Concepts and Applications of Finite Element Analysis, 4th Edition

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It is now 2.5 years after the OP asked this question, so my answer is probably more for anyone who has followed a link here, hoping for some insight. On the grounds that FEM programming is special,0 I will try to answer this question rather than flag it as off-topic. Anyway, some of my answer is applicable to FEM in general, some is specific to Abaqus.

Quick check: If you're only asking for the specific numerical value to use for the (usual or standard) location of integration points, then the answer is that it depends. Luckily, standard values are widely available for a variety of elements (see resources below).

However, I assume you're asking about writing a User-Element (UEL) subroutine but are not yet familiar with how elements are formulated, or what an integration point is.

The answer: In the standard displacement-based FEM the constitutive response of an individual finite element is usually obtained by numerical integration (aka quadrature) at one or more points on or within the element. How many and where these points are located depends on the element type, certain performance tradeoffs, etc, and the particular integration technique being used. Integration techniques that I have seen used for continuum (solid) finite elements include:

• More Common: Gauss integration -- the number & position of sampling points are determined by the Gauss quadrature rule used; nodes are not included in the sampling domain of [-1,1].
• Less Common: Newton-Cotes integration -- evenly spaced sampling points; includes the nodes in the sampling domain of (-1,1).

In my experience, the standard practice by far is to use Gauss quadrature or reduced integration methods (which are often variations of Gauss quadrature). In Gauss quadrature, the location of the integration points are taken at special ("optimal") points within the element known as Gauss points which have been shown to provide a high level of reliably accurate solutions for a given level of computational expense - at least for the typical polynomial functions used for many isoparametric finite elements. Other integration techniques have been found to be competitive in some cases1 but Gauss quadrature is certainly the gold standard. There are other techniques that I'm not familiar with.

Practical advice: Assuming an isoparametric formulation, in the UEL you use "element shape functions" and the primary field variables defined by the nodal degrees of freedom (with a solid mechanics focus, these are typically the displacements) to calculate the element strains, stresses, etc. at each integration point. If this doesn't make sense to you, see resources below.

Note that if you need the stresses at the nodes (or at any other point) you must extrapolate them from the integration points, again using the shape functions, or calculate/integrate directly at the nodes.

Suggested resources: Please: If you're writing a user subroutine you should already know what an integration point is. I'm sorry, but that's just how it is. You have to know at least the basics before you attempt to write a UEL.

That said, I think it's great that you're interested in programming for FEA/FEM. If you're motivated but not at university where you can enroll in an FEM course or two, then there are a number of resources available, from Massive Open Online Courses (MOOCs), to a plethora of textbooks - I generally recommend anything written by Zienkiewicz. For a readable yet "solid" introduction with an emphasis on solid mechanics, I like Concepts and Applications of Finite Element Analysis, 4th Edition, by Cook et al (aka the "Cook Book"). Good luck!

0 You typically need a lot of background before you even ask the right questions.

1 Trefethen, 2008, "Is Gauss Quadrature Better than Clenshaw-Curtis?", DOI 10.1137/060659831