Abaqus Integration Point

[Marthe Bernskoetter]

Itis 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).

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:

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!

Do you mean in the python environment? You have section points for shell elements which are trough thickness you set these through your shell section. The amount of integration points depend on your element type.

An integration point in FEM where the primary variables are solved. Just keep that in mind. In user subroutines in Abaqus, the calculation takes place at each integration point. Remember that and go forward. If you are unsatisfied, take a look at any FEM book for the definition/explanation of the integration point. It is not dependent on subroutines.

Also within a subroutine other variables such as state variables, SVARS, or stored at the integration points so if your element has say 4 SVARS you need to keep track of then there will 8 * 4 = 32 SVARS in the entire 8 node element.

The beam is 150 mm long, 2.5 mm wide, and 5 mm deep; built-in at one end; and carrying a tip load of 5 N at the free end. The material has a Young's modulus, , of 70 GPa and a Poisson's ratio of 0.0. Using beam theory, the static deflection of the tip of the beam for a load is given as

Several different finite element meshes were used in ABAQUS/Standard simulations of the cantilever beam problem, as shown in Figure 4–3. The simulations use either linear or quadratic, fully integrated elements and illustrate the effects of both the order of the element (first versus second) and the mesh density on the accuracy of the results.

The linear elements CPS4 and C3D8 underpredict the deflection so badly that the results are unusable. The results are least accurate with coarse meshes, but even a fine mesh (8 24) still predicts a tip displacement that is only 56% of the theoretical value. Notice that for the linear, fully integrated elements it makes no difference how many elements there are through the thickness of the beam. The underprediction of tip deflection is caused by shear locking, which is a problem with all fully integrated, first-order, solid elements.

As we have seen, shear locking causes the elements to be too stiff in bending. It is explained as follows. Consider a small piece of material in a structure subject to pure bending. The material will distort as shown in Figure 4–4. Lines initially parallel to the horizontal axis take on constant curvature, and lines through the thickness remain straight. The angle between the horizontal and vertical lines remains at 90.Figure 4–4 Deformation of material subjected to bending moment .

For visualization, dotted lines that pass through the integration points are plotted. It is apparent that the upper line has increased in length, indicating that the direct stress in the 1-direction, , is tensile. Similarly, the length of the lower dotted line has decreased, indicating that is compressive. The length of the vertical dotted lines has not changed (assuming that displacements are small); therefore, at all integration points is zero. All this is consistent with the expected state of stress of a small piece of material subjected to pure bending. However, at each integration point the angle between the vertical and horizontal lines, which was initially 90, has changed. This indicates that the shear stress, , at these points is nonzero. This is incorrect: the shear stress in a piece of material under pure bending is zero.

This spurious shear stress arises because the edges of the element are unable to curve. Its presence means that strain energy is creating shearing deformation rather than the intended bending deformation, so the overall deflections are less: the element is too stiff.

Shear locking only affects the performance of fully integrated, linear elements subjected to bending loads. These elements function perfectly well under direct or shear loads. Shear locking is not a problem for quadratic elements since their edges are able to curve (see Figure 4–6). The predicted tip displacements for the quadratic elements shown in Table 4–1 are close to the theoretical value. However, quadratic elements will also exhibit some locking if they are distorted or if the bending stress has a gradient, both of which can occur in practical problems.Figure 4–6 Deformation of a fully integrated, quadratic element subjected to bending moment .

Fully integrated, linear elements should be used only when you are fairly certain that the loads will produce minimal bending in your model. Use a different element type if you have doubts about the type of deformation the loading will create. Fully integrated, quadratic elements can also lock under complex states of stress; thus, you should check the results carefully if they are used exclusively in your model. However, they are very useful for modeling areas where there are local stress concentrations.

Only quadrilateral and hexahedral elements can use a reduced-integration scheme; all wedge, tetrahedral, and triangular solid elements use full integration, although they can be used in the same mesh with reduced-integration hexahedral or quadrilateral elements.

Neither of the dotted visualization lines has changed in length, and the angle between them is also unchanged, which means that all components of stress at the element's single integration point are zero. This bending mode of deformation is thus a zero-energy mode because no strain energy is generated by this element distortion. The element is unable to resist this type of deformation since it has no stiffness in this mode. In coarse meshes this zero-energy mode can propagate through the mesh, producing meaningless results.

The quadratic reduced-integration elements available in ABAQUS/Standard also have hourglass modes. However, the modes are almost impossible to propagate in a normal mesh and are rarely a problem if the mesh is sufficiently fine. The 1 6 mesh of C3D20R elements fails to converge because of hourglassing unless two elements are used through the width, but the more refined meshes do not fail even when only one element is used through the width. Quadratic reduced-integration elements are not susceptible to locking, even when subjected to complicated states of stress. Therefore, these elements are generally the best choice for most general stress/displacement simulations, except in large-displacement simulations involving very large strains and in some types of contact analyses.

The incompatible mode elements, available only in ABAQUS/Standard, are an attempt to overcome the problems of shear locking in fully integrated, first-order elements. Since shear locking is caused by the inability of the element's displacement field to model the kinematics associated with bending, additional degrees of freedom, which enhance the element's deformation gradients, are introduced into the first-order element. These enhancements to the deformation gradients allow a first-order element to have a linear variation of the deformation gradient across the element's domain as shown in Figure 4–9(a). The standard element formulation results in a constant deformation gradient across the element as shown in Figure 4–9(b), resulting in the nonzero shear stress associated with shear locking.Figure 4–9 Variation of deformation gradient in (a) an incompatible mode (enhanced deformation gradient) element and (b) a first-order element using a standard formulation.

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