# Finite Element Method Questions and Answers – Numerical Integration

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

1. Which method of approach is useful for evaluating four noded quadratic elements?
a) Numerical integration
b) Penality approach method
d) Rayleighs method

Explanation: Gaussian quadrature is to select the n Gauss points and n weights such that provides an exact answer for polynomials f(ξ) of as large degree as possible. In other words, the Idea is that if the n-point integration formula is exact for all polynomials up to as high a degree as possible, then the formula will work well even if f is not a polynomial.

2. One point formula in quadratic approach is ____
a) w1f(ξ1)
b) σ=εD
c) Nt=(1-ξ)(1-η)
d) Constant matrix

Explanation: In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. This is seen to be the familiar midpoint rule.

3. Two point formula of a quadratic approach is _____
a) X direction
b) w1f(ξ1)+w2f(ξ2)
c) Nt=(1-ξ)(1-η)
d) σ=D

Explanation: In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. From this solution, we can conclude that n-point Gaussian quadrature will provide an exact answer if f is a polynomial of order (2n – 1) or less.

4. The extension of Gaussian quadrature to two-dimensional integrals of the form of _____
a) I≈$$\sum_{i=1}^{n}\sum_{j=1}^{n}$$wiwjf(ξij)
b) Natural co-ordinates
c) w1f(ξ1)+w2f(ξ2)
d) w1f(ξ1)

Explanation: In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. An n-point Gaussian quadrature rule, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i=1,…, n. The domain of integration for such a rule is conventionally taken as [−1, 1].

5. Stiffness integration for quadratic element for 2*2 matrix is ____
a) Nt=(1-ξ)(1-η)
b) kij=$$\sum_{IP=1}^{4}$$WIPIP
c) Nt=(1-η)
d) Nt=$$\frac{1}{4}$$(1-ξ)(1-η)

Explanation: Stiffness is the rigidity of an object, the extent to which it resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small.

6. The stresses in the quadratic element are not ______
a) Linear
b) Uniform
c) Constant
d) Undefined

Explanation: The stress applied to a material is the force per unit area applied to the material. The maximum stress a material can stand before it breaks is called the breaking stress or ultimate tensile stress. Tensile means the material is under tension. The forces acting on it are trying to stretch the material.

7. The stresses are evaluated at the __________
a) Nodal points
b) Nodal displacements
c) Gauss points
d) Elements

Explanation: The stress applied to a material is the force per unit area applied to the material. The maximum stress a material can stand before it breaks is called the breaking stress or ultimate tensile stress. The forces acting on it are trying to stretch the material. In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.

8. For quadrilateral with 2X2 integration gives _____ sets of stress values.
a) One
b) Two
c) Three
d) Four

Explanation: The stress applied to a material is the force per unit area applied to the material. The maximum stress a material can stand before it breaks is called the breaking stress or ultimate tensile stress. Tensile means the material is under tension. The forces acting on it are trying to stretch the material.

9. For degenerate four noded quadrilateral element the errors are _____
a) Constant
b) Uniform
c) Higher
d) Lesser

Explanation: A degenerated element is an element whose characteristic face shape is quadrilateral, but is modeled with at least one triangular face. Degenerated elements are often used for modeling transition regions between fine and coarse meshes, or for modeling irregular and warped surfaces.

10. Gauss points are also the points used for numerical evaluation of _____
a) Surfaces
b) ke
c) Elements
d) Planes