Robust Adaptive High-Order RANS Methods
MASSACHUSETTS INGTWFUTE
OF TECHNOLOGY
by
Jun Kudo
OCT 0 9 201
Sc.B., Brown University (2010)
LIBRARIES
Submitted to the School of Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Computation for Design and Optimization
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
Author ...............................
Scho$l of Engineering
June 20, 2014
Signature redacted
Certified by.............................
David L. DAofal
Professor of Aeronautics and Astronautics
Thesis Supervisor
Signature redacted
Accepted by ...............................
adjiconstantinou
Ni
Professor of Mechanical Engineering
Co-Director, Computation for Design and Optimization
2
Robust Adaptive High-Order RANS Methods
by
Jun Kudo
Submitted to the School of Engineering
on June 20, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Computation for Design and Optimization
Abstract
The ability to achieve accurate predictions of turbulent flow over arbitrarily complex
geometries proves critical in the advancement of aerospace design. However, quantitatively accurate results from modern Computational Fluid Dynamics (CFD) tools are
often accompanied by intractably high computational expenses and are significantly
hindered by the lack of automation. In particular, the generation of a suitable mesh
for a given flow problem often requires significant amounts of human input. This
process however encounters difficulties for turbulent flows which exhibit a wide range
of length scales that must be spatially resolved for an accurate solution. Higherorder adaptive methods are attractive candidates for addressing these deficiencies by
promising accurate solutions at a reduced cost in a highly automated fashion. However, these methods in general are still not robust enough for industrial applications
and significant advances must be made before the true realization of robust automated
three-dimensional turbulent CFD.
This thesis presents steps towards this realization of a robust high-order adaptive
Reynolds-Averaged Navier-Stokes (RANS) method for the analysis of turbulent flows.
Specifically, a discontinuous Galerkin (DG) discretization of the RANS equations and
an output-based error estimation with an associated mesh adaptation algorithm is
demonstrated. To improve the robustness associated with the RANS discretization,
modifications to the negative continuation of the Spalart-Allmaras turbulence model
are reviewed and numerically demonstrated on a test case. An existing metric-based
adaptation framework is adopted and modified to improve the procedure's global convergence behavior. The resulting discretization and modified adaptation procedure
is then applied to two-dimensional and three-dimensional turbulent flows to demonstrate the overall capability of the method.
Thesis Supervisor: David L. Darmofal
Title: Professor of Aeronautics and Astronautics
3
4
Acknowledgments
I would like to thank all those who have made this thesis possible. I would first like
to thank my adviser, Prof. David Darmofal, for giving me the opportunity to work
with him and for all his encouragement throughout my graduate study. I would also
like to thank Dr. Steven Allmaras for his help and insight along the way. In addition,
I would like to recognize Marshall for teaching me various useful tools and for his
unwavering dedication to our software maintenance.
This thesis would not have been possible without the help of the past and present
generations of the ProjectX team. I would like to thank them all for their many
contributions that formed the foundation of ProjectX that enabled this work. I would
specifically like to thank: Huafei, for helping me get started; Masa, for laying down
the foundation of the adaptive framework; Josh, for helping me set up my threedimensional meshes and geometries; my office mates, Steven and Phil, for always
being available for discussion on any topic; and Carlee, Jeff, Savi, and Yixuan, for
bringing in some much needed fresh energy into the lab.
On a more personal note, I would also like to thank all of my friends that have
helped make my life during my graduate studies more fun than it might otherwise
have been. Special thanks are directed towards Dan and Kenny who helped me stay
sane when the going got tough and Allison who acted like my personal graduate
adviser when I became frustrated. I would like to send my most sincere thanks to
Melissa for all her love and support over the past two years. I do not know how to
fully express my gratitude and am looking forward to spending more time together
in Boston in the upcoming years.
Finally, I would like to acknowledge the financial support of the DOD NDSEG Fellowship program and NASA (NASA Cooperative Agreement #NNX12AJ75A,
nical monitor Dr. Harold Atkins).
5
tech-
6
Contents
1
Introduction
13
1.1
M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2
O bjective
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.3.1
High-Order Methods and Discontinuous Galerkin Methods . .
15
1.3.2
Error Estimate and Adaptation
. . . . . . . . . . . . . . . . .
16
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4
2
Discretization of the RANS Equations
21
2.1
The RANS Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2
The SA Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
Baseline Model . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2.2
Modifications to Baseline Model . . . . . . . . . . . . . . . . .
24
Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3.1
Inviscid Discretization
. . . . . . . . . . . . . . . . . . . . . .
28
2.3.2
Viscous Discretization
. . . . . . . . . . . . . . . . . . . . . .
28
2.3.3
Source Discretization . . . . . . . . . . . . . . . . . . . . . . .
29
Temporal Discretization and Solution Technique . . . . . . . . . . . .
29
2.4.1
30
2.3
2.4
3
Numerical Results
. . . . . . . . . . . . . . . . . . . . . . . .
Output Error Estimation and Continuous Mesh Framework
33
3.1
Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.1.1
34
Dual-Weighted Residual Method and Localization . . . . . . .
7
3.2
4
Continuous Mesh Framework
. . . . . . . . . . . . . . . . . . . . . .
35
3.2.1
Metric-Conforming Meshes . . . . . . . . . . . . . . . . . . . .
36
3.2.2
Mesh-Conforming Metric Fields . . . . . . . . . . . . . . . . .
37
Metric Field Optimization using Local Error Sampling and Synthesis
Framework
4.1
4.2
4.3
5
6
39
Model Definition
. . . . . . . . . . . . . . . .. . . . . . . . . . . . . .
39
4.1.1
Continuous Relaxation . . . . . . . . . . . . . . . . . . . . . .
39
4.1.2
Metric Manipulation Framework . . . . . . . . . . . . . . . . .
41
4.1.3
Cost M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.1.4
Surrogate Error Model from Local Error Sampling and Synthesis 42
Optimization of Surrogate Model
. . . . . . . . . . . . . . . . . . . .
44
4.2.1
Heuristic Optimization of Surrogate Model . . . . . . . . . . .
45
4.2.2
Modifications to Metric Optimization Procedure . . . . . . . .
47
Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.3.1
r'-Type Corner Singularity
51
4.3.2
RAE 2822 Transonic RANS-SA
4.3.3
Supersonic flow over NACA 0012
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
52
. . . . . . . . . . . . . . . .
55
Turbulent Aerodynamic Problems
63
5.1
Three-element MDA 30P-30N, Subsonic
. . . . . . . . . . . . . . . .
63
5.2
3D Zero Pressure Gradient Flat Plate, Subsonic . . . . . . . . . . . .
66
5.3
3D Duct, Subsonic
70
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion
77
6.1
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
A Adaptation From Under-resolved Meshes
A.1
Test Cases and Initial Meshes
A.2
Solution on Initial Meshes
A.3
Adaptive Results
81
. . . . . . . . . . . . . . . . . . . . . .
81
. . . . . . . . . . . . . . . . . . . . . . . .
82
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
8
List of Figures
1-1
General Framework for PDE Solver with Mesh Adaptation . . . . . .
16
2-1
Comparison of Modified Diffusion Coefficients
27
2-2
Comparison of convergence for flow over a NACA 4412 using negative
. . . . . . . . . . . . .
v variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
Element size h versus the distance of the element centroid from the
corner for optimized meshes for r' singularity problem with a = 2/3 .
4-2
53
Comparison of error indicator and degrees of freedom history for transonic flow over a RAE 2822 Airfoil
4-3
31
. . . . . . . . . . . . . . . . . . .
54
Cumulative computation time taken to obtain primal solution and error
indicator estimate on a given iterate's mesh for transonic flow over a
RAE 2822 airfoil
4-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mesh adaptation history for transonic flow over a RAE 2822 Airfoil
using the heuristic metric optimization method
4-5
. . . . . . . . . . . .
58
Adaptation history for supersonic flow over a NACA 0012 Airfoil using
the gradient-based optimization method
4-8
57
Adaptation history for supersonic flow over a NACA 0012 Airfoil using
the heuristic optimization method . . . . . . . . . . . . . . . . . . . .
4-7
56
Mesh adaptation history for transonic flow over a RAE 2822 Airfoil
using the gradient-based metric optimization method . . . . . . . . .
4-6
55
. . . . . . . . . . . . . . . .
58
Select adapted meshes (iterations 17 and 18) and respective Mach number distributions generated with the heuristic optimization method,
NACA0012 supersonic flow.
. . . . . . . . . . . . . . . . . . . . . . .
9
59
4-9
Select adapted meshes (iterations 19 and 20) and respective Mach number distributions generated with the heuristic optimization method,
NACA0012 supersonic flow.
. . . . . . . . . . . . . . . . . . . . . . .
60
4-10 Select adapted meshes and respective Mach number distributions generated with the gradient-based optimization method, NACA0012 supersonic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5-1
Initial mesh, Three-element MDA flow. . . . . . . . . . . . . . . . . .
63
5-2
Convergence of drag output for the three-element MDA flow; reference
value obtained from p = 3,
5-3
doftarget
= 250k adapted solution
. . . . .
Comparison of adapted p = 1 and p = 2 meshes with similar error
levels, Three-element MDA flow . . . . . . . . . . . . . . . . . . . . .
5-4
. . . . . . . . . .
65
p = 2, doftarget = 120k adapted mesh and respective Mach number
distribution, overview, Three-element MDA flow.
5-6
65
Pressure coefficient comparison of adapted p = 1 and p = 2 meshes
with similar error levels, Three-element MDA flow.
5-5
64
p = 2,
doftarget
. . . . . . . . . . .
66
= 120k adapted mesh and respective Mach number
distribution, far field, Three-element MDA flow. . . . . . . . . . . . .
67
5-7
Initial structured mesh, 3D flat plate flow.
67
5-8
Convergence of drag output for the 3D flat plate flow; reference value
5-9
. . . . . . . . . . . . . . .
obtained from two-dimensional simulations . . . . . . . . . . . . . . .
68
P=2, 25k dof optimized mesh, 3D flat plate flow.
69
. . . . . . . . . . .
5-10 Mach number distribution of 2D center slice, initial mesh, 3D flat plate
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
5-11 Mach number distribution of 2D center slice, P=2, 25k dof optimized
mesh, 3D flat plate flow
. . . . . . . . . . . . . . . . . . . . . . . . .
70
5-12 Initial structured mesh, subsonic duct flow. . . . . . . . . . . . . . . .
71
5-13 Drag adaptation history for p = 2, subsonic duct flow . . . . . . . . .
72
5-14 P=2, 100k dof optimized mesh, subsonic duct flow.
73
10
. . . . . . . . . .
5-15 Mach number distribution, 2D view along duct at X = 0.25, subsonic
duct flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5-16 Mach number distribution, 2D cross sectional view at outflow, 2.0 units
downstream from leading edge, subsonic duct flow . . . . . . . . . . .
A-1 Initial isotropic mesh with no boundary layer resolution - 460 Elements
75
82
-
A-2 Initial anisotropic mesh with significant boundary layer resolution
460 Elements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
A-3 Initial pu, T wall normal profile, 1.5 units downstream of the leading
edge, isotropic mesh
. . . . . . . . . . . . . . . . . . . . . . . . . . .
83
A-4 Initial pu, T wall normal profile, 1.5 units downstream of the leading
edge, anisotropic mesh . . . . . . . . . . . . . . . . . . . . . . . . . .
84
A-5 Drag adaptation history for turbulent boundary layer flow starting with
isotropic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
A-6 DOF adaptation history for turbulent boundary layer flow starting
with isotropic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
A-7 Drag adaptation history for turbulent boundary layer flow starting with
anisotropic m esh
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
A-8 DOF adaptation history for turbulent boundary layer flow starting
with anisotropic mesh
. . . . . . . . . . . . . . . . . . . . . . . . . .
86
A-9 Heat flux adaptation history for scalar boundary layer flow starting
with isotropic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
A-10 DOF adaptation history for scalar boundary layer flow starting with
isotropic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
A-11 Heat flux adaptation history for scalar boundary layer flow starting
with anisotropic mesh
. . . . . . . . . . . . . . . . . . . . . . . . . .
87
A-12 DOF adaptation history for scalar boundary layer flow starting with
anisotropic mesh
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
88
12
Chapter 1
Introduction
1.1
Motivation
With the evolution of numerical algorithms and substantial advancements in computational power, the presence of Computational Fluid Dynamics (CFD) tools throughout industry and academia has steadily increased in the past few decades. These
numerical tools promise the ability to rapidly and accurately simulate flows that may
otherwise be prohibitively expensive to test experimentally. Despite this potential,
many prevalent CFD software packages lack efficiency when high accuracy simulations
are required and require a significant amount of user involvement for mesh generation. Aerodynamic flow, and in particular turbulent flow, can exhibit features with a
wide range of lengths scales and singularities that can potentially interfere with the
realization of accurate and robust CFD.
