A mean value formula for elliptic curves

It is proved in this paper that for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate and that of y-coordinates of its n-division points is n times of its y-coordinate

Enriched finite element methods : advances & applications

This thesis presents advances and applications of the eXtended Finite Element Method (XFEM). The novelty of the XFEM is the enrichment of the primary variables in the elements intersected by the discontinuity surface by appropriate functions. The enrichment scheme carries the local behaviour of the problem and the main advantage is that the method does not require themesh to conform to the internal boundaries. But this flexibility comes with associated difficulties: (1) Blending problem; (2) Numerical integration of enrichment functions and (3) sub-optimal rate of convergence. This thesis addresses the difficulty in the numerical integration of the enrichment functions in the XFEM by proposing two new numerical integration schemes. The first method relies on conformal mapping, where the regions intersected by the discontinuity surface are mapped onto a unit disk. The second method relies on strain smoothing applied to discontinuous finite element approximations. By writing the strain field as a non-local weighted average of the compatible strain field, integration on the interior of the finite elements is transformed into boundary integration, so that no sub-division into integration cells is required. The accuracy and the efficiency of both the methods are studied numerically with problems involving strong and weak discontinuities. The XFEM is applied to study the crack inclusion interaction in a particle reinforced composite material. The influence of the crack length, the number of inclusions and the geometry of the inclusions on the crack tip stress field is numerically studied. Linear natural frequencies of cracked functionally graded material plates are studied within the framework of the XFEM. The effect of the plate aspect ratio, the crack length, the crack orientation, the gradient index and the influence of cracks is numerically studied. LATEX-ed Friday, October 14, 2011; 10:55am © Sundararajan NatarajanEThOS - Electronic Theses Online ServiceGBUnited Kingdo

Enriched finite element methods: advances & applications

This thesis presents advances and applications of the eXtended Finite Element Method (XFEM). The novelty of the XFEM is the enrichment of the primary variables in the elements intersected by the discontinuity surface by appropriate functions. The enrichment scheme carries the local behaviour of the problem and the main advantage is that the method does not require themesh to conform to the internal boundaries. But this flexibility comes with associated difficulties: (1) Blending problem; (2) Numerical integration of enrichment functions and (3) sub-optimal rate of convergence. This thesis addresses the difficulty in the numerical integration of the enrichment functions in the XFEM by proposing two new numerical integration schemes. The first method relies on conformal mapping, where the regions intersected by the discontinuity surface are mapped onto a unit disk. The second method relies on strain smoothing applied to discontinuous finite element approximations. By writing the strain field as a non-local weighted average of the compatible strain field, integration on the interior of the finite elements is transformed into boundary integration, so that no sub-division into integration cells is required. The accuracy and the efficiency of both the methods are studied numerically with problems involving strong and weak discontinuities. The XFEM is applied to study the crack inclusion interaction in a particle reinforced composite material. The influence of the crack length, the number of inclusions and the geometry of the inclusions on the crack tip stress field is numerically studied. Linear natural frequencies of cracked functionally graded material plates are studied within the framework of the XFEM. The effect of the plate aspect ratio, the crack length, the crack orientation, the gradient index and the influence of cracks is numerically studied. LATEX-ed Friday, October 14, 2011; 10:55am © Sundararajan Nataraja

2010-2011, University of Memphis bulletin

University of Memphis bulletin containing the graduate catalog for 2010-2011.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1430/thumbnail.jp

2009-2010, University of Memphis bulletin

University of Memphis bulletin containing the graduate catalog for 2009-2010.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1429/thumbnail.jp

NASA Tech Briefs, January 1987

Topics include: NASA TU Services; New Product Ideas; Electronic Components and Circuits; Electronic Systems; Physical Sciences; Materials; Computer Programs; Mechanics; Fabrication Technology; Machinery; Mathematics and Information Sciences; Life Sciences

1958-1959 CATALOG ISSUE- BULLETIN

Course catalog for 1958-1959https://digitalrepository.unm.edu/course_catalogs/1064/thumbnail.jp

2014-2015, University of Memphis bulletin

University of Memphis bulletin containing the graduate catalog for 2014-2015.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1433/thumbnail.jp