1. III Solution of pde’s using variational principles
4.1 Introduction
Introduction
Euler-Lagrange equations
Method of Ritz for minimising functionals
Weighted residual methods
The Finite Element Method

2. Introduction Variational principles
Variational principles are familiar in mechanics
the ‘best’ approximate wave function for the ground state of a quantum system is the one with the minimum energy
The path between two endpoints (t1, t2) in configuration space taken by a particle is the one for which the action is minimised
Energy or Action is a function of a function or functions
Wave function or particle positions and velocities
A function of a function is called a functional
A functional is minimal if its functional derivative is zero
This condition can be expressed as a partial differential equation

3. Introduction Hamilton’s principal of least action
L = T – V is the Lagrangian
The path actually taken is the one for which infinitesimal variations in the path result in no change in the action

4. Introduction Hamilton’s principal of least action
The condition that a particular function is the one that minimises the value of a functional can be expressed as a partial differential equation
We are therefore presented with an alternative method for solving partial differential equations besides directly seeking an analytical or numerical solution
We can solve the partial differential equation by finding the function which minimises a functional
Lagrange’s equations arise from the condition that the action be minimal

5. 4.2 Euler-Lagrange Equations
Let J[y(x)] be the functional
Denote the function that minimises J[y] and satisfies boundary conditions specified in the problem by
Let h(x) be an arbitrary function which is zero at the boundaries in the problem so that eh(x) is an arbitrary function that satisfies the boundary conditions
e is a number which will tend to zero

6. Euler-Lagrange Equations Functionals
Boundary conditions
y(a) = A
y(b) = B
Function

7. Euler-Lagrange Equations Functionals
y is the solution to a pde as well as being the function which minimises F[x,y,y’]
We can therefore solve a pde by finding the function which minimises the corresponding functional

8. 4.3 Method of Ritz for minimising functionals
Electrostatic potential u(x,y) inside region D SF p 362
Charges with density f(x,y) inside the square
Boundary condition zero potential on boundary
Potential energy functional
Euler-Lagrange equation D

9. Method of Ritz for minimising functionals Electrostatic potential problem
Basis set which satisfies boundary conditions

10. Method of Ritz for minimising functionals Electrostatic potential problem
 Series expansion of solution
Substitute into functional
Differentiate wrt cj

11. Method of Ritz for minimising functionals Electrostatic potential problem
Functional minimised when
Linear equations to be solved for ci
Aij.cj = bi
where

12. 4.4 Weighted residual methods
For some pde’s no corresponding functional can be found
Define a residual (solution error) and minimise this
Let L be a differential operator containing spatial derivatives
D is the region of interest bounded by surface S
An IBVP is specified by

13. Weighted residual methods Trial solution and residuals
ui(x) are basis functions
Define pde and IC residuals
RE and RI are zero if uT(x,t) is an exact solution

14. Weighted residual methods Weighting functions
The weighted residual method generates and approximate solution in which RE and RI are minimised
Additional basis set (set of weighting functions) wi(x)
Find ci which minimise residuals according to
RE and RI then become functions of the expansion coefficients ci

15. Weighted residual methods Weighting functions
Bubnov-Galerkin method
wi(x) = ui(x) i.e. basis functions themselves
Least squares method
Positive definite functionals u(x) real
Conditions for minima 

16. 4.5 The Finite Element Method
Variational methods that use basis functions that extend over the entire region of interest are
not readily adaptable from one problem to another
not suited for problems with complex boundary shapes
Finite element method employs a simple, adaptable basis set

17. The finite element method Computational fluid dynamics websites
Gallery of Fluid Dynamics
Introduction to CFD
CFD resources online
CFD at Glasgow University
Computational fluid dynamics (CFD) websites
Vortex shedding illustrations by CFDnet
Vortex Shedding around a Square Cylinder
Centre for Marine Vessel Development and Research Department of Mechanical Engineering Dalhousie University, Nova Scotia

18. The finite element method Mesh generation
Finer mesh elements in regions where the solution varies rapidly
Meshes may be regular or irregular polygons
Local coordinate axes and
node numbers
Global coordinate axes 1 2 3
Definition of local and global coordinate axes and node numberings

19. The finite element method Example: bar under stress
Define mesh
Define local and global node numbering
Make local/global node mapping
Compute contributions to functional from each element
Assemble matrix and solve resulting equations

20. The finite element method Example: bar under stress
Variational principle
W = virtual work done on system by external forces (F) and load (T)
U = elastic strain energy of bar
W = U or (U – W) = P = 0

21. The finite element method Example: bar under stress
Eliminate dh/dx using integration by parts

22. The finite element method Example: bar under stress
Boundary
conditions
Differential equation being solved

23. The finite element method Example: bar under stress
Introduce a finite element basis to solve the minimisation problem P[u(x)] = 0
Assume linear displacement function  
u(X) = a1 + a2 X
ui(X) = a1 + a2 Xi
uj(X) = a1 + a2 Xj
u(X) i j X
Solve for coefficients a
X is the local displacement variable

24. The finite element method Example: bar under stress
Substitute to obtain finite elements
u(X) = N1u1 + N2 u2 N1 N2
u(X) = [N1 N2] (u)
u1 and u2 are coefficients of the
basis functions N1 and N2

25. The finite element method Example: bar under stress
Potential energy functional Grandin pp91ff

26. The finite element method Example: bar under stress
Strain energy
per element

27. The finite element method Example: bar under stress
Node force potential energy
Distributed load potential energy

28. The finite element method Example: bar under stress
Energy functional for one element
Equilibrium condition for all i

29. The finite element method Example: bar under stress
Equilibrium condition for one element
Assemble matrix for global displacement vector

30. The finite element method Example: bar under stress
Solve resulting linear equations for u