FEM

Question 1

What is the purpose of FEM?

Answer 1

To find the displacement, stress, and/or strain in a structure.

Question 2

What is Lagrangian interpolation? How is it defined in FEM?

Answer 2

It is the theory of constructing curves using polynomials. In FEM, a Lagrangian Interpolating Function is one which is equal to the desired function at every node.

Question 3

What is a basis function?

Answer 3

A function over an element with the value of one above its node and zero everywhere else.

Question 4

What is the expansion coefficient to a basis function?

Answer 4

The coefficient that when multiplied by the basis function, gives the value of u(x).

Question 5

What is a Stiffness Matrix?

Answer 5

It relates the local nodal displacements, {d}, to the local forces on the element, {f}, by {f}=[k]{d}.

Question 6

What is the general form of FEM?

Answer 6

The expansion of f(x) through a finite set of expansion functions.

Question 7

Why is FEM so popular?

Answer 7

It’s very adaptable to arbitrary structures and therefore flexible to many problems.

Question 8

What is the stiffness matrix for a 1D problem

Answer 8

[k] = AInt([B]^t[D][B])dx

Question 9

What is the form of the body forces matrix for a 1D problem

Answer 9

{f} = A*Int([N]^t{f(x)}dx

Question 10

What is the Jacobian and why is it useful in FEM?

Answer 10

The Jacobian maps a local element to an arbitrary element. Useful because the integrations can be performed on a very simple local element, and easily be mapped to the real, global element.

Question 11

I(x) = ?

Answer 11

I(x) = [(x-S_0)(x-S_1)…(x-S__k-1)(x-S_k+1)…(x-S_n)]/

[(S_k-S_0)(S_k-S_1)…(S_k-S_k-1)(S_0-S_k+1)…(S_k-S_n)]

Question 12

What is the Jacobian for a quadratic element?

Answer 12

|J| = det |a b|
|c d|

where:
a = dx/ds
b = dy/ds
c = dx/dt
d = dy/dt

Question 13

Define Total Potential Energy and it’s components

Answer 13

It is define as the sum of the Internal Strain Energy, U, and the Potential Energy of the external forces, Omega. Where:

U = 0.5Int({strain}^t{stress}dV

Omega = -Int({f}{d}dV

Question 14

How do you improve the accuracy of an approximation? What are the trade-offs?

Answer 14

Increase polynomial order, decrease mesh size, use quadratic mesh type. They are more expensive as they become more complex

Question 15

State the Equilibrium Equations and where they come from.

Answer 15

d/dx(Normal stress_x)+d/dx(Shear Stress_yx) + Body Force_x = 0

d/dy(Normal stress_y)+d/dx(Shear Stress_xy) + Body Force_y = 0

Tau_xy=Tau_yx

Question 16

State the Compatibility Equations and where they come from.

Answer 16

Strain in x = du/dx
Strain in y = dv/dy
Angular Strain = du/dx + dv/dy

Question 17

State the steps to FEM

Answer 17

1) Define Mesh
2) Form Basis Functions
3) Define Strain/Disp and Stress/Strain Relations
4) Derive Stiffness Matrix
5) Form Global Stiffness Matrix
6) Apply Boundary Conditions
7) Solve

Question 18

Define plane stress and plane strain problems and write [D] for each.

Answer 18

Plane Stress - A state of stress in which the normal stress and the shear stresses directed perpendicular to the planes are assumed to be zero. Typically have to be thin and loads only in x-y plane.

[D] = E/(1-v^2)*[a]

a = 1 v 0
      v 1 0
      0 0 (1-v)/2

Plane Strain - Strain and shear strain in the z direction is zero. Typically have to be long with constant cross-sectional area with loads only in the x or y direction and do not vary in the z.

[D] = E/{(1+v)(1-2v)}*[b]

b = 1-v  v  0
      v   1-v  0
      0    0   (1-2v)/2

Question 19

Define the General Displacement Function for a triangular and quadrelateral element

Answer 19

Triangle:

Phi = u(x,y) = a1 + a2x + a3y
v(x,y) a4 + a5x +a6y

Quad:

Phi = u(x,y) = a1 + a2x + a3y + a4xy
v(x,y) a5 + a6x +a7y +a8xy

Question 20

Explain what is meant by CST

Answer 20

Constant Strain Triangle - Strain in the triangular element is not spatially dependent

Question 21

State the Total Potential Energy Equation in terms of displacement

Answer 21

TPE = 0.5{d}^t * Int([B]^t[D][B])dV{d} - {d}^t * {f}

Where:

{f} = Int([N]^t*{X})dV + {P} + Int([N]^t{T}dS

Question 22

Explain the key difference between CST and Quad elements.

Answer 22

Quadrilateral element have another bilinear term which improves the accuracy of the approximation but is more complex to solve.

Question 23

Why is it advantageous to evaluate at local regions and map to arbitrary ones?

Answer 23

It makes the integral of [k] much easier, as it can be done over a very simple shape and then mapped to an arbitrary shape

Question 24

What is the Newton-Cotes formula? Why is it used in FEM? What are the steps? How accurate is it?

Answer 24

This is a numerical integration method used in the calculation of integrals on a computer, as regular integration can be expensive and inefficient if elements have spatially dependent strains. Steps are:

1) Split function into intervals equally space apart
2) Create Lagrangian Interpolating function. - we know how to use L.I. functions
3) f(x) = SUM[I(x)*(f(xi)]

Where I is a basis function.

4) Int(f(x)) = SUM[Int([I(x)*(f(xi)]]

= SUM((Int(I(x)) between -1 and 1)*f(xi))

Exactly accurate if the number of points is equal to the order of the polynomial

Question 25

Recall the points and weights for 1-, 2-, and 3-Point Gaussian Quadrature over the interval [-1, 1]

Answer 25

1-Point:
x=0
w = 2

2-Point:
x1,2 = +-1/SQRT(3)
w1,2 = 1

3-Point:
x1,2,3 = 0, +-SQRT(3/5)
w1,2,3 = (8/9), (5/9), (5/9)

Question 26

State the formula for Gaussian Quadrature in 1D and 2D

Answer 26

1D:
Int(f(x)) = SUM(wi*f(xi))

2D
Int(f(x,y)) = SUM[SUM(wiwjf(xi,yj))]

Question 27

State the compatibility equations for an axisymmetric element

Answer 27

Strain_x = du/dr
Strain_z = dw/dz
Strain _theta = u/r
Shear strain_rz = du/dz + dw/dr

Question 28

State [B] for an axisymmetric element

Answer 28

[B_i] = 1/(2*A)*M_i
M = :
|a 0|
|0 b|
|c 0|
|d e|

Where a = beta_i
b = gamma_i
c = (a_i+gamma_i*z)/r + beta_i
d = alpha_i
e = beta_i

[B] = [B_i][B_j][B_m]