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Transcript of PDEs
Airflow around an aircraft.
Structural dynamics of a bridge.
Movement of money in financial markets.
Heat conduction across the earth
where u(t,x,y,z) is temperature, K, , and are material properties and x, y, z and t are space locations and time. Temperature on a computer board
The generalised 2nd order linear PDE can be written:
where a, b, c, d, e, f, g are constants (some may be 0). Elliptic PDE:
Temperature, u(x,y) profile around two computer chips on a printed circuit board.
Q is the power source and K is the thermal conductivity
Elliptic PDEs represent phenomena that have already reached a steady state and are, hence, time independent.
2 spacial dimensions considered: x, y Parabolic PDEs describe time dependent phenomena, such as conduction of heat, that are evolving towards steady state.
One dimensional heat or diffusion equation One-dimensional heat diffusion along a pipe. Pipe is heated from one end For example, a continuously-vibrating undamped Violin or Guitar string.
Example – wave equation
(Parabolic & Hyperbolic) u(x,y) = c Mixed: n equations with n unknowns
- matrix system, banded, sparse Gaussian Elimination, etc.
Disadvantage: Need to store whole matrix. Slow, especially for large matrices.
Advantage: Robust even with ill-conditioned matrices. Jacobi, Gauss Seidel, etc.
Advantage: Good for large matrix systems. No need to store whole matrix. Fast, even for large matrices.
Disadvantage: Poor for ill-conditioned matrices. May converge only slowly. A = D+L+U (Diagonal + Lower triangle + Upper triangle) eg., use central difference (need to introduce artificial nodes outside the boundary, but note error)
so that Set up the parameters of the problem such as FLAG, h, k, g
NB when solving a Laplace equation it is better to set up Excel to solve Poisson and then just set g = 0
Set up the domain (size, x/y values)
Fill in the known (Dirichlet) boundary values
Engineering simulation – Multiphysics, structural mechanics, fluid dynamics, ...
Incorporates pre-processing (geometry creation, meshing), solver and post-processing modules in a graphical user interface.
Finite element modelling, finite volume packages for numerically solving problems. http://www.csc.fi/english/pages/elmer
Open source – free to download – but huge!
PDEs solved numerically by finite element method
Non-standard boundary conditions
CAD – MESHING – ANALYSIS - VISUALISATION
Bandwidth - Rate at which data can be transferred.
Latency - The time between a process being ready to transmit data and the time at which it is transmitted (often arises due to synchronisation issues).
Speed up - How fast a parallel simulation is compared to a serial one, i.e. serial runtime/parallel runtime.
Scalability - How large a system can be made before performance starts to drop off.
Parabolic Explicit & Implicit schemes Von-Neumann Stability Method
.... leads to
Explicit: r <= 0.5
Implicit: no restriction Integrated Circuit temperature example with COMSOL
http://www.uk.comsol.com/ The convection equation (or ‘one-way wave equation’) can be used to predict how a wave is propagated in a fluid at speed a
We assume a > 0
[If not, the wave will travel in the opposite direction and we will need to use a ‘downwind’ scheme.] Wave equation explicit scheme Convection equation
explicit & implicit schemes divergence theorem ‘relates’ what is happening in the interior of a domain to what is happening on the boundary east or west
north or south east or west
north or south Linear function
shape functions Uniform refinement - Local refinement