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.

Relaxation system based sub-grid scale modelling for large eddy simulation of Burgers' equation

An implicit sub-grid scale model for large eddy simulation is presented by utilising the concept of a relaxation system for one dimensional Burgers' equation in a novel way. The Burgers' equation is solved for three different unsteady flow situations by varying the ratio of relaxation parameter (epsilon) to time step. The coarse mesh results obtained with a relaxation scheme are compared with the filtered DNS solution of the same problem on a fine mesh using a fourth-order CWENO discretisation in space and third-order TVD Runge-Kutta discretisation in time. The numerical solutions obtained through the relaxation system have the same order of accuracy in space and time and they closely match with the filtered DNS solutions.

A new parallel overlapped domain decomposition method for nonlinear dynamic finite element analysis

In this paper a new parallel algorithm for nonlinear transient dynamic analysis of large structures has been presented. An unconditionally stable Newmark-beta method (constant average acceleration technique) has been employed for time integration. The proposed parallel algorithm has been devised within the broad framework of domain decomposition techniques. However, unlike most of the existing parallel algorithms (devised for structural dynamic applications) which are basically derived using nonoverlapped domains, the proposed algorithm uses overlapped domains. The parallel overlapped domain decomposition algorithm proposed in this paper has been formulated by splitting the mass, damping and stiffness matrices arises out of finite element discretisation of a given structure. A predictor-corrector scheme has been formulated for iteratively improving the solution in each step. A computer program based on the proposed algorithm has been developed and implemented with message passing interface as software development environment. PARAM-10000 MIMD parallel computer has been used to evaluate the performances. Numerical experiments have been conducted to validate as well as to evaluate the performance of the proposed parallel algorithm. Comparisons have been made with the conventional nonoverlapped domain decomposition algorithms. Numerical studies indicate that the proposed algorithm is superior in performance to the conventional domain decomposition algorithms. (C) 2003 Elsevier Ltd. All rights reserved.

Surrogate based design optimisation of composite aerofoil cross-section for helicopter vibration reduction

Design optimisation of a helicopter rotor blade is performed. The objective is to reduce helicopter vibration and constraints are put on frequencies and aeroelastic stability. The ply angles of the D-spar and skin of the composite rotor blade with NACA 0015 aerofoil section are considered as design variables. Polynomial response surfaces and space filling experimental designs are used to generate surrogate models of the objective function with respect to cross-section properties. The stacking sequence corresponding to the optimal cross-section is found using a real-coded genetic algorithm. Ply angle discretisation of 1 degrees, 15 degrees, 30 degrees and 45 degrees are used. The mean value of the objective function is used to find the optimal blade designs and the resulting designs are tested for variance. The optimal designs show a vibration reduction of 26% to 33% from the baseline design. A substantial reduction in vibration and an aeroelastically stable blade is obtained even after accounting for composite material uncertainty.

Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions

The paper discusses the frequency domain based solution for a certain class of wave equations such as: a second order partial differential equation in one variable with constant and varying coefficients (Cantilever beam) and a coupled second order partial differential equation in two variables with constant and varying coefficients (Timoshenko beam). The exact solution of the Cantilever beam with uniform and varying cross-section and the Timoshenko beam with uniform cross-section is available. However, the exact solution for Timoshenko beam with varying cross-section is not available. Laplace spectral methods are used to solve these problems exactly in frequency domain. The numerical solution in frequency domain is done by discretisation in space by approximating the unknown function using spectral functions like Chebyshev polynomials, Legendre polynomials and also Normal polynomials. Different numerical methods such as Galerkin Method, Petrov- Galerkin method, Method of moments and Collocation method or the Pseudo-spectral method in frequency domain are studied and compared with the available exact solution. An approximate solution is also obtained for the Timoshenko beam with varying cross-section using Laplace Spectral Element Method (LSEM). The group speeds are computed exactly for the Cantilever beam and Timoshenko beam with uniform cross-section and is compared with the group speeds obtained numerically. The shear mode and the bending modes of the Timoshenko beam with uniform cross-section are separated numerically by applying a modulated pulse as the shear force and the corresponding group speeds for varying taper parameter in are obtained numerically by varying the frequency of the input pulse. An approximate expression for calculating group speeds corresponding to the shear mode and the bending mode, and also the cut-off frequency is obtained. Finally, we show that the cut-off frequency disappears for large in, for epsilon > 0 and increases for large in, for epsilon < 0.