• Proceedings of the Korean Nuclear Society Conference
• /
• 1996.05a
• /
• pp.131-136
• /
• 1996

An analytic nodal expansion method has been derived for the multigroup neutron diffusion equation in 2-D cylindrical(R-Z) coordinate. In this method we used the second order Legendre polynomials for source, and transverse leakage, and then the diffusion eqaution was solved analytically. This formalism has been applied to 2-D LWR model. $textsc{k}$$_{eff}$, power distribution, and computing time have been compared with those of ADEP code(finite difference method). The benchmark showed that the analytic nodal expansion method in R-Z coordinate has good accuracy and quite faster than the finite difference method. This is another merit of using R-Z coordinate in that the transverse integration over surfaces is better than the linear integration over length. This makes the discontinuity factor useless.s.

• Nuclear Engineering and Technology
• /
• v.55 no.8
• /
• pp.2864-2872
• /
• 2023

The criticality problem in this study is studied with the recently investigated the Anlı-Güngör scattering function. The scattering function depends on the Legendre polynomials as the Mika scattering function, but it includes only one scattering parameter, t, and its orders. Both Mika and Anlı-Güngör scattering are the same for only linear anisotropic scattering. The difference appears for the quadratic scattering and further. The analytical calculations are performed with the H_N method, and the numerical results are calculated with Wolfram Mathematica. Interpolation technique in Mathematica is also used to approximate the isotropic scattering results when t parameter goes to zero. Thus, the calculated results could be compared with the literature data for isotropic scattering.

• Asian-Australasian Journal of Animal Sciences
• /
• v.25 no.6
• /
• pp.764-771
• /
• 2012

Direct and maternal genetic heritabilities and their correlations with body weight at 5 stages in the life span of purebred Berkshire pigs, from birth to harvest, were estimated to scrutinize body weight development with the records for 5,088 purebred Berkshire pigs in a Korean farm, using the REML based on an animal model. Body weights were measured at birth (Birth), at weaning (Weaning: mean 22.9 d), at the beginning of a performance test (On: mean 72.7 d), at the end of a performance test (Off: mean 152.4 d), and at harvest (Finish: mean 174.3 d). Ordinary polynomials and Legendre with order 1, 2, and 3 were adopted to adjust body weight with age in the multivariate animal models. Legendre with order 3 fitted best concerning prediction error deviation (PED) and yielded the lowest AIC for multivariate analysis of longitudinal body weights. Direct genetic correlations between body weight at Birth and body weight at Weaning, On, Off, and Finish were 0.48, 0.36, 0.10, and 0.10, respectively. The estimated maternal genetic correlations of body weight at Finish with body weight at Birth, Weaning, On, and Off were 0.39, 0.49, 0.65, and 0.90, respectively. Direct genetic heritabilities progressively increased from birth to harvest and were 0.09, 0.11, 0.20, 0.31, and 0.43 for body weight at Birth, Weaning, On, Off, and Finish, respectively. Maternal genetic heritabilities generally decreased and were 0.26, 0.34, 0.15, 0.10, and 0.10 for body weight at Birth, Weaning, On, Off, and Finish, respectively. As pigs age, maternal genetic effects on growth are reduced and pigs begin to rely more on the expression of their own genes. Although maternal genetic effects on body weight may not be large, they are sustained through life.

• KSCE Journal of Civil and Environmental Engineering Research
• /
• v.14 no.4
• /
• pp.729-740
• /
• 1994

In the shape optimal design based on h-version of FEM, the ideal mesh for the initial geometry most probably will not be suitable for the final analysis. Thus, it is necessary to remesh the geometry of the model at each stage of optimization. However, the p-version of FEM appears to be a very attractive alternative for use in shape optimization. The main advantages are as follows; firstly, the elements are not sensitive to distortion for interpolation polynomials of order $p{\geq}3$; secondly, even singular problems can be solved more efficiently with p-version than with the h-version by proper mesh design; thirdly, the initial mesh design are identical. The 2-D p-version model for shape optimization is presented on the basis of Bezier's curve fitting, gradient projection method, and integrals of Legendre polynomials. The numerical results are performed by p-version software RASNA.

• Journal of Welding and Joining
• /
• v.30 no.6
• /
• pp.74-79
• /
• 2012

It depends on the joint configuration, dimensions and constraints of the joint whether the residual stress at the root of single-sided butt weld is tensile or not. Therefore, recommendation is generally made that high R ratio should be used in the fatigue test of welded joint in order to prevent excessively long life caused by compressive residual stress. In this research, the residual stress profile in butt weld joint was obtained through compliance method, using successive extension of a slot and measurement of the variation of strain during the slot extension. The residual stress profile was firstly assumed to be the linear summation of Legendre polynomials up to 9th order excluding 0th and 1st order. Strain variation on the surface was measured while the slot was being extended by cutting to find out the 8 unknown coefficients of each polynomial term. The cut was made by the electric discharge machine. It was concluded that the residual stress near the surface is positive valued, however, it turned into the negative value as soon as it passed through 2 or 3 mm of the depth.

