chapter 2 isoparametric element

Iso-parametric FormulationM . E. Engineering Design

Natural Co-ordinate Systems Lagrangian Interpolation Polynomials Isoparametric Elements Formulation Numerical Integration Gauss quadrature one-, two- and three-dimensional triangular elements formulation rectangular elements Serendipity elements Illustrative Examples.Iso-parametric FormulationUnit iii

http://www.civilengineeringterms.com/structural-engineering/local-coordinates-global/Local CoordinatesA local coordinates system whose origin is located within the element in order to simplify the algebraic manipulations in the derivation of the element matrix.The use of natural coordinates in expressing approximate functions is advantageous because special integration formulas can often be applied to evaluate the integrals in the element matrix.Natural coordinates also play a crucial role in the development of elements with carved boundaries.Natural CoordinatesIt is a local coordinate system that permits the specification of a point within the element by a dimensionless parameter whose absolute magnitude never exceeds unity.It is dimension less.They are defined with respect to the element rather than with reference to the global coordinates.Global CoordinatesGlobal coordinates are convenient for specifying the location of each node, the orientation of each element, the boundary conditions and the loads for the entire domain.The solution to the field variable is generally represented with respect to the global coordinates.

http://www.answers.com/Q/What_is_isoparametric_elementWhat is isoparametric element?

Isoparametric elements use the same set of shape functions to represent both the element geometry and displacement interpolations (ux, uy). The shape functions are defined by natural coordinates, such as triangle coordinates for triangles and square coordinates for any quadrilateral. The advantages of isoparametric elements include the ability to map more complex shapes and have compatible geometries.

http://imechanica.org/node/8325Lagrangian Vs. Serendipity Finite ElementsLagrangian elements have two disadvantages, It will give one additional internal node(centre node) and Incomplete polynomial.Serendipity elements don't have centre node and gives complete polynomial. see Pascal triangle.refer "Energy and fem in structural mechanics" by Irving H.Shames.

In finite element analysis the following types of integrations are widely used in one dimensional and two dimensional problems especially for the computation of element stiffness or for the element nodal vector.Several methods are available to solve the integration among the methods Gauss quadrature method is most widely used. Another method is Newtons cotes quadrature.courtesy S.Md. Jalaludeen FEA Engg

courtesy S.Md. Jalaludeen FEA Engg

Example of one point approximation i.e. n=1 courtesy S.Md. Jalaludeen FEA Engg

Example of one point approximation i.e. n=2 courtesy S.Md. Jalaludeen FEA Engg

Gauss quadrature method of integration for rectangular element courtesy S.Md. Jalaludeen FEA in EnggMostly, this Gauss-quadrature method is used to find the approximate value of integral function similar to above when the variables are in natural coordinate system as in the case of stiffness matrix calculation. If E and 11are considered as natural coordinates for the two dimensional rectangular element as shown in the above fig then the integral function and its approximating method will be given as

Tutorial 3.1 courtesy S.Md. Jalaludeen FEA in EnggDetermine the Cartesian coordinates of the point P which has local coordinates as follows.The Cartesian coordinates of the element at its nodes 1,2,3,5 units in the table and the local coordinates of the point P is

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Now, once again by substituting the values of shape functions and nodal coordinates in Eqn. (1), we can get the values of Cartesian coordinates of the point P.Tutorial 3.1 courtesy S.Md. Jalaludeen FEA in Engg

Tutorial 3.2 courtesy S.Md. Jalaludeen FEA in EnggThe Cartesian coordinates of the corner nodes of a quadrilateral element are given by (0, -1), (- 2,3), (2, 4) and (5, 3). Find the coordinate transformation between the global and local coordinates. Using this, determine the Cartesian coordinates of the point defined by (r, s) = (0.5, 0.5) in the local coordinate system.For the given quadrilateral element the Cartesian coordinates are shown in fig,

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Tutorial 3.2 courtesy S.Md. Jalaludeen FEA in Engg

Tutorial 3.4 courtesy S.Md. Jalaludeen FEA in EnggFor the element shown in figure .Determine the Jacobian matrix.The Cartesian coordinates of the given element are,

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Tutorial 3.5 courtesy S.Md. Jalaludeen FEA in EnggEvaluate the Jacobian matrix at the local coordinates for the linear quadrilateral element with its global coordinates as shown in fig .Also evaluate the strain-displacement matrix.

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Tutorial 3.6 courtesy S.Md. Jalaludeen FEA in EnggEstablish the strain-displacement matrix for the linear quadrilateral element as shown in fig. at Gauss point in local coordinate system.


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