# Finite Element Method Questions and Answers – Three Dimensional Models – Alternative Formulations & Nonlinear Problems

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional Models – Alternative Formulations & Nonlinear Problems”.

1. What are the requirements of a weight function?
a) It should be an integrable function and belonging to a linearly independent set
b) It should be an integrable function and belonging to a linearly dependent set
c) It should be an integrable function and belonging to a non linearly independent set
d) It should be an integrable function and belonging to a non linearly dependent set

Explanation: There are two main requirements of a weight function. First and foremost, it should be integral in the desired form for solving the weighted residual problems. It is also mandatory that the weight function belongs to a linearly independent set.

2. Which of the following method can be used to solve weighted residual formulations?
a) Rayleigh Ritz method
b) Petrov – Galerkin method
c) Newton method
d) Euler method

Explanation: Petrov – Galerkin method is used to solve weighted residual formulations. Some of the other methods in use include – Bubnov – Galerkin method, Collocation method, Sub domain method, and Least squares method. These methods can solve both one and two dimensional residual formulations.

3. Which of the following equation is an example of the mixed formulations?
a) Timoshenko beam theory
b) Newton’s law of forces
c) Euler – Bernoulli beam theory
d) Galerkin method

Explanation: The Euler – Bernoulli beam theory is an example of mixed formulation in finite element analysis. The weak form makes use of Hermite cubic interpolation functions; whereas the weighted residual method requires it to have a nonzero fourth order derivative. To simplify calculations, the equation is decomposed into simpler forms.

4. Under certain assumptions, non linear problems can be described by linear differential equations.
a) True
b) False

Explanation: The given statement is true. Under certain specific assumptions, nonlinear problems that are usually described by non linear differential equations can be described by making use of linear differential equations. These assumptions could be about the variation of slope, neglecting of certain terms etc.

5. Which of the following method is explicitly used for solving nonlinear algebraic equations?
a) Picard method
b) Euler method
c) Ritz method
d) Method of least squares

Explanation: Picard method, also referred to as direct iteration method is used for solving nonlinear algebraic equations. This method helps arrive at an approximate solution to algebraic equations by linearization.
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6. Which of the following method makes use of Taylor series expansion?
a) Bernoulli method
b) Newton – Raphson Method
c) Poisson method
d) Naveli method

Explanation: Newton – Raphson method is based on the Taylor series expansion. This method is used for solving nonlinear algebraic equations by making use of iterative methods. The iteration goes on until it satisfies one of the following conditions –

• Convergence criteria is satisfied
• Residual value is lesser than the preselected value

7. “Tangent matrix for the Navier – Stokes equation is unsymmetrical.” Why is this so?
a) Presence of element coefficients
b) Absence of element coefficients
c) Presence of variational principle
d) Absence of variational principle

Explanation: Due to the absence of variational principle, the tangent matrix of the Navier – Stokes equation is unsymmetrical. A variational principle is basically a scalar quantity which is defined in the integral form either in terms of the dependant variable or its derivatives.

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