My research interests include development of robust solution
techniques for computational fluid dynamics, error estimation,
computational geometry management, parallel computation,
large-scale model reduction, and design under uncertainty. Some
current and recent research projects are:
Machine learning anisotropy detection
Hybridized and embedded discontinuous Galerkin methods
Output-based error estimation and mesh adaptation
Adaptive RANS calculations with the discontinuous Galerkin method
Unsteady output-based adaptation
Entropy-adjoint approach to mesh refinement
Contaminant source inversion
Cut-cell meshing
Nonlinear model reduction for inverse problems
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Machine learning anisotropy detection
Numerical simulations require quality computational
meshes, but the construction of an optimal mesh, one that
maximizes accuracy for a given cost, is not trivial. In this
work, we tackle and simplify one aspect of adaptive mesh
generation, determination of anisotropy, which refers to
direction-dependent sizing of the elements in the mesh.
Anisotropic meshes are important for efficiently resolving
certain flow features, such as boundary layers, wakes, and
shocks, that appear in computational fluid dynamics.
To predict the optimal mesh anisotropy, we use machine learning
techniques, which have the potential to accurately and
efficiently model responses of highly-nonlinear problems over a
wide range of parameters. We train a neural network on a large
amount of data from a rigorous, but expensive, mesh optimization
procedure (MOESS), and then attempt to reproduce this mapping
from simpler solution features.
Ongoing research directions in this are include:
- Extension to three dimensions
- Testing on a wider variety of fluid flows
- Prediction of sizing as well as anisotropy
Relevant Publications
Krzysztof J. Fidkowski and Guodong Chen.
A machine-learning anisotropy detection algorithm for output-adapted
meshes.
AIAA Paper 2020--0341, 2020.
[ bib |
.pdf ]
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Hybridized and embedded discontinuous Galerkin methods
Although discontinuous Galerkin (DG) method have enabled
high-order accurate computational fluid dynamics simulations,
their memory footprint and computational costs remain large.
Two approaches for reducing the expense of DG are (1) modifying
the discretization; and (2) optimizing the computational mesh.
We study both approaches and compare their relative benefits.
Hybridization of DG is an approach that modifies the high-order
discretization to reduce its expense for a given mesh. The high
cost of DG arises from the large number of degrees of freedom
required to approximate an element-wise discontinuous high-order
polynomial solution. These degrees of freedom are
globally-coupled, increasing the memory requirements for solvers
Hybridized discontinuous Galerkin (HDG) methods reduce the
number of globally-coupled degrees of freedom by decoupling
element solution approximations and stitching them together
through weak flux continuity enforcement. HDG methods introduce
face unknowns that become the only globally-coupled degrees of
freedom in the system. Since the number of face unknowns is
generally much lower than the number of element unknowns, HDG
methods can be computationally cheaper and use less memory
compared to DG. The embedded discontinuous Galerkin (EDG)
methodis a particular type of HDG method in which the
approximation space of face unknowns is continuous, further
reducing the number of globally-coupled degrees of freedom.
We have developed mesh optimization approaches for hybridized
discretizations. In addition to reducing computational costs,
the resulting methods improve (1) robustness of the solution
through quantitative error estimates, and (2) robustness of the
solver through a mesh size continuation approach in which the
problem is solved on successively finer meshes.
Ongoing research directions in this are include:
- Extension to three dimensions
- Efficient solvers for HDG and EDG
- Application to unsteady problems
Relevant Publications
Krzysztof J. Fidkowski and Guodong Chen.
Output‐based mesh optimization for hybridized and embedded
discontinuous Galerkin methods.
International Journal for Numerical Methods in Engineering,
121(5):867--887, 2019.
[ bib |
DOI |
.pdf ]
Krzysztof J. Fidkowski.
A hybridized discontinuous Galerkin method on mapped deforming
domains.
Computers and Fluids, 139(5):80--91, November 2016.
[ bib |
DOI |
.pdf ]
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Output-based error estimation and mesh adaptation
Computational Fluid Dynamics (CFD) has become an indispensable
tool for aerodynamic analysis and design. Driven by increasing
computational power and improvements in numerical methods, CFD
is at a state where three-dimensional simulations of complex
physical phenomena are now routine. However, such capability
comes with a new liability: ensuring that the computed solutions
are sufficiently accurate. CFD users, experts or not, cannot
reliably manage this liability alone for complex simulations.
The goal of this research is to develop methods that will assist
users and improve the robustness of these simulations. The two
key directions of these research are:
- Developing appropriate error estimators using
adjoint-based output error calculations.
- Developing robust mesh adaptation strategies for complex
geometries.
An illustration of a typical CFD analysis cycle augmented with
error estimation and mesh adaptation is given below.
Relevant Publications:
Krzysztof J. Fidkowski and David L. Darmofal.
Review of output-based error estimation and mesh adaptation in
computational fluid dynamics.
AIAA Journal, 49(4):673--694, 2011.
