# Finite Element Method Questions and Answers – Isoparametric Formulation and Numerical Integration

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Isoparametric Formulation and Numerical Integration”.

1. Why is numerical integration used?
a) Exact evaluation of integrals not possible
b) Exact evaluation of integrals possible
c) Approximate evaluation of integrals not possible
d) Approximate evaluation of integrals possible

Explanation: Exact evaluation of the integrals that appear in element matrices is not possible. This is due to algebraic complexity associated with the same. This complexity is mainly due to the coefficients that are present in the differential equations. To overcome this issue, numerical integration is made use of.

2. Numerical integration involves approximation of integrand by making use of trigonometric functions.
a) True
b) False

Explanation: The given statement is false. Numerical integration involves approximation of integrand by making use of polynomial functions of a suitable degree. This is because integration of polynomials always yields at exact solutions.

3. Which of the following is an advantage of the local coordinate used in Gauss Legendre quadrature?
a) Convenience in construction of numerical functions
b) Inconvenience in construction of interpolation functions
c) Convenience in construction of interpolation functions
d) Inconvenience in construction of numerical functions

Explanation: The local/natural coordinate that is used in Gauss Legendre quadrature, serves two purposes –

• It is convenient to construct the interpolation functions.
• It is a pre requisite for integration in the Gauss Legendre quadrature.

4. Which of the following is true for a sub parametric formulation?
a) Geometry cannot be represented
b) Geometry is represented by same order elements
c) Geometry is represented by higher order elements
d) Geometry is represented by lower order elements

Explanation: In the case of sub parametric formulation, geometry is represented by lower order elements in comparison with the elements used to approximate dependant variables. The Euler Bernoulli beam element falls under this category.

5. Which of the following corresponds to isoparametric formulation?
a) Same elements are used for representation of geometry and approximation functions
b) Different elements are used for representation of geometry and approximation functions
c) Higher order elements are used for representation of geometry
d) Higher order elements are used to approximate dependant variables

Explanation: In isoparametric formulation, the same element is used for approximation of geometry as well as the dependant variables. Displacements are represented using the natural coordinates; whereas they are differentiated with respect to global coordinates.

6. Which of the following is a difference between Newton Cotes method and the Gauss quadrature method?
a) Both the methods are similar
b) Newton Cotes formulae is used for equally spaced base points
c) Gauss quadrature formulae is used for equally spaced base points
d) Newton Cotes formulae is used for unequally spaced base points

Explanation: The Newton Cotes method and Gauss quadrature method are two methods available for numerical integration. While the Newton Cotes method corresponds to equally spaced base points, the Gauss quadrature method corresponds to unequally spaced base points.

Sanfoundry Global Education & Learning Series – Finite Element Method.