Isoparametric Elements and Numerical Integration

Most finite elements used in commercial codes are either isoparametric, super-parametric or sub-parametric elements. These elements can map highly distorted curvilinear geometries. An isoparametric element maps the geometry with the same shape functions defining the displacement function. A super-parametric element maps the geometry with a higher order shape function than that which defines the displacement function. Finally, a sub-parametric element maps the geometry with a lower order shape function than that which defines the displacement function. Super-parametric and isoparametric representation is illustrated for the 3-noded triangle. The super-parametric representation is presented as follows.

In the super-parametric representation, the triangular coordinates define the element geometry through the first set of relationships. The second set of relationships show the displacement expansion defined by the shape functions, which are in turn expressed in terms of the triangular coordinates. Evidently the element geometry and element displacements are not treated equally. If we proceed to higher order triangles with straight sides, only the displacement expansion is refined whereas the geometry definition remains the same. Elements built according to this prescription are called super-parametric, a term that emphasizes that unequal treatment.

The key idea is to use the shape functions to represent both the element geometry and the problem unknowns, which in structural mechanics are displacements. Hence the name isoparametric element (“iso” means equal), often abbreviated to iso-P element.

To generalize to include other two-dimensional elements, the term triangular coordinates are replaced by the general term natural coordinates. Natural coordinates (triangular coordinates for triangles, quadrilateral coordinates for quadrilaterals) appear as parameters that define the shape functions. The shape functions then connect the geometry with the displacements.

The generalization to an arbitrary two-dimensional element with n nodes is thus presented. Two sets of relations, one for the element geometry and the other for the element displacements, are required. Both sets exhibit the same interpolation in terms of the shape functions.

Combining both sets of expressions

Thus for the linear (3-noded) triangular element

The shape functions are simply the triangular coordinates

Since the resulting expression is the same as that produced by the super-parametric approach, the linear triangle is actually both super-parametric and isoparametric. It is in fact the only triangle that is both super-parametric and isoparametric. The 6-noded triangular element has the following geometric and displacement interpolation

where the shape functions are

The isoparametric version can thus have curved sides defined by the location of the mid-side nodes.

The natural coordinates for a triangular element are the triangular coordinates ζ1, ζ2 and ζ3. The natural coordinates for a quadrilateral element are ξ and η. These are called quadrilateral coordinates. These coordinates vary from -1 on one side to +1 at the other. This particular variation range (instead of, say, 0 to 1) was chosen by the investigators who originally developed isoparametric quadrilaterals to facilitate the use of the standard Gauss integration formulas. The isoparametric 4-noded quadrilateral has the following geometric and displacement interpolation

where the shape functions are

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The element equilibrium equations can be written for a geometrically linear element to be

Exact integration of the above stiffness matrix and the fixed end forces often cannot be done analytically. Constant strain triangles (CST) have constant [B] and [D] matrices and are a rare exception to the rule since their stiffness and fixed end force matrices can be integrated in closed form. For the quadrilateral, the matrix [B] often depends on the location of the coordinates. This occurs because the shape functions once differentiated are still a function of the natural coordinates.

The derivation of the stiffness terms is presented for the isoparametric quadrilateral. Partial derivatives of shape functions with respect to the Cartesian coordinates x and y are required for the strain and stress calculations ultimately to derive the stiffness matrix. Since the shape functions are not directly functions of x and y but of the natural coordinates ξ and η, the determination of Cartesian partial derivatives is not trivial. We require the Jacobian of two-dimensional transformations that connect the differentials of {x, y} to those of {ξ,η} and vice-versa.

Hence, the shape function derivatives in terms of the quadrilateral coordinates are

or in matrix form

These terms feature in the strain matrix [B] which ultimately yields the stiffness matrix [K] for the element. The symbolic inversion of J for arbitrary ξ and η in general leads to extremely complicated expressions unless the element has a particularly simple geometry. This complexity was one of the factors that motivated the use of numerical integration techniques. The use of numerical integration is thus essential for evaluating element integrals of isoparametric elements. The standard practice has been to use Gauss integration (also known as Gauss-Legendre quadrature) because such rules use a minimal number of sample points to achieve a desired level of accuracy.

The Gauss integration rules in one dimension are

where p is the number of Gauss integration points (equal of more than one), wi are the integration weights and ξi

are the sample point abscissas in the interval –1 to 1. Hence the first four one dimensional Gauss rules are

The four rules integrate exactly polynomials in a ξ of orders up to 1, 3, 5 and 7, respectively. In general a one-dimensional Gauss rule with p points integrates exactly polynomials of order up to 2p – 1. This is called the degree of the formula.

The simplest two-dimensional Gauss rules are called product rules. They are obtained by applying the one-dimensional rules to each independent variable in turn. To apply these rules we must first reduce the integrand to the canonical form.

Once this is done we can process numerically each integral in turn.

where p1 and p2 are the number of Gauss points in the ξ and η directions, respectively. Usually the same number p

= p1 = p2 is chosen if the shape functions are taken to be the same in the ξ and η directions.

Hence, if the stiffness matrix is

Reducing the integrand into canonical form

This matrix function can be numerically integrated over the domain –1 ≤ ξ ≤ +1, –1 ≤ η ≤ +1 by an appropriate Gauss product rule. For square 4-noded quadrilaterals, the integrand h BT EBJ is at most quadratic in ξ and η, and hence 2 x 2 Gauss integration suffice to compute the integral exactly (i.e. the element is fully integrated and not under-integrated). Using a higher order Gauss integration rule, such as 3 x 3 and 4 x 4, reproduces exactly the same stiffness matrix produced by the 2 x 2 integration rule. Using a 1 x 1 rule yields a rank-deficiency matrix. For a non-square quadrilateral, there is little difference in the stiffness matrix beyond the 2 x 2 Gauss integration, with higher Gauss integration orders producing a slightly stiffer response (and hence slightly higher element natural frequencies).

As for the 6-noded triangle, the same stiffness matrix is obtained for integration rule 3, rule -3 or rule 7 as long as the triangular shape is maintained and the mid-side nodes are exactly at the mid-point between corner nodes. A highly distorted triangle on the other hand will return varying stiffness matrices for different Gauss integration rules.

A reduced integrated element (Gauss integration of a lower p order) may suffer from matrix singularities as there are insufficient stiffness terms unlike a fully integrated element which will have all its stiffness terms calculated.

Under-integrated elements may thus suffer from hourglassing where the deformed shape shows clear zig-zag patterns indicating DOFs which are not stiff.

However, often it is usually advantages to use A REDUCED INTEGRATION TECHNIQUE (i.e. the minimum integration requirements that PRESERVES THE RATE OF CONVERGENCE which would result if exact integration were used) as, for very good reasons, A CANCELLATION OF ERRORS due to discretization and due to inexact integration can occur ⁵.

5 ZIENKIEWICZ, O.C. & TAYLOR, R.L. The Finite Element Method. The Basis. Volume 1. 5^th Edition.

Butterworth-1.3.26 Element and Nodal Stress Recovery ⁶