Higher Order Isoparametric Elements Research Articles

High order isoparametric elements in boundary element method—Smooth spheroidal element

The smooth spheroidal element, a high order isoparametric element, is constructed and applied in the boundary element method (BEM) for modeling closed surface of spheroid as well as the field variables using a single element. The smoothness of the element is realized by repeated use of real nodes as auxiliary nodes of interpolation along both the latitude and meridian directions to remove the end node/line effects existing in the conventional closure element. The special techniques of shape function manipulations are suggested and presented to deal with the integrals with hyper- or near hyper-singularities in a uniform and simple way encountered in the BEM. A combined technique, the distance transformation with possible subdivision is proposed to improve the evaluation of near singular integrals. Through a number of numerical examples, the accuracy and efficiency of using high order smooth elements are demonstrated by checking the fitting accuracy of geometrical parameters, the computation costs and some bench mark tests in elasticity.

High order isoparametric elements in boundary element method—Smooth elliptical element

The smooth elliptic element, a high order isoparametric element, is constructed and applied in the boundary element method for modeling closed geometries of ellipses as well as the field variables using a single element. The smoothness of the element is realized by repeated use of real nodes as auxiliary nodes along the circumferential direction of ellipse to remove the end node effect. By writing Lagrange polynomials into the summation of monomials in descending order then into the nested products, the coefficients of shape functions can be generated automatically to get rid of manual work encountered in the construction of a variety of high order elements with different node numbers and distributions. Special techniques of shape function manipulations are suggested and presented in dealing with the integrals with hyper- or near hyper-singularities in a uniform and simple way. A combined technique, the distance transformation with possible element subdivision is proposed to improve the evaluation of near singular integrals. The accuracy and efficiency of using high order elements are demonstrated through a number of numerical examples, by checking the fitting accuracy of geometrical parameters, the computation costs and some simple bench mark tests in elasticity.

Electromagnetic Scattering by Isoparametric Elements of Higher Order

Abstract. A boundary element method with higher order isoparametric elements allows for the simulation of scattering problems given by the electric field integral equation. This paper describes the implementation and provides striking numerical evidence of the higher order convergence rates for the approximation of the electric surface current. One key issue is the necessity of higher order geometrical approximation which leads to an extreme efficiency of some mesh-dependent variable-order approximation.

Mesh-based digital image correlation method using higher order isoparametric elements

By using the 8-node-quadrangular isoparametric element to discretize the speckle image, a 8-node-quadrangular mesh-based digital image correlation method is proposed. The model of 8-node-quadrangular mesh digital image correlation and its solving scheme based on the Newton–Raphson iteration have been constructed. The method is realized and then verified with speckle images produced by numerical simulations and real experiments. The results show that the 8-node-quadrangular mesh digital image correlation has a similar accuracy on a simple deformation measurement (translation, rotation and constant strain deformation) compared with mesh digital image correlation using the simple 4-node-quadrangular element. But for a heterogeneous deformation field measurement, the 8-node-quadrangular mesh digital image correlation has a higher accuracy because the 8-node isoparametric element used has the second-order interpolation accuracy, which is higher than that in 4-node-quadrangular mesh digital image correlation and in the conventional subset-based digital image correlation. Moreover, the 8-node-quadrangular mesh digital image correlation has the advantage for the measurement of a specimen with circular edges; the 8-node-quadrangular mesh digital image correlation can measure the displacement of all points close, even on, the curve edges because the 8-node-quadrangular isoparametric element is able to be mapped onto an element with second-order curved edges.

A study on the solution stability by the position of internal nodes in hihger order isoparametric elements

Higher order isoparametric elements are usually used in the finite element analysis because they can represent easily the geometric shape of a complex structure ad can improve the solution quality. When these elements are used, the position of internal nodes affects greatly on the solution accuracy. Decreasing of the accuracy related to the position of internal nodes is due to non-conformal mapping often results in an unstable Jacobian value. This paper, in order to remove this difficulty, suggests a modified shape function which can establish conformal mapping between two coordinate systems. Numerical experiments with the proposed shape function show that a stable solution can be obtained without respect to the position of internal nodes, and offer convenience that one can take arbitrarily the position of internal nodes considering only the geometric shape of element boundaries.

Effect of Element Distortion on Accuracy of Flow Calculations by Using Higher-Order Isoparametric Elements.

Effect of Element Distortion on Accuracy of Flow Calculations by Using Higher-Order Isoparametric Elements.

Numerical evaluation of stress intensity factor and J integral in three‐point bend specimen

AbstractThe present paper attempts to evaluate the fracture mechanics parameters, the stress intensity factor (K) and Rice's energy integral (J) in plane strain conditions for three‐point bend specimens. Both the parameters have been evaluated by the FEM using higher order isoparametric elements (i.e. quadratic elements). The crack tip elastic singularity (1/√r) has been taken into account by the use of the special crack tip elements of degenerate triangular element type as well as the fine eight‐noded isoparametric plane elements. The stress distribution has been compared with the Westergaard solution in the vicinity of the crack. The K and J values have also been‐compared with the theoretical results.