# Finite Element Method Questions and Answers – Error Analysis – Solution Convergence and Accuracy

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Error Analysis – Solution Convergence and Accuracy”.

1. Which of the following restriction is required for establishment of convergence for finite element models?
a) Element interpolation must be capable of modeling the dependant variables and its derivatives
b) Element interpolation must be capable of modeling the dependant variables only
c) Element interpolation must be capable of modeling the derivatives only
d) Convergence is not applicable

Explanation: For establishment of convergence, there are two necessary conditions. The element interpolation functions must be capable of modeling both the dependant variables and their derivatives respectively. As the element size decreases, this becomes even more important for convergence to be established.

2. When is the finite element model expected to converge in the energy norm?
a) When |e| > Chp
b) When |e| < Chp
c) When |e| = Chp
d) When |e| != Chp

Explanation: Based on various studies in the interpolation theory, the finite element approximation is expected to converge in the energy norm; if and only if |e| < Chp.
where,
e = error energy norm
C = point of convergence
h = characteristic element length
p = rate of convergence

3. When is the slope of line = rate of convergence?
a) In the case of complex elements
b) In the case of all types of elements
c) In the case of simplex elements
d) This condition cannot be satisfied

Explanation: Simplex elements always make use of complete polynomials by Lagrangian form. This is contrary to that of serendipity elements which make use of incomplete polynomials. If this is the case, the equation results in a straight line plot for a log – log plot of error versus mesh size. At this point, the slope of line is considered equivalent to the rate of convergence.

4. Why is a hp adaptive system not preferred, even though it corresponds to optimal convergence?
a) Extensive errors
b) User involvement is minimal
c) Requirement of skilled labor
d) Programming and database difficulties

Explanation: It is very difficult to program the hp adaptive system. One needs to be very careful while planning its database structure. In this system, we need to select the item that needs to be changed first. There is extensive shift in solution values for very minimal changes in input.

5. Mesh convergence corresponds to time taken for mesh refinement to take place.
a) True
b) False

Explanation: The given statement is false. Mesh convergence is defined as the procedure via which, the number of elements required in a model to ensure that there is no change in the solution, even with changes in the mesh size. The responses of a system tend to converge to recurring solutions with diminishing element size.
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6. Which of the following is a valid convergence test?
a) Patch test
b) Catch test
c) Mesh procedure
d) Constraint procedure

Explanation: Patch test is a mathematically valid convergence test. A patch of elements containing at least one internal node each is considered. To be a convergent formulation, the internal node values must match the values of assumed functions at the respective points. Calculated nth order derivatives should correspond to the values of constants in the equation.

7. Accuracy of the solution has no relation with the number of elements.
a) True
b) False

Explanation: The given statement is false. Accuracy of the solution is directly related to the number of elements used in the finite element method. More elements correspond to better accuracy, while lesser number of elements do not offer satisfactory results.

8. Which of the following is a disadvantage of element division?
a) Increased computational cost
b) Better accuracy
c) Decreased computational cost

Explanation: Even though usage of higher order elements, and element division enhance accuracy of the finite element solution, they pose some disadvantages. In order to tackle the higher order elements, computation costs get higher to support the high end calculations involved. Time taken for computation is also increased drastically.

9. Which of the following accuracies is preferred in the industry today?
a) Obsolete accuracy
b) Absolute accuracy
c) Reasonable accuracy
d) Periodic accuracy

Explanation: In the industry, getting quick results with reasonable accuracy is preferred over getting the exact or absolute accuracy. This is because, absolute accuracy is very cumbersome with high computational costs and run time. However, absolute accuracy is preferred in industries of great precision; such as aerospace, aeronautics etc.

10. How to enhance the accuracy of elements in the critical region?
a) Spend more time solving numerically
b) Make use of high end finite element packages
c) Decrease the number of elements in that region
d) Increase the number of elements in that region

Explanation: While meshing a structure with a critical region, extra caution needs to be taken. The critical region needs to have more number of elements in order to enhance the accuracy of the solution. However, it is not preferred to increase the elements in the entire cross section of the structure under discussion.

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