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Summer Projects Presentation

  • brief introduction to exterior calculus

    • a k-form is a thing that can be integrated over a k-manifold
    • in 3D 1-forms and 2-forms have three components at each point, and are often treated as vector fields.

  • how to discretize?

    • we want to keep the integrable structure
    • sampling the field at some points is most obvious, but it's unclear how to integrate
    • instead, put components on respective pieces of a cell complex. now integration is clear.
    • this can also be thought of as sampling by integrating over the piece, instead of just sampling a point.
    • this discretization preserves stokes theorem (give an example of the exterior derivative)
    • this can be thought of an approximation of the continuum where we take the limit as the grid gets fine. or, this can be thought of as an interesting discrete structure in its own right.
    • other EC operations like hodge duality and the laplacian can be formulated in DEC, but are less natural.

  • fourier methods

    • start by introducing the point charge potential on a grid problem.

      • trick: a function on Z^n is just a sum of delta functions, so we can use the normal fourier transform.
      • in fact, a discrete function is just something times the comb function, and a periodic function is something convolved with the comb, explaining why FT sends periodic to discrete and discrete to periodic.
      • we get this expression for the FT of the potential, which we can expand into a series, or, more usefully, integrate over one period to recover the values.

    • next example: div free flow in 2d with point source

      • can just take your favorite continuous streamfunction and sample and take exterior derivative. eg, theta
      • alternatively, you can impose a gauge condition, and then we get this system of eqns whose solution looks very similar to the first problem.

    • a more useful example: stokes

      • like before we get a small system of eqns in freq space, which we can invert
      • I got a computer to do this. Here's a graph of the speed compared with a standard linear solver method.
      • Since there are N^3 cells and factoring an mxm matrix generally takes O(m^3) time, the graph agrees with the theoretical O(N^9) growth for setting up QR, O(N^6) for solving with QR, and O(N^3 log N) for both parts of fourier.

  • and also: here's an electromegnetism simulation. It looks cool!