1. Abstracts of the 23rd UK Conference of the
Association for Computational Mechanics in Engineering
8 - 10 April 2015, Swansea University, Swansea
High fidelity methods for compressible flow problems on moving domains
Michael Weberstadt, Antonio J. Gil and Rub´en Sevilla
Zienkiewicz Centre for Computational Engineering, College of Engineering
Swansea University, Singleton Park, SA2 8PP, United Kingdom
{439409, a.j.gil, r.sevilla}@swansea.ac.uk
ABSTRACT
Stabilised finite element methods, such as the Streamlined/Upwind Petrov Galerkin, have been exten-
sively used in the solution of flow [1], and fluid-structure interaction problems [2]. In this paper, the
SUPG formulation is extended to higher degrees of approximation for moving domains and combined
with a second order temporal integrator, the generalised alpha-method [3]. The so called geometric con-
servation law [4], which arises from inexact integration of the Jacobian field in time for moving domains,
is satisfied by solving numerical involutions.
This formulation has some distinct advantages. 1) Geometric conservation law is decoupled from flow
variables, enhancing the e ciency of the scheme whilst maintaining second order accuracy in time for
static and dynamic domains. 2) Implicit temporal integration allows the flow solver to be weakly coupled
with a solid dynamics solver inside an iterative scheme for application to fluid structure interaction
problems, where highly non-linear e↵ects must be properly resolved between fluid and solid. 3) For a
given required level of accuracy, it can be shown that higher order degrees of spatial approximation and
second order temporal accuracy lead to reduced computational cost when simulating transient problems.
4) Advantages 1) and 3) lead to preservation of flow e↵ects such as vortices over large distances and time
periods for reduced computational e↵ort, ideal for the simulation of impinging wind gusts.
Test problems are presented to demonstrate the advantages of the outlined formulation. Solutions to the
Burgers’ equation are evaluated in 1D to ascertain the robustness and optimal convergence of the scheme.
The evaluation is then extended to the solution of the 2D Euler equations, where the performance is
evaluated. In both cases, evaluations are made for both static and dynamic domains.
References
[1] R. Sevilla, O. Hassan and K. Morgan, “An analysis of the performance of a high-order stabilised
finite element method for simulating compressible flows,” Computer Methods in Applied Mechanics
and Engineering, vol. 253, pp. 15–27, 2013.
[2] W. Dettmer and D. Peri´c, “A computational framework for fluid-structure interaction: Finite element
formulation and applications,” Computer Methods in Applied Mechanics and Engineering, vol. 195,
pp. 5754–5779, 2005.
[3] K. E. Jansen, C. H. Whiting and G. M. Hulbert, “A generalized-alpha method for integrating the
filtered Navier-Stokes equations with a stabilized finite element method,” Computer Methods in Ap-
plied Mechanics and Engineering, vol. 190, pp. 305–319, 1999.
[4] A. J. Gil, J. Bonet, J. Silla and O. Hassan, “A discrete geometric conservation law (DGCL) for a cell
vertex finite-volume algorithm on moving domains,” International Journal for Numerical Methods
in Biomedical Engineering, vol. 26, pp. 770–779, 2010.