amr_guaily--directionally adaptive least squares finite element method

8/14/2019 Amr_guaily--directionally Adaptive Least Squares Finite Element Method

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DIRECTIONALLY ADAPTIVE LEAST SQUARES FI-

NITE ELEMENT METHOD FOR THE COMPRESSIBLEEULER EQUATIONS

By

Amr Gamal Mohammad Guaily

B.Sc. in Aerospace Engineering, 2002

A Thesis Submitted to the

Faculty of Engineering, Cairo University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

in

Engineering Mechanics

FACULTY OF ENGINEERING, CAIRO UNIVERSITY

GIZA, EGYPT

2006


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i

ACKNOWLEDGMENTS

All gratitude is due to Allah almighty.

The author wishes to express his gratitude to all those who provided help in

various ways at the different stages of this work.

I wish to express my deepest and sincere gratitude and appreciation to my

main supervisor, prof. Dr. A. A. Megahed, professor of engineering mechanics,

for consideration, for suggesting the problem, and for his sincere guidance during

this work.

I also wish to express my gratefulness to my supervisor Prof. Dr. M. M.

Abd-El-Rahmann, professor of aerodynamics, for his continuous support and

guidance during this work.

Assist. Prof. M. W. El-Mallah has played an important role in this work

through his valuable discussions that have been very useful in overcoming tech-

nical difficulties encountered during the work, so I wish to express my deepest

and sincere gratitude to him.

My family has always played an important role in my studies, so I would

like to express my deep gratitude and appreciation to my parents and my elder

brothers for their continuous support.


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ii

ABSTRACT

The least-squares finite element method is used to solve the compressible

Euler equations in both 2-D Cartesian and axisymmetric forms. Since the

method is naturally diffusive, no explicit artificial viscosity is added to the for-

mulation. The inherent artificial viscosity, however, is usually large and hence

does not allow sharp resolution of discontinuities unless extremely fine grids are

used. To remedy this problem, while retaining the advantages of the least

squares method, a moving-node grid adaptation technique is used. The out-

standing feature of the adaptive method is its sensitivity to directional features

like shock waves, leading to the automatic construction of adapted grids where

the element edge(s) are strongly aligned with such flow phenomena.

Using well-known transonic and supersonic test cases, it is demonstrated

that by coupling the least squares method with a robust adaptive method, shocks

can be captured with high resolution despite using relatively coarse grids.

A paper extracted from the thesis was accepted to be presented at the

IASTED international conference on modeling and simulation (MS 2006), which

will be held May 24 to May 26, 2006, at Montreal, Canada.

Paper title

Enhanced Adaptive Finite Element Method for The Cartesian and Axi-

symmetric Inviscid Compressible Flows


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iii

Table of Contents

ACKNOWLEDGMENTS

ABSTRACT

Table of Contents iii

List of Figures v

Nomenclature

Chapter 1 Introduction and Literature Review 1

1.1 Introduction 1

1.2 The Finite Element Method 1

1.3 Advantages of The Finite Element Method 2

1.4 Approaches of the FEM Formulation 3

1.4.1 Direct Approach 3

1.4.2 Variational Approach 3

1.4.3 Weighted residual Approach 3

1.5 Literature Review of the Compressible Euler FEM Schemes 3

1.6 Literature Review of the Adaptation Techniques 5

1.7 Current Work 8

Chapter 2 Least Squares FEM for Inviscid Compressible Flows 10

2.1 Introduction 10

2.2 Least Squares Formulation 10

2.3 Interpretation of the Inherent Viscosity 14

2.4 Finite Element Approximation 15

2.5 Boundary Conditions 17

2.6 Solution Method 19


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2.7 Numerical Results 20

2.7.1 Planer Problems 20

2.7.1.1 Shock-Reflection Problem (SRP) 20

2.7.1.2 Supersonic Channel Flow (SCP) 24

2.7.1.3 Airfoil-Spoiler Configuration 33

2.7.2 Axisymmetric Problems 39

2.7.2.1 Nozzle Flow (Shock Free) 39

2.7.2.2 Nozzle Flow (With Shock) 45

2.7.2.3 Jet Flow (Nearly Fully Expanded) 49

2.7.2.4 Jet Flow (Fully Expanded) 53

2.7.2.5 Under Expanded Jet Flow (UEJF) 54

Chapter 3 Directionally Adaptive Technique for FEM 60

3.1 Introduction 60

3.2 Mathematical Analysis 60

3.2.1 Edge-Based Error Estimate 60

3.2.2 Moving-Node Scheme 64

3.2.3 Grid Smoothening 66

3.2.4 The Grid Adaptation Procedure 67

3.3 Numerical Results 68

3.3.1 Analytical Test Case 68

3.3.2 Shock Reflection Problem (SRP) 71

3.3.3 Supersonic Channel Problem (SCP) 94

Chapter 4 Summary and Conclusions 100

4.1 Thesis Summary 100

4.2 Conclusions 101

4.3 Recommendations for future work 103

References


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List of Figures

Figure 1-1 Original mesh to be adapted 6Figure 1-2 r-method used in adaptation 6

Figure 1-3 h-method used in adaptation 6

Figure 2-1 Element shape in global and local coordinates 16

Figure 2-2 Discontinuous angle at wall nodes 18

Figure 2-3 Computational domain (SRP) 20

Figure 2-4 Pressure contours for different time steps, (SRP) 21

Figure 2-5 Pressure distribution at y= 0.05 (SRP) 22

Figure 2-6 Density contours for different time steps, (SRP) 22

Figure 2-7 Mach number contours for different time steps, (SRP) 23

Figure 2-8 Convergence history of the flow solver (SRP) 23

Figure 2-9 Computational domain (SCP) 24

Figure 2-10 Computational domain and grid (SCP) 26

Figure 2-11 Density contours, t=0.15 (SCP) 26



Figure 2-14 Pressure contours, t=0.15 (SCP) 28



Figure 2-17 Mach number contours, t=0.15 (SCP) 29



Figure 2-20 Mach number distribution for the upper wall (SCP) 31

Figure 2-21 Mach number distribution for the lower wall (SCP) 31

Figure 2-22 Convergence history of the flow solver (SCP) 32

Figure 2-23 Velocity vector plot (SCP) 32

Figure 2-24 Zoom on the lower boundary (SCP) 33

Figure 2-25 Oil drop flow visualization 34

Figure 2-26 Airfoil-spoiler details 35


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Figure 2-27 Grid used in airfoil-spoiler configuration 35

Figure 2-28 Zoom on the grid near the airfoil 36

Figure 2-29 Pressure coefficient contours (airfoil-spoiler problem) 36

Figure 2-30 Coefficient of pressure on the upper surface 37

Figure 2-31 Coefficient of pressure on the lower surface 37

Figure 2-32 Growth of wake in (airfoil-spoiler problem) 38

Figure 2-33 Physical nozzle and the used grid (shock free nozzle flow) 40

Figure 2-34 Mach number contours (shock free nozzle flow) 40

Figure 2-35 Pressure contours (shock free nozzle flow) 41

Figure 2-36 Radial velocity contours (shock free nozzle flow) 41

Figure 2-37 Pressure distribution (shock free nozzle flow) 42

Figure 2-38 Mach number distribution (shock free nozzle flow) 43

Figure 2-39 Temperature distribution (shock free nozzle flow) 43

Figure 2-40 Density distribution (shock free nozzle flow) 44

Figure 2-41 Velocity vector plot (shock free nozzle flow) 44

Figure 2-42 Zoom on the lower boundary (shock free nozzle flow) 45

Figure 2-43 Mach number contours (nozzle flow with shock) 46

Figure 2-44 Mach number distribution (Nozzle Flow With Shock) 46

Figure 2-45 Pressure contours (Nozzle Flow With Shock) 47

Figure 2-46 Pressure distribution (Nozzle Flow With Shock) 47

Figure 2-47 Convergence history (Nozzle Flow With Shock) 48

Figure 2-48 Velocity vector plot (Nozzle Flow With Shock) 48

Figure 2-49 Supersonic free jet main regions 50

Figure 2-50 Axisymmetric jet flow boundaries 50

Figure 2-51 Computational domain and center line Mach number

distribution (jet flow problem) 51

Figure 2-52 Axial Mach number distributions (jet flow problem) 52

Figure 2-53 Mach number contours (jet flow problem) 52

Figure 2-54 Mach Number Contours (Jet Flow Full Expanded) 53

Figure 2-55 Radial velocity Contours (Jet Flow Full Expanded) 54

Figure 2-56 Computational domain (UEJF) 55

Figure 2-57 Contour plot for the Mach number (UEJF) 56


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Figure 2-58 Axial Mach number distribution (UEJF) 56

