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Vectors with random orientation

Posted Feb 19, 2016, 5:01 p.m. EST Parameters, Variables, & Functions 8 Replies

Youness Alvandi
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Hey guys,

I'm interested to generate random vectors for the initial condition of my 3D model which is basically for solving time-dependent PDEs. The problem is the components of these vectors are random but sum of squares (which is the magnitude of the vectors) should be constant. In other words, I need to generate vectors with same magnitude but with random orientations. So the question is if I can do this in COMSOL using built in random function or any other ways without going through the hassle of using MATLAB.
8 Replies Last Post Aug 16, 2016, 9:23 p.m. EDT
Youness Alvandi
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Posted: 8 years ago Feb 21, 2016, 10:53 a.m. EST
Hello? Can somebody help?
Hello? Can somebody help?
Youness Alvandi
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Posted: 8 years ago Feb 23, 2016, 9:29 a.m. EST
I guess I found how to do this. You need to generate three random numbers using the random function in COMSOL. Let's say these numbers are a, b, and c. Using the expression bellow you can generate components of unit vectors with random orientations. Multiply them with the magnitude of your desired vector and you get vectors with same magnitude but random orientations.

a'=a/sqrt(a^2+b^2+c^2)
b'=b/sqrt(a^2+b^2+c^2)
c'=c/sqrt(a^2+b^2+c^2)
I guess I found how to do this. You need to generate three random numbers using the random function in COMSOL. Let's say these numbers are a, b, and c. Using the expression bellow you can generate components of unit vectors with random orientations. Multiply them with the magnitude of your desired vector and you get vectors with same magnitude but random orientations. a'=a/sqrt(a^2+b^2+c^2) b'=b/sqrt(a^2+b^2+c^2) c'=c/sqrt(a^2+b^2+c^2)
Gunnar Andersson COMSOL Employee

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Posted: 8 years ago Feb 24, 2016, 2:12 a.m. EST
This is a decent approximation, but you don't get a uniform distribution over set of directions.
This is a decent approximation, but you don't get a uniform distribution over set of directions.
Youness Alvandi
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Posted: 8 years ago Mar 2, 2016, 3:42 p.m. EST

This is a decent approximation, but you don't get a uniform distribution over set of directions.


Do you know any alternatives?
[QUOTE] This is a decent approximation, but you don't get a uniform distribution over set of directions. [/QUOTE] Do you know any alternatives?
Gunnar Andersson COMSOL Employee

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Posted: 8 years ago Mar 3, 2016, 2:31 a.m. EST
Do you know any alternatives?


For a mathematical question such as yours I think that Google is a faster way to find an answer than this forum: Search for e.g. "random vector sphere".

[QUOTE]Do you know any alternatives?[/QUOTE] For a mathematical question such as yours I think that Google is a faster way to find an answer than this forum: Search for e.g. "random vector sphere".
CHEN
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Posted: 8 years ago Aug 16, 2016, 11:48 a.m. EDT
I'm having the same problem. So, let me give it a try.

Let's say the magnitude of the vector is equal to N and the direction of the vector varies with time.

For 2D simulation, I think one can create a random function (named rn1) with uniform distribution that ranges from -pi to pi (or 0 to 2*pi). By doing so, the x-compnent and y-compoent of the vector correspond to N*cos(rn1(t)) and N*sin(rn1(t)).

As for 3D simulation, I think it is just in analogy to spherical coordinate. If one creates two random functions both ranging from -pi to pi, then
x-component: N*sin(rn1(t))*cos(rn2(t))
y-component: N*sin(rn1(t))*sin(rn2(t))
z-component: N*cos(rn1(t))

Am I right about this?
I'm having the same problem. So, let me give it a try. Let's say the magnitude of the vector is equal to N and the direction of the vector varies with time. For 2D simulation, I think one can create a random function (named rn1) with uniform distribution that ranges from -pi to pi (or 0 to 2*pi). By doing so, the x-compnent and y-compoent of the vector correspond to N*cos(rn1(t)) and N*sin(rn1(t)). As for 3D simulation, I think it is just in analogy to spherical coordinate. If one creates two random functions both ranging from -pi to pi, then x-component: N*sin(rn1(t))*cos(rn2(t)) y-component: N*sin(rn1(t))*sin(rn2(t)) z-component: N*cos(rn1(t)) Am I right about this?
Edgar J. Kaiser Certified Consultant
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Posted: 8 years ago Aug 16, 2016, 4:47 p.m. EDT

No, that's not correct. Things are different on a sphere!

Being curious I followed Gunnar's advice and in less than a minute I found: mathworld.wolfram.com/SpherePointPicking.html

wolfram.com is frequently a very good resource for mathematics.

Cheers
Edgar

--
Edgar J. Kaiser
emPhys Physical Technology
www.emphys.com
No, that's not correct. Things are different on a sphere! Being curious I followed Gunnar's advice and in less than a minute I found: http://mathworld.wolfram.com/SpherePointPicking.html wolfram.com is frequently a very good resource for mathematics. Cheers Edgar -- Edgar J. Kaiser emPhys Physical Technology http://www.emphys.com
CHEN
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Posted: 8 years ago Aug 16, 2016, 9:23 p.m. EDT
Hi Edgar,

Thanks for your reply. I finally know how to solve problem correctly.
Hi Edgar, Thanks for your reply. I finally know how to solve problem correctly.

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