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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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