Figure 2-59 Axial pressure distribution (UEJF) 57

Figure 2-60 Axial density distribution (UEJF) 57

Figure 2-61 Axial distribution of radial-velocity (UEJF) 58

Figure 2-62 Radial distribution of axial Mach number (UEJF) 58

Figure 2-63 Radial distribution of pressure (UEJF) 59

Figure 2-64 Radial distribution of radial-velocity (UEJF) 59

Figure 3-1 Transformation of a unite circle to an ellipse byS 63

Figure 3-2 Spring analogy for a patch of elements. 64

Figure 3-3 A node on the grid and its surroundings 66

Figure 3-4 Initial mesh (analytical test case). 69

Figure 3-5 Isocontours for the initial mesh (analytical test case). 69

Figure 3-6 Adapted mesh (analytical test case). 69

Figure 3-7 Adapted isocontours (analytical test case). 70

Figure 3-8 Magnification of grid (analytical test case). 70

Figure 3-9 The computation domain (SRP). 71

Figure 3-10 The initial and 1st adapted pressure (SRP) 72

Figure 3-11 The 2nd and 3rd adapted pressure (SRP) 72

Figure 3-12 The initial grid and the and the pressure contours (SRP) 74

Figure 3-13 The 1st adapted grid and the pressure contours (SRP) 74

Figure 3-14 The 2nd adapted grid and the pressure contours (SRP) 75

Figure 3-15 The 3rd adapted grid and the pressure contours (SRP) 75

Figure 3-16 The 4th adapted grid and the pressure contours (SRP) 76

Figure 3-17 The 5th adapted grid and the pressure contours (SRP) 76

Figure 3-18 Zoom on the grid in the incident shock region SRP 77

Figure 3-19 Zoom on the grid in the reflected shock region (SRP) 77

Figure 3-20 Pressure distribution at y=0.5 (SRP) 78

Figure 3-21 Convergence history (SRP) 78

Figure 3-22 The initial grid and the Mach number contours (SRP) 79

Figure 3-23 The 1st adapted grid and Mach number SRP 79

Figure 3-24 The 2nd adapted grid and Mach number (SRP) 80

Figure 3-25 The 3

rd

adapted grid and Mach number (SRP) 80


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Figure 3-26 The 4th adapted grid and Mach number (SRP) 81

Figure 3-27 The 5th adapted grid and Mach number (SRP) 81

Figure 3-28 The initial grid and pressure (density adapted) SRP 82

Figure 3-29 The 1st adapted grid and pressure (density adapted) SRP 82

Figure 3-30 the 2nd adapted grid and pressure (density adapted) SRP 83

Figure 3-31 The 3rd adapted grid and pressure (density adapted) SRP 83

Figure 3-32 The 4th

adapted grid and pressure (density adapted) SRP 84

Figure 3-33 The 5th

adapted grid and pressure (density adapted) SRP 84

Figure 3-34 Comparison of pressure adapted (upper) and density adapted

(lower) pressure contours for the 1st

adp. SRP 85


(lower) pressure contours for the 2nd

adp. SRP 85


(lower) pressure contours for the 3rd

adp. SRP 86


(lower) pressure contours for the 4th

adp. SRP 86


(lower) pressure contours for the 5th adp. SRP 87


(lower) grid for the 1st

adp. SRP 88


(lower) grid for the 2nd

adp. SRP 88


(lower) grid for the 3rd

adp. SRP 89


(lower) grid for the 4th

adp. SRP 89


(lower) grid for the 5th

adp. SRP 90

Figure 3-44 Comparison of pressure distribution at y=0.5 for SRP 90


(lower) Mach number contours for the 1st

adp. SRP 91


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(lower) pressure contours for the 2nd

adp. SRP 91


(lower) pressure contours for the 3rd

adp. SRP 92



adp. SRP 92



adp. SRP 93

Figure 3-50 Comparison of pressure contours (SRP) 93

Figure 3-51 The initial grid and the corresponding pressure contours 95

Figure 3-52 The 1st adapted grid and the corresponding pressure cont. 95

Figure 3-53 The 2nd adapted grid and the corresponding pressure cont. 96

Figure 3-54 The 3rd adapted grid and the corresponding pressure cont. 96

Figure 3-55 The 4th adapted grid and the corresponding pressure cont. 97

Figure 3-56 The 5th adapted grid and the corresponding pressure cont. 97

Figure 3-57 Evolution of density contours during adaptation 98

Figure 3-58 Magnification of grid in reflected shock region 99

Figure 3-59 Convergence history 99


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Nomenclature

xA , yA , axisA Jacobian matrices

E Nodal error

g Adaptation parameter

H Hessian matrix

h The element edge length

I Functional to be minimized

k Spring constant

L2 Residual error

P Potential energy

Q Vector of unknowns

Residual vector (scalar)

T Transformation matrix

t Time

W Weight function

x, y Cartesian coordinates

Solution domain

Basis function

Dirac delta function

Local element coordinate

Contours of the elements

Diagonal matrix of the eigen-values

Relaxation parameter.


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Subscripts

i, j Nodal indices

Free stream value

dim Dimensional value

t, n Normal and tangential coordinates

Superscripts

T Transposes

n Time level

e Element

dn Number of nodes per element

m Level of iteration


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1.1 Introduction

This research aims at building a least squares finite element Euler solver

for the transonic regime combined with an error estimate and a directionally

adaptive grid algorithm, allowing the use of anisotropic (stretched) elements.

Adaptive finite element methods place more fine scale elements where more

resolution is needed. Isotropic or shape regular adaptive methods use only ele-

ments with bounded aspect ratio (stretched elements are avoided). Anisotropic

adaptive methods fit high aspect ratio elements (highly stretched elements) along

the regions of rapid variation of the solution for situations like shocks or bound-

ary layers. Anisotropic adaptive methods give a bigger saving in terms of com-

putational cost (number of elements and degrees of freedom) than the isotropic

ones if stretched elements are placed appropriately [1].

The current interest in the area of high-speed flows has increased the need

for advanced computational fluid dynamics (CFD) codes, which have become

the primary tools for the prediction of aero-thermal loads. Such flows are charac-

terized by regions with steep directional gradients of flow variables, embedded in

regions where the flow variables vary more smoothly. One approach [2] for im-

proving the solution accuracy of such problems is to apply grid adaptation tech-

niques.

1.2 The Finite Element Method

It was originally introduced by civil engineers: Harold Martin at the uni-

versity of Washington (also a Structural Analysis Consultant to the Boeing Com-

pany), and H. Argyris working at Stuttgart and at Imperial Collage.

Chapter 1 Introduction and Literature Review


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The essence of the method was certainly available for a long time, but the

above two names made it a practical proposition. In addition both names were

among the first to suggest and actually use the method for fluid dynamics prob-

lems.

The FEM is essentially a numerical method for solving Partial Differential

Equations (PDE's) [4].

In general, if an analytical (exact) solution cannot be found for the PDEs

governing a given continuum problem, then a numerical solution must be at-

tempted at a discrete number of points.

The continuum field in fluid mechanics is hence subdivided into a grid at

which the dependent variable(s) is (are) to be calculated. This dependent vari-

able(s) is (are) assumed to behave locally in a given way (linearly, quadratically,

according to a spline, etc.).

