Variational Methods

Question 1

what is the goal of using variational methods

Answer 1

• to minimise a functional

Question 2

say we have a functional J that depends on the function u(x). What would you (simply) do to find the u that minimises J

Answer 2

• take the directional derivative of J and set it to 0

Question 3

what is the euler lagrange equation

Answer 3

• dF/dy - d/dx*(dF/dy’) = 0

Question 4

what is the arc length element ds in cylindrical coordinates

Answer 4

• ds = sqrt(dr^2 + r^2*dθ^2 + dz^2)

Question 5

what is the arc length element ds in spherical coordinates

Answer 5

• ds = sqrt(dr^2 + r^2dθ^2 + r^2sin(θ)^2*dφ^2)

Question 6

what are the three steps for solving constrained problems via lagrange problems

Answer 6

• multiply constraints G_i = 0 by lagrange multipliers λ and add these to the functional J that is to be minimised subject to the constraints, to form the functional I = J + Σλ_i*G_i
• take directional derivatives of I wrt lagrange multipliers and variables in the problem
• set each of the directional derivatives to 0 and solve to find the unknown functions

Question 7

what are the three steps for deriving the weak form of a differential equation

Answer 7

• multiply the differential equation by an arbitrary weight function
• integrate over the domain
• apply integration by parts if possible and insert Nuemann boundary conditions

Question 8

what is the dirichlet condition for the poisson equation

Answer 8

• u = 0 on dV
• dV refers to the subscript of the integral wrt dS

Question 9

what is the neumann condition for the poisson equation

Answer 9

• ∇u.n = h on the boundary
• if u or v = 0 at some point on ‘dV’ then h = 0

Question 10

what is the form of the trial function youre looking for an approximate solution for in numerical methods

Answer 10

• u_bar = Σ(i=1 to n) c_i*φ_i
• φ_i = pre-chosen basis function
• c_i and n = unknowns to be found

Question 11

if a differential equation cant be solved in closed form, what are the two techniques used to obtain approximate solutions

Answer 11

• rayleigh-ritz method
• galerkin method

Question 12

what are the steps for the rayleigh-ritz method

Answer 12

• define the functional I for which you wish to find stationary points
• choose a combination of linearly independent functions that will be used to approximate the solution (basis functions)
• the basis functions must satisfy the dirichlet condition (u=0 on S_g)
• insert the approximate solution into the functional that is now denoted by I_h
• take the directional derivative of I_h wrt the unknown amplitudes of the basis functions
• determine the amplitudes of the basis functions which yield a stationary point of I_h

Question 13

when does using the galerkin method work

Answer 13

• it works on the weak form of PDE which is restricted to the space of finite dimension
• its a methods of weighted residuals where weight function = basis function

Question 14

when does using the rayleigh-ritz method work

Answer 14

• it works on the variational form which is restricted to the space of finite dimension