# Finite Element Method Questions and Answers – Need for Weighted Integral Forms

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Need for Weighted Integral Forms”.

1. Which of the following is a step involved in solving using the weighted residual method?
a) Solving the system of equations based on approximate solutions assumed
b) Solving the system of equations based on actual solutions assumed
c) Solving the system of equations based on direction
d) Solving the system of equations based on magnitude

Explanation: The weighted residual method has two important steps. First step corresponds to the assumption of approximate solution based on the nature of the equation. The second step involves solving of a system of equations arising from the first step; subject to corresponding boundary conditions.

2. What are the different types of approximation methods?
a) Vector and scalar methods
b) Polynomial and Trigonometric methods
c) Polynomial and Integral methods
d) Trigonometric and Differential methods

Explanation: There are two types of approximation methods. The polynomial method is apt for simple, linear one dimensional problems. This is very similar to the Lagrange method. The trigonometric method is based on the Fourier series and has the inclusion of sine and cosine terms alike.

3. Which of the following corresponds to residual in the weighted residual method?
a) Non computable quantity
b) Computable quantity
c) Error that arises
d) Diminished value

Explanation: The word residual corresponds to the error that arises while solving using the weighted residual method. This is an approximation method, making way for inaccuracy in terms of result obtained. The difference in the solution obtained via this method and the finite element method is termed as residual.

4. Which of the following is an example of the weighted residual method?
a) Euler Bernoulli Method
b) Newton method
c) Reynolds method
d) Point Collocation method

Explanation: Point collocation method is a type of weighted residual method. This method involves selection of weighted function in a manner wherein the residual/error is set to zero value at different locations. This weight function is then selected as the displaced Dirac delta function.

5. Which of the following is an advantage of point collocation method?
a) Computational effort is minimal
b) Computational effort is maximum
c) Time consuming
d) Not suitable for complex calculations

Explanation: The point collocation method is considered advantageous due to various reasons. The computational effort required to solve a problem using this method is minimal. By setting the residual to zero at various points, the error at that location doesn’t become zero; it helps ease calculations by getting rid of various unknown variables.

6. In sub domain collocation method, the physical domain is divided into many overlapping sub domains.
a) True
b) False

Explanation: The given statement is false. In sub domain collocation method, the physical domain is divided into many non-overlapping sub domains. The number of sub domains is always equal to the number of unknown coefficients in the approximating function.

7. Least square method is a type of weighted residual solving procedure.
a) True
b) False

Explanation: The given statement is true. Least square method is one of the four types of weighted residual solving procedures. In this method, weight functions are chosen to be derivatives of residual functions with respect to their unknown fitting coefficients of the approximate solution.

8. Which of the following is a disadvantage of the Least square method?
a) Time consuming
b) Less cost of operation
c) Easy to handle
d) Computationally expensive

Explanation: Because of the involvement of many partial derivatives and other complex numerical expressions, the least square method is considered to be computationally expensive. However, this method never fails to produce symmetric coefficient matrices regardless of the approximate solution and differential operator.

9. Which of the following corresponds to the Galerkin method?
a) Requires lesser computational effort when compared to least square method
b) Requires lesser computational effort when compared to point collocation method
c) Requires more computational effort when compared to least square method
d) Requires more computational effort when compared to point collocation method

Explanation: The Galerkin method is generally considered for developing finite element equations for field associated problems. It requires lesser computational effort when compared to the least square method. This is due to the absence of complex numerical expressions and differential equations.

10. When does the Galerkin method produce symmetric, positive and a definite coefficient matrix?
a) When the coefficient is unknown
b) When the differential operator is self ad-joint
c) When the coefficient is known
d) When the differential operator is determinant to itself

Explanation: The Galerkin method succeeds in producing symmetric, positive and definite coefficient matrix only when the corresponding differential operator is self ad-joint. This method is also the most preferred; owing to its accuracy and lesser time consumption.

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