The American Institute of Aeronautics and Astronautics (AIAA) organizes drag
prediction workshops (DPW) with the purpose of evaluating the capability of current
state of the art CFD for turbulence modeling on industrially relevant geometries and
flow conditions. The starting meshes for the geometries of interest are generated based
on industry's best practices and these meshes are uniformly refined in an attempt
to achieve mesh independence.
The results from the workshops display that the
current CFD capabilities are insufficient for engineering applications; the uncertainties
associated with the drag results are unacceptably high and to properly reduce the
13
spread of the results, the starting meshes must be uniformly refined to levels that
make the simulations intractably large for routine analysis [32, 30, 52, 53].
Further troubling are the discoveries presented by Mavriplis [36] who demonstrated
that even for very large grids, asymptotic results appear to be different for different
families of self-similar grids.
Adequate resolution of the relevant flow features by
uniform refinement proves difficult when the flow exhibits a wide range of scales
as seen in high-speed turbulent flows. Therefore, even for very large best practice
meshes, the flow solutions may contain significant discretization errors if important
regions of the flow are not well resolved.
These results demonstrate that there is a significant need for improvements in
the automation and efficiency of the current CFD methods.
Spatial discretization
error can significantly contribute to the inaccuracies in turbulent flow simulations
and worse yet, can prove to be very elusive to even expert grid generators. Global
refinement can potentially reduce this error but proves inefficient due to the very
large grid sizes required to resolve the relevant flow features. High-order, adaptive
RANS methods present an attractive path to alleviate these issues by automatically
and efficiently resolving turbulent flows.
1.2
Objective
The objective of this work is to develop a robust high-order, adaptive method for the
simulation of high-speed turbulent flows and to demonstrate the performance of this
method on two-dimensional and three-dimensional aerodynamic cases.
14
1.3
1.3.1
Background
High-Order Methods and Discontinuous Galerkin Methods
The goal of high-order methods is to achieve higher fidelity solutions at a lower
cost. Generally, a discretization method has an error convergence behavior of E oc
0(h') where E is a measure of error, h is a measure of the mesh size, and r is the
error convergence rate.
In the aeronautical industry, the most prevalent methods
for simulating flows are second-order finite volume methods (FVM) which exhibit a
convergence rate of r = 2. However, as accuracy requirements become more stringent,
second-order accurate methods may prove to be insufficient as they begin to require
intractable amounts of degrees of freedom. High-order methods attempt to alleviate
this issue by improving the simulation efficiency by increasing the convergence rate
r. In this work, high-order methods are those that achieve r > 2 (for L 2 error).
In finite-volume and finite-difference schemes, higher order accuracy is generally
achieved by extending the approximation stencil. However, this extension introduces
many complications including difficulties in parallelization and boundary condition
treatment and the need for time-integration methods with stronger stability properties.
Finite element schemes on the other hand provide a particularly attractive framework for achieving high-order convergence. For these schemes, the convergence rate
r can be improved by increasing the degree of the basis polynomials. The methods
are also naturally applicable to unstructured meshes which are useful for the tessellation of more complex geometries of interest. While finite element methods offer a
conceptually simple path to high-order accuracy, the continuous Galerkin method is
unstable for use on convection-dominated problems. As such, in this work, a discon.
tinuous Galerkin (DG) finite element method is used
The first DG method was introduced by Reed and Hill for scalar hyperbolic equations [44] in 1973. The error analysis for this work was later provided by LeSaint
15
and Raviart [31] and Richter [45]. This DG method was then extended to nonlinear
hyperbolic problems by Chavent and Salzano [14] by incorporation of the Godunov's
flux. Cockbur, Shu, and co-authors [17, 16, 15, 18] extended the method to non-linear
systems of hyperbolic equations by combining the DG spatial discretization with a
Runge-Kutta explicit time integration.
The development of the DG methods for elliptic problems began with the interior
penalty methods by Arnold [3] and Wheeler [57]. More recently, Bassi and Rebay
developed the so-called BRI [6] and later the BR2 [7] DG discretization of the diffusive
operator.
1.3.2
Error Estimate and Adaptation
Mesh adaptation methods present an attractive alternative to current prevalent CFD
practices in which a mesh is painstakingly created by empirical best practices. Adaptation promises to significantly reduce the amount of human intervention to produce
a more reliable output prediction. These methods fundamentally rely on a definition
of an error indicator which localizes and identifies the regions that need discretization
modifications. With these error indicators, the process attempts to reduce the output
error estimates via the adaptation machinery. The generalized adaptive framework
can be seen in Figure 1-1.
Problem
Compute flow
Estimate
,definition
and outputs
output error
0 Outputs
o Error Estimate,
Adapt mesh
to control error
Figure 1-1: General Framework for PDE Solver with Mesh Adaptation
The process is initialized with a problem definition which involves an initial mesh,
relevant boundary and initial conditions, and an output function of interest. With
these inputs, the PDE is solved on the current grid and an output error is estimated.
If the error is larger than a provided tolerance, an adaptation algorithm is used
16
to generate a new mesh according to the localized error estimates. The process is
then repeated on this adapted mesh and continued until the target error tolerance is
achieved.
Error Estimation
Error estimation forms the foundation for any adaptation framework by properly
identifying the elements with large error contributions.
Various a posteriori error
estimation techniques have been developed in an attempt to best achieve these goals.
These methods, which include estimates based on gradients [4], energy [1], and interpolation error [21], do not generally work for hyperbolic problems where upstream
perturbations and errors can pollute the solution downstream [50].
For these convection dominated flows, output-based error estimation techniques
which explicitly incorporate the adjoint associated with the output have been developed. The adjoint (or dual) solution represents the sensitivity of an output with
respect to residual perturbations and as such, helps to identify the flow regimes that
are critical for accurate prediction of the output.
In this work, the dual-weighted
residual (DWR) method proposed by Becker and Rannacher [9, 10] is used.
Adaptation
Given a measure of error, the goal of the adaptation procedure is to modify the
current discretization to decrease the given output error estimate. For finite element
methods, these modifications can largely be classified into three types: h-adaptation,
p-adaptation, and hp-adaptation.
In h-adaptation, the elements' shape and size are modified; with this process,
elements can be moved, sub-divided or agglomerated to achieve different spatial distributions of shape and size.
For h-adaptation methods, the size and orientation
information of each element is often formulated as a metric tensor field [23, 55].
In p-adaptation, the polynomial approximation order of the elements is modified
while keeping a constant triangulation. Although p-adaptation has been shown to
be more efficient for sufficiently smooth flows, these methods are not particularly
17
well-suited for triangulations that currently contain insufficient resolution of the flow.
The hp-adaptation methods modify both the triangulation and the polynomial
approximation order and is attractive in that these methods can potentially combine
the benefits of both h and p adaptation. However, the choice between resizing the
element and locally changing the scheme's discretization order is far from trivial
and has been the focus of much research [28, 48, 13]. For simplicity and with the
goal of automatic mesh generation in mind, this work focuses solely on h-adaptation
techniques.
Anisotropic h-adaptation methods were originally largely driven by estimates of
the directional interpolation error of a representative scalar. Peraire et al. [42] determined the desired mesh space based on estimating the Hessian of the density field.
Similarly, Venditti and Darmofal [55] used the Hessian of the Mach number coupled with the DWR method to construct an anisotropic adaptation algorithm for
the Navier-Stokes equations.
Fidkowski [24] later generalized Venditti's approach
to higher-order discretizations based on high-order derivatives of the Mach number.
While shown to be successful, these adaptation algorithms heuristically assume that
the mesh anisotropy is fundamentally governed by the directional interpolation error
of one scalar quantity and do not take into account the anisotropy of the adjoint.
More recent research has sought to eliminate this heuristic assumption by focusing on anisotropic adaptation algorithms that more directly target the output error.
Formaggia [26, 25] used an output-based a posteriori error analysis combined with
Hessian-based interpolation error estimates to create an indicator that includes the
anisotropy of the element. Park [40] directly targeted the output error through local
mesh operators. Yano and Darmofal [58] proposed the Mesh Optimization via Error
Sampling and Synthesis (MOESS) algorithm, which performs adaptation by solving
a continuous constrained optimization problem that attempts to obtain a metric field
that minimizes the error estimate. In this thesis, we use Yano's MOESS framework
with modifications to the optimization algorithm.
18
1.4
Thesis Overview
This thesis describes the development of a high-order, h-adaptive, DG discretization
for the RANS equations coupled with the negative SA turbulence model. The thesis
makes the following contributions:
" Demonstration of the negative SA turbulence model as described in [2] in a DG
setting
" Modifications to the metric optimization step of the MOESS algorithm for increased robustness in the adaptation procedure
" Application of the adaptation algorithm to high-order discretizations of the
RANS equations for 2D and 3D turbulent flows
Chapter 2 begins with a review of the RANS equations and the SA turbulence model.
This chapter also details several modifications to the SA model recommended by Allmaras [2] to increase the general robustness of its discretization. The chapter concludes with a detailed description of the discretization of the RANS-SA system and
with a test case that demonstrates the increased robustness of the modified negative
SA model. Chapter 3 provides a review of the output error estimation and continuous mesh framework as background for the adaptation algorithm used in this work.
Chapter 4 reviews the MOESS framework and associated mechanisms as introduced
by Yano [58] and details modifications made to the metric optimization portion of the
original algorithm. Chapter 4 concludes with a demonstration of the modified optimization on representative model problems with comparisons to the original heuristic optimization procedure. Chapter 5 concludes with multiple two-dimensional and
three-dimensional turbulent test cases demonstrating the performance of the adaptive
algorithm. Chapter 6 ends with conclusions and suggestions for future work.
19
20
Chapter 2
Discretization of the RANS
Equations
This chapter describes the RANS equations, the baseline and negative SA turbulence
models, and the discretization of the entire coupled system.
2.1
The RANS Equations
The RANS equations are derived by applying Reynolds decomposition to the instantaneous compressible Navier-Stokes equations and then time-averaging the resulting
equation set.
Given a flow variable, f(x, t), we define the time averaging (Reynolds-averaging)
procedure, f(x), as
f(x) = lim
T-
oo Sto+
tO
f (x, t)
(2.1)
The flow variables can then decomposed into mean and fluctuating parts, e.g. the
density can we written as:
+ P
P=
(2.2)
where p' is the fluctuating part.
For compressible flows, we also define a convenient density-weighted time averag21
ing (Favre-averaging) procedure, f(x), as
-
f(x)
rlim
p(x, t)f (x, t)dt
P -oto,
(2.3)
Applying the time averaging procedure to the compressible Navier-Stokes equations, we get the form of the RANS equations used in this work:
(
=0
ai )= 2(p +pt)axj a- Xik ,
(2.5)
~-[
=
axj
[cp
Pr
+
a
Prt axj
(2.4)
3axx
(lk
+
a
axj
(2.6)
2i(p + pt)
s-82
-
-
at (Pii)1i ax3
at a+ axi
3 aXk
'ij
where p is the density, ui is the component of velocity in the i direction, p is the
pressure, e is the internal energy, h is the enthalpy, T is the temperature, sij =
1
Ou+
is the strain rate tensor, p is the dynamic viscosity, pt is the dynamic
eddy viscosity, Pr is the Prandtl number, and Prt is the turbulent Prandtl number.
In these equations, the Reynolds stress tensor that emerges from time-averaging
the flow equations is approximated using the Boussinesq eddy viscosity assumption.
The Boussinesq approximation relates the Reynolds stress to the mean flow viscous
stress tensor and introduces an additional unknown in the form of the eddy viscosity,
pt. Since equations (2.4) through (2.6) contain an additional unknown in the form
of this eddy viscosity, pt, the flow equations cannot be solved without an additional
equation for closure.
2.2
The SA Turbulence Model
The RANS system can be completed with an addition of a turbulence model that
describes the transport of the eddy viscosity, pt.
22
In this work, the SA turbulence
model is used. For reference, we present the baseline model as given in [49] and then
proceed to illustrate the negative variations [38, 2] used and compared in this work.
