Green's Function for Poisson Equation w/ Mixed BCs

In summary, the conversation discusses the difficulty in calculating the Green's function for a 3D Poisson solver with specific boundary conditions on a rectangular box. The person seeking help is advised to consider a simpler 2D case, use separation of variables, consider symmetry, and utilize the method of images to solve the problem. They are also encouraged to seek help from a more experienced individual if needed.
  • #1
duranta23
3
0
Hello

I am trying to build a 3D Poisson solver using method of moments. I need to find out the Green's function for the system. My system is a rectangular box and boundary conditions are as follows:

On all surfaces BC is neumann.
Only on the upper and lower surface, the middle 1/3 region has dirichlet BC. the other two 1/3 regions on both side of it are also neumann.

I am really stuck on how to calculate the Green's function. The equation for Green's function is so simple (Del)2G = delta(r-r')

But I can't incorporate the BCs. Please help!
 
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  • #2




Thank you for reaching out for help with your 3D Poisson solver project. The Green's function is a fundamental tool in solving boundary value problems, and incorporating boundary conditions can be a bit tricky. Here are a few tips that may help you in finding the Green's function for your system:

1. Start with a simpler 2D case: Since your system is a rectangular box, it may be helpful to first consider a 2D version of your problem. This will allow you to better understand the boundary conditions and their effects on the Green's function.

2. Use separation of variables: In many cases, the Green's function can be expressed as a product of two simpler functions. For example, in your case, the Green's function may be expressed as G(x,y,z;x',y',z') = X(x,x')Y(y,y')Z(z,z'), where X, Y, and Z are functions that satisfy the boundary conditions on each of the three surfaces.

3. Consider symmetry: Depending on the symmetry of your system, the Green's function may also exhibit certain symmetries. For example, if your system is symmetric about the x-y plane, then the Green's function should also exhibit this symmetry.

4. Use the method of images: This method involves introducing virtual charges or sources to satisfy the boundary conditions. For example, in your case, you can introduce a virtual charge on the upper surface to satisfy the Dirichlet boundary condition in the middle 1/3 region, and then use the method of moments to solve for the Green's function.

I hope these tips help you in finding the Green's function for your system. If you are still having trouble, I recommend consulting with a colleague or mentor who has experience in solving boundary value problems. Best of luck with your project!
 

What is the Poisson equation and what are mixed boundary conditions?

The Poisson equation is a partial differential equation that describes the distribution of a scalar field in a given region. It is commonly used in physics and engineering to model phenomena such as heat transfer, electrostatics, and fluid flow. Mixed boundary conditions refer to a combination of different types of boundary conditions, such as Dirichlet and Neumann, being applied to different parts of the boundary of a region.

What is the Green's function for the Poisson equation with mixed boundary conditions?

The Green's function for the Poisson equation with mixed boundary conditions is a mathematical function that satisfies the Poisson equation and the given boundary conditions. It is used to solve the Poisson equation in a specific region by representing the solution as a linear combination of the Green's function and the boundary conditions.

Why is the Green's function useful in solving the Poisson equation with mixed boundary conditions?

The Green's function allows for a systematic and efficient way to solve the Poisson equation with mixed boundary conditions. It provides a general solution that can be applied to a variety of boundary conditions without needing to solve the equation each time for a specific set of conditions. This saves time and effort in finding solutions to complex problems.

What are some applications of the Green's function for the Poisson equation with mixed boundary conditions?

The Green's function for the Poisson equation with mixed boundary conditions has many applications in physics and engineering, such as in electrostatics, heat transfer, and fluid mechanics. It is also useful in solving boundary value problems in quantum mechanics and in finding the response of a system to an external force or source.

What are the limitations of using the Green's function for the Poisson equation with mixed boundary conditions?

The Green's function method may not always be applicable, as it relies on the linearity of the Poisson equation and the boundary conditions. It may also be challenging to find the Green's function for complex boundary conditions. In addition, the solution obtained using the Green's function may not always be physically meaningful or unique, and may require additional analysis or adjustments.

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