Have you ever found yourself wondering which Abaqus element formulation is most suitable for your analysis? With the vast options provided within Abaqus Standard and Abaqus Explicit, novice and experienced users alike often find themselves asking this very question, and for good reason: selecting the appropriate element type with respect to your simulation objectives is vitally

important for obtaining accurate results. In this blog series, we aim to highlight some of the key aspects one should consider when determining the optimal Abaqus element formulation for a given analysis.

## Article 1: An Introduction to Abaqus Elements

In order to appreciate the advantages and limitations of each element type, it is important to understand both the nomenclature and underlying fundamentals governing finite element behavior in Abaqus. Towards that goal, there are five aspects of an element that characterize its behavior in Abaqus:

- Family

- Degrees of Freedom

- Number of Nodes

- Formulation

- Integration

The element Family, several of which are shown below, is used to describe the type of element and hint at applications for which it may be suitable. The major distinction between element families is the geometry type that each family assumes.

The Degrees of Freedom define the variables being calculated during an analysis and fundamentally control the element’s behavior. For example, solid elements possess only active translational degrees of freedom, whereas shell elements have both translational and rotational (for a stress/displacement analysis). Other Degrees of Freedom, such as temperature or acoustic pressure, may be relevant based on the element family and analysis techniques being utilized.

The Number of Nodes that an element contains directly impacts the total degrees of freedom and therefore has a significant impact on the element’s ability to deform. As an example, one can consider the behavior of a three-noded triangle versus a four-noded quadrilateral; when deformed, the three-noded triangle results in a state of constant strain and is therefore less compliant than the four-noded quadrilateral. Additionally (and perhaps more importantly), the number of nodes an element has dictates the strategy that will be used to interpolate the degrees of freedom calculated at the nodes to the rest of the element. Within Abaqus (and most other advanced solvers), there are two available orders of interpolation: linear and quadratic.

- First-Order (Linear): Elements that have nodes only at their corners

- Second-Order (Quadratic): Elements that have corner nodes and mid-side nodes

The Formulation of an element dictates the underlying mathematical algorithms governing element behavior. Fundamentally, there are two distinct types of elemental behavior: Lagrangian and Eulerian. The Langrangian model describes elements which deform with the material, whereas Eulerian elements are fixed in space and allow material to flow through them. For obvious reasons, this means that Lagrangian elements are appropriate for stress/displacement analyses and Eulerian elements are typically more suitable for representing fluid mechanics. In addition to the overriding Lagrangian or Eulerian formulation, some element families have several “sub-formulations” intended to accommodate different types of behavior. Shell elements, for instance, have three available formulation classes to choose from: general purpose shells, “thin” shells, or “thick” shells (i.e. plane stress vs. plane strain). Other examples of alternative element formulations can be observed in continuum (solid) elements where Hybrid (H), Incompatible Modes (I), and Modified (M) formulations exist (among others).

To allow for complete material generality, Abaqus uses Integration to determine various quantities throughout the volume of an element. The material response is evaluated at each of the integration points, which are typically defined using a Gaussian quadrature (for most elements). Because each integration point requires mathematical resolution, it logically follows that elements with more integration points are computationally more expensive than those with fewer. To that end, several Abaqus element families provide a “Reduced Integration” option which, as the name suggests, uses fewer integration points than the standard element formulation. When used appropriately, reduced integration elements can substantially improve model efficiency and solver runtime. However, it must be noted that reduced integration elements have limitations which can lead to inaccurate or spurious results if used incorrectly.

Although the number of variables which must be considered when selecting an appropriate finite element in Abaqus may initially seem overwhelming, the process is actually quite simple and, over time, becomes second-nature. By carefully considering the physics of what you are simulating as well as the runtime constraints and desired analytical outputs, there are a number of “rules” which CAE analysts can employ for modeling various phenomena. Stay tuned for future installments of our Element Selection blog series for more tips and tricks that can be used when determining the elemental makeup of your Finite Element Analysis (FEA) model!

## Element Blog Series:

Article 1: An Introduction to Abaqus Elements

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