Finite Element Method Introduction General Principle Variational formulation Discretization Algorithm Examples

Introduction To solve partial differential equations (PDE) we often use the finite element method. This is a numerical technique which approximates the solution of a problem that we first have to transform. This method is frequently used to solve physical problems because these equations can represent the dynamic behavior of physical systems. For example, it can be used to calculate the movement of a cord shaken by one of its ends or the deformation of a metal structure.

General Principle The Finite Element Method consists in finding a discrete mathematic algorithm allowing to search for an approximate solution of a partial differential equation (PDE). Generally this PDE relates to a u function defined on a field and which has conditions on boundary. We usually call Dirichlet conditions (values on boundary) and Neumann conditions (gradient on boundary). Actually we have to solve a problem where, thanks to the variational formulation, solutions check existing conditions weaker than those of the starting problem and where discretization allows to find an approximate solution.

Variational Formulation Let f be a continuous function on ΩU(δΩ) and u the solution on the following PDE: With u checking the Dirichlet's condition: u=0 on δΩ. Let v a function such that v=0 on δΩ. We transform this equation in: Using an integration by parts on the first term we find :

Variational Formulation We have v=0 on δΩ, so we obtain the weak formulation of the problem: If u is twice differentiable there's equivalence between this formulation and the one of the initial problem. So we will solve the weak formulation instead of solving the initial problem. (To have this equivalence we also need to have a field Ω regular enough and f derivable enough.)‏ Notations: So we have:

Discretization To solve the problem we have to choose a mesh (regular or not) of the field Ω (to approximate numerically the solution). The more the mesh will be tightened the more the approximate solution will be close to true. We also have to take a basis of functions adapted to the mesh. We often use polynomials of Lagrange. In this basis the function i equals 1 at the node xi and 0 at other nodes. In the one-dimensional case it looks like :

Discretization Let M be a mesh on Ω and b=(e1,...,en) the basis associated. We're searching for ū the solution of the discretized problem such as ū=0 on δΩ and for all v function such that v=0 on δΩ: However in this discretized space, to say that any vector checks the preceding proposition is equivalent to say that all the vectors of the basis check the proposition. If we break up the solution in the basis of the ei we obtain: If we increase the number of basis functions the solutions un will have to converge towards the solution u of the PDE.

Algorithm With the preceding notations we can define: - the A matrix having for components a(e i ,e j)‏ - the U vector having for components ui which are the coordinates of the solution approached on the basis b - the B vector having for components L(ej)‏ Then this problem is equal to solving linear equations with n equations and n unknowns : AU = B The matrix A is called matrix of rigidity. It's a symmetric positive definite matrix. So it's invertible and we have: U=A-1B

Algorithm Here the order of the steps of the finite element method: 1. To calculate the matrix of rigidity A 2. To caculate the terms L(ej) 3. To solve AU=B according to the selected level of discretization. U is given by U=A-1B . The main problem is to choose the best way to reverse the matrix A because this is the step which needs the most important computing power. 4. We obtain ū thanks to the vector U which contains the coordinates of ū on the basis b and then we have an approximate solution of the problem.

Examples A physical example of situaton where this problem is encountered is that of bending of a beam. Here we consider the beam has a length of 1, with x=0 and x=1 as ends, subjected to a linear density of load and simply supported at its ends. So the value of the bending moment u(x) at the moment x is solution of the problem with k2 = 1/EI(x) where E is the Young modulus of material and I(x) is the principal moment of inertia of the section of the beam.

Examples Finite Element Method allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacement. That's why it's used in aeronautical or automotive industries for example. Digital simulation of a crash test on a car.