While looking into higher-order PDEs, I came across the biharmonic.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Biharmonic operator

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