In the finite difference method (FDM) the PDE is written in discrete form

involving the grid points. For the FEM the PDE is first recast in an integral form

over the entire domain and its residual minimized by several means. Both the

FDM and the FEM result in a large system of simultaneous linear algebraic equa-

tions. The dependent variable is solved for at all grid points simultaneously

through direct or iterative methods.

1.3 Advantages of The Finite Element Method

The FEM approximates complicated geometrical boundaries easily. Almost

invariably the FDM starts by regularizing the domain, i.e. mapping complex re-

gions into regular (rectangular) regions.


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The FEM accounts for boundary conditions in an easy, straightforward

manner. Unlike the FDM, the FEM has no numerical boundary conditions. The

no penetration boundary condition is implemented by a Dirichlet condition even

with a curved boundary.

1.4 Approaches of the FEM Formulation

Given below are a different approaches of the FEM formulation. For more

details see ref. [4]

1.4.1 Direct Approach

This approach is applicable only to simple problems governed by an alge-

braic relationship or a simple first order ordinary differential equation (ODE) for

1-D problems.

1.4.2 Variational Approach

This approach is applicable to physical problems governed by extremiza-

tion (minimum or maximum) laws. While this is mostly the case in structural

analysis, it is not the general case for fluid mechanics where not many variational

principles exist.

1.4.3 Weighted residual Approach

This approach is applicable to problems where a variational principle does

not exist, for nonlinear problems and for unsteady state problems. The residual of

the PDE is minimized, weighted by a certain weight that depends on the particu-

lar residual approach.

1.5 Literature Review of the Compressible Euler FEM Schemes

So far, many different approaches have been adopted in developing numerical

schemes to solve the compressible Euler equations. Using the idea of upwind in

the finite difference method, Brooks and Hughes [5] introduced the Streamline

Upwind Petrov-Galerkin (SUPG) method, in which the weight function is modi-


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fied by adding a perturbation to the standard Galerkin test function. The added

perturbation creates an upwind effect by weighting more heavily the upstream

nodes. Another approach was proposed by Baruzzi et al.[6], where the Laplacian

of dependent variables was added to the continuity and momentum equations.

The amount of artificial viscosity was then controlled by a single parameter as

the coefficient of the Laplacians. They later extended this first-order artificial

viscosity method to second-order.

Another method is based on the least squares weighed residual method. The

method has very good stability properties due to its minimization nature, and has

been applied for the solution of a variety of problems. As one of the earliest ef-

forts in this field one can mention the technique presented by Polk and Lynn [7]

for the solution of unsteady gas dynamic equations, with elements that are con-

structed in both space and time. Fletcher [8] used the least-squares method to

solve the Euler equations for the subcritical compressible flows. The special fea-

ture of his method was to represent groups of variables rather than single vari-

ables. Application of the least squares method to a governing equation of the

form: L(Q)=f leads to the favorable result of a symmetric and positive definite

coefficient matrix, if L is a first-order differential operator.

Taghaddosi et al. [2] applied this approach combined with adaptive grid al-

gorithm to the Euler equations. Also Moussaoui [9] applied this approach to

solve both the compressible and incompressible flows using one formulation. If

L is a higher order operator, this property is completely lost during the integra-

tion by parts and moreover elements with higher order continuity requirements,

must be employed. Lynn and Arya [10] proposed to break down the higher order

system to its first order counterpart as a way of eliminating this disadvantage.

Pontaza et al. [11] used this approach to solve both the Euler and Navier-Stokes

equations for the compressible regime. Bolton and Thatcher [12] used the

LSFEM to solve the Navier-Stokes equations in the form of stress and stream

functions


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1.6 Literature Review of the Adaptation Techniques

For a given mesh as shown in Figure 1-1, grid adaptation methods aremainly composed of an adaptive strategy such as mesh movement (r-method) as

in Figure 1-2; mesh refinement/coarsening (h-method); as in Figure 1-3, higher-

order interpolation (p-method). Even when the same error estimate is used to

assess the accuracy of the solution, the resulting adapted grid depends strongly

on the selected adaptation strategy. Classical methods such as grid refinement

produce isotropic meshes in which the length scales in all directions are essen-

tially the same. These methods are optimal only for those flow regions possess-

ing nearly equal gradients in all spatial directions. As a result, directional flow

features such as shocks, contact discontinuities and boundary layers are not nec-

essarily adapted efficiently and the number of elements needed to represent them

may increase disproportionally with each isotropic refinement.

An alternative approach would be to seek solutions on anisotropic meshes

where more resolution is introduced along those directions with rapidly changing

flow variables. This idea was introduced by Peraire et al. [13], who used an adap-

tive remeshing procedure that incorporated directional stretching for the solution

of the 2D Eulers equations on triangular grids. Another method may be used as

proposed by Fortan et al. [14], who used a metric as a measure of error, coupled

to an h-r strategy, to achieve directionally adapted grids with high aspect ratios.


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Figure 1-1 Original mesh to be adapted

Figure 1-2 r-method used in adaptation

Figure 1-3 h-method used in adaptation


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The above approaches have primarily been used on unstructured meshes.

This trend is mainly driven by the intrinsic ability of triangular elements in 2D

and tetrahedral elements in 3D to deal with arbitrarily complex geometries. Un-

structured adaptation algorithms can yield highly stretched grids as well as lo-

cally refined/coarsened meshes. In contrast, most refinement techniques for

structured grids avoid propagating the refinement to the boundaries by allowing

sides to have hanging nodes.

Despite these advantages for unstructured meshes, structured grids of quad-

rilateral elements in 2D and hexahedral elements in 3D are still used with great

success in CFD. One reason is that the integration of the governing equations on

a structured grid requires less CPU time than on an unstructured one for the same

number of nodes. Structured grids are also more suitable for turbulence model-

ing, particularly near solid walls where normals to the wall may be needed.

Furthermore, a certain degree of grid anisotropy may also be introduced for

structured grids through an improved moving-node scheme. A moving-node

technique was originally introduced by Gnoffo [15], and generalized by Nakaha-

shi and Deiwert [16] in the context of finite volume methods (FVM). All these

schemes are based on spring analogy where the grid is viewed as a network of

springs whose stiffness constants represent a measure of error [15]. The grid ver-

tices are displaced until the equilibrium state of the spring forces is reached. Such

techniques are characterized by their low cost and the conservation of nodal con-

nectivity, but often can stall or diverge and tolerate only a limited range of nodal

movement.

A directionally adaptive FEM using an edge-based error estimate on quad-

rilateral grids was proposed by Ait-Ali-Yahia et al.[18]. In this method the use

of an appropriate error estimate, combined with the vector nature of spring forces

(i.e. their magnitude and direction), permits one to design a convergent adaptive

procedure capable of achieving wider nodal movement and a high degree of grid

anisotropy. The error of the numerical solution is evaluated using a bound avail-


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able from finite element interpolation theory. The Hessian of a selected solution

variable is computed and then modified to produce a positive definite matrix al-

lowing one to define a measure of error, namely a Riemannian metric. The edge-

based error estimate is thus expressed as the length of the edge of the elements in

this Riemannian metric. The construction of an anisotropic mesh may thus be

interpreted as being a uniform mesh in the defined metric. This metric introduces

and controls the magnitude as well as the direction of the grid anisotropy. A

mesh movement scheme is then applied as the adaptive strategy, which in con-

trast to the spring analogy technique used by Nakahashi and Deiwert [16], has no

constraint on grid orthogonality. This leads to a simple and efficient nodal redis-

tribution algorithm offering a greater range of point displacements. In this

method, the optimal grid for a fixed number of nodes is thus defined as one in

which the error is equidistributed over the edges.

1.7 Current Work

The least squares finite element method is used to solve the compressible

Euler's Equations for both the Cartesian and axisymmetric flows in the non-

conservative form. The quality of the numerical results indicates the remarkable

performance of the used technique. This is quite evident from the final results

despite using coarse mesh. Also the robustness of the technique allows one to use

large time step to reach the steady state solution in few iterations, which saves

computational time considerably. But a disadvantage existed; namely the high

value of the inherent artificial viscosity in the method which forbidden the sharpresolution of discontinuities, this disadvantages has been remedied.

