mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

PhD thesis, chapter 4

 2020-02-13 
This is the fourth post in a series of posts about my PhD thesis.
The fourth chapter of my thesis looks at weakly imposing Signorini boundary conditions on the boundary element method.

Signorini conditions

Signorini boundary conditions are composed of the following three conditions on the boundary:
\begin{align*} u &\leqslant g\\ \frac{\partial u}{\partial n} &\leqslant \psi\\ \left(\frac{\partial u}{\partial n} -\psi\right)(u-g) &=0 \end{align*}
In these equations, \(u\) is the solution, \(\frac{\partial u}{\partial n}\) is the derivative of the solution in the direction normal to the boundary, and \(g\) and \(\psi\) are (known) functions.
These conditions model an object that is coming into contact with a surface: imagine an object that is being pushed upwards towards a surface. \(u\) is the height of the object at each point; \(\frac{\partial u}{\partial n}\) is the speed the object is moving upwards at each point; \(g\) is the height of the surface; and \(\psi\) is the speed at which the upwards force will cause the object to move. We now consider the meaning of each of the three conditions.
The first condition (\(u \leqslant g\) says the the height of the object is less than or equal to the height of the surface. Or in other words, no part of the object has been pushed through the surface. If you've ever picked up a real object, you will see that this is sensible.
The second condition (\(\frac{\partial u}{\partial n} \leqslant \psi\)) says that the speed of the object is less than or equal to the speed that the upwards force will cause. Or in other words, the is no extra hidden force that could cause the object to move faster.
The final condition (\(\left(\frac{\partial u}{\partial n} -\psi\right)(u-g)=0\)) says that either \(u=g\) or \(\frac{\partial u}{\partial n}=\psi\). Or in other words, either the object is touching the surface, or it is not and so it is travelling at the speed caused by the force.

Non-linear problems

It is possible to rewrite these conditions as the following, where \(c\) is any positive constant:
$$u-g=\left[u-g + c\left(\frac{\partial u}{\partial n}-\psi\right)\right]_-$$
The term \(\left[a\right]_-\) is equal to \(a\) if \(a\) is negative or 0 if \(a\) is positive (ie \(\left[a\right]_-=\min(0,a)\)). If you think about what this will be equal to if \(u=g\), \(u\lt g\), \(\frac{\partial u}{\partial n}=\psi\), and \(\frac{\partial u}{\partial n}\lt\psi\), you can convince yourself that it is equivalent to the three Signorini conditions.
Writing the condition like this is helpful, as this can easily be added into the matrix system due to the boundary element method to weakly impose it. There is, however, a complication: due to the \([\cdot]_-\) operator, the term we add on is non-linear and cannot be represented as a matrix. We therefore need to do a little more than simply use our favourite matrix solver to solve this problem.
To solve this non-linear system, we use an iterative approach: first make a guess at what the solution might be (eg guess that \(u=0\)). We then use this guess to calculate the value of the non-linear term, then solve the linear system obtained when the value is substituted in. This gives us a second guess of the solution: we can do follow the same approach to obtain a third guess. And a fourth. And a fifth. We continue until one of our guesses is very close to the following guess, and we have an approximation of the solution.

Analysis

After deriving formulations for weakly imposed Signorini conditions on the boundary element method, we proceed as we did in chapter 3 and analyse the method.
The analysis in chapter 4 differs from that in chapter 3 as the system in this chapter is non-linear. The final result, however, closely resembles the results in chapter 3: we obtain and a priori error bounds:
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$
As in chapter 3, The value of the constant \(a\) for the spaces we use is \(\tfrac32\).

Numerical results

As in chapter 3, we used Bempp to run some numerical experiments to demonstrate the performance of this method.
The error of our approximate solutions of a Signorini problem on the interior of a sphere with meshes with different values of \(h\) for two choices of combinations of discrete spaces. The dashed lines show order \(\tfrac32\) convergence.
These results are for two different combinations of the discrete spaces we looked at in chapter 2. The plot on the left shows the expected order \(\tfrac32\). The plot on the right, however, shows a lower convergence than our a priori error bound predicted. This is due to the matrices obtained when using this combination of spaces being ill-conditioned, and so our solver struggles to find a good solution. This ill-conditioning is worse for smaller values of \(h\), which is why the plot starts at around the expected order of convergence, but then the convergence tails off.

These results conclude chapter 4 of my thesis. Why not take a break and snack on the following figure before reading on.
The solution of a mixed Dirichlet–Signorini problem on the interior of an apple, solved using weakly imposed boundary conditions.
Previous post in series
This is the fourth 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 "elbatnuocnu" backwards 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

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

Archive

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