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
                        
(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 "decagon" 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

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

Archive

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