MANE 4240 & CIVL 4240 Introduction To Finite Elements
MANE 4240 & CIVL 4240 Introduction To Finite Elements
MANE 4240 & CIVL 4240 Introduction To Finite Elements
Prof. Suvranu De
Introduction to
differential equations
Reading assignment:
Summary:
x
x
x=0 x=L
u 0 at x 0
du
EA F at x L Neumann/ force bc
dx
Differential equation + Boundary conditions = Strong form
of the “boundary value problem”
Flexible string
S = tensile force in string
p(x) = lateral force distribution
y (force per unit length)
w(x) = lateral deflection of the
x=0 x=L string in the y-direction
x x
S S
p(x)
x
x
x=0 x=L
Q(x)
Differential equation
d dj
k Q 0; 0 xL
dx dx
j 0 at x 0 Known head
dj
k h at x L Known velocity
dx
Observe:
1. All the cases we considered lead to very similar differential
equations and boundary conditions.
2. In 1D it is easy to analytically solve these equations
3. Not so in 2 and 3D especially when the geometry of the domain is
complex: need to solve approximately
4. We’ll learn how to solve these equations in 1D. The approximation
techniques easily translate to 2 and 3D, no matter how complex the
geometry
A generic problem in 1D
d 2u
2
x 0; 0 x 1
dx
u 0 at x 0
u 1 at x 1
Analytical solution
1 7
u ( x) x 3 x
6 6
Solve for unknowns ao, a1, etc and plug them back into
This is your
u( x) a0jo ( x) a1j1 ( x) a2j2 ( x) ...
approximate solution to
the strong form