This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Velocity – Pressure Finite Element Model”.

1. In velocity-pressure formulation in FEM, which step is not used in the development of a weak form?

a) Multiply governing equations with weight functions

b) Integrating over the element domain

c) Integrating by parts

d) Performing coordinate transformation

View Answer

Explanation: In the development of weak form, we consider a typical element and develop the finite element model over it by following the three-step procedure. Firstly, multiply governing equations with weight functions and Integrate over the element domain. Secondly, perform integration by parts and, thirdly, define the coefficient of weight function.

2. In the formulation of governing equations, which option does not signify the characteristics of a weight function?

a) Weight functions are multiplied to governing equations to obtain weak forms

b) Weight functions are interpreted from the physical setup of the problem

c) Weight function must denote a non-dimensional quantity

d) Weight function can be interpreted as a velocity

View Answer

Explanation: In the development of the weak form, we consider a typical element and multiply its governing equations with weight functions. They can be interpreted physically for a given equation; for example, in the momentum equation, the weight function must be interpreted as velocity. For the representing volume change, weight function is like pressure. Thus, it can be a dimensional quantity.

3. For a governing equation, what does one conclude from the weak formulation if it does not contain boundary integral involving weight function?

a) Integration by parts is used

b) Integration by parts is not used

c) The weight function is as a primary variable

d) The weight function is to be made continuous across inter-element boundaries

View Answer

Explanation: For a governing equation, the weak formulation does not contain boundary integral if there is no integration by parts used, this implies that the weight function is not a primary variable but a part of the secondary variables, this, in turn, requires that the weight function is not to be made continuous across inter-element boundary.

4. In FEM, which option is the correct weak form of the following momentum equation?

ρ\(\frac{\partial v_x}{\partial t}-\frac{\partial \sigma _{xx}}{\partial x}-\frac{\partial \sigma_{xy}}{\partial y}-f_x\)=0

a)l\(\int_{\Omega_e}[\rho w_1\frac{\partial v_x}{\partial t}+\frac{\partial w_1}{\partial x}\sigma_{xx}+\frac{\partial w_1}{\partial y}\sigma_{xy}-w_1 f_x]\)dxdy+∮_{┏c} w_{1}(σ_{xx}n_{x}+σ_{xy}n_{y})ds=0

b) l\(\int_{\Omega_e}[\rho w_1\frac{\partial v_x}{\partial t}+\frac{\partial w_1}{\partial x}\sigma_{xx}+\frac{\partial w_1}{\partial y}\sigma_{xy}-w_1 f_x]\)dxdy=0

c) l\(\int_{\Omega_e}[\rho w_1\frac{\partial v_x}{\partial t}+\frac{\partial w_1}{\partial x}\sigma_{xx}+\frac{\partial w_1}{\partial y}\sigma_{xy}-w_1 f_x]\)dxdy+∮_{┏c} w_{1}(σ_{xx}+σ_{xy})ds=0

d) l\(\int_{\Omega_e} w_1[\rho\frac{\partial v_x}{\partial t}-\frac{\partial \sigma_{xx}}{\partial x}-\frac{\partial \sigma_{xy}}{\partial y}-f_x]\)dxdy++∮_{┏c} w_{1}(σ_{xx}n_{x}+σ_{xy}n_{y})ds=0

View Answer

Explanation: The three-step procedure obtains the weak forms of the momentum equation over an element. Firstly, multiply the governing equation with weight function w

_{1}and integrate over the element domain Ω

_{e}. Secondly, perform integration by parts and thirdly define the coefficient of weight function. Thus, the weak form must contain two separate integrals, the terms \(\frac{\partial w_1}{\partial x}\) and σ

_{xx}n

_{x}.

5. Using constitutive relations, what is the value of τ_{xx}if μ=0.3 and v=4x?

a) 0.24

b) 2.4

c) 0.12

d) 1.2

View Answer

Explanation: From constitutive relations, τ

_{xx}=2μ\(\frac{\partial v_x}{\partial x}\), where μ is the coefficient of viscosity.

\(\frac{\partial v_x}{\partial x}\)

= \(\frac{\partial (4x)}{\partial x}\)

=4.

Thus τ

_{xx}=2*μ*4.

Given μ=0.3

τ

_{xx}=2*0.3*4

=2.4.

6. Using constitutive relations, what is the value of τ_{xy}if μ=0.3 and v=4xy-6y?

a) 0.24x

b) 2.4x

c) 0.12x

d) 1.2x

View Answer

Explanation: From constitutive relations, τ

_{xy}=μ\((\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x})\), where μ is the coefficient of viscosity. On comparing v=4xy+6y with v=vx+vy, we get vx=4xy and vy =6y.

\(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\)

=4x+0

=4x.

Thus τ

_{xy}=μ*(4x).

Given μ=0.3

τ

_{xx}=0.3*4x

=1.2x.