To remedy the disadvantage mentioned above in the least-squares finite

element method, an adaptive algorithm with directional features, using an edge

based error estimate on quadrilateral meshes, is used. The error of the numerical

solution is measured by its second derivatives and the resulting Hessian tensor is

used to define a Riemannian metric. A mesh movement strategy with no or-


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thogonality constraints is used to equidistribute the lengths of the edges of the

elements of the edges of the elements in the defined metric. The adaptive proce-

dure has been proven to be effective on analytical test case.

The used technique enforces the solid wall boundary condition since it en-

forces the boundary condition as a Dirichlet type at each node using a transfor-

mation matrix for each node. This results in a faster convergence than other

methods and it can be used for both steady and unsteady flows unlike other tech-

niques.

The flow solver, combined with the proposed grid adaptation method is

then validated on a variety of problems. The quality of the numerical results in-

dicates the remarkable performance of the adaptive method, and demonstrates its

superiority to many existing techniques. This is quite evident in the final adapted

grids.

Overall, the combination of the least-squares method with the directionally

adaptive method has produced promising results, despite using an artificial vis-

cosity mechanism that has no free parameters, on relatively coarse grids.

The rest of the thesis is structured as follows

Chapter 2 deals with the least squares finite element formulation followed

by a variety of internal and external flow problems to validate the method

Chapter 3 deals with the formulation of the adaptation scheme followed by

a variety of flow problems to validate the method.

Chapter 4 concludes the results of preceding chapters and gives recom-

mendations for future work.


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2.1 Introduction

In this chapter the least-squares finite element method is introduced in details.

The least squares finite element method is easy to implement, is naturally diffu-

sive, contains no free parameter(s), is stable thus allowing equal-order interpola-

tion of all variables, and more importantly, results in a symmetric positive defi-

nite coefficient matrix. Application of other finite element methods such as

Galerkin or the SUPG leads to non-symmetric matrices. The present work, there-

fore, is an attempt to investigate the least squares finite element method for com-

pressible flows with shocks.

2.2 Least Squares Formulation

The Euler equations for both Cartesian and axisymmetric flows can be

written in the nondimensional form as a first order system in terms of primitive

variables [2] as:

0 x y axisQ Q Q A A A Qt x y

+ + + =

(2-1)

where

( , , , )TQ u v p= is the vector of unknowns.

Chapter 2 Least Squares FEM for Inviscid

Compressible Flows


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0 0

10 0

0 0 00 0

x

u

uA

up u

=

0 0

0 0 0

10 0

0 0

y

v

vA

v

p v

=

0 0 0

0 0 0 0

0 0 0 0

0 0 0

axis

vy

A

vy

=

with

1 = for axisymmetric flow

0 = for planar flow

in which is the density, ( , )u v the velocity components, p the pressure, and

is the specific heat ratio.

The non-dimensionlization is as follow,

= dim / , u = udim / C, v = vdim / C, p = pdim / ( C2), t = tdim C / L.

where (dim stands for dimensional)

Density.

u Longitudinal velocity.

v Lateral velocity.

p Pressure.

t Time.

L Characteristic length.


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Free stream density.

C Free stream speed of sound.

The system given in equation (2-1) may be linearized using Newton

method, by setting:

1n nQ Q Q+ = +

neglecting the higher order terms, and discratizing the unsteady term using

backward difference as:

Q Q

t t

so, equation (2-1) can be rewritten as:

1n L Q f + = (2-2)

where

1( )

( )

0 0

n n n

x y

n nn n

x y a x i s

Tn n n n

ax i s

L A A I A x y t

Q Q f A A f

x y

v p vf

y y

= + + +

= + +

=


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2

2

( ) 0

1

0

10

0 ( )

u v v

x y y x y y

p u u

x x yA

p v v

y x y

p p p u v v

x y y x y y

+ + +

=

+ + +

Defining the residual vector as

1n

L Q f +

= +

the least squares functional is

1 1( ) ( ) ( )2

n T I Q d +

= (2-3)

Now we can introduce the finite element approximation.

1 1 1

1

enn n n

h j j

j

Q Q QN+ + +

=

= (2-4)

where

en is the number of nodes per element.

j jN IN = is the element shape function.

and the weighting function is

1( )nW L

QN+

= =

so the least squares weak form is


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1( ) ( ) 0T n LN L Q f d +

+ = (2-5)

Using equation (2-4) in equation (2-5) results the linear algebraic equa-

tions:

[ ]{ } { } K Q R = (2-6)

where

( ) ( )e

e T e

i j i j K L L d N N

=

( )e

e T

i ir L fd N

= (2-7)

are evaluated using Gauss-Legendre quadrature.

2.3 Interpretation of the Inherent Viscosity

For the Eulers equation (2-1) the numerical viscosity inherent in the least

squares formulation can be demonstrated as follows [2]. The right-hand side

vector of equation (2-6) as defined in equation (2-7) is

( )

1( )

( )

e

e

e T

i i

T

n n ni i x y i

n nn n

x y ax is

r L fd

A A I A x y t

Q Q A A f d

x y

N

N NN

=

= + + +

+ +


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.. .e

T n nn n n n ei i

x y x y

Q Q A A A A d

x y x y

N N

= + + +

After integrating this equation by parts (Greens formula), the first term on

the right-hand side yields the following term

e

T n nn n n n

x y x y

Q Q A A A A d

x y x y

+ +

which represents an inherent numerical viscosity of the least squares

method. The effect of the time step on the amount of artificial viscosity is exam-

ined for each problem separately by numerical experiment to determine the ap-

propriate time step.

2.4 Finite Element Approximation

In the finite element discritization, both variables and problem geometry

are approximated using shape functions. If the same shape functions are used for

approximation of geometry and variables, the resulting element is called iso-

parametric. In this work iso-parametric quadrilateral bilinear elements will be

used. This type of elements relies on the bilinear local shape function given by

( , ) (1 )(1 ) / 4i i iN = + + (2-8)

(i , i) are the local coordinate of the corner node i of any quadrilateral

element, while ( , ) are the element local axes as illustrated in Figure 2-1. The

integer index i varies between 1 and 4 according to the corresponding node.


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Figure 2-1 Element shape in global and local coordinates

The problem variables are approximated by

iii

NQQ ==

4

1(2-9)

Where Q i is the value ofQ , at node i.

Transformation between local coordinates and global Cartesian coordinates

is governed by

iii

iii

Nyy

Nxx

4

1

4

1

=

=

=

=(2-10)

Partial derivatives w.r.t. x and y are related to partial derivatives w.r.t.

and by


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[ ]

=

=

y

Nx

N

J

y

Nx

N

yx

yx

N

N

i

i

i

i

i

i

(2-11)

We note that the Jacobian matrix J is a function of and only. By inver-

sion of J, derivatives w.r.t. x and y, become functions in and . Area integrals

are transformed according to:

dx dy = dA = det(J) d d. (2-12)

The last relation equation (2-12) is very important since all calculus is done

in terms of the local coordinates.

Integration over each element is done numerically using Gauss-Legendre

quadrature of the 5th order accuracy. This method of integration involves evalua-

tion of the integrand at fixed points, and multiplication of the resulting value by a

corresponding weight, followed by summation ([4], [17]).

2.5 Boundary Conditions

In general we have four types of boundary conditions

1- Dirichlet type in which Q "the dependent variable" is specified.

2- Neumann type in whichQ

n

is specified, n is the direction normal to the

boundary.

3- Cauchy type in which part of the boundary is Dirichlet type and the rest it

Neumann type.


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4- Robbins type in which 1 2 3Q

c Q c cn

+ =

is specified, where,

1 2 3, , andc c c are constants

For Eulers equations, the number of boundary conditions to be imposed on

the domain is determined by the theory of characteristics. In the finite element

method, the numerical boundary conditions are naturally imposed which is con-

sidered one of the most important features of the finite element method. There-

fore no special treatments are required. For our first order system we have

Dirichlet type of boundary conditions except for the non-Dirichlet boundary con-

dition at a solid wall U.n=0 where n is the outward unit normal.

