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 "prime" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

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

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

Archive

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