This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Variational Methods of Approximation”.

1. What is considered as the first step of solving using variational method?

a) Equation is put into weighted integral form

b) Approximate solution is assumed to be the linear combination of chosen functions

c) Equation is put into weighted differential form

d) Exact solution is assumed to be the non linear combination of chosen functions

View Answer

Explanation: While solving equations using the variational approach method, the equations under consideration are first put into a weighted integral form. Following this, the approximate solution is assumed to be the linear combination of chosen functions and undetermined coefficients.

2. What type of equations are solved using the variational method?

a) Integral equations

b) Differential equations

c) Vector equations

d) Scalar equations

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Explanation: The variational method is used to solved differential equations type of problems. While solving differential equations is considered difficult, partial differential equations require much more expertise in order to solve them. To tackle this issue, approximate methods are made use of. This helps cut down on time and effort required to solve the same. This approximation methods are referred to as the variational method.

3. What are the types of variational method in use today?

a) Euler and Lagrange methods

b) Newton and Einstein methods

c) Galerkin and Rayleigh-Ritz methods

d) Euler and Nashville methods

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Explanation: Two types of variational method approaches are in extensive use today – Galerkin method and Rayleigh-Ritz method. In Galerkin method, the weight functions are chosen to be trial functions themselves. In the Rayleigh-Ritz method, both expression of field variables and minimization of potential energy are clubbed together in order to arrive at the solution.

4. Which of the following is a similarity between the Rayleigh-Ritz method and the Finite element method?

a) Both methods use actual solutions as trial functions

b) Both methods are variational methods

c) Both methods take non linear combinations of trial functions

d) Both methods take linear combinations of trial functions

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Explanation: The Rayleigh-Ritz method and Finite element method have many similarities amongst them. Some of them include –

- Both methods take linear combinations of trial functions
- Both use approximating solutions as trial functions
- Completeness condition of function must be satisfied
- Solution is sought only by making the function stationary

5. Which of the following is a difference between the Rayleigh-Ritz method and the Finite element method?

a) Rayleigh-Ritz method assumes trial function over the entire structure

b) Finite element method assumes trial function over the entire structure

c) Rayleigh-Ritz method assumes trial function over an element

d) No differences between the two methods

View Answer

Explanation: The Rayleigh-Ritz method is an example of the variational method of approximating solution; whereas the Finite element method is used for getting to exact solutions. The Rayleigh-Ritz method assumes the trial function over the entire structure, while the finite element method assumes trial function over an element alone.

6. Why is the Finite element method preferred over its variational counterparts?

a) Lesser steps involved

b) Accuracy of solution and ease of solving

c) More steps involved

d) Time taken for computation

View Answer

Explanation: The finite element method is known for its accuracy in solutions and simplified structure of computation. This method encourages discretization, followed by assigning approximating functions to each of the smaller element. These are then interpolated into continuous forms to arrive at the solution.

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