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Numerical Integration

An important aspect of the finite element method is the requirement for numerical integration. For example, the stiffness matrix [k] and load vector {r} needs evaluating for each element as follows

Where [N] and [B] are the shape function and strain-displacement matrices respectively.

It is possible to evaluate such integrals using closed form analytical solutions for a few element types (see explicit integration), but this would, in general, reduce the flexibility of the finite element method both dramatically and unnecessarily. Numerical integration provides a convenient and efficient method to carry out such integration and, with appropriate use, yields extremely accurate results.

There are two numerical integration methods used in LUSAS

  • Newton-Cotes

  • Gaussian

In the Newton-Cotes method the integration points are equally spaced along the natural coordinate axes for the section, typically at the nodal positions. In the Gaussian integration method, the integration points are not equally spaced along the natural coordinate axes for the section, but are at internal element positions called Gauss points. In the latter case, the positions of the integration points have been optimised to give a more efficient method than the Newton-Cotes method, albeit not at as convenient integration positions. The methods are, however, actually very similar in practice.

Consider a line element, in which the one-dimensional integral is required as follows

The numerical integration is given for this as

Where Wi are known as the integration weights and the f(xi) are the values of the function at the pre-determined integration points.

For this line element the integration point positions and the corresponding integration weights for the different integration rules of the Newton-Cotes method are given in the following table

Integration Rule

Integration Point Locations

Integration Weight

1 Point Rule

2.00000

2 Point Rule

1.00000

3 Point Rule

0.33333

1.33333

0.33333

Similarly, the integration point positions and the corresponding integration weights for the different integration rules of the Gaussian method appropriate to this line element are given as follows

Integration Rule

Integration Point Locations

Integration Weight

1 Point Rule

2.00000

2 Point Rule

1.00000

3 Point Rule

0.55556

0.88889

0.55556

Similar rules apply for surface and volume elements. See appendix A of the theory manual for more information.

As an example of the use these numerical integration schemes, consider the following integral

The analytical solution is readily available as

Using the Newton-Cotes integration method we have, for the one-point rule

Where .means the function x2 is evaluated at x=0.

For the two-point rule, this becomes

And for the three-point rule

Using the Gaussian integration method we have, for the one-point rule

For the two-point rule

And. for the three-point rule

It can be seen that the Gaussian method exactly integrates the equation with the 2-point integration rule, whilst the Newton-Cotes method requires a 3-point rule. Specifically, a Gaussian rule of order n integrates exactly a polynomial of order (2n-1) whilst the Newton-Cotes rule of order n integrates exactly a polynomial of order (n-1). In general, Gauss integration is the more efficient since a given order of integration may be performed with less integration points than for the Newton-Cotes methods (a direct result of the optimised Gauss point locations). This is important since such integrals form a large part of the cost of a finite element analysis.

In LUSAS, Gaussian integration is the most prevalent method. The Newton-Cotes method is used, however, where stress results are particularly required at nodal positions. This is the default case for delamination elements and optional for the integration of variables along the length of cross section beams. It is the default case when integrating the rigidity matrix through the depth of cross-section beam elements (the default integration employs a 3x3 Newton Cotes rule for linear materials and a 5x5 rule for nonlinear materials). More information on the cross sectional integration for these elements is available in the Element Reference Guide.

The appropriate order of integration rule to use for an element depends on the matrix that is to be evaluated and the specific element being considered, but it does need to consider the following issues

a)     If a high enough integration rule is used, all matrices will be evaluated very accurately. However, the cost of an analysis increases significantly when higher order rules are used than are strictly necessary. This is because the integration of all the element matrices in the finite element method are performed at each Gauss point – the more Gauss points, the more numerical loops are required

b)    Using too low a rule may integrate the matrices very inaccurately, such that they contain rigid body modes or mechanisms. Such behaviour affects the results to a very large degree

An element is said to be fully integrated if the integration rule used is the lowest order required to evaluate the element matrices as accurately as an analytical integration and also ensure that there are no element mechanisms associated with the element. Conversely, an element is said to be under-integrated if the integration rule used is the lowest order required to evaluate the element matrices as accurately as an analytical integration but retains one or more element mechanisms associated with the element. There are good reasons for under-integration that are discussed further later.

In considering, for example, the correct integration rule required for a 3-noded bar element, the stiffness matrix can be evaluated from the following, between the natural coordinate limits of 1.

Which is clearly a cubic or 3rd order matrix. Since a Gaussian rule of order n integrates exactly a polynomial of order (2n-1), this stiffness matrix will be fully integrated with a Gaussian rule of order 2. It is usually good practice to evaluate the other element matrices with the same order (e.g. the forces).

The mass matrix is, however, dealt with slightly differently. For more information, see Evaluating Mass Matrices.

In the LUSAS Element Reference manual, there is a section entitled “Integration Schemes” that describes the integration rules used for each element. An example is in the following table, from the two-dimensional plane stress element section

Stiffness

Default

1-point (TPM3), 3-point (TPM6)

2x2 (QPM4, QPM8)

Fine (see Options)

3x3 (QPM8), 3-point (TPM3)

 

Mass

Default

1-point (TPM3), 3-point (TPM6)

2x2 (QPM4, QPM8)

Fine (see Options)

3x3 (QPM8), 3-point (TPM3)

 

In this section, the “default” integration rule is specified for each element within this element suite. The integration locations for the 2x2 and 3x3 rules (and others) are described in the appendix of the element reference manual.

The term “fine” integration actually means “full” integration. For some elements, the “fine” integration rule will be the same as the default rule. This means that the default rule used is the “full” integration rule - that is, the integration rule that yields accurate results without element mechanisms present. For others, the default and fine rule will be different. In such cases, the default rule used under-integrates the element matrix and is called a reduced integration rule.

Reduced integration has been found most effective in a number of element types. This is not an uncommon approach and is based on substantial experience. Any mechanisms present from such a usage are not commonly experienced in a typical linear element assembly although they may occur in both materially and geometrically nonlinear analyses. When investigating the analysis results, checks should be performed for spurious stress oscillations and peculiarities in the deformed configuration that may indicate the presence of an element mechanism.

The benefit of reduced-integration is typically that under-integration of the stiffness matrix prevents locking, which may occur with full integration when either the element is subjected to parasitic shear for example or as the material reaches the incompressible limit in the presence of material nonlinearity. Locking produces an over-stiff solution, most significantly affecting the bending and/or shear results. Locking can, however, be significantly suppressed with the higher order rules when using a fine mesh discretisation (unfortunately leading to costly analyses). For elements that use a reduced rule by default, the NOTES section in the Element Reference Manual describes the specific reasons for this choice of integration rule.

The fine integration option for an element may be invoked within MODELLER using File> Model Properties> Solution> Element options.

There are also a number of non-standard integration rules that have been developed for specific elements, these include the 5-point rule for the Semiloof thin shell elements and the 13 and 14-point rules for the 3D continuum solid element.

Implication: Evaluating Mass and Damping Matrices

In LUSAS, for dynamic analyses, fine integration is invoked automatically to ensure that the mass and damping matrices are fully integrated. They are generated as follows

From which it can be seen that the element mass and damping matrices are evaluated directly from the shape functions [N], rather than their derivatives (as in the case of the other element matrices), and therefore requiring a higher order rule. See evaluating mass matrices

Implication: Element mechanisms - What are they?

Reduced or under-integration can provide an improvement to element behaviour. It does pose, however, a numerical difficulty in that one or more element mechanisms may be associated with the element.


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