mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

PhD thesis, chapter 5

 2020-02-16 
This is the fifth post in a series of posts about my PhD thesis.
In the fifth and final chapter of my thesis, we look at how boundary conditions can be weakly imposed on the Helmholtz equation.

Analysis

As in chapter 4, we must adapt the analysis of chapter 3 to apply to Helmholtz problems. The boundary operators for the Helmholtz equation satisfy less strong conditions than the operators for Laplace's equation (for Laplace's equation, the operators satisfy a condition called coercivity; for Helmholtz, the operators satisfy a weaker condition called Gårding's inequality), making proving results about Helmholtz problem harder.
After some work, we are able to prove an a priori error bound (with \(a=\tfrac32\) for the spaces we use):
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$

Numerical results

As in the previous chapters, we use Bempp to show that computations with this method match the theory.
The error of our approximate solutions of a Dirichlet (left) and mixed Dirichlet–Neumann problems in the exterior of a sphere with meshes with different values of \(h\). The dashed lines show order \(\tfrac32\) convergence.

Wave scattering

Boundary element methods are often used to solve Helmholtz wave scattering problems. These are problems in which a sound wave is travelling though a medium (eg the air), then hits an object: you want to know what the sound wave that scatters off the object looks like.
If there are multiple objects that the wave is scattering off, the boundary element method formulation can get quite complicated. When using weak imposition, the formulation is simpler: this one advantage of this method.
The following diagram shows a sound wave scattering off a mixure of sound-hard and sound-soft spheres. Sound-hard objects reflect sound well, while sound-soft objects absorb it well.
A sound wave scattering off a mixture of sound-hard (white) and sound-soft (black) spheres.
If you are trying to design something with particular properties—for example, a barrier that absorbs sound—you may want to solve lots of wave scattering problems on an object on some objects with various values taken for their reflective properties. This type of problem is often called an inverse problem.
For this type of problem, weakly imposing boundary conditions has advantages: the discretisation of the Calderón projector can be reused for each problem, and only the terms due to the weakly imposed boundary conditions need to be recalculated. This is an advantages as the boundary condition terms are much less expensive (ie they use much less time and memory) to calculate than the Calderón term that is reused.

This concludes chapter 5, the final chapter of my thesis. Why not celebrate reaching the end by cracking open the following figure before reading the concluding blog post.
An acoustic wave scattering off a sound-hard champagne bottle and a sound-soft cork.
Previous post in series
This is the fifth post in a series of posts about my PhD thesis.
Next post in series
×3      ×3      ×3      ×3      ×3
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li> <logo>
To prove you are not a spam bot, please type "theorem" in the box below (case sensitive):

Archive

Show me a random blog post
 2024 

Feb 2024

Zines, pt. 2

Jan 2024

Christmas (2023) is over
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

electromagnetic field interpolation datasaurus dozen matrix multiplication realhats rhombicuboctahedron pi curvature draughts inverse matrices fonts advent calendar simultaneous equations binary weak imposition mathslogicbot flexagons christmas pizza cutting world cup live stream error bars propositional calculus golden spiral reuleaux polygons matt parker estimation graph theory chalkdust magazine gather town numerical analysis matrix of cofactors phd bodmas logic trigonometry triangles turtles wool cross stitch statistics folding tube maps crochet harriss spiral edinburgh royal baby machine learning books hexapawn logs fractals latex recursion nine men's morris ternary london underground gaussian elimination numbers oeis puzzles stirling numbers polynomials ucl game show probability mathsjam noughts and crosses go pi approximation day christmas card exponential growth menace weather station folding paper gerry anderson boundary element methods squares misleading statistics finite group mean hats python map projections frobel correlation dates people maths preconditioning newcastle anscombe's quartet data visualisation standard deviation accuracy wave scattering tennis pythagoras guest posts programming sobolev spaces bubble bobble pac-man fence posts chebyshev stickers countdown captain scarlet sport databet a gamut of games big internet math-off braiding martin gardner cambridge palindromes youtube asteroids mathsteroids determinants hyperbolic surfaces errors golden ratio sound zines game of life manchester quadrilaterals computational complexity national lottery games dinosaurs talking maths in public chess pascal's triangle runge's phenomenon radio 4 raspberry pi probability matrices sorting php plastic ratio logo javascript coins approximation arithmetic geometry light data crossnumber news royal institution rugby matrix of minors hannah fry manchester science festival european cup finite element method dragon curves graphs reddit final fantasy speed video games inline code london bempp craft geogebra convergence the aperiodical 24 hour maths dataset football tmip platonic solids signorini conditions

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2024