Flapping wings present a challenging but likely rewarding approach for
achieving efficient flight at the scales of typical micro aerial vehicles
(MAVs). In an effort to use computational approaches to study the design of
efficient flapping wings, we have developed a multi-fidelity framework to
automatically generate energetically optimized wing kinematics. Traditional
engineering tools are combined with the high-fidelity DG solvers, which allow
us to determine the actual performance of each proposed design.

Two optimal wing designs are shown to the right. The first one (top) has no
cambering, which results in significant flow separation and large
deviations between the low-fidelity solvers and the Navier-Stokes results.
By introducting a dynamic cambering (bottom), the separation is almost
eliminated which leads to a better wing design.

Micromechanical resonators:

In wireless communication systems, there is a significant interest in
high-quality electromechanical resonators. These are very difficult to
simulate computationally, due to the low losses and the semi-infinite
domains. We have developed a time-domain high-order DG scheme that can
accurately model full-scale three-dimensional devices. The corresponding
resonant properties can be extracted using a filter diagonalization
process. The example shows a double-disk resonator, driven by a
Gaussian force applied radially on the left disk.

IMEX timestepping for LES problems:

In the numerical simulation of turbulent flows, the large
variations in grid size introduces stiffness to the systems.
This imposes unreasonable high restrictions on the timesteps
taken by explicit solvers. Using Implicit-Explicit Runge-Kutta
methods, we are able to use explicit solvers in more than 90%
of the domains, and implicit solvers only in the boundary layers.
In addition, a quasi-Newton solver integrates the implicit
equations highly efficiently at these relatively short, time-accurate
timesteps.

Unstructured Mesh Generation:

My DistMesh mesh generator is a widely used software for generation
of unstructured simplex meshes on geometries described by implicit
functions. The algorithm is simple and produces meshes of very high
quality. The implicit domains allow for applications such as moving
meshes, coupling with level set methods, and meshing of objects in
images and MRI scans. It is available as free software at the webpage below:

High-order methods require accurate curved unstructured meshes, which
are hard to generate. We have proposed a new approach based on a
non-linear elasticity analogy. By solving for an equilibrium
configuration, the method produces curved meshes automatically from
existing straight-sided meshes. The produced meshes are highly resistant
to element inversion, and the framework can also be used for generation
of deforming and stretched meshes.

Kelvin-Helmholtz Instability:

This example, inspired by Munz et at (2003), shows how a large
scale acoustic wave interacts with small scale flow features, leading
to vorticity generation. The problem is solved using a Discontinuous
Galerkin method with polynomials of degree 7, and it is a good example
of the importance of high-order discretizations in order to accurately
capture and propagate the acoustic waves. Also, due to the highly
nonlinear behaviour it is clear that simplifications based on linearized
Euler or the Lighthill Analogy would give incorrect results.

Volumetric Membrane Models:

By using a high-order discontinuous Galerkin formulation for
dynamic analysis of solids, highly anisotropic tetrahedral elements
can be used without reducing the accuracy or introducting locking.
This allows for modeling of thin structures such as membranes without
specialized models. The example shows a square membrane which is
clamped at one edge and subject to a gravitational force. The
aspect ratio of the elements is 1/200.