Figure 2-2 Discontinuous angle at wall nodes

An interesting procedure is the one known as coordinate rotation method.

In this formulation ([9],[19]), the velocity components at the solid wall are ro-

tated from the global Cartesian coordinate to a local coordinate system which

permits the tangency condition to be imposed as a Dirichlet boundary condition

as follow. Let the normal direction at a solid wall point A be given by

(cos,sin), and denote by un and ut the normal and tangential components of

the velocity U, respectively. Now, we have the following relation:


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cos sin

sin cos

n

t

uu

uv

=

For nodes on solid wall boundary, a new nodal vector of unknowns is de-

fined by replacing the Cartesian components (u, v) by the normal and tangential

components in the global matrix[ ]K . The resulting matrix is nonsymmetrical.

To restore the symmetry we pre-multiply the corresponding rows by the matrix T

defined as:

1 0 0 0

0 cos sin 0T=

0 sin cos 0

0 0 0 1

Due to the discrete representation of the computational domain, the angle

at a typical wall node A (Figure 2-2) can have two different values. It is neces-

sary to assign a unique angle for such nodes. To accomplish this, a weighted av-

erage of the angles from the two adjacent wall edges can be used:

2 1 1 2

1 2

A

L L

L L

+=

+

2.6 Solution Method

The resulting system of linear algebraic equations (2.6) is solved as follows:

we first compute a U DU

T

factorization of the coefficient matrix[ ]K . D is ablock diagonal matrix with blocks of order 1 or 2, and Uis a matrix composed of

the product of a permutation matrix and a unit upper triangular matrix. The solu-

tion of the linear system is then found.

The previous method is implemented using a built in subroutine in the used

language (Compaq Visual FORTRAN 6.5).


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2.7 Numerical Results

2.7.1 Planer Problems

2.7.1.1 Shock-Reflection Problem (SRP)

We consider compressible, inviscid flow along a wall where the reflection

of an oblique shock occurs. The computational domain is a 4.1 X 1.0 rectangle,

which is discretized uniformly into (60 X 20) bi-linear elements. We consider

flow at an inlet Mach number of Min = 2.9 and specify the following boundary

conditions at the inlet (, u, v, p) = (1.0, 2.9, 0.0, 0.7143). To simulate the

oblique shock we specify the following at the top boundary :( , u, v, p) = (1.7,

2.6193, -0.5063, 1.5282) ([20], [21]). The lower boundary is impermeable and

the exit boundary is left free (supersonic exit)

Figure 2-3 Computational domain (SRP)

The effect of the inherent viscosity that is controlled by the time step, on

the shock resolution is shown in Figure 2-4, which shows the pressure contours,

when the time step changes from 0.15 to 0.05 (from top to bottom t=0.15, 0.1,

0.05). As expected oscillations near the shock start growing after reducing the

artificial viscosity (reducing the time step) and the shock becomes more smeared

by increasing it (increasing the time step), as indicated in Figure 2-5. Figure 2-6


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21

shows the density contours while Figure 2-7 shows the Mach number contours

where the time step has changed from 0.15 to 0.05. This shows the effect of the

time step on the solution as the shock becomes more refined for smaller time

step. Figure 2-8 shows the convergence history using different time steps. As

expected as the time step decreased the solution reaches steady state slower.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

Figure 2-4 Pressure contours for different time steps, (SRP)


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Figure 2-5 Pressure distribution at y= 0.05 (SRP)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

Figure 2-6 Density contours for different time steps, (SRP)


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0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

Figure 2-7 Mach number contours for different time steps, (SRP)

Figure 2-8 Convergence history of the flow solver (SRP)


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2.7.1.2 Supersonic Channel Flow (SCP)

A more challenging problem for this algorithm exists in the analysis of

supersonic flow past the parabolic arc bump (Figure 2-9) of height to length

ratio = 4%. This problem creates an interesting shock interaction behind the

bump. The shock from the leading edge of the bump is reflected down off the

ceiling of the tunnel and intersects with the shock formed at the trailing edge of

the bump (see Figure 2-9 for details).

Figure 2-9 Computational domain (SCP)

The curved lower wall for the circular arc is given by

2 2

0 1.5 0.5

0.5 0.5

0 0.5 1.5

x

y R x b x

x

< <

= + <


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The boundary conditions at the inlet are:

=1.0, u=4 uinfy (1-y) , v=0.0

and at the exit, the pressure is specified as 1p

=

On the walls the flow tangency condition is imposed and the exit boundary

is left free (supersonic exit).

The grid consists of 64 X 16 uniformly distributed bi-linear rectangular

elements with 16 elements in the y-direction, 22 elements on the bump, and 21

elements in each side of it as shown in Figure 2-10.

The effect of the inherent viscosity, controlled by the time step, on the

shock resolution is shown in the figures starting from Figure 2-11 to Figure 2-13,

which show the density contours, calculated with time steps ranging from 0.15 to

0.05.

Figure 2-14 to Figure 2-16 show the pressure contours, which assure the

important role of the time step on shock resolution. The leading- and trailing

edge shocks as well as the interaction of the trailing-edge shock with the re-

flected shock are qualitatively well captured.

The Mach number contours are plotted in the figures starting from Figure 2-17 to

Figure 2-19 to assure the ability of the scheme in capturing such a complex shock

structure.

Figure 2-20 and Figure 2-21 show the Mach number distribution of the up-

per and lower walls respectively compared to Eidelman et al. [22] which shows

good agreement. As expected oscillations near the shock start growing after re-

ducing the artificial viscosity and the shock becomes more smeared by increasing

it.


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Figure 2-22 shows the convergence history using different time steps. As

expected as the time step decreased the solution reaches steady state slower.

The quality of the used method in applying the solid wall boundary condi-

tion is demonstrated in the figures starting from Figure 2-23 to Figure 2-24,

which show the velocity vector plot and a zoom on the lower wall. The tangency

condition is applied very well as seen in these figures.

Figure 2-10 Computational domain and grid (SCP)

-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2-11 Density contours, t=0.15 (SCP)


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-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



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-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2-14 Pressure contours, t=0.15 (SCP)

-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



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-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2-17 Mach number contours, t=0.15 (SCP)


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-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



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Figure 2-20 Mach number distribution for the upper wall (SCP)

Figure 2-21 Mach number distribution for the lower wall (SCP)


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Figure 2-22 Convergence history of the flow solver (SCP)

-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2-23 Velocity vector plot (SCP)


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Figure 2-24 Zoom on the lower boundary (SCP)

2.7.1.3 Airfoil-Spoiler Configuration

Spoilers are widely used in aircraft as lateral control devices, as speed breaks,

and as lift dampers during landing. Despite their wide usage, very little theoreti-

cal information exists. Almost all design work is done by comparison with ex-

perimental results followed by wind tunnel tests of trail models. The flow fieldpast an airfoil with spoiler is complex. In fact the flow separated from the upper

airfoil surface, due to the adverse pressure gradient generated by the presence of

the spoiler, then reattaches to the spoiler surface at small angles of attack [23]. A

recirculating bubble is formed upstream of the spoiler hinge. The flow separates

again from the spoiler tip and converts into wake as a free shear layer. The flow

on the lower airfoil surface also leaves the trailing edge and converts into wake.

These shedding vortices make the wake highly turbulent and oscillatory. This

unsteady wake affects the effectiveness of the spoiler.

Extensive experimental investigations on steady spoiler characteristics

have been taken by several authors ([24], [25], [26],[27]). On the theoretical

side, Abdelrahman [24] developed a numerical scheme based on the steady state

incompressible Navier-Stokes equations. Choi et al. [28] studied the transient

response of an airfoil to a rapidly deploying spoiler using turbulent compressible

Navier-Stokes equations with a turbulence model.


