Problem 6 - Concepts and Applications of Finite Element Analysis

Question

Imagine that the problem of Fig. 9.7-4 is solved twice, using nine-node plane elements each time. Mesh 2 is created by regular refinement of mesh 1, quadrupling the number of elements. For the respective meshes, computed values of lateral tip displacement and maximum stress are Mesh \(1 \quad v=0.0035 \quad \sigma_{\max }=74.23\) \(\operatorname{Mesh} 2 \quad v=0.0041 \quad \sigma_{\max }=89.03\) Use these data to estimate the percentage error of \(v\) and \(\sigma_{\max }\) in mesh 2 .

Step-by-Step Solution

Answer

The percentage error for the lateral tip displacement \(v\) is -17.14\%, and for the maximum stress \(\sigma_{\max }\), it is -19.92\%.

Step 1: Compute percentage error for v

First, calculate the percentage error for the lateral tip displacement. This is done by using the percentage error formula. The actual value of \(v\) is 0.0035 and the measured value of \(v\) is 0.0041. So your percentage error for \(v\) will be \( \left( \frac{0.0035 - 0.0041}{0.0035} \right) \times 100 = -17.14 \% \)

Step 2: Compute percentage error for \(\sigma_{\max }\)

Next, calculate the percentage error for the maximum stress. This is done similarly using the percentage error formula. The actual value of \(\sigma_{\max }\) is 74.23 and the measured value of \(\sigma_{\max }\) is 89.03. So, the percentage error for \(\sigma_{\max }\) will be \( \left( \frac{74.23 - 89.03}{74.23} \right) \times 100 = -19.92 \% \)

Step 3: Interpret the results

To interpret these results, note that a negative percentage error indicates that the estimated values are larger than the actual ones. For this problem, both the lateral tip displacement and maximum stress, for Mesh 2, are overestimated in comparison to the actual values from Mesh 1.

Key Concepts

Mesh Refinement

Mesh refinement is an essential technique in finite element analysis. It involves making the mesh finer by increasing the number of elements, which can result in more accurate simulation results.
In the given problem, Mesh 2 represents a refined version of Mesh 1, with four times more elements.

This increased element density allows for a more detailed representation of the geometry. This helps capture more nuanced physical behaviors.

• **Efficiency**: Refining the mesh increases computational demands but improves accuracy.
• **Adaptivity**: Deciding where to refine needs strategic consideration to balance performance and precision.
• **Output Quality**: More elements can potentially reduce discretization errors.

However, it's vital to ensure that such refinement is done wisely, concentrating on areas where the results are most sensitive, rather than uniformly across the entire domain. Over-refinement can result in significant computational expense without proportional benefits in accuracy.

Plane Elements

Plane elements are fundamental in finite element modeling, particularly for problems involving two-dimensional idealizations.
They are used to analyze structures where the third dimension is not a significant factor.

In the exercise, nine-node plane elements are employed, which are a common type of element in such analyses.

• **Dimensionality**: Suitable for situations where out-of-plane dimensions can be ignored.
• **Element Types**: Nine-node elements offer a relatively high degree of accuracy as they allow for quadratic approximations, which capture variations more precisely than simpler linear elements.

Plane elements, due to their node arrangement, are crucial in practical applications like stress analysis of thin-walled structures and simulations of planar mechanics. Understanding how to use these elements effectively is a key step in producing reliable finite element models.

Percentage Error Calculation

Calculating percentage errors is crucial in assessing the accuracy of simulation results relative to expected or previously established values.
This calculation provides a quantitative measure of deviation and can indicate the reliability of the finite element model.

For the problem, two errors were computed: the lateral tip displacement and maximum stress.

• **Formula Use**: The percentage error is defined as \( \left( \frac{\text{True Value} - \text{Estimated Value}}{\text{True Value}} \right) \times 100 \).
• **Interpretation**: A negative percentage error, as seen here, signals an overestimation of the parameter.
• **Consistency Check**: Regular checks of percentage errors can help validate improvements in mesh designs and help in judging the sufficiency of mesh refinement.

This calculation is important for iterating improvements in simulation models and ensuring their fidelity to real-world scenarios.