177 research outputs found
Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation
In this paper three p-adaptation strategies based on the minimization of the truncation error are presented for high order discontinuous Galerkin methods. The truncation error is approximated by means of a ? -estimation procedure and enables the identification of mesh regions that require adaptation. Three adaptation strategies are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. All strategies require fine solutions, which are obtained by enriching the polynomial order, but while the former needs time converged solutions, the last two rely on non-converged solutions, which lead to faster computations. In addition, the high order method permits the spatial decoupling for the estimated errors and enables anisotropic p-adaptation. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier?Stokes equations. It is shown that the two quasi- a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively
Truncation Error-Based Anisotropic -Adaptation for Unsteady Flows for High-Order Discontinuous Galerkin Methods
In this work, we extend the -estimation method to unsteady problems and
use it to adapt the polynomial degree for high-order discontinuous Galerkin
simulations of unsteady flows. The adaptation is local and anisotropic and
allows capturing relevant unsteady flow features while enhancing the accuracy
of time evolving functionals (e.g., lift, drag). To achieve an efficient and
unsteady truncation error-based -adaptation scheme, we first revisit the
definition of the truncation error, studying the effect of the treatment of the
mass matrix arising from the temporal term. Secondly, we extend the
-estimation strategy to unsteady problems. Finally, we present and
compare two adaptation strategies for unsteady problems: the dynamic and static
-adaptation methods. In the first one (dynamic) the error is measured
periodically during a simulation and the polynomial degree is adapted
immediately after every estimation procedure. In the second one (static) the
error is also measured periodically, but only one -adaptation process is
performed after several estimation stages, using a combination of the periodic
error measures. The static -adaptation strategy is suitable for
time-periodic flows, while the dynamic one can be generalized to any flow
evolution.
We consider two test cases to evaluate the efficiency of the proposed
-adaptation strategies. The first one considers the compressible Euler
equations to simulate the advection of a density pulse. The second one solves
the compressible Navier-Stokes equations to simulate the flow around a cylinder
at Re=100. The local and anisotropic adaptation enables significant reductions
in the number of degrees of freedom with respect to uniform refinement, leading
to speed-ups of up to for the Euler test case and for
the Navier-Stokes test case
Mesh adaptation strategies for compressible flows using a high-order spectral/hp element discretisation
An accurate calculation of aerodynamic force coe cients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier-Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coe cients. To reduce error in the calculated force coe cients whilst keeping computational cost minimal, I propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator. We initially calculate a steady-state solution to the governing equations using a low polynomial order and use a goal-based error indicator to identify parts of the computational domain that require improved solution accuracy and increase the approximation order there. We demonstrate the cost-effectiveness of our method across a range of polynomial orders by considering a number of examples in two- and three-dimensions and in subsonic and transonic flow regimes. Reductions in both the number of degrees of freedom required to resolve the force coe cients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.