2.2.1
Baseline Model
The baseline model takes a form of a transport equation for a working variable, v,
which is related to the eddy viscosity by
x3
,
x 3 1=
where v = p/p is the kinematic viscosity.
x
(2.7)
-
1t = pufv 1
The SA working variable i; obeys the
transport equation
-D+T +-
DtP
1
[V.- ((v + 7)Vi) +cb2 (Vi'
2
]
(2.8)
where the production P, wall destruction D and trip term T are
P
= Cb1(I -
ft2) S,
D= (c1f -
ft2
T
[]
=
ft1(aU)2
(2.9)
S is the modified vorticity and is given as
V fv2,
K2d2
The function
fw
j
v2
1
x
+ xfVi
(2.10)
is
F
f g + C6 s
g = r + cw2(r' - r),
S2d2
(2.11)
The trip and laminar suppression terms are
[d 2 + gd2]
23
ft2
= ct3 exp (-ct4X 2
)
fti = ctigt exp (-ct2 IL
(2.12)
where d is the distance to nearest wall,
0.41, Cw1
1.2,c
4
= Cbl/
2
Cbl
+ (1 + Cb2)/O, c,2 = 0.3,
=
Cw3 =
0.1355, - = 2/3, Cb2 = 0.622, K =
2, cui = 7.1, cti
=
1, Ct2 = 2, ct3
=
= 0.5.
An equivalent compressible conservation form can be constructed by combining
the SA transport equation with the mass conservation equation,
a
(p ;)+
-(pUV) = p(P- D +T)+
1
C
]b 2
(+)p-V21
V -[p(v + D)V~] v)
+ ;
p
T
(2.13)
In this work, all cases are run fully turbulent with no effort to model transition
and as such, the trip terms are omitted in our discretization (T = 0).
2.2.2
Modifications to Baseline Model
The discretization of the SA equation given in (2.13) can permit non-physical solutions
that must be handled. Here we present the modifications proposed in [2] to handle
these situations and the final resulting conservation form used in this work.
Modified Vorticity
Physically, the modified vorticity S
S + 9 should always be a positive value such
that the production term is always a positive quantity. However, in a discrete setting,
the baseline modified vorticity can become zero or negative because the fv2 closure
function is negative over a range of X and this numerical phenomena can cause robustness problems. To remedy this possibility, a modified form of S is used in this
work,
I
~S
+9
3 ;>-c
=
S +
_
)S-S' S
2S~c+C,,,S
(c. 3-2c.
2
2
s
>_ -+C2S
< -C,2S
(2.14)
with c, 2 = 0.7 and cv3 = 0.9. This function results in a form of S that is identical to
the original formulation for 5 > 0.3S, is C 1 continuous and positive for all non-zero
S.
24
Negative iY Model
Analytically, the exact solution of (2.13) is non-negative given non-negative boundary and initial conditions. However, this property is not always obtained on coarse
grids and transient states where the discrete solution may exhibit negative turbulence
solutions.
To ensure that the eddy viscosity always obtains a non-negative value, the rela-
tionship in (2.7) is modified to be
>0
{Pvfv1i
0.0
(2.15)
< 0
0,
Since negative i values can emerge on these discrete solutions, a continuation
of the SA equation for negative E; must be constructed to deal with these potential
undershoots. The negative SA model used in this work is given by
DiDDE =P. -Dn
1
+ T+ - [V - ((V + Lfn)WD) + Cb2 (V ;)2]
(.6
where the modified production P.,, is given by
Cb(1
ft 2 ) V
Pn =
1(Cb1
-
ct 3)S
> 0
(2.17)
L; 5 0,
the modified destruction Dn is given by
Dn
(c,1.-%f2)
(Cwifw -
ft2 ) 'j
2
L; > 0
(2.18)
and the diffusion modification term fn is given as
1.0
fn =
0
0
25
(>
(2.19)
with ci = 16. This analytic continuation is C 1 continuous with respect to V at V = 0.
Furthermore, the model ensures that the original positive SA model is unchanged
for V > 0 and forces negative L transients back towards zero.
The model is also
constructed with the goal of mitigating nonlinearity.
Again, an equivalent compressible conservation form can be constructed by combining the equation (2.16) with the mass conservation equation,
C9
-(p)+V
t
1
- (puV) = p(P. - D,)[+ -
Cb.
[V]+
a
where the modified diffusion coefficient, n = pt(1 +
(V)2
21q-
-(v+
a
f,)Vp-VV (2.20)
xfn)
Comparison of Negative SA Models
We highlight the key differences between the model given in (2.20) and the negative
extension given in [38]. This earlier variant of the present formulation presents the
continuation in the same form as (2.20) and uses the same modified definitions of S
and pt given in (2.14) and (2.15) respectively. The only differences lie in the negative
extensions of production term P, and the modified diffusion coefficient q. The variant
provided in Oliver's work defines the production term as
P
C1(1 - f12)
cb1SLgn
>0
(2.21)
V < 0,
where
1000X22
gn = 1 -
(2.22)
1+ x
and defines the modified diffusion coefficient as
{q
tt (I X)
V(2.23)
p(T + X +oiX2) u
< 0,
The modified diffusion coefficients are plotted in Figure 2-1
26
200
-
Negative SA
--- Oliver SA
150-
-
100 -
0
-0
-15
-10
-5
0
5
4u
Figure 2-1: Comparison
of Modified Diffusion Coefficients
at
2.3
Spatial Discretization
We can re-write RANS equations coupled with the SA turbulence equation as a
general system of conservation laws of the form
OU+ V - Tc""(u) - V - (A(u)Vu) = S(u, Vu)
(2.24)
where u = [p, pui, pE, pL;]T is the state vector, Te"" is the inviscid flux, A(u)Vu is
the viscous flux, and S is the source term.
We seek a solution in a finite-dimensional approximation space Vhp defined on a
triangulation
Th
consisting of non-overlapping elements r, of characteristic size h of
the domain Q. Formally, the function space Vhp is defined as,
V P = fv E [L (Q)]''I
oVC
fq E [PP(ref)]', VK E T}
(2.25)
where r is the state rank, PP denotes the space of polynomials of order p and f,, denotes the q-th degree polynomial mapping from the reference element to the physical
element K. The weak form associated with the DG approximation of the conservation
27
laws now follows as, find
uh,p(ti)
Vhp
E
E Vh,p such that
+ lRh,p(Uhp, Vh,p) = 0,
(2.26)
VVhp E V
where Rh,p consists of convective, diffusive, and source contributions
JZh,p (Uh,p, Vh,p
2.3.1
conv(uh,p,
Vh,p)
p
(Uh,1 (Uhp,
Vh,p)
ERh1 purce (Uh,p,
Vh,p)
(2
Inviscid Discretization
The discretization of the inviscid terms is given by
p"(w,
zjvvT.Fcn"(w)+
v) =-
(v+ _
T- H(w+, w-, i+)
vT7b.-i
KETh
(2.28)
where (.)+ and (.) are values on opposite sides of a face f, n+ is the normal vector
pointing from
+ to -, H is the numerical inviscid flux for an interior face, and
yb
is
the numerical inviscid boundary flux. 1i and aQ are the set of interior and boundary
faces, respectively. In this work, the interior face numerical flux function uses Roe's
approximate Riemann solver [46] of the form
W-
+
2yonvw+
oe (W
H(w+, w-, n+)=
1~,,l++ -. ire(W+) + n- - co""(w-))+
|A"'+
-
y
+-I
W
(2.29)
where ARoe is the flux Jacobian matrix computed about the Roe's mean state. The
boundary flux
state
Ub
2.3.2
Fb
is in general a function of the interior state w+ and the boundary
which itself is a function of w+ and the prescribed boundary conditions.
Viscous Discretization
The viscous terms are discretized using the second method of Bassi and Rebay [7, 8].
To simplify the notation, we define the jump, [1, and average, {-}, operators for scalar
28
~)
s and vector F quantities as:
s (+_
)
[S+
(g+ +s-~) [N = (V+-'
6
6=
I(-+
2
+ V'-)
-
-
2
The Bassi and Rebay viscous flux can now written as
Rdiff(w v)
Z
j
Vv T - (A(w)Vw)
KETh
[Iw]W. {A T (w)Vv}
faj [(w+
-
-
[v]T- ({A(w)Vw} + 77
ub)T(AT(ub)v+)
.
-
+
f
ii,(w)})]
(w)
where Fv' is the numerical viscous boundary flux, Fr and rf are the auxiliary variables, and qf is the stabilization parameter. For all cases shown in this work, the
stabilization parameter 7f is set to a conservative value of 6.0.
2.3.3
Source Discretization
The source terms are discretized using the mixed form presented in [5] which is
asymtotically dual-consistent [39]. The discretization is given by
vV'- S(w, Vw + r"(w))
Rr"(wv)
(2.30)
KETh
where the global lifting operator rgo is given as
rhb(w) =
';if,,(w) +
2.4
'(w)
(2.31)
fEon
fEri
Temporal Discretization and Solution Technique
To fully define the spatial discretization, a basis must be selected for the space V[.
In this work, we use a nodal Lagrange basis that is element-wise discontinuous such
29
that the discrete solution has a solution of the form
Us(t)#j(x)
Uh(X, t) =
(2.32)
i=j
where U E RN. The spatially discrete system is now given as:
M-
dU
dt
+ R(U) = 0
(2.33)
where M is the mass matrix and R is the spatial residual vector. Since this work
focuses on steady problems, the temporal discretization is used only to improve the
performance of the solver by marching the solution in time from the initial state
U(0) to the steady state. In particular, a first-order backward Euler is used for time
integration such that the complete discrete system is given by
1
IM(Um+l - Um ) + R(Um+1) = 0
At
(2.34)
Newton's method is applied at each time step and the resulting linear system
is solved using the GMRES algorithm [47] with an in-place block-ILU(0) factorization [22] with minimum discarded fill reordering. For smaller linear systems, a sparse
direct solver (UMFPACK [20]) is used.
2.4.1
Numerical Results
We compare the presented negative SA model against the earlier negative variant
presented in [38] on a model problem from the NASA Langley Turbulence Modeling
Resource website.
Test Case - 2D NACA 4412 Airfoil Trailing Edge Separation
To demonstrate the effects of the modifications, an example case is solved using both
versions of the negative SA model.
The case is M.. = 0.09, Re, = 1.52 x 106,
a = 13.870 flow over a NACA 4412 airfoil. For each model, the RANS-SA system
30
105
-
P1
P2
P3
--- P1
-- -P2
---P3
-
i 10
ca
-0
-
Negative SA
Negative SA
Negative SA
Oliver SA
0UUI.I
-
4000 --
Oliver SA
Oliver SA
3000 .
..
-5
*s 10-
Negative SA
Negative SA
Negative SA
Oliver SA
Oliver SA
Oliver SA
R
E
I-
z
P1
P2
P3
- - - P1
- - - P2
- - - P3
-
20001000
-
-
-Vill
1010
0
20
40
Iteration
60
80
100
0
2
20
40
60
Iteration
80
100
(b) Computation Time
(a) Nonlinear Residual Convergence
Figure 2-2: Comparison of convergence for flow over a NACA 4412 using negative T
variants
is discretized using the discretization presented in Section 2.3 and the solution is
computed using p = 1, 2, 3 polynomials.
To focus only on the dependence on the
modifications to the turbulence model, UMFPACK is used to solve the linear systems.
Figure 2-2a shows the nonlinear residual history of the NACA 4412 case computed on
a coarse mesh (7168 elements). For the p = 2,3 solves, the presented i; modifications
mitigate the overall nonlinearity which helps to decrease the number of nonlinear
iterations required for convergence. Figure 2-2b demonstrates that this modified SA
equation does not noticeably modify the work required at each iteration. The drag
computed using the two models differs by only 0.02 percent.
The presented negative iY modifications have little effect on the converged solution
as evidenced by the final drag values. This is expected as both models preserve the
positive i portion of the SA equation. However, the presented model better mitigates
nonlinearity which results in a more robust scheme.
31
120
32
Chapter 3
Output Error Estimation and
Continuous Mesh Framework
In this chapter, we review the key ingredients required for the mesh adaptation algorithm used in this work. Specifically, we present the dual-weighted residual method
proposed by Becker and Rannacher [9, 10] to estimate the output error and the continuous mesh framework [34, 35] as a means to control this error.
3.1
Error Estimation
Let the output of interest be denoted by J = J(u), where u E V is the exact
solution to the governing PDE, and J(-) : V -+ R is the output functional. Given a
DG solution uhp E Vh,p, an approximation to the desired output is given by
Jh,p = Jh,(Uh&,)
(3.1)
where Jh,p : Vh,p -+ R is the discrete functional. The objective of the error estimation
is to approximate the true error in the output functional,
Etrue
J - Jh, = J(u) - Jh,p(uh,p).
