Finite Element Method Questions and Answers – Two Dimensional Isoparametric Elements – Four Node Quadrilateral

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This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Two Dimensional Isoparametric Elements – Four Node Quadrilateral”.

1. In two dimensional isoparametric elements, we can generate element stiffness matrix by using ____
a) Numerical integration
b) Differential equations
c) Partial derivatives
d) Undefined

Explanation: The term isoparametric is derived from the use of the same shape functions (or interpolation functions) [N] to define the element’s geometric shape as are used to define the displacements within the element.

2. The vector q=[q1,q2………q8]T of a four noded quadrilateral denotes ____
b) Transition matrix
c) Element displacement vector
d) Constant matrix

Explanation: A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point.

3. For a four noded quadrilateral, we define shape functions on _____
a) X direction
b) Y direction
d) Master element

Explanation: Master Element (ME) is the main point of reference in our analysis. The ME represents the person itself, and it gives us a primary layer of our personality. To determine the quality of ME, and overall chart, we have to analyze what kind of connection and access ME has to other Elements. The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes.

4. The master element is defined in ______
a) Co-ordinates
b) Natural co-ordinates
c) Universal co-ordinates

Explanation: Master Element (ME) is the main point of reference in our analysis. The ME represents the person itself, and it gives us a primary layer of our personality. To determine the quality of ME, and overall chart, we have to analyze what kind of connection and access ME has to other Elements.

5. Shape function can be written as _____
a) Nt=(1-ξ)(1-η)
b) Nt=(1-ξ)
c) Nt=(1-η)
d) Nt=$$\frac{1}{4}$$(1-ξ)(1-η)

Explanation: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions.

6. For a four noded element while implementing a computer program, the compact representation of shape function is ____
a) Nt=$$\frac{1}{4}$$(1-ξ)(1-η)
b) Nt=(1-ξ)(1-η)
c) Nt=$$\frac{1}{4}$$(1+ξξi)(1+ηηi)
d) Undefined

Explanation: FourNodeQuad is a four-node plane-strain element using bilinear isoparametric formulation. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot’s theory of porous medium. Each element node has 3 degrees-of-freedom (DOF): DOF 1 and 2 for solid displacement (u) and DOF 3 for fluid pressure (p).

7. For a four noded quadrilateral elements, In uT=[u.v]T the displacement elements can be represented as u=N1q1+N2q3+ N3q5+ N4q7
v= N1q2+N2q4+ N3q6+ N4q8
then the shape function can be represented as _____
a) $$N=\left[\begin{array}{ |c c c c}q_1 & q_5 \\ q_2 &q_6\\q_3 &q_7\\q_4 & q_8\end{array}\right]$$
b) $$N=\begin{bmatrix}q_1 &q_3 &q_5 &q_7 \\ q_2 &q_4&q_6&q_8\end{bmatrix}$$
c) $$N=\begin{bmatrix}q_1 \\ q_2\end{bmatrix}$$
d) $$N=\begin{bmatrix}N_1 & 0 & N_3 & 0&N_5&0&N_7 & 0 \\ 0 & N_2 &0 &N_4&0&N_6&0&N_8\end{bmatrix}$$

Explanation: Displacement function in FEM. When the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation. In other words, displacements of any points in the element will be interpolated from the nodal displacements, and this is the main reason for the approximate nature of the solution.

8. The stiffness matrix from the quadrilateral element can be derived from _____
a) Uniform energy
b) Strain energy
c) Stress
d) Displacement

Explanation: In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to as certain an approximate solution to the differential equation.

9. For four noded quadrilateral element, the global load vector can be determined by considering the body force term in _____
a) Kinetic energy
b) Potential energy
c) Kinematic energy
d) Temperature

Explanation: A body force that is distributed force per unit volume, a vector, many people probably call up Vector’s definition (from Despicable Me). He says: It’s a mathematical term. A quantity represented by an arrow with both direction and magnitude. … Vector: a quantity with more than one element (more than one piece of information).

10. Shape functions are linear functions along the _____
a) Surfaces
b) Edges
c) Elements
d) Planes

Explanation: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions.

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