# Finite Element Method Questions and Answers – Eigen Value and Time Dependent Problems

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Eigen Value and Time Dependent Problems”.

1. Which of the following is an application of the Laplace eigenvalue problem?
a) Vibration modes in acoustics
b) Translation modes in acoustics
c) Buckling of plates
d) Beam deformation

Explanation: The Laplace method is considered as the first model problem to study finite element methods for eigenvalues. This is due to its simplicity in theory and implementation. Some applications of the Laplace method include – vibration modes in acoustics, nuclear magnetic resonance measurements of diffusive transport, electron wave functions in quantum waveguides etc.

2. What is the order of a biharmonic eigenvalue problem?
a) Second
b) Fourth
c) First
d) Third

Explanation: The biharmonic equation comes under the fourth order of eigenvalue problem considerations. It has many applications such as – vibration and buckling of plates, inverse scattering theory etc. Source problem of the biharmonic type of eigenvalue problem is the biharmonic equation.

3. What are the classical methods used to discretize biharmonic equations?
a) Rayleigh, Euler and Finite methods
b) Idler, Ritz and Roman methods
c) Conforming, non conforming and mixed methods
d) Only roman method is used

Explanation: There are three classical approaches for discretization of biharmonic elements. They are – conforming element method; which makes use of partition of unity finite elements. Non conforming method makes use of Adini or Morley elements. The third method is a mixed approach which requires only Lagrange finite element spaces.

4. Which of the following is an alternative for the classical approaches of discretization?
a) C0 exterior penalty Galerkin method
b) Lagrange method
c) Laplace method
d) C0 interior penalty Galerkin method

Explanation: The C0 interior penalty Galerkin method (C0 IPG) is an alternative for the classical approaches of discretization used for biharmonic eigenvalue problems. This is a discontinuous Galerkin method that is used for second order elliptical problems. It is based on standard continuous Lagrange finite element spaces.

5. Which of the following is a disadvantage of using the mixed classical approach of discretization?
a) Can result in spurious results
b) Computation time
c) Cost of handling
d) Able workforce

Explanation: For boundary conditions of simply supported plates, mixed finite element practices can result in spurious solutions on non convex domains. It’s a similar case for boundary conditions that arise in models of phase separation phenomena.
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6. Which of the following is not a type of eigenvalue problem?
a) Transmission eigenvalue problem
b) Newton’s eigenvalue problem
c) Maxwell’s eigenvalue problem
d) Schrodinger eigenvalue problem

Explanation: Newton’s eigenvalue problem does not exist. Equations corresponding to macroscopic electromagnetic field are referred to as Maxwell’s equations. Transmission type of eigenvalue problems arise in inverse computations. Schrodinger’s equations correspond to motion of a particle in an external potential.

7. What is considered as the first step in finite element formulation of time dependant problems?
a) Spatial approximation
b) Temporal approximation
c) Nominal approximation
d) Final approximation

Explanation: Spatial approximation is considered as the first step in finite element formulation of time dependant problems. In this, the solution (u) of the equation under consideration is approximated using a preset equation. Spatial model is then found out by making use of static/steady state model procedures.

8. The semi discrete formulation involves approximation of partial variation of the dependant variable.
a) True
b) False

Explanation: The given statement is false. The semi discrete formulation involves approximation of spatial variation of the dependant variable. Initially, the weak form is constructed by making use of integration by parts. The equation so obtained is interpolated and approximated to arrive at the desired result.

9. When is the solution scheme termed as unstable?
a) If error is constant
b) If error grows bounded with time
c) If error grows unboundedly with time
d) If error is variable

Explanation: As the method used for solving time dependant problems is an approximation method, errors are bound to arise. Truncating and round off errors are incontrollable. As solution is dependent on time, error can grow with time. If it grows unboundedly, it is referred to as unstable.

10. What is the difference between stability and accuracy of the solution?
a) There is no difference
b) Stability is higher in value
c) Accuracy is measure of boundedness of approximate solution with time
d) Stability is measure of boundedness of approximate solution with time

Explanation: Accuracy and Stability are two important parameters used for determining efficiency of the solving procedure. While accuracy refers to the closeness of approximate solution to the exact solution; stability is a measure of boundedness of the approximate solution with time.

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