This document discusses isoparametric finite elements, which are used to analyze systems with curved boundaries. Isoparametric elements describe both the geometry and field variables (e.g. displacements) using the same interpolation functions. This allows curved elements to be approximated as straight-sided elements defined in local natural coordinates. The actual curved geometry defined in global coordinates is then transformed to the reference configuration of the isoparametric element. Natural coordinate systems use dimensionless parameters from -1 to 1 to specify points within an element.

1. UNIT V

2. INTRODUCTION
• Some of the systems like machine tool tables
and beams will be of straight boundaries
• But the pressure vessels and chimney are
bounded by curved surfaces.
• The straight sided elements are analysed using
one, two or three dimensional linear element
techniques.
• Curved boundary shell type elements are
analysed using axi-symmetric concepts
• For other types of curved boundary systems
which are not looking like shell, isoparametric
elements have been employed

3. ISOPARAMETRIC ELEMENT
FORMULATION
• The element whose shape and field variable are
described by the same interpolation functions of
same order are called as isoparametric elements.
• The element coordinates and element
displacements are expressed in the form of
interpolations using the natural coordinates of the
system of the element.

4. • These isoparametric elements of simple elements
which are expressed in local coordinates which are
referred as master elements are the transformed
shapes of curved sided finite elements of actual
system expressed in global coordinate system as
shown in Figure.
• The curved sided elements are mostly
approximated to triangular or quadrilateral
elements and these approximated shapes, specified
by global coordinate system are converted to
isoparametric element whose geometry and
displacements are specified by natural coordinate
system

11. • We know that for CST element,
u = N1u1+N2u2+N3u3 and
v = N1v1+N2v2+N3v3
For the triangular element, the coordinates x,y
of point P can also be represented in terms of
nodal coordinates using the same shape
functions.
That is the coordinates of P are given by,
x = N1x1+N2x2+N3x3 and --- (1)
y = N1y1+N2y2+N3y3
Also N1+N2+N3 = 1
Therefore, N3 = 1-N1-N2

12. Substituting the above equation in (1) we get
x = N1x1+N2x2+(1-N1-N2)x3
x = (x1 – x3)N1 + (x2 – x3)N2+x3
Similarly y = (y1 – y3)N1 + (y2 – y3)N2+y3

22. Natural Coordinates
• A simple natural coordinate system is a kind of
local coordinate system that permits the
specification of a point within the element by
a set of dimensionless numbers whose
magnitude varies from -1 to +1.
• The rectangular or quadrilateral or any curved
sided actual finite element specified by global
coordinate system, the element will be
described by straight sided square shape or
rectangle only

43. Element Force Vector
• The element Force vector is given by,

98. Pascal’s Triangle
• One of the important consideration in
choosing the polynomial expansion is the
displacement shape should not change with a
change in local coordinate. This property is
known as geometric isotropy.
• Geometric isotropic or geometric invariance is
achieved if the polynomial is of balanced. The
balanced polynomial can be achieved by
Pascal Triangle