On the application of two Gauss-Legendre quadrature rules for composite numerical integration over a tetrahedral region

In this paper we first present a Gauss-Legendre quadrature rule for the evaluation of I = ∫ ∫ T ∫ f (x, y, z) d x d y d z, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z){divides}0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ) to change the integral into an equivalent integral {Mathematical expression} over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ){divides} -1 ≤ ξ, η, ζ ≤ 1}. We then apply the one-dimensional Gauss-Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss-Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra T i c (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. By use of the affine transformations defined over each T i c and the linearity property of integrals leads to the result:I = underover(∑, i = 1, 4) ∫ ∫ Tic ∫ f (x, y, z) d x d y d z = frac(1, 4) ∫ ∫ T ∫ G (X, Y, Z) d X d Y d Z,where{Mathematical expression}refer to an affine transformations which map each T i c into the standard tetrahedral region T. We then write{Mathematical expression}and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p 3 tetrahedra T i (i = 1(1)p 3) each of which has volume equal to 1/(6p 3) units. We have again shown that the use of affine transformations over each T i and the use of linearity property of integrals leads to the result:{Mathematical expression}where{Mathematical expression}refer to the affine transformations which map each T i in (x (α,p), y (α,p), z (α,p)) space into a standard tetrahedron T in the (X, Y, Z) space. We can now apply the two rules earlier derived to the integral ∫ ∫ T ∫ H (X, Y, Z) d X d Y d Z, this amounts to the application of composite numerical integration of T into p 3 and 4p 3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals. © 2006 Elsevier Inc. All rights reserved

The influence of the microstructural shape on the mechanical behaviour of interpenetrating phase composites

The microstructure-property relationship for interpenetrating phase composites (IPCs) is currently poorly understood. In an attempt to improve this understanding this study focused on one particular part of this relationship: the effect of phase shape on the elastic and plastic behaviour. A review of previous research showed that investigations had linked phase shape to the elastic and plastic behaviour of various inclusion reinforced composites, but that no similar work had been completed for IPCs. To study the complex response of the IPC microstructure under load, a numerical modelling analysis using the finite element method (FEM) was undertaken. Two three-dimensional models of IPCs were created, the first consisting of an interconnected spherical phase with the interstitial space forming the other interconnected phase, and the second replacing the spherical phase with an interconnected cylindrical phase. With the simulation of a uniaxial tension test under elastic and plastic conditions, these two models exhibited different responses based on the shape of the phases. Results from an analysis of the macroscopic behaviour identified that the cylindrical model produced greater effective properties than the spherical model at the same volume fraction. The influence of phase shape was connected to the increased contiguity of the superior phase within the IPC for the cylindrical model, which allowed similar levels of long-range continuity with smaller amounts of the superior phase (compared to the spherical model). An examination of microstructural stress distributions showed that preferential stress transfer occurred along paths of low compliance. This provided an explanation of how the improved contiguity of the stiffer (or stronger) phase could enhance the macroscopic effective properties of an IPC. Contiguity of the stronger phase was particularly important for plastic behaviour, where early yielding of the weaker phase requires the stronger phase to carry nearly all the load within itself

On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface

This paper first presents a Gauss Legendre quadrature rule for the evaluation of I = ∫ ∫T f (x, y) d x d y, where f (x, y) is an analytic function in x, y and T is the standard triangular surface: {(x, y) | 0 ≤ x, y ≤ 1, x + y ≤ 1} in the two space (x, y). We transform this integral into an equivalent integral ∫ ∫S f (x (ξ, η), y (ξ, η)) frac(∂ (x, y), ∂ (ξ, η)) d ξ d η where S is the 2-square in (ξ, η) space: {(ξ, η) | - 1 ≤ ξ, η ≤ 1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules with new weight coefficients and new sampling points. Then a second Gauss Legendre quadrature rule of composite type is obtained. This rule is derived by discretising T into three new triangles TiC (i = 1, 2, 3) of equal size which are obtained by joining centroid of T, C = (1 / 3, 1 / 3) to the three vertices of T. By use of affine transformations defined over each TiC and the linearity property of integrals leads to the result:I = underover(∑, i = 1, 3) ∫ ∫TiC f (x, y) d x d y = frac(1, 3) ∫ ∫T G (X, Y) d X d Y,where G (X, Y) = ∑i = 1n × n f (xiC (X, Y), yiC (X, Y)) and x = xiC (X, Y) and y = yiC (X, Y) refer to affine transformations which map each TiC into T the standard triangular surface. We then write ∫ ∫T G (X, Y) d X d Y = ∫ ∫S G (X (ξ, η), Y (ξ, η)) frac(∂ (X, Y), ∂ (ξ, η)) d ξ d η and a composite rule of integration is thus obtained. We next propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti(i = 1 (1) n2) each of which has an area equal to 1 / (2 n2) units. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result:∫ ∫T f (x, y) d x d y = underover(∑, i = 1, n × n) ∫ ∫Ti f (x, y) d x d y = frac(1, n2) ∫ ∫T H (X, Y) d X d Y,where H (X, Y) = ∑i = 1n × n f (xi (X, Y), yi (X, Y)) and x = xi (X, Y), y = yi (X, Y) refer to affine transformations which map each Ti in (x, y) space into T a standard triangular surface T in the (x, y) space. We can now apply the two rules earlier derived to the integral ∫ ∫T H (X, Y) d X d Y, this amounts to application of composite numerical integration of T into n2 and 3n2 triangles of equal sizes respectively. We can now apply the rules, which are derived earlier to the evaluation of the integral, ∫ ∫T f (x, y) d x d y and each of these procedures converges to the exact value of the integral ∫ ∫T f (x, y) d x d y for sufficiently large value of n and the convergence is much faster for higher order rules. We have demonstrated this aspect by applying the above composite integration method to some typical integrals. © 2007 Elsevier Inc. All rights reserved

Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface

This paper first presents a Gauss Legendre quadrature method for numerical integration of View the MathML source, where f(x, y) is an analytic function in x, y and T is the standard triangular surface: {(x, y)∣0 ⩽ x, y ⩽ 1, x + y ⩽ 1} in the Cartesian two dimensional (x, y) space. We then use a transformation x = x(ξ, η), y = y(ξ, η ) to change the integral I to an equivalent integral View the MathML source, where S is now the 2-square in (ξ, η) space: {(ξ, η)∣ − 1 ⩽ ξ, η ⩽ 1}. We then apply the one dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti (i = 1(1)n2) each of which has an area equal to 1/(2n2) units. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result: View the MathML source Turn MathJax on where View the MathML source and x = xi(X, Y) and y = yi(X, Y) refer to affine transformations which map each Ti in (x, y) space into a standard triangular surface T in (X, Y) space. We can now apply Gauss Legendre quadrature formulas which are derived earlier for I to evaluate the integral View the MathML source. We observe that the above procedure which clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integral View the MathML source, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We have demonstrated this aspect by applying the above explained Composite Numerical Integration method to some typical integrals

Gauss Legendre quadrature over a triangle

This paper presents a Gauss Legendre quadrature method for numerical integration over the standard triangular surface: (x, y) | 0 â x, y â 1, x + y â 1 in the Cartesian two-dimensional (x, y) space. Mathematical transformation from (x, y) space to (ξ, η) space map the standard triangle in (x, y) space to a standard 2-square in (ξ, η) space: (ξ, η)|-l â ξ, η â 1. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points and yields results which are accurate and reliable. Results obtained with new formulae are compared with the existing formulae. © Indian Institute of Science

Moment method analysis of linearly tapered slot antennas: Low loss components for switched beam radiometers

A Moment Method Model for the radiation pattern characterization of single Linearly Tapered Slot Antennas (LTSA) in air or on a dielectric substrate is developed. This characterization consists of: (1) finding the radiated far-fields of the antenna; (2) determining the E-Plane and H-Plane beamwidths and sidelobe levels; and (3) determining the D-Plane beamwidth and cross polarization levels, as antenna parameters length, height, taper angle, substrate thickness, and the relative substrate permittivity vary. The LTSA geometry does not lend itself to analytical solution with the given parameter ranges. Therefore, a computer modeling scheme and a code are necessary to analyze the problem. This necessity imposes some further objectives or requirements on the solution method (modeling) and tool (computer code). These may be listed as follows: (1) a good approximation to the real antenna geometry; and (2) feasible computer storage and time requirements. According to these requirements, the work is concentrated on the development of efficient modeling schemes for these type of problems and on reducing the central processing unit (CPU) time required from the computer code. A Method of Moments (MoM) code is developed for the analysis of LTSA's within the parameter ranges given

Isogeometric analysis: an overview and computer implementation aspects

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. The tight coupling of CAD and analysis within IGA requires knowledge from both fields and it is one of the goals of the present paper to outline much of the commonly used notation. In this manuscript, through a clear and simple Matlab implementation, we present an introduction to IGA applied to the Finite Element (FE) method and related computer implementation aspects. Furthermore, implemen- tation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. The open source Matlab code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The Bezier extraction concept that allows FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA