Finite Element Method Multiple Choice Questions on “Eigen Value and Time Dependent Problems – 1”.
1. The simultaneous linear equations used in FEM for solution of static problems are KX=F, the methods available for solving these equations are divided into two types: direct and iterative.
a) True
b) False
Answer: a
Clarification: When FEM is used for solution of static problems, we deal with a set of simultaneous Linear Equations of the form KX=F, where K is stiffness Matrix, X is displacement matrix and F is load vector. The order of matrix K is very large and the methods available for solving the equation are divided into two types: direct and iterative. Direct methods are used for equations without any round of error and iterative methods are used for the equations which start with an initial approximation.
2. For the following equations, what is the value of x2 using Gaussian elimination method?
x1-x2+3x3=10——– (i)
5x2-5x3=-5———— (ii)
-7x3=-28 —————- (iii)
a) 1
b) 2
c) 3
d) 4
Answer: c
Clarification: From the 3rd equation it’s seen that x3=(frac{-28}{-7})=4. Using x3 in 2nd equation we get
5*x2-5*4=-5
5*x2=15
x2=3
3. For the following equations, what is the value of x1 using Gaussian elimination method?
x1-x2+3x3=10——– (i)
5x2-5x3=-5———— (ii)
-7x3=-28—————- (iii)
a) 1
b) 2
c) 3
d) 4
Answer: a
Clarification: From the 3rd equation it’s seen that x3=(frac{-28}{-7})=4. Using x3 in 2nd equation we get
5*x2-5*4=-5
5*x2=15
x2=3
Using x3, x2 in 1st equation
x1-3+3*4=10
x1-3+12=10
x1=1
4. Which option is not correct about direct methods for solving system of linear equations?
a) In the absence of errors it yields exact solution
b) Errors arising from round off and truncation may give useless results
c) Gaussian elimination method is an example
d) Starts with an initial approximation
Answer: d
Clarification: The methods used for solving a system of linear equations are classified as: direct and iterative. Direct methods are those, which, in the absence of round-off and other errors, yield an exact solution in a finite number of elementary arithmetic operations. Indeed the errors arising from round-off and truncation may lead to extremely poor or even useless results. The fundamental method used for direct solutions is Gaussian elimination.
5. Which option is not correct about iterative methods for solving system of linear equations?
a) Convergence yields a good approximate solution
b) Insensitive to the growth of round-off errors
c) Gaussian elimination method is an example
d) Starts with an initial approximation
Answer: c
Clarification: The methods available for solving a system of linear equations can be divided into two types: direct and iterative. Iterative methods are those, which start with an initial approximation. When the process converges, we can expect to get a good approximate solution. The main advantages of iterative methods are the simplicity and uniformity of the operations to be performed, which make them well suited for use on computers and their relative insensitivity to growth of round-off errors.
6. In structural mechanics, which option is not correct about linear analysis?
a) Displacements are infinitesimally small
b) Material is linearly elastic
c) Externally applied loads are a function of time
d) Applied loads are not a function of time
Answer: c
Clarification: In a linear analysis, we assume that the displacements of a finite element assemblage are infinitesimally small and the material is linearly elastic. In addition, we also assume that a nature of boundary conditions remains unchanged during application of loads on Finite element assemblage. Loads are constant with respect to time.
7. A generalized Eigen value problem [K- ω2M]X=0 has a non-zero solution for X. What can be the value of determinant of the matrix [K- ω2M]?
a) Any integer
b) 0
c) +1
d) Positive integer
Answer: b
Clarification: A generalized Eigen value problem is represented by homogeneous matrix equation, [K- ω2M]X=0. From matrix equations methods, the equation has a non-zero solution for X if the determinant of the matrix [K-ω2M] equals to zero.
8. In FEM, the forced vibrations equation after Finite Element discretization of a structure can be expressed as which option?
a) Mẍ+Kẋ=F
b) Mẍ+Kẋ=0
c) Mẍ+Kx=F
d) Mẍ+Kx=0
Answer: c
Clarification: The forced vibrations equation after Finite Element discretization of a structure can be expressed as Mẍ+Kx=F where M and K are the mass and stiffness matrices of the structure, F is the external load vector; x and ẍ are the displacement and acceleration vectors. In the forced vibration equation the force vector is non-zero.
9. The free vibrations equation after Finite Element discretization of a structure is expressed as Mẍ+Kx=0. Which option is not correct about the free vibration case?
a) Displacements are harmonic
b) x=Xeiωt where X is amplitude
c) [K-ω2M]X=0
d) KX=Mω2
Answer: d
Clarification: In a free vibration analysis, the external load vector is zero and the displacements, x are harmonic x=Xeiωt where X is amplitude, on substituting x in governing equation we get [K-ω2M]X=0 or KX=Mω2X.
10. The generalized Eigen value problem [K-ω2M]X=0 has a non-zero solution for X. What is the value of natural frequency, ω if K=(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix}),
M=(begin{pmatrix}9&9&9\9&9&9\9&9&9end{pmatrix})?
a) 3
b) 1/9
c) 9
d) 1/3
Answer: d
Clarification: The generalized Eigen value problem [K-ω2M]=0 has a non-zero solution for X if the determinant of the matrix [K-ω2M] equals zero,
K=ω2M
(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix})=ω2*(begin{pmatrix}9&9&9\9&9&9\9&9&9end{pmatrix})
Equating corresponding elements, we get 9*ω2=1
ω2=1/9
Natural frequency, ω =1/3.
11. Which option is not correct about free vibration analysis problem KX= λMX, where X represents the amplitude of displacement x?
a) The displacements are harmonic
b) X represent mode shapes or Eigen vectors
c) λ represent Eigen value
d) ω represents Eigen value
Answer: d
Clarification: In a free vibration analysis KX= λMX, the external load vector is zero and the displacements are harmonic x=Xeiωt where X represents the amplitude of displacement x called Eigen vectors and λ= ω2 represent Eigen value.
12. After Finite Element discretization of a structure, which option expresses the free vibrations equation?
a) Mẍ+Kẋ=F
b) Mẍ+Kẋ=0
c) Mẍ+Kx=F
d) Mẍ+Kx=0
Answer: d
Clarification: After Finite Element discretization of a structure, the free vibrations equation can be expressed as Mẍ+Kx=0 where M and K are the mass and stiffness matrices of the structure; x and ẍ are the displacement and acceleration vectors respectively. In a free vibration analysis, the external load vector is zero.
13. For the eigenvalue problem of the form A(u) = λB(u), which option is not correct about the parameters used in the equation below?
(-frac{d^2x}{dx^2}=lambda u)
a) A=(frac{d^2x}{dx^2})
b) B=1
c) B=0
d) λ is called eigenvalue
Answer: c
Clarification: For the eigenvalue problem of the form A(u) = λB(u), A and B denote linear differential operators, has nontrivial solutions u. The values of λ are called eigenvalues and the associated functions U are called Eigen functions. For example, the given equation has A=(frac{d^2x}{dx^2}) and B=1.
14. The generalized Eigen value problem [K- λM]X=0 has a non-zero solution for X. What is the value of λ if K=(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix}), M=(begin{pmatrix}4&4&4\4&4&4\4&4&4end{pmatrix})?
a) 1
b) (frac{1}{2})
c) 4
d) (frac{1}{4})
Answer: d
Clarification: The generalized Eigen value problem [K- λM]X=0 has a non-zero solution for X if the determinant of the matrix [K- λM] equals zero or K=λM
(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix})=λ(begin{pmatrix}4&4&4\4&4&4\4&4&4end{pmatrix})
Equating corresponding elements, we get 1=4*λ
λ=(frac{1}{4}).
15. For the following eigenvalue equation to represent a heat transfer problem, a=kA and C=ρcA.
(-frac{d}{dx}(afrac{dU}{dx}))=λCU
a) True
b) False
Answer: a
Clarification: For the given eigenvalue equation the quantities a and C depend on the physics of problem. For a heat transfer problem, a=kA and C=ρcA where k is thermal conductivity, A is cross-sectional area and c is specific heat.