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Flow Description

The flow was examined using both water table and oil streaks on a flow-

splitter plate mounted on a wind tunnel model [23].

These examinations indicate that the wake has two main characteristics.

First it has an unsteady nature due to a shed vortex street. Second, despite this

periodic behavior there is a region of wake that remains nearly constant in shape

and closes a short distance downstream of the trailing edge. Figure 2-25 repre-

sents the flow on the splitter plate as investigated by Wentz and Ostowari [25] at

zero angle of attack, 40o splitter deflection angle, 2.2x106 Reynolds number, and

0.13 Mach number. It clearly shows two basic regions: first, an outer essentially

potential flow region, and second, a near wake region behind the spoiler. Also

Figure 2-25 shows that the near wake may be subdivided into two more parts.

The part upstream of the wing trailing edge shows very little fluid motion and

thus may be considered at constant pressure. The part downstream of the trailing

edge is characterized by a pair of vortices; the upper one rotating clockwise (for

left to right flow) and smaller one below it rotating counterclockwise

Figure 2-25 Oil drop flow visualization


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Figure 2-26 Airfoil-spoiler details

Figure 2-26 describes the details of the used configuration, while Figure 2-27

shows the O-type grid of [65x20] elements used to solve the problem of the air-

foil-spoiler configuration, followed be a zoom on the grid near the airfoil surface

in Figure 2-28, where the parameters in Figure 2-26 are as follows for NACA

23012:

d = 0.75 c, t= 0.01 c, = 30o, and L=0.15 c. The inlet Mach number used was

M= 0.2.

Figure 2-27 Grid used in airfoil-spoiler configuration


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Figure 2-28 Zoom on the grid near the airfoil

Figure 2-29 shows the pressure coefficient isocontours, while Figure 2-30 and

Figure 2-31 show the distribution of the pressure coefficient over the upper and

lower surfaces compared to the experimental results of [24]. The coefficient of

pressure suddenly decreases to a negative value due to the presence of the

spoiler.

Figure 2-29 Pressure coefficient contours (airfoil-spoiler problem)


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-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

X

Cp

Exp. Num.

Figure 2-30 Coefficient of pressure on the upper surface

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

X

Cp

Exp. Num.

Figure 2-31 Coefficient of pressure on the lower surface


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Figure 2-32 Growth of wake in (airfoil-spoiler problem)

Figure 2-32 shows the growth of the wake behind the spoiler which in

qualitative agreement with the experimental results of [25].


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2.7.2 Axisymmetric Problems

2.7.2.1 Nozzle Flow (Shock Free)

Nozzle flow will be considered here as it is one of the most important test

cases used to show code robustness. In particularly it tests the efficiency of the

flow tangency condition, which is considered one of the most difficult aspects

when talking about Euler's equations.

Nozzle Geometry

( )( )( )

( )( )( )

2

2

0.1398 0.0347 tanh 9 2.5 0.0 0.5

0.1398 0.0347 tanh 9 1 2.5 0.5 1.0

x xA x

x x

+ + = + +

where A is the nozzle cross-sectional area and the radius, r=(A / )0.5.

The computational domain is discretized uniformly into [60 X 20] bi-linear

elements as shown in Figure 2-33. We consider flow at an inlet Mach number of

Min = 0.224 and specify the following boundary conditions at the inlet "subsonicinlet" (, u, v) = (1.0, 0.224, 0.0). The lower and upper boundaries are imperme-

able and the exit boundary is left free (supersonic exit).

Figure 2-34 shows the Mach number contours through the nozzle. Despite

solving the whole domain, the full symmetry is evident from the Mach number

contours as well as the pressure contours shown Figure 2-35. Figure 2-36 shows

the contours of the radial velocity, which assures the capability of the used tech-

nique to capture the symmetry of the flow.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

x

r

Figure 2-33 Physical nozzle and the used grid (shock free nozzle flow)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 2-34 Mach number contours (shock free nozzle flow)


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 2-35 Pressure contours (shock free nozzle flow)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 2-36 Radial velocity contours (shock free nozzle flow)


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The flow accelerates (Mach number increases) as it goes through the noz-

zle as shown in Figure 2-37 which presents the pressure distribution of the nozzle

center line as well as the nozzle wall compared to the exact one dimensional so-

lution [29]. Figure 2-38 shows the Mach number distribution for both the nozzle

wall and the centerline compared to the one-dimensional solution. The wall ef-

fect is evident in the distribution of the wall Mach number as it starts to decrease

at an axial location of 0.7 due to the change in the area (acts like a compression

corner). Figure 2-39 shows the distribution of the temperature across the nozzle

in which again the two dimensional effect is clear. The density distribution is

shown in Figure 2-40.

Figure 2-37 Pressure distribution (shock free nozzle flow)


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Figure 2-38 Mach number distribution (shock free nozzle flow)

Figure 2-39 Temperature distribution (shock free nozzle flow)


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Figure 2-40 Density distribution (shock free nozzle flow)

To show how far the flow tangency condition is imposed, Figure 2-41

shows the velocity vector plot of the flow inside the nozzle. Figure 2-42 is a

zoom on the lower nozzle wall, which assures the robustness of the used method

in enforcing the solid wall boundary condition.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

x

r

Figure 2-41 Velocity vector plot (shock free nozzle flow)


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Figure 2-42 Zoom on the lower boundary (shock free nozzle flow)

2.7.2.2 Nozzle Flow (With Shock)

The previous nozzle shape is solved but with a predetermined back pres-

sure to generate a normal shock inside the divergent part.

So now we have a different exit boundary condition, which is a subsonic

exit. According to the theory of characteristics we should define only one de-

pendent variable (pressure) at that boundary. The used backpressure is = 0.5.

Figure 2-43 shows the Mach number contours of the resulting solution. In this

figure the two dimensional shock is evident and as expected the shock takes the

shape of a cap. Figure 2-44 shows the Mach number distribution for the center-

line and for the wall compared to the exact 1-D solution as illustrated in [29]. As

seen the shock is well captured despite using coarse grid [60X20].


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Figure 2-45 shows the pressure contours in which again the shock is evident

Figure 2-46 shows the pressure distribution, which assures the robustness of the

used method despite using coarse grid. Figure 2-47 shows the convergence his-

tory of the flow solver. In this figure we started the solution by large time step to

reach the steady state solution faster then the time step was reduced to refine the

solution. In this figure, the quadratic convergence feature is evident which as-

sures the robustness of the flow solver. Figure 2-48 shows a velocity vector plot,

which again assures the shock existence.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 2-45 Pressure contours (Nozzle Flow With Shock)

Figure 2-46 Pressure distribution (Nozzle Flow With Shock)


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Figure 2-47 Convergence history (Nozzle Flow With Shock)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Velocity Vector Plot

Figure 2-48 Velocity vector plot (Nozzle Flow With Shock)


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2.7.2.3 Jet Flow (Nearly Fully Expanded)

The current test case is that of a supersonic jet flow. An experimental investiga-

tion of flow of a moderate Reynolds-number and Mach-number axisymmetric,

near isentropically expanded, cold jet has been performed by Troutt and

McLaughlin [30], with the following jet conditions

Mach number (Mj) 2.1

Reynolds number (Re) 70,000

Jet / Nozzle Radius (rj) 0.005 mm

Total Temperature (To) 294

Static Pressure (P) 5060 N/m2

Center line Jet Temperature (Tj) 156.4 K

Center line Sonic Speed (Cj) 250 m/s

Center line Jet Speed (Uj) 525 m/s

Center line Jet Density (j) 0.1114 Kg/ m3

Before we go further we have to clear some definitions related to the mean

flow field shown in Figure 2-49.

Potential Core Region:

It is the region where the axial velocity equals the jet velocity and no ra-

dial velocity exists.

The Shear Mixing Region:

It is the region where the axial velocity is less than half of its centerline

value.

