Problem 49 - Concepts and Applications of Finite Element Analysis

Question

In three dimensions, which of the cubic terms would you add to a complete quadratic field (10 terms), if there are to be no preferred directions and the total number of terms in the expansion is to be (a) \(11,(b) 13,(c) 14,(d) 16\), (e) \(17,(\mathrm{f}) 19 ?\)

Step-by-Step Solution

Answer

For case (a), no cubic terms can be added. For case (b), 3 cubic terms can be added. For case (c), no cubic terms can be added. For case (d), 6 cubic terms can be added. For case (e), no cubic terms can be added. For case (f), 9 cubic terms can be added.

Step 1: Solution for case (a)

In the first case we are allowed to add 1 term (11-10). Since it should not favour any direction, we can't add any terms (as one term would favour one direction over the others). Thus, we can't add any cubic terms to the quadratic field in this case.

Step 2: Solution for case (b)

In the second case we are allowed to add 3 terms (13-10). Again, to ensure no preferred directions, we would need to add one cubic term for each dimension. Therefore, we can add 3 cubic terms (one for x, one for y and one for z) to the quadratic field in this case.

Step 3: Solution for case (c)

In the third case we are allowed to add 4 terms (14-10). As we can't add 4 terms without favouring a direction (since there would be an imbalance in terms), we can't add any cubic terms to the quadratic field in this case.

Step 4: Solution for case (d)

In the fourth case we are allowed to add 6 terms (16-10). Since we can add two cubic terms for each dimension, we can add 6 cubic terms (two for x, two for y and two for z) to the quadratic field in this case.

Step 5: Solution for case (e)

In the fifth case, we are allowed to add 7 terms (17-10). Here we can't add 7 terms without favouring a direction, so we can't add any cubic terms to the quadratic field in this case.

Step 6: Solution for case (f)

In the last case, we are allowed to add 9 terms (19-10). As we can add three cubic terms for each dimension, we can add 9 cubic terms (three for x, three for y and three for z) to the quadratic field in this case.

Key Concepts

Cubic Terms

Cubic terms in mathematics refer to terms in a polynomial that have variables raised to the third power. For example, terms such as \(x^3, y^3, \text{and} \ z^3\) are considered cubic. In the context of finite element analysis, these terms are crucial when expanding a quadratic field into a higher order polynomial.

• Adding cubic terms allows us to better model and fit the complexity of three-dimensional spaces.
• Unlike linear or quadratic terms, cubic terms account for more complex interactions between variables, such as twisting or curving in 3D.
• When adding cubic terms, it's important that they do not introduce a preference for any particular direction, which can distort results.

When incorporating cubic terms, consider balancing them evenly across dimensions. This ensures no single direction is emphasized too heavily, maintaining the integrity of the model in three-dimensional analyses.

Quadratic Field

A quadratic field is essentially a polynomial representation where the highest degree of any term is two. For instance, terms in a quadratic field include \(x^2, xy,\) and \(y^2\). These types of fields are commonly used in finite element analysis because they provide a balance between computational complexity and flexibility.

• They're primarily employed to simulate phenomena where interactions are not linear but also don't need the complexity of higher-degree terms.
• In the exercise, the quadratic field started with 10 terms. This basis is crucial as it includes all necessary interactions without any directional biases.
• Quadratic fields provide a clear and straightforward representation, making them practical for initial approximations in analyses.

Extending a quadratic field by incorporating cubic terms can offer increased detail and accuracy, especially in capturing the nuances of phenomena with slight deviations in linearity.

Preferred Directions

In the context of finite element analysis and the given exercise, avoiding preferred directions is essential to maintain objectivity and accuracy in results. Preferred directions occur when the extension of a field, like the addition of cubic terms, inadvertently makes the model favor one spatial orientation over others.

• This occurs when terms are unevenly distributed among the dimensions \(x, y, \text{and} \ z\).
• A model with preferred directions can lead to misleading conclusions and incorrect analysis.
• Ensuring balance across dimensions is vital. For instance, if adding cubic terms, ensure each dimension receives an equal number of terms.

In the exercise, adding terms needed to be done carefully to prevent any directional bias, aligning actions with the requirement that no preferred directions be established. This balance ensures that results accurately reflect the physical phenomena without skewing towards any particular directional influence.