Problem 8 - Concepts and Applications of Finite Element Analysis

Question

Addition to an element of internal d.o.f., such as \(a_{1}\) and \(a_{2}\) in Eq. 8.1-4, can be regarded as a device that permits better approximation of equilibrium. equations within the element, without affecting interelement compatibility. Accordingly, do you think the constant-strain triangle (Section 5.4) would be improved by addition of the bubble function modes \(u=\xi_{1} \xi_{2} \xi_{3} a_{1}\) and \(v=\xi_{1} \xi_{2} \xi_{3} a_{2}\) ? Why or why not?

Step-by-Step Solution

Answer

Adding the bubble function modes \(u=\xi_{1}\xi_{2}\xi_{3}a_{1}\) and \(v=\xi_{1}\xi_{2}\xi_{3}a_{2}\) would indeed improve the ability for the CST element to approximate the exact solution. However, this would come at the expense of increasing complexity and losing the main advantage of the CST (its simplicity). Therefore, whether this is beneficial or not can depend on the specific requirements of the model.

Step 1: Understand the concept of the bubble function mode

A bubble function is a type of shape function used in FEA. Its value is maximum at the centroid of an element and zero at the edges. The given equations for \(u\) and \(v\) are examples of bubble functions.

Step 2: Analyse the improvement in element approximation

Applying the bubble function modes implies adding an interior node to the element. Through this added internal node, the shape functions experience an alteration enabling them to better approximate the exact solution within the element. The concept is basically enabled for refining the model without adding more elements, and it's a common approach to improve solution quality in areas with high stress gradients.

Step 3: Consider the impact on interelement compatibility

Even though the approximation is improved within the element through the use of bubble functions, the interelement compatibility remains unaffected. This is because the bubble functions still become zero at the edges (nodes), thus fulfilling the compatibility (continuity) requirements at the element interfaces.

Step 4: Evaluate the use for the Constant-Strain Triangle (CST)

The CST (also known as a linear triangle) is the simplest form of a 2-dimensional element with its displacement field linear. In theory, the use of bubble functions could enhance the approximation quality within the CST element. However, the CST element’s main advantage of simplicity (only requiring 3 nodes) will be lost, turning it into a higher order triangle.

Key Concepts

Bubble Function Mode

The bubble function mode is a particularly useful technique within Finite Element Analysis (FEA) for enhancing the abilities of elements to represent complex physical behaviors. It consists of a shape function that possesses a unique characteristic: it reaches its maximum value at the interior of the element and tapers to zero at the element's edges.

This pattern allows the element to capture internal stress variations more accurately without modifying the external shape or size of the element, which could be crucial in areas subject to high stress gradients. For instance, if you consider a rubber balloon being inflated, the bubble function would help to simulate the changing internal pressures while the surface remains mostly uniform.

The bubble function mode, when applied, adds degrees of freedom internal to the element. These are essentially additional points within the element where the system's response can be calculated, known as 'internal nodes'. They enhance the model's precision by providing a more intricate approximation of the state of equilibrium within the element.

Interior Node Approximation

Interior node approximation is a technique that involves adding nodes within the elements, separate from the corner or edge nodes. This approach can significantly refine the solution within each element by capturing local effects that would otherwise be missed with a simpler element node configuration.

In the context of FEA, the introduction of interior nodes through functions such as the bubble function mode enables the element to approximate complex internal stress patterns. This capacity is particularly useful in modeling scenarios where deformation or stress concentration occurs within the elements, not just at their boundaries.

However, while augmenting the element with interior nodes can offer more accurate results, it also increases computational complexity. For this reason, interior node approximation is usually reserved for areas in the model that are expected to experience significant variations in behavior or where extreme precision is required.

Constant-Strain Triangle (CST)

The constant-strain triangle (CST) is one of the simplest elements used in Finite Element Analysis. As the name suggests, within its bounds, the strain is assumed to be constant. A CST typically requires only three nodes, one at each vertex of the triangle, and the strain is computed based on the displacements at these nodes.

CSTs are particularly advantageous when dealing with problems that can be approximated using a straight-forward linear displacement field. Their simplicity makes them computationally efficient and easy to implement. However, this simplicity comes with limitations, especially in complex problems where the strain within the element might fluctuate significantly.

Adding bubble function modes to a CST aims to infuse the benefits of interior node approximation, potentially capturing the non-uniform strain states within the element. However, by doing so, the fundamental nature of the CST – its simplicity and ease of computation – might be compromised. The decision to enrich a CST with bubble functions requires careful consideration of the balance between solution accuracy and computational simplicity.

Inter-Element Compatibility

Inter-element compatibility is a critical principle in Finite Element Analysis, ensuring that the approximation of the solution is continuous across element boundaries. In other words, the neighboring elements must align precisely along their shared edges, and their displacements should be compatible at their nodes.

This compatibility is essential for accurately modeling the behavior of the entire structure. If it fails, the model could exhibit artificial cracks or overlaps at the interfaces of elements, leading to incorrect predictions of stresses and deformations.

Fortunately, bubble functions maintain inter-element compatibility as they taper to zero at the edges. This means that although the functions introduce additional complexity within the element, they do not affect the displacement field at the boundaries, thus preserving the continuity of the model. It’s a method allowing enriched local approximations while safeguarding the global integrity of the mesh in a simulation.