# Finite Element Method Questions and Answers – Element Formulation – Compatibility and Completeness Requirements

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Element Formulation – Compatibility and Completeness Requirements”.

1. What are the two types of mesh refinement methods in use today?
a) h & p refinement
b) l & m refinement
c) i & j refinement
d) d & e refinement

Explanation: The two types of mesh refinement are h & p refinement. In h refinement, the number of elements are increased that are used to model a particular domain. Whereas, in p refinement the order of polynomials used as interpolating functions is increased.

2. Which of the following defines the compatibility condition?
a) Field variables and their partial derivatives must be discrete
b) Field variables and their partial derivatives must be continuous
c) Only field variables must be continuous
d) Only partial derivatives must be continuous

Explanation: According to the compatibility condition, along any element boundaries, the field variable and its respective partial derivatives whose order is one lesser than the highest order derivative that is appearing in the integral formulation must be continuous.

3. What is the significance of the compatibility condition in structural problems?
a) Doesn’t have any significance in structural problems
b) Ensures formation of voids and gaps
c) It ensures no formation of voids and gaps
d) Ensures boundary discontinuity

Explanation: The significance of the compatibility condition varies from one application to the other. For instance, in case of structural problems; requirement of displacement continuity along the element boundary ensures that there is no formation of voids and gaps during the modeling procedure.

4. What are the uses of compatibility condition with respect to beam elements and heat transfer problems?
a) No uses in heat transfer applications
b) No uses in beam elements
c) Aids formation of kinks and helps in formation of jump discontinuities
d) Omits formation of kinks and gets rid of jump discontinuities

Explanation: Incase of beam elements, the compatibility condition by making use of the requirement of slope continuity omits the formation of kinks. Whereas, with respect to heat transfer problems, compatibility prevents the formation of jump discontinuities in temperature distribution.

5. Which of the following defines the completeness condition?
a) As element size shrinks to zero, field variable and its derivatives must be capable of assuming constant values
b) As element size shrinks to unity, field variable and its derivatives must be capable of assuming constant values
c) As element size expands to infinity, field variable and its derivatives must be capable of assuming constant values
d) Field variable and its derivatives must become zero

Explanation: While computing limits, especially during the mesh refinement process; as the size of element approaches zero, the corresponding field variables and their derivatives must be able of assuming constant values. Highest order derivatives that are present in the integral under discussion should also be capable of doing the same.
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6. Completeness condition has no significance in various applications of the finite element problems.
a) True
b) False

Explanation: The given statement is false. In a structural element, it ensures that the displacement field takes a constant value; corresponding to rigid body motion. In a similar manner, constant slope of beam elements corresponds to rigid body motion, whereas a state of constant temperature corresponds to no heat flux in the element.

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