Extending lagrange interpolation to develop a 4-node quadrilateral element

Today finite element method is a well established tool in engineering analysis and design. Though there axe many two and three dimensional finite elements available, it is rare that a single element performs satisfactorily in majority of practical problems. The present work deals with the development of 4-node quadrilateral element using extended Lagrange interpolation functions. The classical univariate Lagrange interpolation is well developed for 1-D and is used for obtaining shape functions. We propose a new approach to extend the Lagrange interpolation to several variables. When variables axe more than one the method also gives the set of feasible bubble functions. We use the two to generate shape function for the 4-node arbitrary quadrilateral. It will require the incorporation of the condition of rigid body motion, constant strain and Navier equation by imposing necessary constraints. The procedure obviates the need for isoparametric transformation since interpolation functions are generated for arbitrary quadrilateral shapes. While generating the element stiffness matrix, integration can be carried out to the accuracy desired by dividing the quadrilateral into triangles. To validate the performance of the element which we call EXLQUAD4, we conduct several pathological tests available in the literature. EXLQUAD4 predicts both stresses and displacements accurately at every point in the element in all the constant stress fields. In tests involving higher order stress fields the element is assured to converge in the limit of discretisation. A method thus becomes available to generate shape functions directly for arbitrary quadrilateral. The method is applicable also for hexahedra. The approach should find use for development of finite elements for use with other field equations also.

The knight of sugar industry: T. S. Venkatraman (1884-1963)

Tiruvadi Sambasiva Venkatraman (TSV) was a plant breeder. In response to a call from Pundit Madan Mohan Malaviya, he made it his mission to develop high-yielding varieties of sugarcane for manufacturing sugar and making it available as a sweetening agent and an energy source for the malnourished children of India. Using Saccharum officinarum, then under cultivation in India, as the female parent, he artificially fertilized it with pollen from S. barberi, which grew wild in Coimbatore. After 4-5 recurrent backcrossings of S. officinarum Chi wild Sorghum spontaneum with S. officinarum as the female parent, TSV selected the `rare' interspecies hybrid cane varieties that resembled sugarcane and had approximately 2.5 cm thick juicy stems containing 16-18% sucrose - nearly 35 times more than what occurred in parent stocks. The hybrid canes matured quickly, were resistant to waterlogging, drought, and to the red-rot disease caused by Glomerella tucumanensis (Sordariomycetes: Glomerellaceae), and to the sereh-virus disease. Most importantly, they were amenable for propagation using stem cuttings. In recognition of the development of high-yielding sugarcane varieties, TSV was conferred the titles Rao Bahadur, Rao Sahib, and Sir by the British Government, and Padma Bhushan by the Republic of India. In the next few decades, consequent to TSV's work, India turned into the second largest sugar producer in the world, after Brazil. The hybrid sugarcane varieties developed are the foundational stocks for new sugarcane x bamboo hybrids, and for possible resistance to Puccinia megalocephala (Pucciniomycetes: Pucciniaceae) and Ustilago scitaminea (Ustilaginomycetes: Ustilaginaceae) using molecular techniques.