mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 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
PhD thesis, chapter 4
This is the fifth post in a series of posts about my PhD thesis.
Next post in series
PhD thesis, chapter ∞

Similar posts

PhD thesis, chapter 4
PhD thesis, chapter 3
PhD thesis, chapter ∞
PhD thesis, chapter 2

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>
To prove you are not a spam bot, please type "zero" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

May 2021

Close encounters of the second kind

Jan 2021

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

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021