08:30
Computational Fluid and Solid Mechanics: Computational Fluid Dynamics (Part 1)
Chair: Alf Gerisch
08:30
25 mins
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Adaptive time stepping methods within a data assimilation framework applied to non-isothermal flow dynamic
Ferdinand Uilhoorn
Abstract: Numerical modeling of unsteady compressible viscous flows in pipelines plays an important role in engineering and scientific problems. These models are described by a nonhomogeneous hyperbolic system of partial differential equations based on conservation laws for continuity, momentum and energy. The numerical approximation is often done by using the method of lines, which leads to a large sparse system of ordinary differential equations. Efficient time integration of these semi-discretized equations is utmost important here because, in the prediction step, all model states or particles are passed through the numerical model to generate a set of prior samples. As a result, the large sparse set of ordinary differential equations with additive noise is sampled many times. The aim of this study is to seek a time integration scheme that is efficient and robust in doing multiple model realizations within the particle filter framework. Since it is a priori unknown if the problem can be called stiff and no accepted definition of stiffness in a mathematically rigorous manner exist, both stiffness criteria and efficiency of nonstiff solvers were examined. The variable step-size solvers that were implemented are embedded explicit Runge-Kutta schemes, a second order diagonally implicit Runge-Kutta scheme, a Krylov-Rosenbrock method and an implicit Runge-Kutta scheme with coefficients from Radau IIA quadrature formulas. Besides evaluating the performance of the time-stepping schemes for multiple model realizations in the particle filter, the analysis was also done for a
model integration without the presence of noise. For the test problem, a real pipeline configuration was selected, namely, the natural gas pipeline between the UK and continental Europe. Results showed that the explicit Runge-Kutta schemes performed better in terms of efficiency in case the grid density and tolerance are properly selected. The diagonally implicit Runge-Kutta scheme appeared to be most robust if the tolerance was not set too tight, otherwise, it will result in high computation times. This can be resolved by exploiting the sparsity pattern of the system of ordinary differential equations. Nonphysical oscillations might appear in certain situations. Based on the stiffness measures and superior efficiency of the explicit Runge-Kutta schemes the problem can be classified as nonstiff.
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08:55
25 mins
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A hybrid high-order method for flow simulations in discrete fracture network
Florent Hédin, Géraldine Pichot, Alexandre Ern
Abstract: In fractured rocks, fluid flows mostly within a complex arrangement of fractures, classically
modeled as a Discrete Fracture Network (DFN) [1]. In this model, the fractures are
distributed in a three-dimensional domain and are modeled as ellipses whose position and
orientation are given by statistical laws given by geological studies. For large size production
networks provided by external industrial partners, the network under study may contain
millions of fractures generating millions of intersections. Solving flow within
these large 3D DFNs requires robust and efficient numerical methods and software.
So far, the hydraulic head and velocity fields have been computed using the Mixed-Hybrid
Finite Element (MHFEM) method, thanks to the software NEF-Flow [2]. As the flow in
fractured rocks is highly chanelled, a reduction of the computational cost is achieved by
meshing the fractures independently, the fractures with the fine meshes are the ones that
carry most of the flow. It yields non matching meshes at the intersections between fractures
[3]. To further gain in flexibility in the mesh generation, in the management of non conforming
meshes and in the number of degrees of freedom, we propose to solve the same problem
with a HHO (Hybrid High Order) method. HHO is a Hybridizable Discontinuous Galerkin
Method (HGM)[4], allowing general meshes (including polytopal cells and nonmatching interfaces)
and arbitrary polynomial orders.
A HHO method for DFNs has been implemented in the C++ software library HHO-DFN.
HHO capabilities are provided by the Disk++ library [5], the meshes are produced by the
BLSURF_FRAC software developed by the Inria GAMMA3 team [6]. Tests case of increasing
complexity will be shown to compare the results obtained with the two software, NEF-Flow
and HHO-DFN, both in terms of performance and accuracy.
References :
[1] J. Erhel, J. de Dreuzy, and B. Poirriez. Flow simulation in three-dimensional discrete
fracture networks. SIAM Journal on Scientific Computing, 31(4):2688–2705, 2009.
[2] P. Laug and G. Pichot. Mesh Generation and Flow Simulation in Large Tridimensional
Fracture Networks. https://hal.inria.fr/hal-02102811, 2019.
[3] G. Pichot, J. Erhel, and J.-R. de Dreuzy. A generalized mixed hybrid mortar method
for solving flow in stochastic discrete fracture networks. SIAM Journal on Scientific
Computing, 34(1):B86–B105, 2012.
[4] B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous
galerkin, mixed, and continuous galerkin methods for second order elliptic problems.
SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009.
[5] M. Cicuttin, D.A. Di Pietro, and A. Ern. Implementation of discontinuous skeletal
methods on arbitrary-dimensional, polytopal meshes using generic programming. Journal
of Computational and Applied Mathematics, 344:852 – 874, 2018.
[6] H. Borouchaki, P. Laug, and P. L. George. Parametric surface meshing using a com-
bined advancing-front generalized-delaunay approach. International Journal for Numer-
ical Methods in Engineering, 49(1-2):233–259, 2000.
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09:20
25 mins
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Time-Dependent Two-Dimensional Fourth-Order Problems: Optimal Convergence
Jean-Pierre Croisille, Dalia Fishelov
Abstract: Here we present a new approach for the analysis of high-order compact schemes
for the buckling plate problem, which models the 2D incompressible Navier-Stokes system.
In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013,
we have suggested fourth-order compact schemes for the Navier-Stokes equations.
The same type of schemes may be applied to the clamped plate problem.
For these methods the truncation error is only of first-order at near-boundary points,
but is of fourth order at interior points.
It is proven that the rate of convergence is actually four, thus the error
tends to zero as $O(h^4)$.
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09:45
25 mins
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A CSCM approximation of steady MHD flow and heat transfer between parallel plates with hydrodynamic slip and convective boundary conditions
Münevver Tezer-Sezgin, Önder Türk
Abstract: The steady magnetohydrodynamic (MHD) flow together with its heat
transfer between parallel plates is considered in which the electrically
conducting fluid has temperature dependent properties such as viscosity, thermal and electrical conductivity.
The fluid is driven by a constant pressure gradient, and a uniform external transverse magnetic field is applied perpendicular to the plates.
The effects of viscous and Joule dissipations are considered in the energy equation, and the fluid is assumed to be slipping in the vicinity of the plates.
The effects of the magnetic field, the hydrodynamic slip, and convective thermal boundary conditions on the flow and heat transfer are investigated as well as the temperature dependent parameters.
The Chebyshev spectral collocation method which is easy to implement is presented for the approximation of the solutions to the governing equations. The velocity and the temperature of the fluid are obtained with a cheap computational expense.
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