[ bib |
DOI |
.pdf ]
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Adaptive RANS calculations with the discontinuous Galerkin method
The accuracy of aerospace computational simulations depends
heavily on the amount of numerical error present, which in turn
depends on the allocation of resources, such as mesh size
distribution. This is especially true for Reynolds-averaged
Navier-Stokes (RANS) calculations, which possess a large range
of spatial scales that make a priori mesh construction
difficult. In this project a high-order CFD code based on the
discontinuous Galerkin discretization is used to adaptively
resolve two and three-dimensional RANS cases. Research areas
include robust solvers on under-resolved meshes, output-based
anisotropy detection, and efficient meshing. Sample results
obtained thus far are shown below.
Relevant Publications and Presentations:
M.A. Ceze and K.J. Fidkowski.
Output-Driven Anisotropic Mesh Adaptation for Viscous Flows Using Discrete Choice Optimization.
AIAA Paper Number 2010-0170, 2010.
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Unsteady output-based adaptation
The objective of this project is to improve the robustness and
efficiency of unsteady CFD simulations using adjoint-based
adaptive methods. While output error estimation has received
considerable attention for steady problems, its application to
unsteady simulations remains a challenging problem in
computation. This project addresses the theoretical and
implementation hurdles of applying output-based methods to
unsteady simulations. Topics addressed include development of a
suitable variational space-time discretization and solver,
solution of the unsteady adjoint problem, and combined spatial
and temporal mesh adaptation on dynamic resolution meshes. A
sample space-time adaptive result of a gust encounter is shown
below.
Relevant Publications:
Krzysztof J. Fidkowski and Yuxing Luo.
Output-based space-time mesh adaptation for the compressible
Navier-Stokes equations. Journal of Computational Physics, 2011.
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Entropy-adjoint approach to mesh refinement
Joint work with Philip Roe
When only a handful of engineering outputs are of interest, the
computational mesh can be tailored to predict those outputs
well. The process requires solutions of auxiliary adjoint
problems for each output that provide information on the
sensitivity of the output to discretization errors in the mesh.
This information guides mesh adaptation, so that after a few
iterations of the process, the engineer receives an accurate
solution along with error bars for the outputs of
interest. However, the extra adjoint solutions add a non-trivial
amount of computational work. It turns out for many equations,
including Navier-Stokes, there exists one "free" adjoint
solution that is related to the amount of entropy generated in
the flow. This adjoint is obtained by a simple variable
transformation and is therefore quite cheap to implement. An
example case adapted using such an entropy adjoint, along with
other adaptive indicators for comparison, is presented
below. This indicator is particularly well-suited for capturing
vortex structures, such as those that persist for extended
lengths in rotorcraft problems. Ongoing research is
investigating the applicability of the entropy adjoint and to
unsteady aerospace engineering simulations.
Relevant Publications:
K.J. Fidkowski, and P.L. Roe.
An Entropy
Adjoint Approach to Mesh Refinement. SIAM Journal on
Scientific Computing, 32(3), 2010, pp 1261-1287.
K.J. Fidkowski, and P.L. Roe.
Entropy-based Refinement I:
The Entropy Adjoint Approach
2009 AIAA Computational
Fluid Dynamics Conference, June 2009.
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Contaminant source inversion
Joint work with Karen Willcox, Chad Lieberman, Bart van Bloemen Waanders
The scenario of interest in this project is that of a
contaminant dispersed in an urban environment: the
concentration diffuses and convects with the wind. The
challenge is to use limited sensor measurements to
reconstruct where the profile came from and where it is
going. Such a large-scale inverse problem quickly becomes
intractable for real-time results that could be vital for
decision-making. The animation to the right illustrates a
forward simulation starting from one possible initial
concentration -- the forward problem alone took 1 hour to
run on 32 processors.
Two solution approaches are pursued in this project:
- Deterministic inversion using offline-precomputed
reduced models.
- Statistical inversion using Markov-chain
Monte-Carlo accelerated with adjoint-based output
calculation.
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Deterministic inversion using model reduction
One way to solve the problem in real time is to build a reduced
model of the unsteady system. This reduced model (typically a
couple hundred unknowns) is then used to invert measured
concentrations into an approximation for the initial conditions,
which can then be immediately run forward in time for
prediction. The time-consuming model reduction can be run ahead
of time on a supercomputer, while the real-time inversion can be
performed with the reduced model on laptops in the field. Shown
below is a comparison between true and inverted (using the full
and reduced models) initial conditions for a sample problem in
which full time histories from 36 randomly-placed sensors were
used to invert for the initial concentration field.
Statistical inversion with adjoint-based output
calculation.
Single-point deterministic inverse calculations can be
ill-conditioned when the measurement data are limited. More
robust in such cases is a probabilistic approach that provides
statistical information about where the contaminant could have
originated. However, obtaining this information generally
requires a very time consuming sampling process. The goal of
this work is to dramatically speed up the probabilistic
inversion by combining Markov-chain Monte Carlo (MCMC) sampling
with adjoint-based output calculations. A probabilistic
inversion result is shown below in terms of MCMC sample traces
for the same geometry as discussed above. The inverse
calculation assumed one contaminant source with an unknown
position. The calculation of tens of thousands of samples was
rapid enough to be performed in real time.
