Structure preserving discretization for CFD
Structure-preserving numerical methods aim to preserve physical conservation laws exactly in a discrete setting. Examples are: exact conservation of mass, conservation of kinetic energy and helicity in time. In order to establish this one needs a formulation (Lagrangians, Hamiltonians) which includes the conservation properties and appropriate finite dimensional sub-spaces in which the conserved quantities can be represented.
Spatial discretization
- Mimetic spectral element methods: Higher order methods which satisfy a discrete De Rham sequence. This allows for the conservation of many physical properties at the discrete level, see https://arxiv.org/abs/1604.00257 or https://arxiv.org/abs/1707.00346
- The use of algebraic dual polynomial spaces: In finite element methods one looks for the (optimal) solution in a linear vector space. Associated to linear vector spaces are their algebraic dual spaces. The combination of primal and dual spaces leads to:
- very sparse spectral element methods;
- parts of the system matrix which are independent of the size and shape of the mesh;
- low condition number of the resulting system
For more information see: https://arxiv.org/abs/1712.09472
![](https://filelist.tudelft.nl/LR/Organisatie/Afdelingen/Aerodynamics__Wind_Energy__Flight_Performance_and_Propulsion/Aerodynamics/Research/02_Struc.jpg)
- Hybrid finite element methods: Hybrid finite element methods are based on domain decomposition techniques. These methods were proposed midway the 1960’s. By solving variables in a suitably chosen trace space, the method allows for full parallelization of the solution in the sub-domains. Furthermore, hybrid methods admit a larger class of admissible boundary conditions. In solid mechanics, for instance, both the displacements and the forces at the boundary are available. Such methods will play a role in multi-physics applications such as fluid-structure interaction or conjugate heat transfer problems.
![](https://filelist.tudelft.nl/LR/Organisatie/Afdelingen/Aerodynamics__Wind_Energy__Flight_Performance_and_Propulsion/Aerodynamics/Research/01_Struc.jpg)
In order to conserve certain physical quantities in time special time stepping procedures need to be developed. In the Aerodynamics group research in time-stepping techniques entails:
- Investigation of the application of Spectral Deferred Corrections (SDC), Integral Deferred Corrections (IDC) and the Picard Integral Exponential Solver (PIES) as a means to perform time integration for unsteady flow and fluid-structure interaction problems. By dividing a time interval into a number of sub-steps, an arbitrary order of accuracy can be achieved by performing a number of sweeps with a low order method over the time interval. Each sweep results in a correction to the solution. This allows the use of parallel-in-time discretization that can potentially speed up unsteady simulations on cluster computers. A second benefit is the possibility to reconstruct the solution at any point in the interval, which can be beneficial when coupling to another solver that uses a more refined time discretization, as one could encounter in fluid-structure interaction or fluid-acoustic coupling.
- Dual time integration methods to ensure conservation in time. Staggered time stepping schemes, https://arxiv.org/abs/1604.00257, energy-preserving time-stepping and symplectic time-stepping methods, https://arxiv.org/abs/1505.03422
![](https://filelist.tudelft.nl/LR/Organisatie/Afdelingen/Aerodynamics__Wind_Energy__Flight_Performance_and_Propulsion/Aerodynamics/Research/time_stepping.png)
For more information:
![](https://filelist.tudelft.nl/_processed_/6/8/csm_M_I_Gerritsma_58bd681917.jpg)
Dr.ir. M.I. (Marc) Gerritsma
![](https://filelist.tudelft.nl/_processed_/6/9/csm_A_H_vanZuijlen_c466b50e5c.jpg)