To well capture the free jet phenomenon the solution domain is used as shown in

Figure 2-50


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Figure 2-51 (top) shows the domain of solution used in the current study

which extended 70Rj in the axial direction and 20Rj in the radial direction using

clustering in the normal direction near the nozzle end to well capture the rapid

variations in the mixing layer region. Bi-linear elements are used with 60*26

elements in the axial and normal directions respectively.

Figure 2-49 Supersonic free jet main regions

Figure 2-50 Axisymmetric jet flow boundaries


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Figure 2-51 Computational domain and center line Mach number distribution

(jet flow problem)

The centerline Mach number distribution for both the experiment and the

current work is shown in Figure 2-51 (bottom). The numerical results are in

good agreement with the experiment despite using coarse grid. The figure shows

the potential core length of the jet to be 10 radii while that of the experiment be-

tween 8 and 10 radii. The sonic point in the jet is reached between 22 and 24 ra-

dii while that of the experiment is between 18 and 20 radii.

Figure 2-52 shows the axial Mach number distributions for different radial

positions in which the jet decelerates and the outer stagnant air is accelerated.

Figure 2-53 shows the Mach number contours. In this figure we can see the ge-

neric features of the supersonic jet flow; e.g. the potential core region. The sym-

metry of the flow is depicted in the figure, which assures the quality of the used

method despite using coarse grid.


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Figure 2-52 Axial Mach number distributions (jet flow problem)

Figure 2-53 Mach number contours (jet flow problem)


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53

2.7.2.4 Jet Flow (Fully Expanded)

The current test case is that of a supersonic full expanded jet flow with Mj = 2.1.

The importance of this test case is that we know the theoretical solution. To well

capture the free jet phenomenon the solution domain shown in Figure 2-50 is

used again. Figure 2-51 (top) shows the domain of solution used in the current

study which extended 70Rj in the axial direction and 20Rj in the radial direction

using clustering in the normal direction near the nozzle end to well capture the

rapid variations in the mixing layer region. Bi-linear elements are used with

40*20 elements in the axial and normal directions respectively. Figure 2-54

shows the contour plot for the Mach number, which supports the expected theo-

retical solution, i.e. the expected cylindrical contours. Figure 2-55 shows the con-

tour plot for the radial velocity. The symmetry of the solution is evident (the full

domain was solved not half) which assures the quality of the used technique.

Figure 2-54 Mach Number Contours (Jet Flow Full Expanded)


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Figure 2-55 Radial velocity Contours (Jet Flow Full Expanded)

2.7.2.5 Under Expanded Jet Flow (UEJF)

The current test case is that of a supersonic under expanded jet flow with

Mj = 2.1 and pressure ratio pj/pamb. = 1.7. The importance of this test case is that

tests the capability of the scheme to capture the well-known cell-structure. To

well capture the free jet phenomenon the solution domain shown in Figure 2-56

is used with a [40 X 36] bi-linear elements which extended 50Rj in the axial di-

rection and 20Rj in the radial direction using clustering in the normal direction

near the nozzle end to well capture the rapid variations in the mixing layer re-

gion. Figure 2-57 shows the contour plot for the Mach number, which supports

the expected solution, i.e. the expected cell structure (since the jet flow has a

higher pressure value than the ambient, an expansion fan is created at the nozzle

exit. This expansion fan is reflected when it reaches the centerline which acts

like a wall. The reflected fan turns the slip line inwards while being reflected

again and when it reaches the centerline again, it is reflected again to meet the

slip line again and turns it outward, thus forming the first cell and so on) in con-


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clusion the expansion fan and the slip line affects each other to make the slip line

takes the shape of a cell.

The symmetry of the solution is evident (the full domain was solved not

only one half) which assures the quality of the used technique. Figure 2-58

shows the axial Mach number distribution at different radial locations. As ex-

pected the Mach number increases till the pressure reaches the ambient pressure

as shown in Figure 2-59. Figure 2-60 shows the axial distribution of the density.

Figure 2-61 shows the axial velocity distribution at different radial location,.

Figures starting from Figure 2-62 to Figure 2-64show the radial variations of the

Mach number, pressure, and the radial velocity respectively, which again assures

the capability of the technique to capture the symmetry despite solving the whole

domain.

Figure 2-56 Computational domain (UEJF)


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Figure 2-57 Contour plot for the Mach number (UEJF)

Figure 2-58 Axial Mach number distribution (UEJF)


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Figure 2-59 Axial pressure distribution (UEJF)

Figure 2-60 Axial density distribution (UEJF)


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Figure 2-61 Axial distribution of radial-velocity (UEJF)

Figure 2-62 Radial distribution of axial Mach number (UEJF)


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Figure 2-63 Radial distribution of pressure (UEJF)

Figure 2-64 Radial distribution of radial-velocity (UEJF)


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3.1 Introduction

The current interest in the area of high-speed flows has increased the need

for advanced computational fluid dynamics (CFD) codes, which have become

the primary tools for the prediction of aero-thermal loads. Such flows are charac-

terized by regions with steep directional gradients of flow variables, embedded in

regions where the flow variables vary more smoothly. One approach for improv-

ing the solution accuracy of such problems is to apply grid adaptation techniques.

3.2 Mathematical Analysis

The mathematical analysis is divided into two sections as follow

3.2.1

Edge-Based Error Estimate

Consider a 1D element in which the solution variable g is approximated by

gh with linear interpolation [18]. A local error Ee defined over an element e can

be estimated as the difference between a quadratic interpolation gq and the actual

linear one provided that the error is zero at the nodes and maximum at the middle

of the element

2

2( )

2

qe e

e

gdE h

dx

= (3-1)

where

is the local element coordinate and he the element length.

Chapter 3 Directionally Adaptive Technique for FEM


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A measure of the overall error in the element is then considered to be the

root-mean-square value of Ee

1

2222

2

1

1200

eqeRMS

e e

ee

h gdEdE h

dxh

= =

(3-2)

If we define an optimal mesh as one for which the error is equidistributed

over elements, the following should hold for each element [18]:

2

2

2

q

e

e

gdch

dx= (3-3)

where c is a positive constant. The second derivative in equation (3-2) is based

on gq, which is the solution being sought and hence not available. So it is here

approximated by the second derivative of the numerical solution, i.e.,

22 /hd g dx .

The above methodology can be extended to 2D based on the fact that each edge

of a 2D element can be considered as a 1D element ([18], [31], [32], [33]). So the

second derivative in equation (3-2) may be replaced with the Hessian matrix as

follows

2 2

2

2 2

2

h h

h h

g g

x x yH

g g

y x y

=

(3-4)


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The second derivative in equation (3-4) will vanish since we use a bi-linear

element. However, a mass lumping (center of mass) [18], can be applied to re-

cover an estimate of the second derivative. This yields the expression

,

,I

I

Ih ijA

h ij IIA

N d Agg

N d A

=

(3-5)

where AI represents the elements sharing node I. After integration of equation (3-

5) by parts, the nodal values of the Hessian reduce to

,, ,

,

I I

I

I j I jh i h iA

h ij IIA

N n d N dAg gg

N dA

=

(3-6)

whereI

represents contours of the elements sharing node I.

The Hessian matrix given by equation (3-4) may be diagonalized as fol-

lows:

( ) ( )T H R R = (3-7)

where is the diagonal matrix of the eigen-values ofHandR is the matrix of the

eigenvectors. The transformation is a scaling in the direction of the axes andR

is a rotation with angle that the eigenvector corresponding to the smallest ei-

gen-value makes with the x-axis.

In order to obtain a symmetric, positive definite matrix, the Hessian ismodified by taking the absolute value of its eigenvalues. This results in

( ) ( ) ( ) ( )T T H R R S S = = (3-8)

where ( ) ( )S R = . The transformation Sof a unite circle would be an el-

lipse, rotated through an angle , whose semi-major axis is the reciprocal of the


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square root of the smallest eigenvalue, and semi-minor axis is the reciprocal of

the square root of the largest eigenvalue [18], as shown in Figure 3-1. Therefore

one can obtain a directionally stretched grid by mapping a uniform mesh using

the transformation S. However, in the current approach a mesh with edges of

equal length is sought in the transformed plane ST, where the length of a curveB

is given by

1

0

( ) ` ( ) ( ) ` ( )Td B s l H l s l d l = (3-9)

and ( )s l is a parametric representation of the curveB.

