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.