Problem 46 - Concepts and Applications of Finite Element Analysis

Question

If each of the two coefficients \(x y\) in Eq. 4.2-12 is replaced by \(x^{2}+y^{2}\), the element becomes incompatible. Why? Suggestion: Let two adjacent elements have two corner nodes in common. Along the boundary between elements, d.o.f. of these corner nodes must produce the same boundary displacement in each element if the elements are to be compatible. But how many d.o.f. are needed to define a quadratic curve? And what does this imply?

Step-by-Step Solution

Answer

The replacement of coefficients \(x y\) with \(x^2 + y^2\) in the equation changes the requirement to define a quadratic curve, which needs three degrees of freedom. However, as the two adjacent elements only share two corner nodes (equivalent to two degrees of freedom), they cannot match their boundary displacements, resulting in incompatibility of the structure.

Step 1: Replacing coefficients

Let's start by replacing the coefficients \(xy\) in Eq. 4.2-12 with \(x^2 + y^2\). This operation changes the linear shape functions in the equation to quadratic. This implies that there's now a necessity to define a quadratic curve.

Step 2: Defining a Quadratic Curve

Now, to define a quadratic curve, we need at least three degrees of freedom: two for the end points and one for the curvature of the curve. Using only two degrees of freedom, which are the DOFs for the corner nodes, is insufficient for this quadratic curve.

Step 3: Implication on Compatibility

Keeping in mind the necessity to produce the same boundary displacement in each element to maintain compatibility, it becomes clear why replacing the coefficients with \(x^2 + y^2\) leads to incompatibility. The two corner nodes shared by the adjacent elements provide only two degrees of freedom, which is not enough to define the now required quadratic curve. As a result, the adjacent elements cannot match their boundary displacements, leading to incompatibility.

Key Concepts

Shape Functions

In the finite element method, shape functions play a critical role. These functions describe how the movement or deformation of an element occurs. When you make changes to the coefficients in an equation, such as replacing a linear shape function term like \(xy\) with \(x^2 + y^2\), the shape function transitions from linear to quadratic. Consequently, this change affects the way the function interpolates values between nodes.

• Linear shape functions imply a straight line interpolation.
• Quadratic shape functions suggest a curve, adding complexity to the element representation.

Switching to quadratic shape functions means more information or data points are needed to describe the element accurately, impacting not only the element itself but also how it connects with its neighbors.

Degrees of Freedom

Degrees of Freedom (DOF) in a finite element are the independent values that define the system's state. When shifting to quadratic coefficients in shape functions, you're explicitly demanding more degrees of freedom. This is because defining a quadratic curve requires at least three DOF. In a simple context:

• Linear elements use two DOF, usually positioned at the end points.
• Quadratic elements demand three DOF—two for the endpoints, and one for the curvature control.

Thus, if you only provide two DOF as in typical linear elements, the quadratic curve cannot be fully defined, leading to inadequate representation of the element behavior.

Element Compatibility

Element compatibility ensures that adjacent finite elements connect seamlessly, sharing boundary conditions and displacements without abrupt changes. This continuity is vital for accurate simulations of physical systems. When elements are adjusted from linear to quadratic, they require additional data points or degrees of freedom for continuity. Simply put, quadratic curves need more specific information to ensure fluid transition between elements. For example:

• Linear elements can easily match displacement with two DOF at corner nodes.
• Quadratic changes necessitate three DOF, which aren't met by linear configurations.

As a result, the boundary displacements don't align, causing a breakdown in compatibility between the adjacent elements.

Quadratic Coefficients

Quadratic coefficients in a shape function modify how an element behaves and interacts with its surroundings. These coefficients represent more complex relationships than linear ones. With a quadratic approach, we have terms like \(x^2 + y^2\) instead of simpler \(xy\), leading to a need for more sophisticated modeling.For a given boundary:

• Quadratic representations define a curve.
• This requires precise control over the bend, shape, and connection of elements.

Replacing simple coefficients with quadratic ones enhances the potential accuracy of a simulation but demands a greater number of DOF. Without these additional DOF, continuity and coherence between elements are disrupted, making it crucial to adjust the framework to accommodate these higher-order terms effectively.