In addition to the adjoint-based p-adaptation strategy, I propose a mesh deformation strategy that relies on a thermo-elastic formulation. The thermal-elastic formulation is initially used to control mesh validity. Two mesh quality indicators are proposed and used to illustrate that by heating up (expanding) or cooling down (contracting) the appropriate elements, an improved robustness of the classical mesh deformation strategy is obtained. The idea is extended to perform shock wave r-adaptation (adaptation through redistribution) for high Mach number flows. The mesh deformation strategy keeps the mesh topology unchanged, contracts the elements that cover the shock wave, keeps the number of elements constant and the computation as e cient as the unrefined case. The suitability of r-adaptation for shock waves is illustrated using internal and external compressible flow problems.Open Acces
Comparison of the ENATE approach and discontinuous Galerkin spectral element method in 1D nonlinear transport equations
In this paper a comparison of the performance of two ways of discretizing the nonlinear convection-diffusion equation in a one-dimensional (1D) domain is performed. The two approaches can be considered within the class of high-order methods. The first one is the discontinuous Galerkin method, which has been profusely used to solve general transport equations, either coupled as the Navier-Stokes equations, or on their own. On the other hand, the ENATE procedure (Enhanced Numerical Approximation of a Transport Equation), uses the exact solution to obtain an exact algebraic equation with integral coefficients that link nodal values with a three-point stencil. This paper is the first of thorough assessments of ENATE by comparing it with well-established high-order methods. Several test cases of the steady Burgers' equation with and without source have been chosen for comparison
SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES
Crack propagation in thin shell structures due to cutting is conveniently simulated
using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell
elements are usually preferred for the discretization in the presence of complex material
behavior and degradation phenomena such as delamination, since they allow for a correct
representation of the thickness geometry. However, in solid-shell elements the small thickness
leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new
selective mass scaling technique is proposed to increase the time-step size without affecting
accuracy. New ”directional” cohesive interface elements are used in conjunction with selective
mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile
shells
Python framework for HP adaptive discontinuous Galerkin methods for two phase flow in porous media
In this paper we present a framework for solving two-phase flow problems in porous media. The discretization is based on a Discontinuous Galerkin method and includes local grid adaptivity and local choice of polynomial degree. The method is implemented using the new Python frontend Dune-FemPy to the open source framework Dune. The code used for the simulations is made available as Jupyter notebook and can be used through a Docker container. We present a number of time stepping approaches ranging from a classical IMPES method to a fully coupled implicit scheme. The implementation of the discretization is very flexible allowing to test different formulations of the two-phase flow model and adaptation strategies
Python Framework for HP Adaptive Discontinuous Galerkin Method for Two Phase Flow in Porous Media
In this paper we present a framework for solving two phase flow problems in
porous media. The discretization is based on a Discontinuous Galerkin method
and includes local grid adaptivity and local choice of polynomial degree. The
method is implemented using the new Python frontend Dune-FemPy to the open
source framework Dune. The code used for the simulations is made available as
Jupyter notebook and can be used through a Docker container. We present a
number of time stepping approaches ranging from a classical IMPES method to
fully coupled implicit scheme. The implementation of the discretization is very
flexible allowing for test different formulations of the two phase flow model
and adaptation strategies.Comment: Keywords: DG, hp-adaptivity, Two-phase flow, IMPES, Fully implicit,
Dune, Python, Porous media. 28 pages, 9 figures, various code snippet
Accelerating high order discontinuous Galerkin solvers using neural networks: 3D compressible Navier-Stokes equations
We propose to accelerate a high order discontinuous Galerkin solver using
neural networks. We include a corrective forcing to a low polynomial order
simulation to enhance its accuracy. The forcing is obtained by training a deep
fully connected neural network, using a high polynomial order simulation but
only for a short time frame. With this corrective forcing, we can run the low
polynomial order simulation faster (with large time steps and low cost per time
step) while improving its accuracy.
We explored this idea for a 1D Burgers' equation in (Marique and Ferrer, CAF
2022), and we have extended this work to the 3D Navier-Stokes equations, with
and without a Large Eddy Simulation closure model. We test the methodology with
the turbulent Taylor Green Vortex case and for various Reynolds numbers (30,
200 and 1600). In addition, the Taylor Green Vortex evolves with time and
covers laminar, transitional, and turbulent regimes, as time progresses.
The proposed methodology proves to be applicable to a variety of flows and
regimes. The results show that the corrective forcing is effective in all
Reynolds numbers and time frames (excluding the initial flow development). We
can train the corrective forcing with a polynomial order of 8, to increase the
accuracy of simulations from a polynomial order 3 to 6, when correcting outside
the training time frame. The low order correct solution is 4 to 5 times faster
than a simulation with comparable accuracy (polynomial order 6).
Additionally, we explore changes in the hyperparameters and use transfer
learning to speed up the training. We observe that it is not useful to train a
corrective forcing using a different flow condition. However, an already
trained corrective forcing can be used to initialise a new training (at the
correct flow conditions) to obtain an effective forcing with only a few
training iterations
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