7. If a governing equation represents volume change in an element, then the weight function in its weak form must be like a force that causes the volume change.

a) True

b) False

View Answer

Explanation: The governing equation \(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\) represents the volume change in an element of dimensions dx and dy. Therefore, the weight function (w) in its weak form must be like a force that causes the volume change. Volume changes occur under the action of hydrostatic pressure, hence w=-P.

8. Which option is not correct concerning the pressure variable, P in the weak form of the momentum and continuity equation?

a) P is a primary variable

b) P is a part of the secondary variables

c) P=constant is the minimum continuity requirement for interpolation

d) It is discontinuous across inter-element boundaries

View Answer

Explanation: In the weak form of the continuity equation, there is no boundary integral involving weight function because no integration by parts is used, this implies that P is not a primary variable; it is a part of the secondary variables, this, in turn, requires that P not be made continuous across inter-element boundaries. The minimum continuity requirement for interpolation of P is that P=constant.

9. Which option is not correct concerning the velocity variables, v_{x} and v_{y} in the weak form of the momentum and continuity equation?

a) They are primary variables

b) The minimum continuity requirement for interpolation is that they are linear in x and y

c) The minimum continuity requirement for interpolation is that they are constant

d) They are continuous across the inter-element boundary

View Answer

Explanation: In the weak form of the continuity equation, the boundary integral involving weight function is present by the use of integration by parts. It implies that v

_{x}and v

_{y}are primary variables; this, in turn, requires that v

_{x}and v

_{y}to be continuous across inter-element boundaries. The minimum continuity requirement for interpolation of v

_{x}and v

_{y}is that they are linear in x and y.

10. Which equation is the correct vector form of the finite element model of momentum and continuity equations in the flow domain?

a) MΔ+K^{11}Δ+K^{12}P=0

b) MΔ+K^{11}Δ+K^{22}P=F^{1}

c) MΔ+K^{22}Δ+K^{12}P=F^{1}

d) MΔ+K^{11}Δ+K^{12}P=F^{1}

View Answer

Explanation: The equation MΔ+K

^{11}Δ+K

^{12}P=F

^{1}gives the vector form of the finite element model of momentum and continuity equations in the flow domain, where

**F**is force matrix,

**M**is a mass matrix,

**K**is related to velocity term,

^{11}**K**is related to the pressure term,

^{12}**P**is the pressure matrix, and

**Δ**is the velocity matrix.

11. For the vector form of the finite element model of momentum and continuity equations MΔ+K^{11}Δ+K^{12}P=0, what is the correct expression for mass matrix M?

a) M=∫_{Ωc}ρψ^{T}ψdx

b) M=∫_{Ωc}fψ^{T}ψdx

c) M=∫_{Ωc}tψ^{T}ψdx

d) M=∫_{Ωc}Pψ^{T}ψdx

View Answer

Explanation: The vector form of the finite element model in the flow domain is , MΔ+K

^{11}Δ+K

^{12}P=F

^{1}where F is force matrix, M is a mass matrix, K

^{11}is related to velocity term, K

^{12}is related to the pressure term, P is the pressure matrix, and Δ is the velocity matrix. M is a mass matrix as it contains the mass density values of elements, thus M=∫

_{Ωc}ρψ

^{T}ψdx, where ρ denotes the mass density.

12. For the vector form of the finite element model of momentum and continuity equations MΔ+K^{11}Δ+K^{12}P=0 what is the order of matrix F if the order of M is 2n x 2n?

a) 2n x 1

b) 1 x 2n

c) m x 2n

d) 2n x m

View Answer

Explanation: In the vector form, the terms

**M**and

**K**are of the order 2n x 2n,

^{11}**K**is of the order 2n x m,

^{12}**K**is of the order m x 2n. The equation MΔ+K

^{21}^{11}Δ+K

^{12}P=F

^{1}gives the vector form, where

**F**is force matrix,

**M**is a mass matrix,

**K**is related to velocity term,

^{11}**K**is related to the pressure term,

^{12}**P**is the pressure matrix, and

**Δ**is the velocity matrix. Note that

13. In matrix algebra, if a matrix is positive definite, then all its eigenvalues are greater than zero.

a) True

b) False

View Answer

Explanation: A positive definite matrix is a symmetric matrix with all eigenvalues greater than zero. The FEM model of the continuity equation does not contain the pressure term, P. Therefore, the assembled equations also have zero in diagonal elements corresponding to the nodal values of P (i.e., the system of equations is not positive-definite).

14. What is the physical interpretation of the weight function w3 in the following weak form of the continuity equation?

-∫_{Ωe}w_{3}\((\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y})\)dxdy=0

a) Hydrostatic pressure

b) Axial force

c) Surface traction

d) Body force

View Answer

Explanation: The weak form of the continuity equation represents the volume change in an element of dimensions dx and dy. Therefore, the weight function (w3) must be like a force that causes the volume change. Volume changes occur under the action of hydrostatic pressure. Hence, w3 is like -P.

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