components, special attention must be paid to their interaction. Whereas the assumption of complete compatibility between adjacent elements usually is sufficient in a global analysis, a more detailed calculation requires the realistic modeling of the contact interface. For this purpose an isoparametric contact element has been developed which is compatible to the two- and three-dimensional elements available in ADINA. The element can be used with a linear, quadratic, or cubic interpolation function. According to the isoparametric concept, curved element boundaries can be modeled respectively. At first, the element has been developed for the investigation of bond between concrete and reinforcement. Various bond stress-slip relations have been programmed including effects which significantly influence the bond behaviour, for example transversal pressure, although the application of the element is not restricted on bond problems. Due to the modular structure of ADINA, which has been maintained in the programming of the contact element, additional stress-slip relations can be incorporated on the level of the contact model. The application of the element with different contact models is shown in three different examples. The end anchorage of a reinforcement bar in a concrete block, the shear transfer across discrete cracks, and the joints of prefabricated wall structures are three examples which show the scale of application of the element.

Finite element solution of one-equation models of turbulent flow

A description is given of the application of the finite element method to solve two-dimensional, incompressible turbulent flow problems using a one-equation turbulence model. The velocity components and the turbulence kinetic energy are approximated over each element by quadratic interpolation functions, while linear interpolation is used for the pressure. The length scale of the turbulence is specified algebraically. The problem is then reduced to the solution of a set of nonlinear algebraic equations and the method of solution is described. Illustrative examples are fully developed turbulent flow in pipes and the plane turbulent shear layer.

A general isoparametric finite element program SDRC SUPERB

SDRC SUPERB is a general purpose finite element program that performs linear static, dynamic and steady state heat conduction analyses of structures made of isotropic and/or orthotropic elastic materials having temperature dependent properties. The finite element library of SUPERB contains isoparametric plane stress, plane strain, flat plate, curved shell, solid type curved shell and solid elements in addition to conventional beam and spring elements. Linear, quadratic and cubic interpolation functions are available for all isoparametric elements. Independent parameters such as displacements and temperatures are obtained from SUPERB using the stiffness method of analysis. The remaining dependent parameters, such as stresses and strains, are evaluated at element gauss points and extrapolated to nodal locations. Averaged values are given as final output. The graphic capabilities of SUPERB consists of geometry and distorted geometry plotting, and stress, strain and temperature contouring. Contours are plotted at user defined cutting planes for solids and at top, middle or bottom surfaces for plate and shell types of structures. In the first part of this paper, the program capabilities of SUPERB are summarized. Extrapolation techniques used for determining dependent nodal parameters and for contour plotting are explained in the second part of the paper. Behavior of standard, wedge and transition type isoparametric elements and the effect of interpolation function orders on accuracy are discussed in the third part. The results of several illustrative problems are included.

CURRENT DEPTH REFRACTION USING FINITE ELEMENTS

A computational model is presented for the steady state prediction of currents, waves, and sea surface elevations in a coastal region outside the surf zone. Irrotational flow of surface gravity waves on large-scale steady currents over a gently sloping sea bed is considered. A second order Stokian approach is used, and all dynamic quantities are integrated over depth and averaged over time, in that order. The flow equations and the boundary conditions are presented. A method is developed for the solution of the non-linear steady model by introducing a sequence of two-level calculations, viz. a 'wave level' and a 'current level'. The variables are split on the two levels. The wave field is found, using that the flux of wave action is constant between adjacent wave rays. The current field and the mean sea surface elevation are determined using a Galerkin finite element method. The current field is approximated by triangular elements with linear interpolation functions, and the mean sea surface elevation is approximated by a triangular element with quadratic (parabolic) interpolation functions. A quasi-twodimensional test solution is tabulated.

Improved numerical integration of thick shell finite elements

AbstractA quadratic thick shell element derived from a three‐dimensional isoparametric element was first introduced by Ahmad and co‐workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided.The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems. Editors' note: A similar development was outlined by O. C. Zienkiewicz and co‐workers in lnt. J. num. Meth. Engng, 3, 275–290 (1971). Some important details differ between the two papers which are thus complementary.

Analysis Of Machine Foundations On Undrained(incompressible) Soils

A boundary element formulation, incorporating an infinite element approach to describe the far-field behaviour outside the immediate vicinity of the loaded area and coupled with the asymptotic form of the full-space Green's function for harmonic point loading in incompressible media, is used to analyse the machine foundation problem. The infinite elements assumed in the analysis are based on Rayleigh wave attenuation away from a source located at the centroid of the loaded area, while in the finite regions, tractions and displacements are described by quadratic interpolation functions between the element nodes, and the integrations are carried out by means of triangular subelement mappings into Gauss quadrature space to reduce the order of the singularities. A combined analytical/numerical approach is used to evaluate the integrals over the infinite free surface. Illustrative impedance plots for shallow foundations are presented in order to illustrate the potential of the formulation for incompressible media. The results for vertical loading can be described accurately using the simple, participating mass concept.