Gauss Legendre Quadrature Formulae for Tetrahedra

In this paper we consider the Gauss Legendre quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)| 0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: {(ξ, η, ζ)| − 1 ≤ ξ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight co-efficients and sampling points. The effectiveness of the formulae is demonstrated by applying them to the integration of three nonpolynomial and three polynomial functions

Foot structure, pain and functional ability in people with gout in primary care: cross-sectional findings from the Clinical Assessment Study of the Foot

Background Gout frequently affects the foot yet relatively little is known about the effects of gout on foot structure, pain and functional ability. This study aimed to describe the impact of gout in a UK primary care population. Methods A cross-sectional study was nested within an observational cohort study of adults aged 50years with foot pain. Participants with gout were identified through their primary care medical records and each matched on age (2years) and gender to four participants without gout. Differences in person-level variables (SF-12 Physical Component Score, Manchester Foot Pain and Disability Index and Short Physical Performance Battery) between gout and non-gout participants were determined using regression models. Differences in foot-level variables (pain regions, skin lesions, deformities, foot posture, and non-weightbearing range of motion) were determined using multi-level regression models. All models were adjusted for body mass index. Means and probabilities with 95% confidence intervals were calculated. Results Twenty-six participants with gout were compared to 102 participants without gout (77% male; mean age 66years, standard deviation 11). Subtalar joint inversion and eversion and 1st metatarsophalangeal joint (MTPJ) dorsiflexion range of motion were significantly lower in the gout participants compared to the non-gout participants. Gout participants were more likely to have mallet toes and less likely to have claw toes compared to non-gout participants. There were no statistically significant differences in person-level variables, foot posture, ankle dorsiflexion range of motion, hallux valgus, pain regions, or skin lesions. Conclusions Non-weightbearing range of motion at the subtalar joint and 1st MTPJ was reduced in people with gout. Patients with gout who present with chronic foot problems should therefore undergo appropriate clinical assessment of foot structure

The Use of Parabolic Arc in Matching Curved Boundary by Point Transformations for Sextic Order Triangular Element

Abstract This paper is concerned with curved boundary triangular element having one curved side and two straight sides. The curved element considered here is the 28-node (sextic) triangular element. On using the isoparametric coordinate transformation, the curved triangle in the global ) , ( y x coordinate system is mapped into a standard triangle: { } . Under this transformation curved boundary of this triangular element is implicitly replaced by sextic arc. The equation of this arc involves parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in sextic arc in such a way that each arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangle will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides

Genetic evaluation of Kankrej bulls by different sire evaluation methods

The genetic evaluation of Kankrej bulls was done by using three different methods viz., BLUP, Wombat sire, and Wombat animal models. First lactation 305-days yields of 348 daughters born to 9 sires and calved from 2016 to 2020 were used for the analyses. For BLUP analysis, season and year of calving were considered as fixed effects, and sire was considered as random effect. The overall average estimate of 2414.40 kg was obtained in BLUP method while 2430.29 kg was obtained in both Wombat sire and animal models. The results revealed that the Wombat animal model discriminated against the sires to the maximum extent followed by Wombat sire and BLUP models. Based on the results, two Kankrej bulls viz., K.8.32 and K.8.62 can be selected as proven bulls for nominated mating for improving the milk productivity of Kankrej cattle.