33
(3.2)
In this work, the output error estimation is achieved using the the dual-weighted
residual (DWR) method proposed by Becker and Rannacher [9, 10]
3.1.1
Dual-Weighted Residual Method and Localization
In the dual-weighted residual method, the output error can be expressed as
Etrue = J - Jh,p = -lZhp(UhP,
V),
(3.3)
Vw E W.
(3.4)
where 7P E W = V + Vhp is the adjoint satisfying
j'[u, uh,],w,
Here, 1'
W x W -+ R and
7
=
W
-
[u,
[U) Uh,p](W),
R are the mean-value linearizations defined
by
j
[u, Uhp] (W, v) =
1Z'[( - 9)u + Ouhp(W, v)dO
[u, Uhp](W) = 1
'(1 -
)u + OUh,pI(w)dO,
where 1'[z](-,-) and J'[z](.) denote the Fr'chet derivative of
lh,p(,)
and Jhp()
with respect to the first argument evaluated about z. As equation (3.4) involves an
infinite dimensional space W and the exact solution u, the adjoint solution '0 E W
is not computable in general. In this work, for the purpose of error estimation, the
true dual solution 0 is estimated with an approximate adjoint
7ph,f
E Vh,p computed
Vvh, E Vf
(3.5)
from a linearization about Uhp,
1R',Z[uh,p](Vh,, #L'h,,3)
where Vp D Vhp where
=
J,f[u,],p][Vh,],
P = p + 1 is the enriched space. The DWR error estimate
using this surrogate adjoint is then given by
Etrue
~
-Rh~p
34
(Uh,p, VNh,fi)
(3.6)
For the purpose of adaptation, a localized error estimate is also defined for each
element r,
Rhp
(Uh~p,7
(3.7)
h,pi 1
A conservative error estimate for the output of interest is then obtained by the summation of the locally positive error estimates:
e
q
(3.8)
rET
3.2
Continuous Mesh Framework
While the DWR method gives a localized output error for each element, adaptation
further requires connecting this elemental error contribution to the element's size and
orientation. For this effort, we reformulate the anisotropic information for a simplex
K as a metric tensor MK, which is a symmetric positive definite (SPD) matrix that
encodes the element's size and orientation [27, 54].
elemental metric tensors, {MK}KETh,
metric field {M()} E-E
From the collection of these
a continuous spatially varying Riemannian
can be constructed [12, 34] which provides a continuous
interpretation of the discrete mesh.
Given a mesh Th, the field {MK}KET
is uniquely defined and reversely, given
a metric tensor field, a family of non-unique metric-conforming triangulations can
be constructed.
The output error for the DG discretization can be shown to be a
function of the metric tensor field [58] and this fact completes the foundation for the
adaptation process. A metric-based adaptation algorithm can now be constructed
that strives to reduce the output error by manipulating the continuous metric tensor
field and constructing the corresponding metric-conforming discrete meshes.
In the following two subsections, we review the continuous mesh framework [34, 35]
by more rigorously defining the duality between the Riemannian metric field and the
corresponding discrete mesh.
35
3.2.1
Metric-Conforming Meshes
A Riemannian metric field {M(x)}E 0 is a smoothly varying field of symmetric positive definite matrices on Q C Rd. The metric field introduces a distance function such
that the length of a segment ab from point a E Q to point b E Q under the metric is
given by:
fM(ab) =
/
abT M(a + bs)b ds
(3.9)
With this definition of length under the metric, we can now formally define what
it means for a discrete mesh to be metric-conforming.
A mesh conforms to a metric if each edge, e, of the triangulation is close to unit
length under the metric field {M()}xE-
and if every element,
ib,
satisfies a measure
of quality. Specifically, a metric-conforming mesh satisfies the edge-length condition,
I
fM(e) < vf2,
Ve E Edges(Th)
(3.10)
/d Eet(M(X))dX)
E [a, 1] with a > 0
(3.11)
and the element-quality condition,
2
()=
ZeEEdges(rc)
M
where QM is the element quality measure.
As noted before, for a given {M(x)}XE, a family of non-unique metric-conforming
meshes with similar geometric characteristics can be generated. In this work, we use
the Bidimensional Anisotropic Mesh Generator (BAMG) [11] developed by INRIA to
generate all two-dimensional metric-conforming meshes and Edge Primitive Insertion
and Collapse (EPIC) [37] developed by The Boeing Company for three dimensions.
For problems with curved geometries, the linear mesh is globally curved using linear
elasticity to properly represent high-order geometric information [38, 43].
36
3.2.2
Mesh-Conforming Metric Fields
Conversely, given a mesh Th, the discontinuous field {M},,ETh can be uniquely defined which then can be used to reconstruct a continuous metric field, {M(x)}XEQ,
represented by metrics associated with the vertices of the triangulation, {MV}VEv
where V is the set of vertices. That is, for a given tessellation, it is possible to find a
continuous metric field that conforms to the mesh.
We start with the construction of {M}KETh which is termed the element-implied
metric. The element-implied metric MK of a simplex element K is a metric under
which each edge of the element is unit length. Specifically, the element-implied metric,
M., is a SPD matrix such that,
'eTMe = 1,
Ve E Edges(r,)
(3.12)
For higher-order curved elements for which the implied metric spatially varies within
the element, we formulate a singular element-implied metric by taking the value from
the centroid of the element.
The vertex-based metrics are then calculated by performing an affine-invariant
average of the elemental metrics of the elements around the vertex in question, i.e.
MV = mean afm({M
re
(3.13)
(v)),
where w(v) is the set of elements surrounding the vertex v and the affine invariant
mean [41] is defined as,
meanaffinv ({M}KEw(v)) = arg min
M
E
log (M-1/ 2 MM-1/ 2 ) 2|$
(3.14)
KEw(v)
Finally, we define a continuous metric field over an element K as a weighted affine
invariant mean of the vertex metrics,
M(x)
=
w,(x)I|log (M-1/2MM-1/ 2 ) 1|2,
argmin
M
vEV(s)
37
x E
K
(3.15)
where w,(x) is the barycentric coordinate corresponding to the vertex v.
With this recovery algorithm, we are now able to reconstruct a continuous metric
field given a discrete mesh. The geometric duality described here serves as the foundation for the adaptation algorithm detailed in Chapter 4 in which we manipulate the
continuous metric description of the current mesh to generate a metric-conforming
mesh for the next iteration in an attempt to lower the output error.
38
Chapter 4
Metric Field Optimization using
Local Error Sampling and
Synthesis Framework
In this chapter, we first review the metric optimization framework proposed by Yano
and Darmofal [59] used in this work.
We then present modifications to the opti-
mization process to increase the robustness of the global adaptation process and
demonstrate the modifications on some simple 2D test cases.
4.1
Model Definition
The adaptation framework used in this work is the Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm developed by Yano and Darmofal [59]. In the
following sections, we review the MOESS algorithm with a focus on the optimization.
of the resulting statement.
4.1.1
Continuous Relaxation
The overarching goal of mesh adaptation is to iteratively improve the mesh or triangulation Th to obtain a better output prediction. To eliminate trivial solutions of
39
arbitrarily refined meshes, we are only interested in "better" triangulations smaller
than a specified degrees of freedom (DOF). This goal can be stated as an optimization
problem to find the optimal triangulation
7* that minimizes the error subject to a
degree of freedom constraint:
T* = arg min e(Th)
subject to C(*Th) < doftarget
(4.1)
7h
where e(-) is the error functional and C(-) is the cost functional that computes the
number of degrees of freedom for a given Th. Since a mesh is defined by nodes and
their connectivity, the optimization problem as formulated above is a discrete problem
and as a result is largely intractable. Here we use a continuous relaxation technique of
this problem proposed by Loseille and Alauzet [33] in which the discrete triangulation
is described by a continuous metric field A4
{M(x)}E.-. The relaxed optimization
problem's objective now becomes, find the optimal metric field A4* that minimizes
the error subject subject to the same DOF constraint:
M4* = argmine(.A4)
subject to C(M) < doftarget
(4.2)
In view of the continuous mesh framework, the cost functional C(-) is now given as
C(M) = jf
det(M(x))dx,
(4.3)
where cp is the degree of freedom associated with a reference element normalized
by the size of the reference element. The coefficients associated with triangular and
tetrahedral elements are
2(p + 1)(p + 2)
and
c" = / (p+ 1)(p + 2)(p + 3)
(4.4)
To estimate the error functional, e(M), we use a locality assumption such that the
total error functional results from the sum of the local elemental error contributions
40
in
and that each of these contributions is a function of the elemental metric tensor:
E(A4) ~:- E 77r(Mr.)
4.1.2
(4.5)
Metric Manipulation Framework
In our mesh adaptation procedure, we are fundamentally interested in manipulating
the metric field in order to obtain a mesh that better approximates the true solution.
More specifically, we are interested in controlling the change in the approximation in
a given direction. In the metric framework, this approximation difference can be seen
as the change in the directional lengths measured under the metric fields:
h(e;M) =
h(e; Mo)
/eTM1/2e\
1/2
(4.6)
eTM1/2e
Standard Euclidian manipulation of the entries of the metric tensors (i.e.
M
M =
6M) proves unsuitable as the updates 6M do not strongly correlate with
changes in this approximability. We instead use a tangent vector S E Symd which
arises from endowing the tensor space with an affine-invariant Riemannian metric [41]
and allows us to more strongly control the modifications to the directional lengths.
The change in the metric tensor from M 0 to a new configuration M under this affine
invariant framework is given as,
M(S) =M1/ 2 exp(S)M
1 2
(4.7)
/
where exp(-) is the matrix exponential. The fractional change in the directional length
is bounded by the magnitude of S,
exp (-
\(41Amhe;n(()S))
IISIi2) <{exp1 (-Amax(S)
2
2
~h(e; M0)
< exp
(2(
IAmin(S)
< exp
/1iiIF
(4.8)
where 11-I|F, is the Frobenius norm, such that this choice of S (referred to as the step
matrix in this work), allows us to control the change in the directional approximability
41
SF
by controlling its magnitude.
4.1.3
Cost Model
With the metric manipulation framework introduced in 4.1.2, an element-wise cost
model p,, in terms of the step matrix is constructed by directly integrating the continuous local cost function over an element:
p.(S.) =
c,
det (M2 exp(S-)M 2 = pK 0 exp
tr (S.)
(4.9)
The global cost is now simply the sum of these elemental cost contributions:
C(.M) = C(S.) = E
4.1.4
p. (Sn)
(4.10)
Surrogate Error Model from Local Error Sampling and
Synthesis
As the error functional is generally not known, a surrogate model is constructed from
a sampling procedure. This construction is achieved by sampling how the elemental
error changes with respect to changes in the element's configuration. For an element
eE
(2, we consider i E [1, ..., nconig] configurations formed by locally splitting the
edges to obtain subdivided meshes ni. By convention, no corresponds to the original
configuration.
For each ith configuration, an elemental-wise local problem is then
solved: find U" E Vhp(i) such that
, , ='0, VVri E Vh,p(ri),
where the local semilinear form
(,
(4.11)
prescribes the boundary fluxes on ri by
assuming the solution on the neighboring elements does not change. We then prescribe
a localized error estimate corresponding to the subdivided mesh ri by recomputing
42
the localized DWR error estimate as
(4.12)
94 lht,p(Uh,,
1=
V#h,P1.e) I
Each subdivided mesh is associated with an elemental metric M, by performing an
affine-invariant average of the implied elemental metrics of the newly formed elements.
As a result, the sampling procedure generates a set of metric-error pairs, {MKq,
7}.
As introduced in 4.1.2, we can now characterize the change from the original metric
tensor Mno to the new configuration MK, using the affine invariant framework:
=
log (M-1/ 2MniMA
1 /2
)
,
i
=
1,..., nconfig
(4.13)
Similarly, we measured the associated changes in the error as
fa, = log (77,, /q
)
i = 1, ... , nconfig
(4.14)
Thus with the affine invariant framework, we can construct step matrix and error
change pairs {Sri,
fa}
which now be used to construct our surrogate error model.
The MOESS algorithm constructs a linear error function of the form,
fr(Sx) = tr(RnS.).
where RK is synthesized from the pairs {S,.,
f,,,}
(4.15)
through least-squares regression.
The local error model in terms of S, is then given as:
7 (S.) = qA exp(tr(RS))
(4.16)
This local error model effectively represents how the local error changes with respect
to local element shape and size changes as encoded by SK.
43
4.2
Optimization of Surrogate Model
The last step of the adaptation process is to perform an optimization of the Riemannian metric field {M}sEQ with the constructed surrogate cost and error models
to minimize the error. Consistent with the continuous mesh description provided in
Section 3.2.2, {M}XEQ is described by the vertex metric tensors {M}vEV. We again
describe the change from the original vertex metric tensor by using the affine invariant
framework such that,
M,(S,) = M0KJ exp(S )M
,
(4.17)
where S, E Symd is the vertex step matrix. Thus the objective becomes to find the
vertex step matrices that minimize the estimated error. As the surrogate error and
cost models given in (4.16) and (4.19) respectively are in terms of the element step
matrix SK, we require a relationship between S,, and S, to transform these models
to be in terms of the the design variables. For this work, we construct the elemental
step matrix SA, via an arithmetic average of vertex step matrices:
(4.18)
V
Sr = {SV}vEv() =
VEV(rK)
Substitution of this relationship into the cost model yields the cost constraint in terms
of the vertex step matrices:
Pr. ({Sv}vEv(K))
C({=
(4.19)
r.E Th
Similarly, the error model used as our objective function becomes
6(f{SV}VEV)
=
E
7
(f{SV}VEV~r)
(4.20)
Since this error model is constructed through local samples taken from the original
mesh, the allowable change in the metric field in one adaptation iteration must be
limited. As specified in (4.8), the choice of the step matrices as the design variables
44
allows us to control the change in the approximability in any direction by controlling the magnitude of the matrix. For this effort, we apply constraints to limit the
magnitude of S,:
ISvlF 5a
v
(4.21)
Since the presented local splitting procedure effectively explores fractional directional
length changes of 2, we require that the requested edge length h(e; M (S)) satisfies:
h(e; M (S)) <2
2
h(e;kMo)
-
(4.22)
-
1
Therefore, we limit the change in approximability to 2 in any direction by setting
a = 2 log(2)
The surrogate optimization problem for the optimal Riemannian metric field is
now given as:
{S*}
= arg min E({SV} EV)
s.t. C({Sv}ve)
ISvIIF < a,
(4.23)
dOftarget
(4.24)
Vv E V
(4.25)
The gradient of the error and cost functions with respect to a given vertex step
matrix S, is
6C
je
r.
({S V~ ovE()
r.Ew(v)-
Vr)
R.
(4.26)
.
6SV
{S}VEv(x)) 21V(r,)l
(4.27)
where w(v) is the set of elements that have v as one of their vertices.
4.2.1
Heuristic Optimization of Surrogate Model
We now review the heuristic optimization method presented by Yano [58] and identify
some limitations of the method that this work later addresses.
45
Summary of Algorithm
For convenience, the step matrix S, is decomposed into the trace and trace-free parts,
S, = s,I + S
where sv = tr(S,)/d and
5,
(4.28)
is the trace-free part of S,.
The heuristic method approaches the optimization problem by initially assuming
that the current configuration is sufficiently close to the optimal configuration such
that the constraints given in (4.25) are inactive. Via steepest descent type updates,
the approach attempts to distribute the available mesh degrees of freedom such that
the investment to any element results in the same marginal improvement .in the error
and proceeds to change the trace-free part of the step matrix in an attempt to achieve
stationarity with respect to shape change.
The specifics of the heuristic algorithm to approximately solve the optimization
problem are as follows:
0. Set Js = a/ntep
/65
1. Compute vertex derivatives, &/6s,
1
6,s, local Lagrange multiplier
A_ = (6,/6sv)/(6C/6sv) about {Sv}VEv
2. Update the isotropic part of Sv according to:
" Refine the top 30% of the vertices v with the largest AV by setting Sn+1/3
=
Sv +6sI
"
Coarsen the top 30% of the vertices v with the smallest AV by setting
Sn+1/3 = S-
-6sI
3. Update the anisotropic part of S, according to
+
4. Rescale Sh+21 3 according to Svn
where 0 is selected to obtain a
metric field with
5. Set n
=
n +1.
=
S n+2/3 +
doftarget
Ifn < nstep go to step 1.
46
01
S+ 1 /3-6s(6/65)/(6/6s)
Limitations
The algorithm as presented above has been shown to automatically and efficiently
generate metric requests to better resolve outputs of interest with no a priori assumptions on a wide range of problems [19, 56, 60]. However, this procedure exhibits
a few limitations which can inhibit the realization of robust automated adaptation.
The heuristic isotropic update shown in step 2 of the algorithm performs well
when the Lagrange multipliers are well distributed such that the vertices with high
sensitivity in the local error with respect to added degrees of freedom are refined and
the vertices with low sensitivity are coarsened. However, since the selection process
for coarsening and refining are independent of the actual distribution, this procedure
can exhibit adverse behavior when the error is dominated by a small percentage of
the total vertices.
In this case, the vertices with high sensitivity will be properly
selected for refinement but the 30% selection protocol will also assign refinement to
vertices that should either be coarsened or untouched. The converse effect can occur if
there are many vertices with large Lagrange multipliers such that the process coarsens
vertices with high sensitivity.
Along these lines, in the extreme case where R, = 0 such that A, = 0
Vv E V,
the presented heuristic algorithm will still apply coarsening and refinement factors to
the current metric even when there is no sensitivity.
Another potential robustness issue is the fact that the updates and the final resulting step matrices do not necessarily remain within the error sampling space. Although
the Js factor is sized such that the trace of every step matrix remains bounded, the
resulting step matrix after the anisotropic update, Sj
2 3
1
, has no guarantee to satisfy
the edge length constraint given in (4.22). The final scaling step further exacerbates
the problem if the initial mesh is much smaller with respect to the doft.get.
4.2.2
Modifications to Metric Optimization Procedure
To alleviate some of the identified issues with the heuristic optimization procedure,
we choose to employ a formal gradient-based optimization algorithm in lieu of the
47
heuristic method to exactly solve the non-linear optimization problem. Here we are
only interested in modifying the optimization process and not the underlying surrogate modeling. However, a straight forward application of a nonlinear optimization
algorithm on the problem as posed in (4.23) proves to be largely intractable as the
number of non-linear constraints scales with the number of vertices. In this section,
we explore alternate optimization statements that satisfy the necessary requirements.
We are specifically interested in the following properties for our optimization for
practical and robust output-based mesh adaptation:
" The problem must be able to be solved in a reasonable amount of time. Since the
number of design variables S, will naturally scale with the number of vertices
of the initial configuration, this property requires that the number of non-linear
constraints remain constant with increasing mesh size.
" For robustness, the design space of the optimization must be within the error
sampling space. That is to say, the final resulting metric request must satisfy
the edge length constraint given in (4.22).
" To minimize the number of adaptation iterations necessary to obtain an optimal
mesh, the design space of the optimization should be as large as possible while
again satisfying the edge constraints.
Metric Optimization with Global Penalty Term
To avoid the issue of a growing number of non-linear Frobenius norm constraints, we
replace the individual vertex norm constraints with a penalty term that is added to the
objective function. This penalty term consists of a penalty parameter p, Vv E V that
is multiplied by the measure of the constraint violation. To avoid possible numerical
issues with the derivatives when
IISVIIF
=
0, the penalty is placed on the square of
the Frobenius norm. The surrogate optimization problem for the optimal Riemannian
48
metric field using this penalty method is now given as:
= argmin ({SV}
{S}vk
) +
{Sv}Vv
s.t.
pvO(| Sv|1
- (2 log(a)) 2 )
(4.29)
VEV
C({Sv}VEv)
< doftarget
(4.30)
where p is a penalty vector of length IVI and 4(x) is a simple quadratic penalty term
given as:
O(W =
0,
S2,
if
<0
(4.31)
if X > 0
This penalty function penalizes the objective function whenever a magnitude of a
vertex step matrix grows larger than the allowed bounds.
However, a formal optimization of (4.29) can result in spurious shape change requests
which do not noticeably improve the solution and can introduce unnecessary difficulties to the mesher. According to the MOESS surrogate error model, if R, =
0, the
error can always be reduced by choosing a 'k such that tr(RKSK) < 0. In practice,
this characteristic of the error model leads to maximum possible shape changes in
almost every vertex metric even when the respective error reduction is insignificantly
small. As such, we include an additional global constraint that forces the optimization
process in some sense to focus on metric changes to those vertices with the highest
error sensitivities:
||SvlI < #|VJ(2log(a))2
(4.32)
vEV
If 3 < 1, this constraint introduces a global limit to how much the original configuration can change measured by the sum of squares of the Frobenius norms and
effectively forces the optimization to focus only on prescribing changes that most
reduce the error.
49
The final optimization statement is now:
{sv*}Isy= arg min E({S}vE) + E
{SV}vEV
s.t.
pVq(HISv |12 - (2log(a)) 2)
(4.33)
vEV
C({S}VEv)
5 doftarget
S S<I
I VI(2log(a))
2
(4.34)
(4.35)
VEV
With this method, we now solve a series of unconstrained (with respect to the
edge length constraints) optimization problems and increase the penalty parameter
if needed after every iteration. The algorithm to solve the optimization problem is as
follows:
0. Set p = pinit and {Sv}vE
= 0
1. Solve (4.33) to obtain {S*} Ev
2. Post-process {S*}VEv to test if (4.22) is satisfied with the resulting step matrices.
Here we present two possible post-processing tests:
(a) Eigenvalue Method: Calculate A (Sv) and test if IA (S,)
< 2 log (a).
(b) Edge-Based Method: Calculate h(ei; Mv) = h(ei; MOJ
exp(S *MOK)
where ej are the eigenvector directions of MO,v and test if the requested
edge lengths satisfy (4.22).
We note here that the edge-based method is less conservative in that the limit
on the change in approximability might not be satisfied in every direction e.
The eigenvalue method however does ensure this constraint is satisfied in every
direction.
3. For each S* that breaks the constraint, increase the respective penalty term by
a prescribed factor: pv = 0 pv where 0 > 1.
If no S* breaks the constraint, exit with {MI}Mv
=
4. Initialize {SV}vEV = {Sv*}VEV and go back to step 1.
50
{M
/2
exp(S* )M1/2}v-
For the adaptive cases presented in this work, at each iteration, the initial total error
is normalized to unity and the initial penalty vector pi
is set to 1 x 10-. Unless
otherwise noted, the eigenvalue post-processing method is used.
For solving the actual optimization problem for a given p, we use the globallyconvergent method-of-moving-asymptotes (MMA) algorithm [51] as implemented in
NLopt [29]. For convenience, in this work, we refer to the optimization statement
as given in (4.33) and the respective solution procedure as the gradient-based metric
optimization method.
4.3
Numerical Results
We present numerical examples of applying the MOESS adaptation algorithm with
the gradient-based optimization method to select problems. We first verify the ability
of the modified adaptation algorithm to produce optimal meshes in the L 2 error
control setting. We then move to two-dimensional aerodynamic problems with which
a comparison between the presented algorithm and the heuristic optimization method
as presented in Section 4.2.1 is made.
4.3.1
r'-Type Corner Singularity
We start by applying the gradient-based optimization method to a simple L 2 -projection
problem with a canonical singularity for verification of the modifications to the
adaptation algorithm. The L 2 approximation problem consists of finding a solution
Uh,p
E Vh,p that minimizes the square of the L 2 projection error:
Uh,p= arg min
Vh,p
(U - vh, )2dx.
(4.36)
f
For this case, we consider a general form of the singularities found at geometric corners
of solutions to elliptic equations given by
u(r, 0) = r' sin [a(O + Oo)]
51
(4.37)
where r is the Euclidean distance from the corner, a > 0 is the singularity strength,
and 0 is the offset angle. The optimal mesh for this r' function for a degree-p
polynomial approximation as shown by Yano [58] consists of isotropic elements with
a size distribution given by
h(r) = Cr-
(4.38)
We apply the gradient-based adaptation algorithm to the L 2 projection problem of
the r' corner singularity function with a = 2/3 and compare the resulting optimized
h distributions to the analytically derived distributions. For each solution order p,
the numbers of degrees of freedom considered are:
p = 1,
doftarget
p = 3,
dOftarget =
= {600
900},
{2000
3000},
For the case presented here, the adaptation process is started on a square isotropic
mesh with the solution's L 2 error as the output of interest.
Figure 4-1 shows the resulting distribution of h against r for the optimized meshes.
Here, the element size h is calculated based on the volume (h = det(M )- 1 / 4 ) and the
distance r is measured from the corner to the centroid of the element. The optimal
valies of h and r vary linearly in the log-log space with an optimal grading coefficient
of kan" = 0.44 and 0.67 for p = 1 and 3, respectively.
The meshes produced
through the adaptation procedure exhibit grading factors of k = 0.45 and 0.66 for
p = 1 and p = 3, respectively with 300 elements and demonstrate that the modified
adaptive algorithm is able to obtain the optimal p-dependent grading automatically.