Relevant Publications and Presentations:
C. Lieberman, K. Fidkowski, K. Willcox, and B. van Bloemen Waanders.
Hessian-based model reduction: large-scale inversion and prediction.
International Journal for Numerical Methods in Fluids, 2012.
[ bib |
DOI |
.pdf ]
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Cut-cell meshing
Mesh generation around complex geometries can be one of the most
time-consuming and user-intensive tasks in practical numerical
computation. This is especially true when employing high-order
methods, which demand coarse mesh elements that have to be
shaped (i.e. curved) to represent surface features with an
adequate level of accuracy. Requirements of positive element
volumes and adequate geometry fidelity are difficult to enforce
in standard boundary conforming meshes.
Boundary-conforming mesh |
Cut-cell mesh |
In cut-cell meshing, the requirement that mesh elements conform
to the geometry boundary is relaxed, allowing for simple
volume-filling background meshes in which the geometry is
submerged or "embedded". The airfoil figure on the right above
shows an example of such a situation. The difficulty of
boundary-conforming mesh generation has been exchanged for a
cutting problem, in which arbitrarily-shaped cut cells
arise from intersections between the background mesh elements
and the geometry.
For the geometry, splines are used in 2D and curved triangular
patches are used in 3D, as illustrated above. Key to the
success of the DG high-order finite element is element
integration rules, which are derived automatically using Green's
theorem. Triangular and tetrahedral background elements are
used as they can be stretched to resolve anisotropic features.
Shown above are Mach number contours from a subsonic Euler
simulation around a wing-body configuration. 10,000 curved
surface patches were used to represent the geometry and the
final, solution-adapted background mesh for a
p=2
solution contained 85,000 elements. Below are
boundary-conforming and cut-cell meshes from a viscous
simulation over an airfoil. Anisotropic mesh refinement was
driven by a drag output error estimate.
Boundary-conforming mesh |
Cut-cell mesh |
Relevant Publications and Presentations:
K.J. Fidkowski and
D.L. Darmofal. A
triangular cut–cell adaptive method for high–order
discretizations of the compressible Navier–Stokes
equations. Journal of Computational Physics. 225, 2007,
pp 1653-1672.
K.J. Fidkowski and D.L. Darmofal. An
adaptive simplex cut–cell method for discontinuous Galerkin
discretizations of the Navier–Stokes equations. AIAA
Paper Number 2007-3941, 2007.
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Nonlinear model reduction for inverse problems
Joint work with Karen Willcox, David Galbally
In model reduction, a large parameter-dependent system of
equations is replaced by a much smaller system that accurately
approximates outputs over a certain range of parameters. Many
systematic techniques exist for performing such reduction; this
work used standard Galerkin projection with proper orthogonal
decomposition (POD) for basis construction. To treat the
nonlinearity efficiently, a masked-projection technique (similar
to gappy POD, missing point estimation, and coefficient function
approximation) was used.
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To demonstrate the model reduction technique, a scalar
convection-diffusion-reaction problem was considered. The
scenario consists of fuel injected into a combustion
chamber and left to react with a surrounding oxidizer as
it convects downstream. A 2D unsteady simulation is shown
at left, for a pulsating injection concentration.
Reduction of a steady 3D combustion chamber was performed
in parallel, reducing the degrees of freedom (DOF) from 8.5
million to 40. Sample fuel concentration profiles are
illustrated below.
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Full system: 8.5 million DOF, 13h CPU time |
Reduced system: 40 DOF, negligible CPU time |
In these simulations, the outputs consisted of average fuel
concentrations downstream, while the parameters were those
entering into the nonlinear reaction rate expression. The
parameters of interest remained adjustable in the reduced model,
and the reduced model was verified to accurately reproduce
outputs over a bounded input parameter set.
One application of such a reduced model is for solving inverse
problems via a Bayesian inference approach. The inverse problem
considered consisted of estimating reaction rate parameters from
measured fuel concentrations. The small size of the reduced
model made Markov-Chain Monte Carlo (MCMC) sampling feasible
(equivalent sampling with the full system would take almost 8
years of CPU time). The MCMC sample histories for two reaction
rate parameters and the resulting histograms after 5000 samples
are shown below.
MCMC samples |
Posterior histogram |
Relevant Publications and Presentations:
D. Galbally, K. Fidkowski, K. Willcox, and O. Ghattas,
Nonlinear Model
Reduction for Uncertainty Quantification in Large-Scale
Inverse Problems. International Journal for Numerical
Methods in Engineering. 81(12), 2009, pp 1581-1603.
Nonlinear Model Reduction for
Uncertainty Quantification in Large-Scale Inverse
Problems
Computational Aerospace Sciences Seminar,
October 2008.
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