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[maggiegold]

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Question on Numerical methods- How do I use programming to solve discretion problems?

I am given this question- I'm stuck with using programming to solve discretion problems Question 1a) Express Tij @ t+1 = F(Ti+1,j, Ti-1,j, Ti,j+1, Ti,j-1) at t explicitly, then advance in time Question 1b) Suppose, there is no t variable, now what? Make a guess! Ideally: F(Ti,j, Ti+1,j , Ti-1,j...

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I am given this question- I'm stuck with using programming to solve discretion problems

For Question 1a), I am given that

The governing equation for the temperature distribution with time on a 2D square plate measuring 1 unit by1 unit is

∂T/∂t = ∂2T/∂x2 + ∂2T/∂y2 ,

subjected to the Dirichlet boundary conditions for T provided in Fig.1.

I am supposed to obtain the temperature contour plot on the square plate with time, say at t=0.01, 0.1 and at steady state.

For Question 1b), I am given that

∂2T/∂x2 + ∂2T/∂y2 = 0,


Question Requirement:

Question 1a) Express Tij @ t+1 = F(Ti+1,j, Ti-1,j, Ti,j+1, Ti,j-1 at t explicitly, then advance in time

Question 1b)

Suppose, there is no t variable, now what? Make a guess!

Ideally: F(Ti,j, Ti+1,j , Ti-1,j, Ti,j+1, Ti,j-1)=0

With guessed values: F(Ti,j, Ti+1,j , Ti-1,j, Ti,j+1, Ti,j-1)= err(/=0)

Objective is to reduce error as much as possible

Requirements

Write down discretized equations(interior, boundary)- explicit, implicit?

Specify what algorithms are used and how to program?

 

[Deleted member 4993]

I am given this question- I'm stuck with using programming to solve discretion problems

For Question 1a), I am given that

The governing equation for the temperature distribution with time on a 2D square plate measuring 1 unit by1 unit is

∂T/∂t = ∂2T/∂x2 + ∂2T/∂y2 ,

subjected to the Dirichlet boundary conditions for T provided in Fig.1.

I am supposed to obtain the temperature contour plot on the square plate with time, say at t=0.01, 0.1 and at steady state.

For Question 1b), I am given that

∂2T/∂x2 + ∂2T/∂y2 = 0,


Question Requirement:

Question 1a) Express Tij @ t+1 = F(Ti+1,j, Ti-1,j, Ti,j+1, Ti,j-1 at t explicitly, then advance in time

Question 1b)

Suppose, there is no t variable, now what? Make a guess!

Ideally: F(Ti,j, Ti+1,j , Ti-1,j, Ti,j+1, Ti,j-1)=0

With guessed values: F(Ti,j, Ti+1,j , Ti-1,j, Ti,j+1, Ti,j-1)= err(/=0)

Objective is to reduce error as much as possible

Requirements

Write down discretized equations(interior, boundary)- explicit, implicit?

Specify what algorithms are used and how to program?

View attachment 24899

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Please share your work/thoughts about this problem.

 

[maggiegold]

I am stuck at the first step. Don't know where to start. We are supposed to write the code to derive the temperature contour plot. I need some pointers to start.

 

[Deleted member 4993]

I am stuck at the first step. Don't know where to start. We are supposed to write the code to derive the temperature contour plot. I need some pointers to start.

Since you are stuck at the beginning and do not know where to begin, let us make sure - we have common understanding about definitions.

Please define for us - Dirichlet boundary conditions?

How is that mathematically expressed for the given problem?