4.3.2
RAE 2822 Transonic RANS-SA
To characterize the differences between the two presented metric optimization methods on aerodynamic problems, we first consider a transonic RANS flow that exhibits
various flow features with a wide range of scales.
Specifically, we consider turbu-
lent transonic flow over an RAE 2822 airfoil with a freestream Mach number of
52
100
10
k =0.47
0.45
kanalt= 0.44
1010-2
10 -1
,2.
10
*.
-elem
10
34-
10
= 200
elem = 300
.--
100
1
10-
r
-elem
-
10-3
elem = 200
= 300
-
10
10-
k= 0.66
k =0.66
kanayli = 0.67
10
100
r
(a) p =1
(b) p = 3
Figure 4-1: Element size h versus the distance of the element centroid from the corner
for optimized meshes for r' singularity problem with a
=
2/3
M... = 0.737, a Reynolds number of Rec = 6.5 x 106, with an angle of attack of 2.79'.
The RANS equations are solved using a p = 2 DG discretization and the meshes are
adapted for drag on the surface with a doftarget = 80, 000.
Figure 4-2 shows the error estimate and degrees of freedom for 20 iterations of
adaptation using the two optimization methods. Both procedures are able to significantly lower the error estimate over the course of the adaptation history to effectively
obtain the same estimated error level but do so along very different trajectories.
The progression of meshes generated from the heuristic and the gradient-based optimization methods can be seen in Figures 4-4 and 4-5 respectively.
The heuristic
optimization method very aggressively refines the elements in the locations where the
MOESS error model at that iteration detects high localized error contributions.
It
is clear from visual inspection of the initial mesh to the mesh of iteration 2, that
the edge lengths of the triangulation change much more rapidly than the factor of 2
sampled during the local solve procedure.
Due to this nature of the metrics extending beyond the sampling space, the transition meshes more frequently contain spurious shape features that significantly change
over multiple adaptive iterations. This phenomena is exemplified by meshes 3 through
15 on which the spatial location of the clustered anisotropy for the shock significantly
53
10,
x10
Gradient-Based
Heuristic2
\-
10-
10'
10
10
-
o4
o-e
.
10-5F0
2-x
-
10'
0
2
4
6
8
10
12
14
16
18
20
0
Adaptation Iteration
2
4
6
8
10
12
Adaptation Iteration
(a) Error Indicator vs Adaptation Iteration
-
-0-
0
14
Gradient-Based
Heuristic
16
18
(b) Degrees of Freedom vs Adaptation Iterati on
Figure 4-2: Comparison of error indicator and degrees of freedom history for transonic
flow over a RAE 2822 Airfoil
shifts as flow becomes better resolved. This aggressive nature of the heuristic updates
also pushes the global degrees of freedom to rapidly increase as seen in Figure 4-2b.
Furthermore, since each iteration's primal initialization is performed with a projection of the previous iterate's converged primal state, the rapid changes in the
triangulation can result in significant difficulties during the primal solves. Figure 43a displays the cumulative computation time taken to obtain a primal solution on a
given mesh; this quantity includes the time taken for every primal and dual solve for
the previous iterations. At adaptation iteration 3 for the heuristic process, this graph
reveals that the primal solve requires a significant amount of computational effort for
convergence due to a poor initialization that stems from rapid changes in the mesh.
The mesh sequenced generated from the gradient-based optimization method is
instead characterized by smooth transitions from mesh to mesh as a result of the strict
enforcement of the edge length constraints. Although the process takes a significantly
larger number of iterations, the lower degrees of freedom during most of the adaptation
history coupled with better initializations due to slower changes in the mesh allow
for all 20 iterations to finish before the completion of the 13th iterate of the heuristic
process. As shown in Figure 4-3b, both methods result in similar error indicator levels
with roughly the same amount of computation time.
54
20
10
14
-
-0;
12
Gradient-Based
Heuristic
-
--
Gradient-Based
-0- Heuristic
10
0d
E
100
10
-
S20
4-
10-
E
20
e
2
4
6
8
10
12
14
Adaptation iteration
16
18
20
0
2
4
6
8
10
12
Cumulative Computation Time (s)
(a) Cumulative Time vs Adaptation Iteration
(b) Error Indicator vs Cumulative Time
Figure 4-3: Cumulative computation time taken to obtain primal solution and error
indicator estimate on a given iterate's mesh for transonic flow over a RAE 2822 airfoil
4.3.3
Supersonic flow over NACA 0012
To quantify the importance of finely controlled mesh adaptation for aerodynamic
flows, we consider a supersonic Euler flow over a NACA 0012 airfoil. The freestream
Mach number is Mo, = 1.3, and the airfoil is at 0' angle of attack.
equations are solved using a p
=
The Euler
1 DG discretization and the meshes are adapted for
drag on the surface with a dOftarget = 20, 000.
Figure 4-6 shows the drag and drag estimate behavior for 20 iterations of adaptation using the heuristic optimization method. The solutions generated through this
method exhibit significant oscillations in both the actual Cd and the Cd error indicator
values; the last five solutions of this sequence exhibit a
Cd
spread of over eight drag
counts. The lack of convergence in the error estimate indicates that the adaptation
process is unable to settle on a family of optimized meshes characterized by similar
metric fields.
Figure 4-7 shows the history that results from starting the adaptation procedure
from the same initial mesh but using the gradient-based metric optimization method.
Here, the generated sequence of solutions progressively reduces the error estimate
until a family of optimal meshes with similar error levels is obtained. These results
demonstrate a distinct lack of Cd oscillations relative to the heuristic results with a
55
14
X1
Cd
(a) Initial Mesh
(b) Iteration 1
(c) Iteration 2
(d) Iteration 3
(e) Iteration 4
(f) Iteration 5
flll 11
ME-
(g) Iteration 10
777
(h) Iteration 15
(i) Iteration 20
Figure 4-4: Mesh adaptation history for transonic flow over a RAE 2822 Airfoil using
the heuristic metric optimization method
spread of less than 0.5 drag counts over the last 5 iterations. Since the formal metric
optimization procedure stays within the error sampling space, the gradient-based
method more robustly settles at this family of optimal meshes.
Select drag-adapted meshes generated through the heuristic optimization method
and their respective Mach number distributions can be seen in Figures 4-8 and 4-9.
The meshes generated through the heuristic method visually demonstrate the inability
56
(a) Initial Mesh
(b) Iteration 1
(c) Iteration 2
(d) Iteration 3
(e) Iteration 4
(f) Iteration 5
(g) Iteration 10
(h) Iteration 15
(i) Iteration 20
Figure 4-5: Mesh adaptation history for transonic flow over a RAE 2822 Airfoil using
the gradient-based metric optimization method
of the process to converge on an optimal mesh for this highly mesh-sensitive problem.
For this particular flow, both the bow shock and the oblique shock contribute to
the difficulties encountered by the mesh adaptation scheme as the meshes exhibit
significant differences in the resolution of the features.
Figure 4-10 displays two
optimized meshes with similar error indicator values generated through the gradientbased method. As exemplified by these very similar meshes and their resulting Mach
57
0.1
0.1
*dolm
dottargal
0.099
0.098
~~~~-~O~04M"-" -"'* M-2' -0000
0,010
0.097
0O
15000
0.096
C
2
2
CP 0.095
U-
0.001.
0
0
0.094
-
*
0.093
0.0001
10000
4000
0.092
0.091
n nQ,
5
10
15
0
20
5
(a)
Cd
10
15
20
Adaptation Iteration
Adaptation Iterations
vs Adaptation Iteration
(b) Cd Error Indicator and Degrees of Freedom vs
Adaptation Iteration
Figure 4-6: Adaptation history for supersonic flow over a NACA 0012 Airfoil using
the heuristic optimization method
0.1
0.1
.
-
-dot
2964
0.099
0.098
0.014
0.097
CU
0.096
Q
LO 0.095
15000
L-
0
0.001
2
0.094
10000
L,
0.093
0.0001
5000
0.092
0.091
"
0.09
W-
5
10
15
20
0
Adaptation Iterations
(a)
Cd
5
10
15
26
Adaptation Iteration
vs Adaptation Iteration
(b) Cd Error Indicator and Degrees of Freedom vs
Adaptation Iteration
Figure 4-7: Adaptation history for supersonic flow over a NACA 0012 Airfoil using
the gradient-based optimization method
number distributions, the gradient-based method is able to reach and settle on this
set of optimized triangulations.
58
(a) Mach number distribution, iteration 17
0.1
0.2 0.3 0.4 0
0.8 0.7 0.8 0,9
1
1.1
(b) Mesh with Eind
4.3 x
(d) Mesh with Eind
4.8 x 10-5, iteration 18
10-4,
iteration 17
1.2 1.3 1.4 1.5
(c) Mach number distribution, iteration 18
Figure 4-8: Select adapted meshes (iterations 17 and 18) and respective Mach number
distributions generated with the heuristic optimization method, NACA0012 supersonic flow.
59
(a) Mach number distribution, iteration 19
0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8
0.9
1
1.1 1.2
1.3
(b) Mesh with
Eind
= 6.5 x
10-4,
iteration 19
1.4 1.5
(d) Mesh with Eind = 6.9 x 10- 5 , iteration 20
(c) Mach number distribution, iteration 20
Figure 4-9: Select adapted meshes (iterations 19 and 20) and respective Mach number
distributions generated with the heuristic optimization method, NACA0012 supersonic flow.
60
0.1
0.2
0.3
04 0.5
0. 0.7 0.8 0.9
1
1.1 1.2 13 1.4 1.5
(a) Mach number distribution, iteration 19
01
02 0.3 0.4
0.5 0.6
07
0.8 09
1
1.1
2 1.
(b) Mesh with Eind
4.5 x
(d) Mesh with
3.9 x 10-5, iteration 20
10-5,
iteration 19
1.4 1.5
(c) Mach number distribution, iteration 20
Eind
Figure 4-10: Select adapted meshes and respective Mach number distributions generated with the gradient-based optimization method, NACA0012 supersonic flow.
61
62
Chapter 5
Turbulent Aerodynamic Problems
In this chapter, we demonstrate the overall robustness of the discretization and modified adaptation procedure by applying the resulting adaptive solver to a range of
two-dimensional and three-dimensional turbulent aerodynamic problems.
5.1
Three-element MDA 30P-30N, Subsonic
We first consider a two-dimensional turbulent flow over a three-element McDonnell
Douglas Aerospace (MDA) airfoil (30P-30N), with a freestream Mach number of
moo= 0.2, an angle of attack of a = 16', and a Reynolds number based on chord of
Re,
=
9 x 106. The initial coarse two-dimensional mesh used for this case is shown in
Figure 5-1. The curved airfoil geometry is represented using q = 3 simplex elements.
Figure 5-1: Initial mesh, Three-element MDA flow.
For this flow, the inflow and outflow boundaries are specified via characteristic
63
farfield boundary conditions. An adiabatic no-slip condition is imposed on the surface
of the airfoil.
For each solution order p, the numbers of degrees of freedom considered are:
p = {1, 2},
dOftarget = {40000
60000
90000
120000},
For each p-doftarget combination, a family of optimized meshes are generated starting
from the shown initial mesh. The performance of each p-doftarget is assessed by averaging the error obtained on five realizations of meshes in the family. Adaptation is
performed on the drag on all three-elements of the airfoil. Figure 5-2 shows the resulting drag error against the number of degrees of freedom. The p
=
2 discretization
0-2
102
-X-P1
-e-P2
10-3w
W-
1.88
10-
10 12.7
10
3.87
'
10
-2.5
10
h = 1/(dof)" 2
10
3
-23
Figure 5-2: Convergence of drag output for the three-element MDA flow; reference
value obtained from p = 3, dOftarget = 250k adapted solution
outperforms the p = 1 discretization for the entire range of
doftarget
considered. The
coarsest p = 2 solution containing approximately 40, 000 degrees of freedom exhibits
a lower error level than the finest p = 1 solution with roughly 120,000 degrees of
freedom. The respective optimized meshes can seen in Figure 5-3; while both meshes
focus on resolving the same physical phenomena, the p = 2 discretization is able
to more accurately calculate the drag on a much coarser mesh. For this relatively
smooth case, the optimal output error convergence rates of c xV
are recovered for
both the p = 1 and p = 2 discretizations.
The adaptation algorithm applied to this complex two-dimensional case not only
64
(a) p
(b) p
1, doftarget
2, doftarget
120k mesh with E = 3.11 x 10-4
40k mesh with
E
3.28 x 10-4
-12
-
Figure 5-3: Comparison of adapted p = 1 and p = 2 meshes with similar error levels,
Three-element MDA flow.