• KSCE Journal of Civil and Environmental Engineering Research
• /
• v.12 no.4_1
• /
• pp.1-8
• /
• 1992

This paper presents a p-version finite element approach for modeling the stress distribution around a circular hole in a finite strip subjected to membrane and flexural behaviors. Also, same problem with a crack emanating from a perforated tension strip was solved by virtual crack extension method. The p-version of the finite element method based on integrals of Legendre polynomials is shown to perform very well for modeling geometries with very steep stress gradients in the vicinity of a circular cutout. Here, the transfinite mapping technique for circular boundaries was used to avoid the discretization errors. The numerical results from the proposed scheme have a good comparison with those by Nisida, Howland, Newman etc. and the conventional finite element approach.

• Kyungpook Mathematical Journal
• /
• v.50 no.2
• /
• pp.177-193
• /
• 2010

Working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the type $F_n(x)=\sum\limits_{k=0}^{n}A^n_kG_k(x)$ and $ G_n(x)=\sum\limits_{k=0}^{n}B_k^nF_k(x)$, where 0, 1, 2,$\cdots$. Here $F_n(x)$, $G_n(x)$ denote special polynomial functions, and $A_k^n$, $B_k^n$ denote coefficients found by use of the orthogonal properties of $F_n(x)$ and $G_n(x)$, or by skillful series manipulations. Typically $G_n(x)=x^n$ and $F_n(x)=P_n(x)$, the n-th Legendre polynomial. We give a collection of inverse series pairs of the type $f(n)=\sum\limits_{k=0}^{n}A_k^ng(k)$ if and only if $g(n)=\sum\limits_{k=0}^{n}B_k^nf(k)$, each pair being based on some reasonably well-known special function. We also state and prove an interesting generalization of a theorem of Rainville in this form.

• International Journal of Aeronautical and Space Sciences
• /
• v.14 no.3
• /
• pp.282-295
• /
• 2013

In order to communicate with a ground station, the tracking profile design problem for a directional antenna system is considered. Because the motions of the gimbal angles in the antenna system affect the image quality, the main object is to minimize the motion of the gimbal angles during the satellite's imaging phase. For this goal, parameter optimization problems in the imaging and maneuver phases are formulated separately in the body-frame, and solved sequentially. Also, several mechanical constraints, such as the limitation of the gimbal angle and rate, are considered in the problems. The tracking profiles of the gimbal angles in the maneuver phases are designed with N-th order polynomials, to continuously connect the tracking profiles between two imaging phases. The results confirm that if the vector trace of the desired antenna-pointing vector is within the antenna's beam-width angle, motions of the gimbal angles are not required in the corresponding imaging phase. Also, through numerical examples, it is shown that motion of the gimbal angles in the imaging phase can be minimized by the proposed design process.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 2007.04a
• /
• pp.481-486
• /
• 2007

The Zienkiewicz-Zhu(Z/Z) error estimate is slightly modified for the hierarchical p-refinement, and is then applied to L-shaped plates subjected to bending to demonstrate its effectiveness. An adaptive procedure in finite element analysis is presented by p-refinement of meshes in conjunction with a posteriori error estimator that is based on the superconvergent patch recovery(SPR) technique. The modified Z/Z error estimate p-refinement is different from the conventional approach because the high order shape functions based on integrals of Legendre polynomials are used to interpolate displacements within an element, on the other hand, the same order of basis function based on Pascal's triangle tree is also used to interpolate recovered stresses. The least-square method is used to fit a polynomial to the stresses computed at the sampling points. The strategy of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed such that a particular element has to be refined automatically to obtain an acceptable level of accuracy by increasing p-levels non-uniformly or selectively. It is noted that the error decreases rapidly with an increase in the number of degrees of freedom and the sequences of p-distributions obtained by the proposed error indicator closely follow the optimal trajectory.

• Journal of The Korean Society of Agricultural Engineers
• /
• v.48 no.1
• /
• pp.61-68
• /
• 2006

The p-version nonlinear finite element model has been developed to analyze the nonlinear behavior of simply supported RC slabs strengthened with carbon fiber reinforced plastic sheets. The shape function is adopted with integral of Legendre polynomials. The compression model of concrete is based on the Kupfer's yield criterion, hardening rule, and crushing condition. The cracking behavior is modeled by a smeared crack model. In this study, the fixed crack approach is adopted as being geometrically fixed in direction once generated. Each steel layer has a uniaxial behavior resisting only the axial force in the bar direction. Identical behavior is assumed fur tension and compression of steel according to the elastic modulus. The carbon fiber reinforced plastic sheets are considered as reinforced layers of equivalent thickness with uniaxial strength and rigidity properties in the present model. It is shown that the proposed model is able to adequately predicte the displacement and ultimate load of nonlinear simply supported RC slabs by a patch with respect to reinforcement ratio, thickness and angles of CFRP sheets.