Since H is a function of the space coordinates, equation (3-9) defines a

Riemannian metric. The modified Hessian is computed and stored on a back-

ground mesh and thus the value ofHat any position of the domain interpolated

during the adaptive process on this mesh. The edge-based error estimate can then

be numerically evaluated from equation (3-9) for each edge of the element.

Figure 3-1 Transformation of a unite circle to an ellipse by S


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3.2.2 Moving-Node Scheme

The adaptive strategy modifies the grid under the guidance of the error es-

timate to improve the quality of the numerical solution. Thus the use of an ap-

propriate adaptive scheme is crucial for achieving the desired directionally

adapted mesh. The used strategy relies on a node-moving scheme, also called

nodal redistribution. As illustrated in Figure 3-2, the mesh may be viewed as a

network of springs [15] whose stiffness constants represent the edge-based error

estimate.

Figure 3-2 Spring analogy for a patch of elements.

The positions of the grid vertices may then be interpreted as the solution of

an energy minimization problem. This yields for each vertex I

2min m in ( ) I J IJ

x I x I IJ

P x x k = (3-10)

where PI denotes the potential energy of the four springs sharing a node I and kIJ

are the associated four stiffness constants. These constants may be specified as


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( )I JIJ

I J

d x xk

x x

=

(3-11)

where

I Jx indicates the Euclidian norm

( )I Jd x x is the length of the edge [ ]I Jx x in the Riemannian metric

defined by equation (3-9). After simplification, equation (3-10) reduces to the

following system describing the equilibrium state of a spring network:

1 1( ) 0mm mI J IJJ

kx x+ + = (3-12)

By lagging xJ and kIJ at the previous iteration m, equation (3-12) becomes

( )mm m

I J IJJ

I m

IJJ

kx xx

k

=

(3-13)

and the position of the vertex I is updated according to the expression

1m mI I I

xx x + = + (3-14)

where is a relaxation parameter (vector). The convergence of this scheme can

be enhanced by using Gauss-Seidel algorithm with the latest values of xJ and kIJwhen using equation (3-13). The iteration process in equation (3-14) can be ap-

plied to all nodes in the domain in order to adapt the mesh to the solution.

Boundary nodes can also move in the same way as internal nodes but they are

then re-projected onto the boundary to maintain the geometric integrity of the

domain. The moving-node scheme is applied to grid points in a sweeping man-

ner. The reason is to allow checking the quality of each newly oriented element

during the mesh movement and thus avoiding formation of elements with a nega-


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tive or nearly zeroing Jacobian. The adaptive method uses the solution of one of

the scalar variables to adapt the mesh. Then goes back to the flow solver with the

adapted mesh. Each mesh adaptation followed by the flow solver is called one

adaptive cycle.

3.2.3 Grid Smoothening

The previous scheme does not guarantee the smoothness of the resulting grid, i.e.

the resulting grid may contain elements with angles greater than 170o

or less than

10o, which may cause ill-posedness of the resulting global matrix. So we pro-

pose to use additional diagonal elements to act as semi-torsional springs, or to

add a grid smoothening step, which guarantees the smoothness of the adapted

grid. Which is performed after each iteration of the adaptation scheme by simply

equating the gradient of the adapted grid lines at each point (see Figure 3-3) as in

equation (3-15) for the y direction and the same applies for the x direction.

Figure 3-3 A node on the grid and its surroundings

yi,j = ((xi,j-xi-1,j)yi+1,j+(xi+1,j-xi,j)yi-1,j)/ (xi+1,j-xi-1,j) (3-15)


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3.2.4 The Grid Adaptation Procedure

The grid adaptation procedure may be summarized in the following steps:

------------------------------------------------------------------------------------------

Read a background mesh and the corresponding solution

Compute Hon the background mesh

Current mesh is initialized by an initial mesh guess (optional)

Move the nodes of the current mesh as follows

DO m=1,MAXITER

DO inod=1,NNODE

DO iedge=1,NEDGE

Determine Hby interpolating on the background mesh (optional)

Compute springs constants by numerical integration of (3.11)

ENDDO

Find new position of inod

Move inod to its new position

Check quality of elements sharing node I

ENDDO

Grid smoothness (optional)

If (MAXDISP .


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3.3 Numerical Results

3.3.1 Analytical Test Case

It is important to examine the behavior of the adaptation algorithm on an

analytical function with strong gradient to represent its quality and robustness.

A function g in the form

( ) ( )( )1 4 4g x,y tan 1000 0.25x y= (3-16)

is used in the solution domain [0, 2] x [0, 1].

The initial coarse mesh [30*20] is shown in Figure 3-4 and the correspond-

ing isocontours of g are shown in Figure 3-5. After 60 iterations (=0.3) of the

mesh movement scheme the adapted mesh shown in Figure 3-6 is obtained. As

illustrated in Figure 3-7 this mesh permits a better representation of the function

g (the contour lines became smoother) using the same number of nodes.

The magnification of the mesh in the discontinuity region presented in

Figure 3-8 shows that the quadrilateral elements are strongly re-oriented in the

direction of the discontinuity with a very high aspect ratio. Also we can see that

the re-oriented element has a very small length in the direction normal to the

shock to represent well the strong gradient normal to the shock, and a very high

length in the shock direction to reduce the used number of elements, which raises

the quality of the used technique.


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Figure 3-4 Initial mesh (analytical test case).

Figure 3-5 Isocontours for the initial mesh (analytical test case).

Figure 3-6 Adapted mesh (analytical test case).


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Figure 3-7 Adapted isocontours (analytical test case).

Figure 3-8 Magnification of grid (analytical test case).


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3.3.2 Shock Reflection Problem (SRP)

The second test case is the shock reflection problem discussed previously

in Chapter 2 (section 2.7.1.1). Figure 3-9 shows the computational domain

again. This example tests certain features of the algorithm including the resolu-

tion of a system of two oblique shocks and their proper angles. This test case

will be solved with and without the proposed smoothening step. Figure 3-10

shows the initial grid (a) and the corresponding pressure contours (b). Also in

this figure the 1st

adapted grid is shown (c) as well as the corresponding pressure

contours (d). Figure 3-11 shows the results for the 2nd and 3rd adaptation cycles.

In these figures one can see that the resulting grids are not adapted well to the

required solution. Therefore in the following figures we will use the proposed

smoothening step.

Figure 3-9 The computation domain (SRP).


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0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

(a)

(b)

(c)

(d)

Figure 3-10 The initial and 1st

adapted pressure (SRP)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

(a)

(b)

(c)

(d)

Figure 3-11 The 2nd

and 3rd

adapted pressure (SRP)


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The initial coarse mesh composed of [61 x 21] nodes and the corresponding iso-

contours are shown in Figure 3-12 . Using t=0.1, the results are adapted through

five cycles using the pressure or the density as the adaptation parameter as speci-

fied in the title of each figure. The artificial viscosity is reduced beginning at the

third cycle by reducing the time step to 0.05, since its amount was very high for

the size of the grid near the shock. Figure 3-13 through Figure 3-17 show the

pressure contours and the grids after each adaptation. The improvement in the

shock resolution after adaptation is quite evident.

Figure 3-18 and Figure 3-19 show how the elements are re-oriented to be

aligned with the shock, creating very high aspect ratio elements, which assures

the robustness of the technique. The pressure distribution at (y=0.5) is shown in

Figure 3-20. The adapted solution captures the shock more sharply. The conver-

gence history is shown in Figure 3-21. Each jump in the Figure corresponds to an

adaptation cycle. The quadratic convergence of the Newton linearization is quite

evident. The Mach number contours for the initial solution and for the adapted

solution are shown in Figure 3-22 through Figure 3-27, which again assures the

importance of the grid adaptation t

amr_guaily--directionally adaptive least squares finite element method

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