Analytical solution of laplace equation 2D

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JITHA K R 2017-11-25

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评论： JITHA K R 2017-11-25

How to find analytical solution of laplace equation in 2D using matlab?

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Birdman 2017-11-25

Can you share the equation?

JITHA K R 2017-11-25

(∂^2 ϕ)/(∂x^2 )+(∂^2 ϕ)/(∂y^2 )=0

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Answer 1

John D'Errico 2017-11-25

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编辑：John D'Errico 2017-11-25

Unless you explain more clearly what you are doing, this is not even a MATLAB question. Really, this feels to me like a question posed by someone who has not done the necessary research (AND THINKING) about what they were tasked with doing. Odds are, your teacher told you this would make a good project, waving their hands, as they said something vague about using Laplace's equation.

Laplace's equation can be used as a mathematical model (or part of a model) for MANY things. Heat flow, diffusion, elastic deformation, etc. In this case, you want to use it for diffusion. But Laplace is not really sufficient. Thus diffusion is a process that happens over time. And a dam is a 3-dimensional thing. Diffusion will operate through the thickness of the dam, but it will vary across the face of the dam, thus the length, width, and thickness of the dam. So realistically, this is a three dimensional problem, that also includes TIME! So the 2-d Laplacian is insufficient. And if you need a time term in there, then you would be looking at something closer to Fourier's law for conduction.

However, mathematics is all about how to reduce a problem that is mathematically intractable. That especially applies to mathematical modeling. So I'll give you a few hints.

First, assume the medium of the dam is homogeneously porous with respect to water seepage. It won't really be so. And a high quality model would probably not assume that to be true. In fact, I might bet that most seepage will probably occur through randomly located micro-cracks. Very tough to model that process though. So homogeneous, isotropic flow makes some sense as an approximation.

Diffusion across the width of a dam will probably not vary significantly except near the edges (unless there is a crack, in which case, get the hell out of there!) So you MIGHT ignore horizontal variation across the width. A high quality model would not do so. But a simple approximation that will let you solve the problem in a finite amount of time would have you assume that derivatives in that direction are essentially zero.

Next, diffusion will show significant differences from top to bottom, since the water pressure will vary by a huge amount from top to bottom. So you cannot ignore that facet.

Finally, diffusion through the thickness is what you are trying to model! So you cannot ignore that dimension.

Next, there is the time term. If you assume that the dam has been standing for a long time with water behind it, this can be viewed as essentially a steady state problem. So we might then assume all time derivatives are zero. That leaves a two dimensional problem.

Thus seepage through a dam could be viewed as a diffusion problem of water through a concrete medium. There are MANY approximations I pointed out above to arrive at this point, not all of which are equally valid.

Laplace is a simple PDE though. The sum of the second partials equals zero. While there would be a multiplicative term out front that would normally important, you set it equal to ZERO. That comes from the assumed steady state nature of the problem. So you don't care how long it takes to get to a steady state. Only that it is there.

There is a problem in this assumption though. The water level in a dam will vary over time. And if you change the water level, then you SIGNIFICANTLY change how the water will seep through! So if it takes more than a few weeks for water to seep through the thickness of the dam, then you cannot realistically drop out that time derivative, as I suggested you might.

Finally, what matters are the boundary conditions. This is a PDE problem, arguably one that may never get too close to a steady state because of continuously changing water levels. But what drives the PDE are the boundary conditions. You can have no solution without defining them. And since I've already written the first chapter of a book on this subject, you need to talk to your instructor. You need to resolve and understand these questions as I have posed them, long before you ever think about modeling this system in MATLAB.

Were you to build a complete model of a dam in three dimensions, incorporating inhomogeneous media with potentially anisotropic flow, you might even get a doctoral thesis out the end. Or a student project, depending on how many simplifying assumptions you throw in there. :)