-10.
-10-
-8-
-8-
-6-
-6-
-4
-4
-2
-2
0-
-0.2
0
0
0.2
0.4
06
0.8
1
1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x/c
x/c
(a) c, distribution of p = 1, doftaget = 120k mesh
(b) c, distribution of p
Figure 5-4: Pressure coefficient comparison of adapted p = 1 and p
similar error levels, Three-element MDA flow.
2, doftarget = 40k mesh
=
2 meshes with
targets the boundary layers formed on every element but also the off-body features
that prove crucial for resolving the interaction between the elements.
As shown
in Figure 5-6, the process results in significant anisotropic refinement in both the
boundary layers and the wakes to effectively capture the interactions of slat with the
65
main element and the main element with the flap. Furthermore, a far field view of
0.75
0.65
0.55
0.45
0.05
(a) Mach number distribution
(b) Optimized mesh
Figure 5-5: p = 2, doftarget = 120k adapted mesh and respective Mach number
distribution, overview, Three-element MDA flow.
this optimized mesh (Figure 5-6) reveals significant refinement along the stagnation
streamline as well as the wake.
These meshes showcase the importance of mesh
adaptation in complex flows; the spatial distribution of the element's size and shape
required to effectively capture the boundary layer phenomena and the flow interaction
of the airfoil elements proves difficult to predict a priori. The combination of mesh
adaptation and the p
=
2 high-order discretization shows significant advantages over
the p = 1 discretization for the flow considered here.
5.2
3D Zero Pressure Gradient Flat Plate, Subsonic
We now consider a M, = 0.2, ReL = 5 x 106 turbulent flow over a three-dimensional
flat plate. The problem is produced by extruding a two-dimensional case into the third
dimension. The baseline two-dimensional problem contains a flat plate of length 2L
66
0.75
0.65
0.55
0.45
0.35
0.25
0.15
0.05
(a) Mach number distribution
(b) Optimized mesh
Figure 5-6: p = 2, doftarget = 120k adapted mesh and respective Mach number
distribution, far field, Three-element MDA flow.
preceded by a short lead-in section of length IL such that the leading edge of the flat
plate does not start directly at the inflow wall. The initial two-dimensional mesh and
the resulting extruded three-dimensional mesh can be seen in Figure 5-7.
Y
LX
(a) 2D side view
(b) 3D view
Figure 5-7: Initial structured mesh, 3D flat plate flow.
67
For the extruded problem in three dimensions, the inflow boundary is specified by
a total temperature, total pressure, and a zero flow angle while the outflow is defined
by a static pressure definition. An adiabatic no-slip condition is imposed on the flat
plate itself while a slip boundary condition is imposed on every other wall.
For each solution order p, the numbers of degrees of freedom considered are:
p = 1,
doftarget ={5000
10000
15000
20000},
p
doftarget
10000
15000
25000},
=
2,
=
{5000
As before, for each p-doftarget combination, a family of optimized meshes are generated and is assessed by averaging the error obtained on five realization. The output
adapted is the drag on the three-dimensional flat plate. Figure 5-8 shows the resulting drag error against the number of degrees of freedom 1 . For this case, the p = 2
102
-6- P1
-e- P2
10-3
10
S10-5
-3.98
r
-
106
10-7
10~8
101
h = 1/(dof)"
3
10 1.2
Figure 5-8: Convergence of drag output for the 3D flat plate flow; reference value
obtained from two-dimensional simulations
discretization outperforms the p = 1 discretization for all but the coarsest doftarget
considered. In fact, the finest p = 1 solution with roughly 30, 000 degrees of freedom exhibits a higher error level than the p = 2 solution containing almost half the
degrees of freedom with approximately 16,000.
For this smooth flow, the optimal
'These results were obtained using the edge-based method instead of the eigenvalue method
during the post-processing step of gradient-based metric optimization as described in Section 4.2.2
68
output error convergence rate of c oc hIP is recovered such that for even more accurate solutions, the p = 2 discretization would exhibit far greater efficiency than the
p
=
1 discretization.
A resulting optimized p
=
2 mesh can be seen in Figure 5-9. As exemplified by this
Y
(a) 2D side view
(b) 3D view
Figure 5-9: P=2, 25k dof optimized mesh, 3D flat plate flow.
optimized mesh, the adaptation process results in optimized meshes with significantly
different characteristics relative to the initial mesh. The algorithm applied to this
case properly detects that the flow exhibits no spanwise variation and automatically
generates meshes with large amounts of anisotropy in this direction; the resulting
optimized meshes contain only 1 to 2 elements along the z-direction while the initial
mesh begins with 6 cells. Furthermore, in an attempt to mitigate the effect of the
leading edge singularity, the adaptation algorithm very aggressively clusters elements
at this transition point. The initial structured mesh fails to properly capture the flow
at the leading edge due to inadequate resolution as shown in Figure 5-10. However,
the adapted meshes with this strong grading of elements properly resolve the initial
formation of the boundary layer as shown in Figure 5-11.
69
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15
0.19
0.01
0.0006
0.008
0.0005
.17
0.19
0.0004
0.006
0.0003
0.004
0.0002
0.002
0.0001
0
II..II,I,II.II
|
I,,
,|
0
.1
0.5
x
i
1
[
,
,
1.5
,
L
|
2
i
-0.06
,
-0.04
-0.02
0
0.02
0.04
0.06
0.08
x
(a) Boundary layer on initial structured mesh
(b) Close up at leading edge
Figure 5-10: Mach number distribution of 2D center slice, initial mesh, 3D flat plate
flow.
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.01
0.008
0.006
0.004
0.002
0
0
1
0.5
1.5
2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
x
x
(a) Boundary layer on p = 2 adapted mesh
(b) Close up at leading edge
0.08
Figure 5-11: Mach number distribution of 2D center slice, P=2, 25k dof optimized
mesh, 3D flat plate flow
5.3
3D Duct, Subsonic
This case presents a M,, = 0.2, ReD = 1 x 106 turbulent flow through a square duct
case of channel height and width D. The flow considered here develops over a length
2D.
70
As in the 3D flat plate case, the three-dimensional initial mesh is produced by
extruding a two-dimensional structured mesh into the third dimension.
Again, we
incorporate a lead-in section of length 'D such that the walls of the duct do not
immediately start at the inflow boundary. Here, we take advantage of the inherent
symmetry of the problem and only consider one quarter of the duct. The initial twodimensional mesh and the resulting extruded three-dimensional mesh can be seen in
Figure 5-12.
KY
X
I 'l ."
(a) 2D view of inlet and outlet
(b) 3D view
Figure 5-12: Initial structured mesh, subsonic duct flow.
For this duct flow, the inflow boundary is specified via a characteristic farfield
boundary condition while the outflow is defined by a static pressure definition. Adiabatic no-slip conditions are imposed on walls of the duct while slip boundary conditions are imposed on every other wall.
We consider the solution order p = 2, with a target degrees of freedom of 100000
with the drag on the walls of the duct as the output of interest and generate a family
of optimized meshes starting from the initial structured mesh. Figure 5-13 shows the
resulting
CD
and
CD
error estimate adaptation histories. For this case, the meshes
generated by EPIC overshoot the requested degrees of freedom by a factor of two to
three.
The adaptation process applied to this problem results in optimized meshes with
71
1-nX
10-3
0.01
1.64e+06
1.960.001
1
.
1.94
0
1.92
I1.28e+06
...
U-
0
1
()0.00
9.20e+05
1.90
1.88
5.60e+05
1.86-
0
2
4
6
8
10
12
14
0
16
2
4
Adaptation Iterations
(a)
cD
6
8
10
12
14
16
Adaptation Iteration
vs. adaptation iteration
(b) CD error estimate convergence
Figure 5-13: Drag adaptation history for p = 2, subsonic duct flow
error estimates over 2 times smaller but with almost 6 times fewer degrees of freedom.
A resulting optimized p = 2 mesh can be seen in Figure 5-14.
Similar to the presented 3D flat plate meshes, the duct meshes generated through
the adaption process contain significant clustering at the leading edges to mitigate
the effect of the edge singularities.
As before, the initial structured mesh fails to
properly capture the flow at the leading edge as shown in Figure 5-15a while the
adapted meshes are able to resolve the initial formation of the boundary layer as seen
.
in Figure 5-15b
Furthermore, the optimized meshes gain higher accuracy per degree of freedom by
effectively aligning the triangulation to better capture the boundary layer phenomena
of the flow.
As can be seen in the outflow comparison presented in Figure 5-16,
the adaptation significantly coarsens the initially fine elements that lie outside the
boundary layer to properly have the mesh reflect the current size of the boundary layer
at that cross section. The optimized mesh also contains significantly coarser elements
at the corner of the duct where the two boundary layers .intersect but still is able to
properly resolve this flow feature. The mesh resolution required to properly resolve
this corner would prove difficult to identify a priori but the adaptation algorithm
autonomously changes the triangulation to efficiently capture this phenomena.
72
X
y
\ it
x
Y
IV
I
_________
~i~I~
~
-- 4I
14
1r."'
(a) 2D view of inlet
(b) 2D view of outlet
Y
(c) 3D view
Figure 5-14: P=2, 100k dof optimized mesh, subsonic duct flow.
73
W-:A
0.02
0.02
0.015
0.015
>0.01
>-0.01
0.005
0.005
I
, ,
0
0
, .
0.5
I II I
I
1
Z
1.5
II . I I .
II .
,
0
2
0
(a) Initial mesh
0.5
, I I, ,
1
z
I,
1.5
,
I
2
(b) p = 2 optimized mesh
Figure 5-15: Mach number distribution, 2D view along duct at X = 0.25, subsonic
duct flow
74
0.06 0.08
0.1
0.12
0.2
0.14 0.16
D.02
0.18 0.04
0.06 0.08 0.1 0.12 0.14 0.16
OAS8
02
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
.I
0
,
, I
0.1
i
I
0.2
0.3
x
I .
0
,
0.4
0.5
,i , ,, ,i ,,
0
0.1
i ,, ,, i ,, ,, i
0.2
x
(a) Initial mesh
0.02 0.04 0.06 0.08 0.1
i
,
I,I
II
I
0.02 0.04
0.5
0.3
0.4
0.5
(b) p = 2 optimized mesh
0.02
0.12 0.14 0.16 0.18 0.2
0.03
0.03
0.02
0.02
0.01
0.01
0
0.04
0.06
0.08
0.1
0.12 0.14 0.16
0.16
0.2
0
0
0.01
0.02
x
0. 03
0
0.01
x
(c) Initial mesh - Close up of duct corner
0.02
0.03
(d) p = 2 optimized mesh - Close up of duct corner
Figure 5-16: Mach number distribution, 2D cross sectional view at outflow, 2.0 units
downstream from leading edge, subsonic duct flow
75
76
Chapter 6
Conclusion
6.1
Summary
This work investigated steps towards the realization of a robust high-order adaptive
Reynolds-Averaged Navier Stokes (RANS) method for the analysis of turbulent flows.
Specifically, modifications were made to the RANS discretization and the Mesh Optimization via Error Sampling and Synthesis (MOESS) framework [58] for increased
robustness of the high-order adaptive solver.
To improve the robustness associated with the RANS discretization, changes to
the negative continuation of the Spalart-Alimaras turbulence model proposed by
Allmaras [2] were reviewed and compared against an earlier variant presented by
Oliver [38].
The introduced negative SA modifications were tested on a subsonic,
turbulent flow over a NACA4412 airfoil with a high angle of attack and shown to
improve the convergence of the nonlinear solver.
In addition, the metric optimization framework proposed by Yano and Darmofal [59] was expanded to better control the mesh-to-mesh transition during the adaptation procedure. For this effort, the metric optimization step of the original algorithm
was changed to a penalty-based statement and a gradient-based solution method was
proposed. We presented numerical examples of applying the MOESS adaptation algorithm with this gradient-based optimization method to select problems. We first
verified the ability of the modified adaptation algorithm to produce optimal meshes
77
in the L 2 error control setting applied to a canonical corner singularity function. We
then moved to two-dimensional aerodynamic problems with the purpose of characterizing the improvements of the modified algorithm. Through a visual comparison of the
intermediate meshes generated through the adaptation procedure for transonic turbulent flow over an RAE2822 airfoil, the modified algorithm was shown to much more
smoothly transition the initial mesh to an optimal mesh with shock and boundary
layer resolution. Next, an inviscid supersonic flow over an NACA0012 airfoil demonstrated the improved mesh-convergence behavior of the gradient-based optimization
method relative to the baseline heuristic algorithm.
