This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Single Variable Problems – Applications – 1”.

1. If the finite element model shown below represents heat conduction in axisymmetric or plane geometries then which option is not true?

\(-\frac{\partial}{\partial x}(a_{11}\frac{\partial u}{\partial x}+a_{12}\frac{\partial u}{\partial y})-\frac{\partial}{\partial y}(a_{21}\frac{\partial u}{\partial x}+a_{22}\frac{\partial u}{\partial y})\) +a_{00}u-f=0

a) u is temperature

b) a_{11}, a_{22} are conductivities in x, y directions

c) f is internal heat generation

d) a_{00}=1

View Answer

Explanation: Given model equation is a 2

^{nd}order partial differential equation. For heat conduction in axisymmetric or plane geometries, the given model equation is applicable for u as temperature, a

_{11}and a

_{22}as conductivities in x and y directions respectively, f as internal heat generation and a

_{00}=0.

2. In steady state heat transfer finite element model [K+H]*{T}={Q}*{P} if convective boundary conditions are neglected then which option is applicable?

a) K=0

b) H=0

c) Q=0

d) T=0

View Answer

Explanation: When convective boundary conditions are neglected in the steady state heat transfer finite element model [K+H]*{T}={Q}*{P}, it reduces to [K]*{T}={Q} form. The terms H and P which are related to convection become zero.

3. For a convective boundary, the natural boundary condition is a balance of energy transfer across the boundary due to conduction and/or convection.

a) True

b) False

View Answer

Explanation: The existence of convection heating or cooling leads to the convection boundary condition, known as Newton boundary condition. For a convective boundary, the natural boundary condition is a balance of energy transfer across the boundary due to conduction and/or convection.

4. In the below equation for steady-state heat transfer in plane systems, what does β stands for?

\(k_x\frac{\partial T}{\partial x}n_x+k_y\frac{\partial T}{\partial y}n_y\)+β(T-T_{∞})=\(\hat{q}\)_{n}

a) Convective heat transfer coefficient

b) Thermal expansion

c) Thermal conductivity

d) Diffusivity

View Answer

Explanation: The existence of convection heating or cooling leads to the convection boundary condition, known as Newton boundary condition. For a convective boundary, the natural boundary condition is a balance of energy transfer across the boundary due to conduction and/or convection. β stands for convective conductance (or the convective heat transfer coefficient).

5. In a steady state heat conduction problem with a thermal conductivity of 22(W/(m*K)), for a typical element of mesh of linear triangular elements, what is the value of a in stiffness matrix,

K=a*\(\begin{pmatrix}

1&-1&0\\-1 & 2 & -1\\0 & -1 & 1\end{pmatrix}\)?

a) 11

b) 12

c) 10

d) 5.5

View Answer

Explanation: For a typical element of mesh of linear triangular elements K=(k/2)*\(\begin{pmatrix}

1&-1&0\\-1 & 2 & -1\\0 & -1 & 1\end{pmatrix}\) denotes the element coefficient matrix. Given thermal conductivity (k) is 22 then

a=k/2

a=22/2

=11.

6. Heat conduction problem over a rectangular domain is shown in figure. Which nodes have temperature value of zero units?

a) Only1

b) Only 7

c) Only 2 and 3

d) 1, 2 and 3

View Answer

Explanation: All the nodes present at the boundary along which temperature value is zero will have T=0. In the given domain, the nodes 1, 2 and 3 are on boundary with temperature value zero. The node 7 is along an insulated boundary where temperature value depends on the temperature of other boundaries.

7. If q denotes the amount of heat flow through any boundary then what is the value of q at node7?

a) Dependent on a,b

b) =0

c) >0

d) <0

View Answer

Explanation: A boundary which is insulated will have no heat flow. Thus All the nodes present at that boundary will have q=0, irrespective of the value of T at other boundaries. In the given domain, the nodes 6, 7 and 8 are on insulated boundary, thus no heat flow.

8. What is the value of T at node 3 if the length of each element is 1/a units?

a) 0.5*b

b) b

c) 2*b

d) b*cos(a)

View Answer

Explanation: Given that the temperature distribution at boundary follows T=bcos\(\frac{\pi ax}{6}\).

At node 3, x=2/a units,

ax=2

T=bcos\(\frac{\pi2}{6}\)

T=bcos\(\frac{\pi}{3}\)

=b*0.5.

9. The velocity field of a fluid flow is denoted by v=2xi+3j. Which option exactly describes the flow?

a) Ideal

b) Compressible

c) Irrational

d) Inviscid

View Answer

Explanation: A fluid is said to be incompressible if ∇.V=0.

∇.V=\(\frac{d(2x)}{dx}+\frac{d(3)}{dy}\)

=2+0

=2.

Thus, the flow is compressible.

10. For a two dimensional problem, the evaluation of boundary integrals amounts to evaluation of line integrals.

a) True

b) False

View Answer

Explanation: The boundary of a 2D element consists of line segments, which are one dimensional elements. Thus, for a two dimensional problem, the evaluation of boundary integrals amounts to evaluation of line integrals (one dimensional elements).

11. A river flows along node 1 through node 4, infiltrating an aquifer at constant rate of 5(m^{3}/day*m^{2}). The length of elements is 0.2m. What is the value of global force due to infiltration of the river at node 1?

a) 0.5

b) 1.0

c) 1.5

d) 2.0

View Answer

Explanation: The value of global force due to infiltration of the river at node 1 is given by 0.5*q*h, where 0.5 denotes the value of the shape function, q is rate of infiltration and h is length of the element.

=0.5*q*h

Given q=5, h=0.2

=0.5*5*0.2

=0.5.

12. In a Heat Transfer problem, which option is used for interpolation of temperature inside a finite element?

a) Natural coordinates

b) Global coordinates

c) Temperature gradients

d) Shape functions

View Answer

Explanation: Shape Functions are used for interpolation of temperature inside a finite element. Global coordinates are used to apply boundary conditions. Natural coordinates are used to derive shape functions. Temperature gradients can’t be used for interpolation.

13. Consider a system where a wall of a tank containing a hot liquid at a temperature T_{0}, with an air stream of temperature T∞ flows on the outside of the tank, maintaining the outside wall temperature of TL. What is the expression for boundary condition of the system?

a) q=h(TL-T∞)

b) T=T∞

c) T=TL

d) TL-T∞

View Answer

Explanation: The boundary conditions are mainly of three kinds: constant temperature, constant heat flux (zero for insulated), and convection. In this problem, the temperature difference between wall and surroundings creates the constant heat flux of q=h(TL-T∞). This heat flux is taken as a boundary condition.

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