Biharmonic Operator: Understanding PDEs for Smooth Meshes

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In summary, the biharmonic equation is a higher-order partial differential equation that involves mixed axes terms. It is used instead of just the fourth-order derivatives because it has the nice property of being "invariant under rigid motions." This makes it useful for obtaining smoothness in meshes.
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shalayka
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While looking into higher-order PDEs, I came across the biharmonic.

Where the biharmonic equation is:

[tex]\left(\frac{\partial^2}{\partial {x}^2} + \frac{\partial^2}{\partial {y}^2} + \frac{\partial^2}{\partial {z}^2}\right)\left(\frac{\partial^2}{\partial {x}^2} + \frac{\partial^2}{\partial {y}^2} + \frac{\partial^2}{\partial {z}^2}\right).[/tex]

Using basic algebra for the multiplication, this works out to include a bunch of terms involving mixed axes:

[tex]\frac{\partial^4}{\partial {x}^4} + \frac{\partial^4}{\partial {y}^4} + \frac{\partial^4}{\partial {z}^4} + \frac{\partial^4}{\partial {x}^2 \partial{y}^2} + \frac{\partial^4}{\partial {x}^2 \partial{y}^2} + \frac{\partial^4}{\partial {y}^2 \partial{z}^2} + \frac{\partial^4}{\partial {y}^2 \partial{z}^2} + \frac{\partial^4}{\partial {x}^2 \partial{z}^2} + \frac{\partial^4}{\partial {x}^2 \partial{z}^2}.[/tex]

Why would one use this instead of:

[tex]\frac{\partial^4}{\partial {x}^4} + \frac{\partial^4}{\partial {y}^4} + \frac{\partial^4}{\partial {z}^4}?[/tex]

Thanks for any help on clarification.

I've found this presentation which shows how the smoothness of meshes is obtained using the biharmonic equation:
http://www.math.bas.bg/or/NATO_ARW/presentations/Ugail.ppt
 
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(I was sorely tempted to say "For the same reason we would use (x+ y)2 instead of x2+ y2, but I will behave!)

Well, one would use one instead of the other because the are different!

In particular, the "harmonic" operator, [itex]\nabla^2[/itex] is [itex]\partial^2/\partial x^2+ \partial^2/\partial y^2+ \partial^2/\partial z^2[/itex] has the nice property that it is "invariant under rigid motions" and, therefore, so is [itex]\nabla^2(\nabla^2 )[/itex] is also "invariant under rigid motions". The second formula you give is not.
 

FAQ: Biharmonic Operator: Understanding PDEs for Smooth Meshes

What is a Biharmonic Operator?

A Biharmonic Operator is a type of differential operator that acts on a function to produce a new function. It is commonly used in partial differential equations (PDEs) to describe the behavior of smooth meshes.

How does the Biharmonic Operator relate to PDEs?

The Biharmonic Operator is a key component in PDEs that involve smooth meshes. It helps to describe the behavior of the mesh by taking into account its curvature and the distribution of forces acting on it.

What are the applications of the Biharmonic Operator?

The Biharmonic Operator has many applications in various fields such as fluid dynamics, structural mechanics, and image processing. It is commonly used to model the behavior of elastic materials and to smooth out noisy images.

Can you explain the concept of "smooth meshes" in relation to the Biharmonic Operator?

Smooth meshes are mathematical representations of physical objects that have a continuous and differentiable surface. The Biharmonic Operator is used to describe the behavior of these meshes, taking into account their curvature and smoothness.

Are there any limitations or challenges associated with using the Biharmonic Operator?

One challenge with using the Biharmonic Operator is that it can be computationally intensive, especially for complex meshes. Additionally, it may not be suitable for describing the behavior of non-smooth or irregular meshes.

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