Finally, to demonstrate the overall robustness of the solver, the resulting discretization and modified adaptation procedure was applied to a range of two-dimensional
and three-dimensional turbulent aerodynamic problems. A subsonic turbulent flow
over a three-element McDonnell Douglas Aerospace airfoil demonstrated the ability
to robustly transition from a initial coarse mesh to a turbulent boundary layer mesh
for both p = 1 and p = 2. The optimal error convergence rates were observed for
both solution orders studied. Next, the flow over a three-dimensional flat plate at
subsonic, turbulent conditions was considered. This case exhibited the capability of
the adaptation process to properly detect the lack of spatial variation in the spanwise
direction and to automatically generate meshes with large amounts of anisotropy in
this direction. Again, optimal error convergence rates were observed for both p = 1
and p = 2. Finally, the ability of the algorithm to capture three-dimensional boundary layer effects was demonstrated in a subsonic turbulent flow through a square
duct.
6.2
Future Work
Improvements to the local error model
Throughout this work, the MOESS framework as presented in Chapter 4 assumes a
linear error model. Potential benefits could be realized with the optimization formulation if a higher-order error model were used instead. This model could be constructed
78
by either obtaining more error samples or by reinterpretation of the samples already
taken during the local solve procedure.
Adaptation for significantly under resolved meshes
The error estimate can significantly underestimate the error on coarse meshes on
which solution features are completed under resolved. On these meshes, the rates calculated through the local sampling procedure can push the triangulation towards nonoptimal metric fields. This problem can be alleviated by either improving the robustness of the error estimate within the current sampling-based adaptation framework
or by incorporating an adaptation strategy that adequately detects under-resolution
and adjusts the error model accordingly. A brief view of this problem is provided in
Appendix A.
Improvements to the metric optimization statement
The presented optimization method and respective solution procedure was shown to
be able to generate optimized metric requests which were then passed to a mesh generator to ultimately lower the error estimate. However, this optimization statement
contains no constraints or controls pertaining to the realizability of the resulting
metric.
Although the Riemannian metric field is represented as a piecewise linear
function on the triangulation, the presented adaptation procedure can still request
metrics that can be arbitrarily difficult for mesh generators to produce corresponding
metric-conforming meshes.
This potential problem may be remedied by formulat-
ing additional constraints or post-processing steps to introduce requirements on the
smoothness of the metric fields.
Three-dimensional RANS simulations
This work demonstrated the capability of a high-order adaptive RANS method to
both efficiently and automatically resolve turbulent flows on select two-dimensional
and three-dimensional aerodynamic flows.
However, the method as presented still
is not robust enough to tackle three-dimensional problems of arbitrary complexity.
79
Discontinuous Galerkin methods still present significant robustness challenges that
must be overcome before they are effectively utilized to solve the more complex threedimensional flows and geometries encountered in industry. Furthermore, robust threedimensional metric-based meshing is still an active problem that also serves as an
existing bottleneck to the realization of high-order adaptive RANS.
80
Appendix A
Adaptation From Under-resolved
Meshes
The turbulent flat plate case presented in Section 5.2 yielded optimized meshes with
significant refinement in the boundary layer and at the leading edge when starting
with an initial mesh with decent boundary layer resolution. However, the characteristic decreases in the error estimates from iteration to iteration observed in this case is
not retained when starting the procedure from a much coarser mesh with significant
under-resolution of the boundary layer. This appendix details the effects of initializing the process with a significantly under-resolved mesh on a similar set of flat plate
problems.
A.1
Test Cases and Initial Meshes
Similar to the case in Section 5.2, we consider a Mo, = 0.2, ReL = 5 x 106 turbulent
flow over a flat plate.
The baseline two-dimensional problem contains a flat plate
of 2L preceded by a short lead-in section of length 0.3L. As before, the respective
three-dimensional meshes are produced by extruding the two-dimensional meshes into
the third dimension. Here, we study both the two-dimensional and three-dimensional
cases.
To study the effects of under-resolution, we initialize the adaptation procedure
81
with an isotropic mesh containing no boundary resolution and an anisotropic mesh
that contains similar normal spacing to the initial mesh shown in Section 5.2. The
isotropic mesh and the anisotropic mesh can see in Figures A-1 and A-2 respectively.
For further comparison, we also study a scalar convection-diffusion problem with
the same ReL on the same meshes where the state on the plate is set to zero and the
freestream state is set to unity.
We consider the solution order p = 1, with a target degrees of freedom equal to
the degrees of freedom of the initial mesh (1380 for 2D and 11040 for 3D) with the
drag (heat flux for the scalar case) on the walls of the plate as the output of interest
and generate meshes starting from these initial structured meshes.
-
1.5
o ->
.0.5-IZZV
0
0
0.5
1
15
2
x
Figure A-1: Initial isotropic mesh with no boundary layer resolution - 460 Elements
A.2
Solution on Initial Meshes
The resulting pu and scalar T wall normal profiles on the two initial meshes can be
seen in Figures A-3 and A-4. For the isotropic mesh, the weak imposition of the
no-slip boundary condition results in essentially slip wall behavior for all four cases.
On the initial anisotropic mesh, the no-slip condition is more clearly enforced.
82
k
1.5
0.0004
0.5
00002
0
0.5
1
1.5
x
2
2.1
(a) Initial Mesh
.
2. j.1
0
2.15
x
2.2
(b) Close up of near-wall resolution
Figure A-2: Initial anisotropic mesh with significant boundary layer resolution - 460
Elements
0.3
0
0.25
E
0.2-
8r
0.15-
0
0
0
3D
2D
3
2D
RANS - pu
RANS - p u
Cony 01 - T
Cony Dili -T
0,
CO0.1
0
Z
0.05-
0
0
0.2
0.4
0.6
0.8
1
p u, T
Figure A-3: Initial pu, T wall normal profile, 1.5 units downstream of the leading
edge, isotropic mesh
A.3
Adaptive Results
The adaptation results for the RANS cases can be seen in Figures A-5 and A-7. The
respective DOF histories can be seen in Figures A-6 and A-8. When starting from
the under-resolved initial mesh, the adaptation severely struggles to reach the optimal
triangulations. Whereas the two-dimensional RANS case eventually flatlines at a minimized error estimate value after 50 adaptation iterations, the three-dimensional case
requires upwards of 70 iterations to reach this similar error level. When starting from
83
0.3
x
0.25E
0
3D RANS - p u
2DRANS-pu
0
3D Conv Diff - T
S20
onv Diff -
3D RANS - p u
2D RANS - p u
3D Conv Difl - T
2D Conv Ditt - T
0 .045 -
0
0
0.04
T
E
0.2
0
S0.15
C
0
0 .035
0.03
-6
40
0 .025
0.02
0
750.1
0 .015
0
z
0.05
0
0.2
0.4
0.8
0.6
0 001
00
02
04
.5.
0
0.2
0.4
0.6
pu, T
1
pu, T
i
(a) Wall profiles
0.8
I
(b) Close up of profiles
Figure A-4: Initial pu, T wall normal profile, 1.5 units downstream of the leading
edge, anisotropic mesh
the anisotropic boundary-layer resolved mesh, the adaptation procedure converges
more rapidly for both the three-dimensional and two-dimensional problems. For both
2D and 3D flows and both initial meshes, the adaptive algorithm is eventually able
to push the triangulations towards optimality.
x 10,
-A
4
10'
30 RANSolrpc
-3D
t
-
3.8
RANS -
-0-- 2D RANS -
Isoropic
Isotropici
10,
3.6
3.4
3.2
0
0
3
10
2.8
2.6
10-s
2.4
2.2
0
10
20
30
40
50
60
70
80
Adaptation Iteration
0
10
20
30
40
50
60
70
Adaptation Iteration
(a) Cd vs. adaptation iteration
(b) Cd error estimate convergence
Figure A-5: Drag adaptation history for turbulent boundary layer flow starting with
isotropic mesh
The adaptation results for the scalar cases can be seen in Figures A-9 and A-11.
The respective DOF histories can be seen in Figures A-10 and A-12.
For the two-
dimensional scalar flat plate cases, starting with the under-resolved mesh does not
84
80
3000
3
x 10,
,'x,
2500-
2.5-
A
-
-
1500
-1.5
o
2
-
A
2000-
0
11
1000 -
0
10
20
30
40
50
60
---
500
d1
70
80
0
10
20
30
Adaptation Iteration
40
50
60
di
70
80
Adaptation Iteration
(a) 2D RANS flow with doftarget
1380
(b) 3D RANS flow with doftarget= 11040
Figure A-6: DOF adaptation history for turbulent boundary layer flow starting with
isotropic mesh
i
X 10-3
[
-
4
e
i
i1
10
RANS - Anisropic
2D RANS - Anisolropic
-3D
-
3.8
-
3D RANS - Anisoropic
2DRANS-Anisotropici
10
3.6
0
3.4
3.2
m
10
2.6-
;
10
2.42.2
2
0
10
20
30
40
50
60
70
80
Adaptation Iteration
10
0
10
20
30
40
50
60
70
Adaptation Iteration
(a) Cd vs. adaptation iteration
(b) Cd error estimate convergence
Figure A-7: Drag adaptation history for turbulent boundary layer flow starting with
anisotropic mesh
seem to adversely affect the mesh convergence as much as seen in the two-dimensional
and three-dimensional turbulent flows. While the error indicator convergence does
still require an additional 5-6 iterations for the isotropic mesh, this increase is significantly smaller than those observed for the RANS cases. In three-dimensions, starting
with the isotropic mesh does require a significant increase in the iterations required
to obtain error indicator convergence relative to starting with the anisotropic mesh.
85
80
-x
10,
II
2.5 J
2500-
I
2000-
I
2
U-
U-
0
0
0
0
1500
-
1.5
1
0
10
0
10
20
30
40
so
60
7
8
Adaptation Iteration
0
01
0
10
20
30
40
50
60
70
80
Adaptation Iteration
(a) 2D RANS flow with doftarget = 1380
(b) 3D RANS flow with doftarget
11040
Figure A-8: DOF adaptation history for turbulent boundary layer flow starting with
anisotropic mesh
2
WI
(.
- 3D Conv Diff -4--~~~ 20
Conv
Diff-Iorpc-
.
1()-2
Isopic
-
-o
10,
10!
10,
1
.
10-2
-+-
3D Conv Di - soropic
2D Conv Difi - :sotropic-
10,
-
10
I
0-100
I
1
5
L
10
15
20
A
25
I
30
35
40
Adaptation Iteration
10 I
0
5
10
15
20
25
30
35
Adaptation Iteration
(a) Heat flux vs. adaptation iteration
(b) Heat flux error estimate convergence
Figure A-9: Heat flux adaptation history for scalar boundary layer flow starting with
isotropic mesh
Further investigation is still required to understand why the adaptation scheme
on the RANS equations (especially in three-dimensions) struggles significantly more
relative to the scalar equations when starting from an under-resolved mesh. The problem is likely to be a combination of the weaker control of DOF in three-dimensional
mesh generation as well as more fundamental robustness issues of the error estimates
for the RANS equations.
86
40
.
x 10
.
suaa.
2500
2.5
2000
2
U-
U-
0
0
0
0
1.5
1000
01
0
5
10
15
20
25
30
35
05
'0
40
Adaptation Iteration
(a) 2D scalar flow with doftarget = 1380
5
10
15
20
25
Adaptation Iteration
30
35
440
(0
(b) 3D scalar flow with doftarget =11040
Figure A-10: DOF adaptation history for scalar boundary layer flow starting with
isotropic mesh
16
x 10
e
-
12
3D Conv Dil - Anisoiropic
-3D Conv
_g.--2D Conv
.
10-2
2D Conv Dili - Anisotropic
Dill - Anisoropic,
Dill - Anisotropic-
-
14-
10~
10
UCO
8
10,
6
4
20
5
10
15
20
25
Adaptation Iteration
30
35
40
10~-I
0
5
10
15
'
20
'
25
i
30
1
35
Adaptation Iteration
(a) Heat flux vs. adaptation iteration
(b) Heat flux error estimate convergence
Figure A-11: Heat flux adaptation history for scalar boundary layer flow starting
with anisotropic mesh
87
40
x
3000
2500-
10
2.5-
2
2000LL
U-
0
0
0
0
1500
1.5
1000-
500
0
10
15
20
25
30
35
40
Adaptation Iteration
0
5
10
15
20
25
30
35
Adaptation Iteration
(b) 3D scalar flow with doftarget
(a) 2D scalar flow with doftarget = 1380
11040
Figure A-12: DOF adaptation history for scalar boundary layer flow starting with
anisotropic mesh
88
40
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