ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 8: 3D FINITE ELEMENT SIMULATIONS IN COMSOL
Consider a heterogeneous bar with a square crosssection of 1m by 1m (in the xy plane) and a length of 3m (in the zdirection). The bar is composed of three different materials, which have propert
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
ME 180 / CE 133, Homework #2
Problem 1
In order to find the weak form, one multiplies the strong form by a test function w:
Z 1
Z 1
00
w( + x) dx = 0 )
(w 0 )0 w0 0 + wx dx = 0
0
0
Because the boundaries at x = 0 and x = 1 are both essential, w(0) = w(1)
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
CE C133 / ME C180
HW #4 Solutions
Problem 1
a) The heat flux is given by
q=
krT =
k
@(T0 + Td r/R)
er =
@r
kTd
er
R
b) The total energy per time moving through the surface of the sphere S is
Z
Z 2 Z
q dS =
qr (R)R2 sin d d = 4kTd R 445 W
S
0
0
The negati
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Spring 2016
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 1: Due Thurday Feb. 4
1. A cylindrical water tank of radius R has a water level of h(t). Water flows i
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
CE C133 / ME C180
HW #3 Solutions
Problem 1
a) The Lagrange shape functions at each node xei of this element are:
4
Y
xej x
.
Ni (x) =
e
e
x
x
j
i
j6=i
b) Since there 4 nodes, the Lagrange polynomial is of order 4 1 = 3.
c) See plot.
d) The LM array is
3
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Spring 2016
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 2: Due Thursday Feb. 11
1. Formulate the weak form problem for the following strong form statement: Fi
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2015
MEC180 HW2
Isaac Finger
2/11/16
1
1.1
Problem 1
Part A
We are rst given dierential equation, and integrated equation and a a solution. We are
rst instructed to nd 0 given the solution u(x) equation.
u(x) = x + 0
u (x) =
(1)
Then introducing these to the
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2015
C180, Finite Element Method
Assignment 2
Due 2/9
1. RayleighRitz method
d2 u
Consider the dierential equation 2 = 1, dened over the interval (0,1), with the
dx
du
= 2 at x = 1, and u = 0, at x = 0. It can be shown the weak
boundary conditions
dx
form co
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2015
C180, Finite Element Method
Assignment 1
Due 2/2/15
1. Consider the dierential equation
du(x)
u(x) = 1, with the condition u(0) = 1.
dx
a) Find the exact solution to this equation analytically.
b) We want to approximate the solution on the (0,1) interval
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2015
C180, Finite Element Method
Assignment 3
Due 2/20/15
1. Consider the following dierential equation:
d
dx
A1
du(x)
dx
= k 2 sin
kx
L
+ 2x,
(1)
Over the domain = (0, L) with boundary conditions
u(0) = 0
(2)
u(L) = 1.
(3)
and
where A = 0.2, and k is a given
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
CE C133 / ME C180
HW #6 Solutions
Problem 1
First we compute the Bmatrix for the linear triangular element:
1/2
0
1/2 0 0 0
1
0
0 0 1
B= 0
1 1/2 0 1/2 1 0
The constitutive relation for plane strain is
2 +
0
2 + 0
D=
0
0
Since B is constant, we have
Z
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Fall 2008
CE C133 / ME C180
HW #1 Solutions
Problem 1
Conservation of mass tells us:
dm(t)
d(V (t)
=
= qin (t) qout (t),
dt
dt
Given incompressible flow, and the volume of the tank V (t) = R2 h(t), we arrive at out
governing law for h(t):
qin qout
dh(t)
=
.
dt
R2
P
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 9: VEHICLE BRACKET ANALYSIS IN COMSOL
Please read the entire assignment before you start generating the model, this will help you plan each step of the process. Consider a hanger bracket used to support an exhaust pipe that is directly after
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 2: HIGHER ORDER ELEMENTS
Consider the followng boundary value problem, with domain = (0, L):
d dx
A1 du = k 2 sin( 2kx ) dx L
A1 = 0.1 k = 16 (1) L=1 u(0) = 1 = given constant = 0 u(L) = 2 = given constant = 1
Solve this with the fini
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 7: THE FINITE ELEMENT METHOD IN 2D
Solve the following boundary value problem, on an archshaped domain, using COMSOL: (kT ) + f = 0, T = T0 along = kT n = q0 (r) along = 0 kT n = 0 along r = ri , ro k = k1 for x  xc  r k = k2
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
PROJECT 4: "POTENTIALS AND EFFICIENT SOLUTION TECHNIQUES"
Solve the following boundary value problem, with domain = (0, L), analytically:
d dx
A1 (x) du = k 2 sin( 2kx ) dx L (1)
A1 (x) = 10 DIF F EREN T SEGM EN T S (SEE BELOW ) k = 22, L = 1, u
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 3: ERROR ESTIMATION AND ADAPTIVE MESHING
d Consider the boundary value problem dx A1 du = f (x), A1 = 1, with dx domain = (0, L), L = 1, and solution u(x) = cos(19x4 ).
Compute the finite element solution uN to this problem using linear
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 5: REVISITING THE BASICS WITH COMSOL
The goal of this assignment is to familiarize you with how to solve a 1D problem in COMSOL. We will revisit the topics that were covered in homework #1 and #2. You should explore all of the menus and set
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
SUPPLEMENTARY HANDOUT: COMSOL GUIDE FOR 1D
This handout is designed to highlight some of the key features and settings that can be found in the menus of COMSOL. Remember that you can always get more background information in the "Help Desk", which c
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
University of California, Berkeley ME 180, Engineering Analysis Using the Finite Element Method Spring 2008 Instructor: T. Zohdi
Quiz 2 Solutions Problem 1 Derive the weak form corresponding to the strong form d 2 du x = 5u + ex dx dx u=4 du = 14
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 6: TIME DEPENDENT PROBLEMS IN 1D
TRANSIENT REACTIONDIFFUSION Derive the weak form of the boundary value problem shown below for a backward (implicit) Euler scheme. Solve the following boundary value problem (L = 1) in COMSOL c=
d dx dc
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
University of California, Berkeley ME 180, Engineering Analysis Using the Finite Element Method Spring 2008 Instructor: T. Zohdi
Quiz 1 Solutions Problem 1 Solve d du 2k A1 = k2 sin x , dx dx L analytically, where A1 , k, and L are constants. Soluti
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
HOMEWORK 1: THE BASICS
Solve the followng boundary value problem, with domain = (0, L), analytically:
d dx
A1 du = k 2 sin( 2kx ) dx L
A1 = given constant = 0.1 k = given constant (1) L=1 u(0) = 1 = given constant = 0 u(L) = 2 = given constant =
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
University of California, Berkeley ME 180, Engineering Analysis Using the Finite Element Method Spring 2008 Instructor: T. Zohdi
Quiz 4 : Solutions Problem 1 a) Derive the approximation functions e () for a master element with nodes at = 1, = 0,
ENGINEERING ANALYSIS USING THE FINITE ELEMENT METHOD
ME 180

Spring 2008
University of California, Berkeley ME 180, Engineering Analysis Using the Finite Element Method Spring 2008 Instructor: T. Zohdi
Quiz 3 Solutions Problem 1 Calculate the stiffness and force matrices corresponding to the problem d du c(x + 1)2 + 3u =