Download Introduction to Differential Equations

MANE 4240 & CIVL 4240
Introduction to Finite Elements
Prof. Suvranu De
Introduction to
differential equations
Reading assignment:
Handout + Lecture notes
Summary:
• Strong form of boundary value problems
•Elastic bar
•String in tension
•Heat conduction
•Flow through a porous medium
•Approximate solution
So far, structural mechanics using Direct Stiffness approach
Finite element method is used to solve physical problems
Solid Mechanics
Fluid Mechanics
Heat Transfer
Electrostatics
Electromagnetism
….
Physical problems are governed by differential equations which satisfy
Boundary conditions
Initial conditions
One variable: Ordinary differential equation (ODE)
Multiple independent variables: Partial differential equation (PDE)
A systematic technique of solving the differential equations
Differential equations (strong) formulation (today)
Variational (weak) formulation
Approximate the weak form using finite elements
Axially loaded elastic bar
y
x
x=0
x=L
A(x) = cross section at x
b(x) = body force distribution
(force per unit length)
x E(x) = Young’s modulus
u(x) = displacement of the bar
at x
Differential equation governing the response of the bar
d 
du 
 AE   b  0;
dx 
dx 
0 xL
Second order differential equations
Requires 2 boundary conditions for solution
y
x
x
x=L
x=0
Boundary conditions (examples)
u0
u 1
at x  0 Dirichlet/ displacement bc
at x  L
u  0 at x  0
du
EA
 F at x  L
dx
Neumann/ force bc
Differential equation + Boundary conditions = Strong form
of the “boundary value problem”
Flexible string
y
x=0
x=L
x
x
S
S = tensile force in string
p(x) = lateral force distribution
(force per unit length)
w(x) = lateral deflection of the
string in the y-direction
S
p(x)
Differential equation governing the response of the bar
d 2u
S 2  p  0;
dx
0 xL
Second order differential equations
Requires 2 boundary conditions for solution
Heat conduction in a fin
y
x
x=0
x=L
A(x) = cross section at x
Q(x) = heat input per unit length
per unit time [J/sm]
x k(x) = thermal conductivity [J/oC
ms]
T(x) = temperature of the fin at x
Q(x)
Differential equation governing the response of the fin
d 
dT 
Ak

  Q  0;
dx 
dx 
0 xL
Second order differential equations
Requires 2 boundary conditions for solution
y
x
x
x=0
x=L
Q(x)
Boundary conditions (examples)
T  0 at x  0
dT
k
 h at x  L
dx
Dirichlet/ displacement bc
Neumann/ force bc
Fluid flow through a porous medium (e.g., flow of water through a
dam)
y
A(x) = cross section at x
Q(x) = fluid input per unit volume
per unit time
x k(x) = permeability constant
x
j(x) = fluid head
x=L
x=0
Q(x)
Differential equation
d  dj 
k
  Q  0;
dx  dx 
0 xL
Second order differential equations
Requires 2 boundary conditions for solution
Boundary conditions (examples)
j  0 at x  0
dj
k
dx
 h at x  L
Known head
Known velocity
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
 x  0;
0  x 1
2
dx
u  0 at x  0
u  1 at x  1
Analytical solution
1
7
u ( x)   x 3  x
6
6
Assume that we do not know this solution.
A generic problem in 1D
A general algorithm for approximate solution:
Guess u( x)  a0jo ( x)  a1j1 ( x)  a2j 2 ( x)  ...
where jo(x), j1(x),… are “known” functions and ao, a1, etc are
constants chosen such that the approximate solution
Satisfies the differential equation
Satisfies the boundary conditions
i.e.,
d 2j o ( x)
d 2j1 ( x)
d 2j 2 ( x)
a0
 a1
 a2
 ...  x  0;
2
2
2
dx
dx
dx
a0j o (0)  a1j1 (0)  a2j 2 (0)  ...  0
0  x 1
a0j o (1)  a1j1 (1)  a2j 2 (1)  ...  1
Solve for unknowns ao, a1, etc and plug them back into
This is your
u( x)  a0j o ( x)  a1j1 ( x)  a2j 2 ( x)  ...
approximate solution to
the strong form