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July 14, 2005, 09:39 
Some discussions about Newton's method

#1 
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Hello,
Does anyone know how to implement the boundary conditions in Newton's method to solve NS equations? In particular: (1) How to do function evaluation on boundaries for the continuity equation, where the pressure is not seen? (2) How to do function evaluation on boundaries for the momentum equations where velocities and pressure all appears? (3) What is the major difference between interior grids and boundaries on function evaluation? I really appreciate if someone can answer this. Thanks in advance. Winston 

July 14, 2005, 11:41 
Re: Some discussions about Newton's method

#2 
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What I currently do is to implement the boundaries in a semiimplicit sense. Since I am solving the equations in a delta form, the boundary changes are held to zero during each iteration of the Newton loop (equivalent to an explicit update). Then the boundary conditions are applied at the end of each Newton iteration. If the BC update occurred at the end of the overall Newton loop, the update would be fully explicit. By applying the BC update at the end of each Newton iteration some "implicitness" is regained.


July 15, 2005, 09:14 
Re: Some discussions about Newton's method

#3 
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Thanks AG. In my case, I am doing fullyimplicit. My problem is that I am not sure how to do function evaluation on boundaries. For continuity equation, we have u and v, for each of the momentum equations, we have u, v, and p. When we do the function evaluation on boundaries, i.e., computing F(u,v,p) on boundaries, should we plug in those boundary conditions into each of continuity equation and momentum equations to get Fs, or there are some particular techniques? Thanks


July 15, 2005, 09:37 
Re: Some discussions about Newton's method

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I use the previous values of p, u, and v on the boundaries for a given Newton iteration. Then I call my BC subroutine after that Newton iteration and update the boundaries with the most recent interior results. Doing this provides a "semiimplicitness" to the boundary updates, and makes handling the boundaries easy within the Newton loop. For each Newton iteration the boundaries don't change until after the iteration is performed.
So  Begin the newton loop compute RHS compute LHS modify LHS matrix assuming all dq's on boundaries are zero during interior update. Solve for interior dq's and update interior q's. Call BC routine to update boundary values. if newton not converged repeat. End Newton loop I've never implemented a scheme where it's fully implicit because I've never seen a case where it was worth the effort. 

July 15, 2005, 12:35 
Re: Some discussions about Newton's method

#5 
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Thanks. In your case, your boundary subroutine has been written, in my case, however, I try to write a boundary subroutine and implement boundary conditions there. So, I need the detailed boundary calculation.
By the way, I compute numerical Jacobian, so I need to do function evaluation F(u,v,p) on the boundary. My real question is how to compute F(continuity), F(umomentum), and F(vmomentum) on boundaries given the boundary conditions of u, v, and p. If you could check your boundary subroutine and give me some hint, that would be great. Thanks Winston 

July 15, 2005, 13:21 
Re: Some discussions about Newton's method

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The boundary conditions I use are based on physical approximations/conditions at the boundary. For example, on a nonmoving solid surface the normal pressure gradient is zero and the velocity is either tangential to the surface (inviscid flow) or zero (viscous flow). So I never explicitly write out the equations of motion on the boundary. But it sounds like you want to write the equations for the fluid motion for the boundary points. Just use ghost points (which get values assigned by extrapolation at each time step) and then simply apply the discrete equations to the boundary. You'll still need to invoke any additional physical constraints, such as the noslip condition, within those equations. For example, the noslip condition implies that velocity derivatives along the surface are all zero, so those terms should be zeroed out in the discrete equations for points on a noslip wall.


July 16, 2005, 21:19 
Re: Some discussions about Newton's method

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Thanks AG. You are absolutely right. I use numerical Jacobian, and I do function evaluation on every grid points, that is why I need to evaluate the continuity and momentum equations on the boundary. But to my dispointment, I cannot get it work.
If you also use Newton's method, how do you compute your Jacobian matrix? Numerically or Analytically? Thanks. Winston 

July 17, 2005, 22:05 
Re: Some discussions about Newton's method

#8 
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Analytically. I don't have any real experience using numerical Jacobians.


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