10.1.1 Applications and Extensions

The most efficient speed for aircraft flight is just below the speed of sound: the transonic regime. Simulations of flight at these speeds consume large quantities of computer time, and are a natural candidate for a DIME application. In addition to the complex geometries of airfoils and turbines for which these simulations are required, the flow tends to develop singular regions or shocks in places that cannot be predicted in advance; the adaptive refinement  capability of a DIME mesh allows the mesh to be fine and detail resolved near shocks while keeping the regions of smooth flow coarsely meshed for economy (Section 12.3).

The version of DIME developed within CP was only able to mesh two-dimensional manifolds. More recent developments are described in Section 10.1.7. The manifold may, however, be embedded in a higher-dimensional space. In collaboration with the Biology  division at Caltech, we have simulated the electrosensory system of the weakly electric fish Apteronotus leptorhynchus. The simulation involves creating a mesh covering the skin of the fish, and using the boundary element method to calculate field strengths in the three-dimensional space surrounding the fish (Section 12.2).

In the same vein of embedding the mesh in higher dimensions, we have simulated a bosonic string of high-energy physics, embedding the mesh in up to 26 spatial dimensions. The problem here is to integrate over not only all positions of the mesh nodes, but also over all triangulations of the mesh (Section 7.2).

The information available to a DIME application is certain data stored in the elements and nodes of the mesh. When doing finite-element  calculations, one would like a somewhat higher level of abstraction, which is to refer to functions defined on a domain, with certain smoothness constraints and boundary conditions. We have made a further software layer on top of DIME to facilitate this: DIMEFEM. With this we may add, multiply, differentiate and integrate functions defined in terms of the Lagrangian finite-element family, and define linear, bilinear, and nonlinear operators acting on these functions. When a bilinear operator is defined, a variational principle may be solved by conjugate-gradient  methods. The preconditioner  for the CG method may in itself involve solving a variational principle. The DIMEFEM package has been applied to a sophisticated incompressible flow algorithm (Section 10.2).