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My current research interests are in high-order accurate numerical methods, unstructured mesh generation, and applications in fluid and solid mechanics such as flapping flight, wind turbines, aeroacoustics, and micromechanical resonators. Some examples are shown below.

Flapping-wing micro-aerial vehicles:

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.

No camber (DivX, 8.3MB)
Dynamic camber (DivX, 7.4MB)

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.

Animation (DivX, 18MB)

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.

Animation (DivX, 11.4MB)

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:

Curved and Deformed Mesh Generation:

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.

Animation (DivX, 5.3MB)

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.

Animation (DivX, 24.0MB)

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.
Animation (DivX, 870kB)

Animation (DivX, 3.2MB)