Problem statement. The pressure distribution on a thin rectangular membrane is expressed by the Laplace Equation as follows: =0 Using the second order discretization method, the above PDE can be...

Problem statement.<br>The pressure distribution on a thin rectangular membrane is<br>expressed by the Laplace Equation as follows:<br>=0<br>Using the second order discretization method, the above PDE can<br>be approximated by its discretized counterpart as:<br>Pi+1j + Pi-1.j + Pi,j+1 + Pij-1 –4P.j = 0<br>+ Pi.j+l<br>Where I and j are the discretization indices in the x and y<br>directions, respectively.<br>

Extracted text: Problem statement. The pressure distribution on a thin rectangular membrane is expressed by the Laplace Equation as follows: =0 Using the second order discretization method, the above PDE can be approximated by its discretized counterpart as: Pi+1j + Pi-1.j + Pi,j+1 + Pij-1 –4P.j = 0 + Pi.j+l Where I and j are the discretization indices in the x and y directions, respectively.

The mesh sizes are given by:<br>Ar = Ay =<br>3<br>The boundary conditions are:<br>p(0, y)= 0; p(2.y)= y(2- y), 0<y<2<br>%3D<br>p(x,0)= 0; p(x,2)=<br>[x,<br>0 <x<1<br>%3D<br>(2-x, 1<x<2<br>1) Why is it that the membrane is assumed to be thin?<br>2) Draw the discretized domain of the rectangular membrane in<br>the x-y coordinate system<br>3) Using the boundary conditions, compute the boundary<br>values at all sides.<br>4) Set up the matrix for the discretized values<br>5) Compute the discretized pressure values using MATLAB or<br>any other software.<br>

Extracted text: The mesh sizes are given by: Ar = Ay = 3 The boundary conditions are: p(0, y)= 0; p(2.y)= y(2- y), 0<><2 %3d="" p(x,0)="0;" p(x,2)="[x," 0=""><><1 %3d="" (2-x,=""><><2 1)="" why="" is="" it="" that="" the="" membrane="" is="" assumed="" to="" be="" thin?="" 2)="" draw="" the="" discretized="" domain="" of="" the="" rectangular="" membrane="" in="" the="" x-y="" coordinate="" system="" 3)="" using="" the="" boundary="" conditions,="" compute="" the="" boundary="" values="" at="" all="" sides.="" 4)="" set="" up="" the="" matrix="" for="" the="" discretized="" values="" 5)="" compute="" the="" discretized="" pressure="" values="" using="" matlab="" or="" any="" other="">

Jun 04, 2022

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Problem statement. The pressure distribution on a thin rectangular membrane is expressed by the Laplace Equation as follows: =0 Using the second order discretization method, the above PDE can be...

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