Mimetic isogeometric modeling and discretization of compressible Euler flows

Mimetic isogeometric modeling and discretization of compressible Euler flows

van Schie, Sebastiaan (TU Delft Aerospace Engineering; TU Delft Electrical Engineering, Mathematics and Computer Science)

Gerritsma, M.I. (mentor)
Möller, M. (mentor)
Toshniwal, D. (graduation committee)

Delft University of Technology

Aerospace Engineering

2021-09-06

This thesis poses a new geometric formulation for compressible Euler flows. A partial decomposition of this model into Roe variables is applied; this turns mass density, momentum and kinetic energy into product quantities of the Roe variables. Lie derivative advection operators of weak forms constructed with this decomposed model naturally follow to be self-adjoint, which results in skew-symmetric discrete advection operators in any number of dimensions. Under certain conditions these conserve products of the Roe variables, leading to a discrete model formulation with advection operators that simultaneously conserve mass, momentum, kinetic energy, internal energy and total energy in compressible Euler flows. While this idea is not new the novelty of this work lies in its extension to mimetic finite element methods and its application to discontinuous compressible Euler flows.The regular geometric Euler model and its Roe variable decomposition have been discretized through mimetic isogeometric analysis. At the core of mimetic discretization methods lies the idea of retaining the De Rham sequence of differential form spaces when projecting these to finite-dimensional approximations and when constructing discrete operators. B-spline differential form spaces have been defined such that the exterior derivative maps in a topologically exact and metric-free way, while the interior product has been discretized in a weak way in order to retain its map between appropriate spaces in the De Rham sequence. Only primal grids are used; the Hodge star operator is discretized through the definition of an L2 inner product to resolve weak forms. Cartan’s homotopy formula allows for a consistent discretization of the Lie derivative through compositions of the interior product and exterior derivative.Several tests were carried out to determine the efficacy and behavior of this regular geometric Euler model, its Roe variable decomposition and the resulting discrete advection operators. Testing one-dimensional linear advection and Burgers’ equation on periodic domains shows that discretizations of the self-adjoint advection operators are consistent with conservative formulations. Sod’s shock tube is used as one-dimensional discontinuous compressible flow test. Both the regular Euler model and its Roe variable decomposition display strong oscillatory tendencies, yet solution convergence is obtained without any issues. While application of a simple moving average-filter removes the worst oscillations more sophisticated methods are necessary for obtaining solutions that are free of unphysical oscillations. The Roe variable decomposition negatively affects the accuracy of shock speed predictions. The presence of strong oscillations likely affects the numerical conservation errors of both methods. Compared to finite volume package Clawpack and the nodal Discontinuous Galerkin (DG) method of Hesthaven & Warburton both models have less numerical diffusion on coarse meshes with the regular model outperforming the Roe variable decomposition. Convergence of momentum conservation error is slow and the errors are large compared to the reference methods. For two-dimensional periodic vortices both the regular geometric Euler model and its Roe variable decomposition outperform both reference methods for stationary and moving vortices. Clawpack displays diffusive behavior, resulting in large L2 errors and high amounts of numerical diffusion. While the L2 errors of the DG method are comparable to those of the two models developed in this work for all basis function orders considered, the resulting DG discretization requires significantly more degrees of freedom to attain this. To obtain similar levels of numerical diffusion the DG method needs up to ten times as many degrees of freedom as the methods presented in the current research.

mimetic discretization
Roe variables
Compressible Flow
Euler equations
Advection
Burgers' equation
Shockwave
Isentropic vortex
isogeometric analysis

http://resolver.tudelft.nl/uuid:57bc1cec-61b4-4431-ab6f-476bc723d916

Double degree in Aerospace Engineering and Applied Mathematics

Student theses

master thesis

© 2021 Sebastiaan van Schie