893 research outputs found
Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods
We present a robust and efficient target-based mesh adaptation methodology,
building on hybridized discontinuous Galerkin schemes for (nonlinear)
convection-diffusion problems, including the compressible Euler and
Navier-Stokes equations. Hybridization of finite element discretizations has
the main advantage, that the resulting set of algebraic equations has globally
coupled degrees of freedom only on the skeleton of the computational mesh.
Consequently, solving for these degrees of freedom involves the solution of a
potentially much smaller system. This not only reduces storage requirements,
but also allows for a faster solution with iterative solvers. The mesh
adaptation is driven by an error estimate obtained via a discrete adjoint
approach. Furthermore, the computed target functional can be corrected with
this error estimate to obtain an even more accurate value. The aim of this
paper is twofold: Firstly, to show the superiority of adjoint-based mesh
adaptation over uniform and residual-based mesh refinement, and secondly to
investigate the efficiency of the global error estimate
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
Deep Learning Closure of the Navier-Stokes Equations for Transition-Continuum Flows
The predictive accuracy of the Navier-Stokes equations is known to degrade at
the limits of the continuum assumption, thereby necessitating expensive and
often highly approximate solutions to the Boltzmann equation. While tractable
in one spatial dimension, their high dimensionality makes multi-dimensional
Boltzmann calculations impractical for all but canonical configurations. It is
therefore desirable to augment the Navier-Stokes equations in these regimes. We
present an application of a deep learning method to extend the validity of the
Navier-Stokes equations to the transition-continuum flows. The technique
encodes the missing physics via a neural network, which is trained directly
from Boltzmann solutions. While standard DL methods can be considered ad-hoc
due to the absence of underlying physical laws, at least in the sense that the
systems are not governed by known partial differential equations, the DL
framework leverages the a-priori known Boltzmann physics while ensuring that
the trained model is consistent with the Navier-Stokes equations. The online
training procedure solves adjoint equations, constructed using algorithmic
differentiation, which efficiently provide the gradient of the loss function
with respect to the learnable parameters. The model is trained and applied to
predict stationary, one-dimensional shock thickness in low-pressure argon
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Space-time discontinuous Petrov-Galerkin finite elements for transient fluid mechanics
Initial mesh design for computational fluid dynamics can be a time-consuming and expensive process. The stability properties and nonlinear convergence of most numerical methods rely on a minimum level of mesh resolution. This means that unless the initial computational mesh is fine enough, convergence can not be guaranteed. Any meshes below this minimum resolution level are termed to be in the ``pre-asymptotic regime.'' This condition implies that meshes need to in some way anticipate the solution before it is known. On top of the minimum requirement that the surface meshes must adequately represent the geometry of the problem under consideration, resolution requirements on the volume mesh make the CFD practitioner's job significantly more time consuming.
In contrast to most other numerical methods, the discontinuous Petrov-Galerkin finite element method retains exceptional stability on extremely coarse meshes. DPG is also inherently very adaptive. It is possible to compute the residual error without knowledge of the exact solution, which can be used to robustly drive adaptivity. This results in a very automated technology, as the user can initialize a computation on the coarsest mesh which adequately represents the geometry then step back and let the program solve and adapt iteratively until it resolves the solution features.
A common complaint of minimum residual methods by computational fluid dynamics practitioners is that they are not locally conservative. In this thesis, this concern is addressed by developing a locally conservative DPG formulation by augmenting the system with Lagrange multipliers. The resulting DPG formulation is then proved to be robust and shown to produce superior numerical results over standard DPG on a selection of test problems.
Adaptive convergence to steady incompressible and compressible Navier-Stokes solutions was explored in Jesse Chan's and Nathan Roberts' dissertations. Space-time offers a natural extension to transient problems as it preserves the stability and adaptivity properties of DPG in the time dimension. Space-time also offers more extensive parallelization capability than problems treated with traditional time stepping as it allows multigrid concurrently in both space and time. A proof of concept space-time DPG formulation is developed for transient convection-diffusion. The robust test norms derived for steady convection-diffusion are extended to the space-time case and proofs of robustness are provided. Numerical results verify the robust behavior and near optimality of the resulting solutions.
The space-time formulation for convection-diffusion is then extended to transient incompressible and compressible Navier-Stokes by analogy. Several numerical experiments are performed, but a mathematical analysis is not attempted for these nonlinear problems. Several side topics are explored such as a study of the compressible Navier-Stokes equations under various variable transformations and the development of consistent test norms through the concept of physical entropy.Computational Science, Engineering, and Mathematic
Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations
Non-linear entropy stability and a summation-by-parts framework are used to
derive entropy stable wall boundary conditions for the three-dimensional
compressible Navier--Stokes equations. A semi-discrete entropy estimate for the
entire domain is achieved when the new boundary conditions are coupled with an
entropy stable discrete interior operator. The data at the boundary are weakly
imposed using a penalty flux approach and a simultaneous-approximation-term
penalty technique. Although discontinuous spectral collocation operators on
unstructured grids are used herein for the purpose of demonstrating their
robustness and efficacy, the new boundary conditions are compatible with any
diagonal norm summation-by-parts spatial operator, including finite element,
finite difference, finite volume, discontinuous Galerkin, and flux
reconstruction/correction procedure via reconstruction schemes. The proposed
boundary treatment is tested for three-dimensional subsonic and supersonic
flows. The numerical computations corroborate the non-linear stability (entropy
stability) and accuracy of the boundary conditions.Comment: 43 page
Methods for Optimal Output Prediction in Computational Fluid Dynamics.
In a Computational Fluid Dynamics (CFD) simulation, not all data is of equal importance. Instead, the goal of the user is often to compute certain critical "outputs" -- such as lift and drag -- accurately. While in recent years CFD simulations have become routine, ensuring accuracy in these outputs is still surprisingly difficult. Unacceptable levels of output error arise even in industry-standard simulations, such as the steady flow around commercial aircraft. This problem is only exacerbated when simulating more complex, unsteady flows.
In this thesis, we present a mesh adaptation strategy for unsteady problems that can automatically reduce errors in outputs of interest. This strategy applies to problems in which the computational domain deforms in time -- such as flapping-flight simulations -- and relies on an unsteady adjoint to identify regions of the mesh contributing most to the output error. This error is then driven down via refinement of the critical regions in both space and time. Here, we demonstrate this strategy on a series of flapping-wing problems in two and three dimensions, using high-order discontinuous Galerkin (DG) methods for both spatial and temporal discretizations. Compared to other methods, results indicate that this strategy can deliver a desired level of output accuracy with significant reductions in computational cost.
After concluding our work on mesh adaptation, we take a step back and investigate another idea for obtaining output accuracy: adapting the numerical method itself. In particular, we show how the test space of discontinuous finite element methods can be "optimized" to achieve accuracy in certain outputs or regions. In this work, we compute test functions that ensure accuracy specifically along domain boundaries. These regions -- which are vital to both scalar outputs (such as lift and drag) and distributions (such as pressure and skin friction) -- are often the most important from an engineering standpoint.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133418/1/kastsm_1.pd
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