##### Document Text Contents

Page 1

Adaptive Finite Element

Solution Algorithm

for the Euler Equations

by Richard A. Shapiro

Page 2

Notes on Numerical Fluid Mechanics (NNFM)

Series Editors: Ernst Heinrich Hirschel, Miinchen

Kozo Fujii, Tokyo

Bram van Leer, Ann Arbor

Keith William Morton, Oxford

Maurizio Pandolfi, Torino

Arthur Rizzi, Stockholm

Bernard Roux, Marseille

(Adresses of the Editors: see last page)

Volume 32

Volume 6 Numerical Methods in Laminar Flame Propagation (N. Peters I J. Warnatz, Eds.)

Volume 7 Proceedings of the Fifth GAMM-Conference on Numerical Methods in Fluid Mechanics

(M. Pandolfi I R. Piva, Eds.)

Volume 8 Vectorization of Computer Programs with Applications of Computational Fluid Dynamics

(W Gentzsch)

Volume 9 Analysis of Laminar Flow over a Backward Facing Step (K. Morgan I J. Periauxl F. Thomasset,

Eds.)

Volume 10 Efficient Solutions of Elliptic Systems (W. Hackbusch, Ed.)

Volume II Advances in Multi-Grid Methods (D. Braess I W Hackbusch I U. Trottenberg, Eds.)

Volume 12 The Efficient Use of Vector Computers with Emphasis on Computational Fluid Dynamics

(W. Schiinauer I W. Gentzsch, Eds.)

Volume 13 Proceedings of the Sixth GAMM-Conference on Numerical Methods in Fluid Mechanics

(D. Rues/W Kordulla, Eds.) (out of print)

Volume 14 Finite Approximations in Fluid Mechanics (E. H. Hirscbel, Ed.)

Volume 15 Direct and Large Eddy Simulation of Turbulence (U. Schumann I R. Friedrich, Eds.)

Volume 16 Numerical Techniques in Continuum Mechanics (W Hackbusch I K. Witsch, Eds.)

Volume 17 Research in Numerical Fluid Dynamics (P. Wesseling, Ed.)

Volume 18 Numerical Simulation of Compressible Navier-Stokes Flows (M. O. Bristeau I R. Glowinski I

J. Periauxl H. Viviand, Eds.)

Volume 19 Three-Dimensional Turbulent Boundary Layers - Calculations and Experiments (B. van den

Bergl D. A. Humphreys I E. Krause I 1. P. F. Lindhout)

Volume 20 Proceedings of the Seventh GAMM-Conference on Numerical Methods in Fluid Mechanics

(M. Deville, Ed.)

Volume 21 Panel Methods in Fluid Mechanics with Emphasis on Aerodynamics (1. Ballmann I R. Eppler I

W. Hackbusch, Eds.)

Volume 22 Numerical Simulation oftbe Transonic DFVLR-F5 Wing Experiment (W. Kordulla, Ed.)

Volume 23 Robust Multi-Grid Methods (W Hackbusch, Ed.)

Volume 24 Nonlinear Hyperbolic Equations - Theory, Computation Methods, and Application (J. Ballmannl

R. Jeltsch, Eds.)

Volume 25 Finite Approximation in Fluid Mechanics II (E. H. Hirschel, Ed.)

Volume 26 Numerical Solution of Compressible Euler Flows (A. Dervieux I B. van Leer I J. Periauxl

A. Rizzi, Eds.)

Volume 27 Numerical Simulation of Oscillatory Convection in Low-Pr Fluids (B. Roux, Ed.)

Volume 28 Vortical Solutions of the Conical Euler Equations (K. G. Powell)

Volume 29 Proceedings of the Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics

(P. Wesseling, Ed.)

Volume 30 Numerical Treatment of the Navier-Stokes Equations (w. Hackbusch I R. Rannacher, Eds.)

Volume 31 Parallel Algorithms for Partial Differential Equations (w. Hackbusch, Ed.)

Volume 32 Adaptive Finite Element Solution Algorithm for the Euler Equations (R. A. Shapiro)

Page 89

A second approach sometimes taken is grid regeneration. In this

method, either the entire computational grid or some smaller portion of

it is regenerated when adaptation occurs. The grid point distribution is

determined by some error indicator based on the solution on the previous

grid. This method is described in detail in articles by Peraire, et al.

[53, 54]. This retains the grid alignment advantages while reducing grid

skewness problems. The main disadvantages are that the cost of grid

regeneration is usually fairly high, the interpolation of the solution from

the old grid to the new grid can be difficult, and the method (so far) has

been limited to triangular and tetrahedral elements.

The third approach is grid enrichment. The survey article by Berger

[4] presents a good overview of grid enrichment methods. In this ap-

proach, additional nodes and elements are inserted into the grid. There

are two main subclasses of these embedded mesh methods: those in

which the embedded mesh is not aligned with the initial mesh [5J and

those in which the mesh is aligned with the initial mesh [20,40, 72, 73J.

Under the latter subclass of embedded mesh methods, one can further

distinguish between methods that result in interface regions [21, 52, 63J

and those that have no interfaces [48, 58J. In this thesis, the grids are

aligned with the initial mesh and have interfaces which require special

treatment.

The next section of this chapter will present an explanation of the

procedure used to determine where and how to adapt. Then the inter-

face treatment will be discussed, and some properties of these interfaces

examined. Some examples demonstrating the utility of adaptation will

be presented, and the computational savings produced by adaptation

will be examined.

6.2 Adaptation Procedure

In a typical problem, one starts out with an idea of what the flow

will look like, but one usually does not know exactly where the features

lie. The adaptive approach starts with an initial grid coarse enough to

be inexpensive to compute, yet fine enough so that most of the essential

features can appear. The first step in the adaptive solution procedure is

to compute a solution on this coarse grid. It is usually not necessary to

77

Page 90

allow this solution to converge completely, and in the method describe

here the solution is allowed to evolve "halfway" to convergence, that is,

until the convergence parameter reaches the square root of its converged

value. Then, an adaptation parameter is calculated, and cells are flagged

either for refinement or unrefinement. After the initial flagging, elements

adjacent to the cells marked for division are marked for division. This

flagging of adjacent cells can be continued for any number of passes to

enlarge the adapted region. In the test cases computed, only the adjacent

cells need to be flagged to obtain good results. The flagged cells at the

coarsest level are divided, then those at the next coarsest, and so on.

After all division is done, unrefinement proceeds from the finest level

to the coarsest. After the initial set of refinements and unrefinements

in performed, the mesh is scanned for "holes". In two dimensions, an

element is considered part of a hole if 3 or more of its faces have been

subdivided. In three dimensions, an element is considered a hole if 4

or more of its faces are subdivided. If an element is a hole, then it

is subdivided to eliminate the hole. This can result in the formation

of more holes, so the process is repeated until the grid stops changing.

The analogous "island" of fine cells in the midst of a region of coarse

cells is not treated explicitly as the adjacent element flagging algorithm

prevents the formation of excessively small islands in most cases. In any

case, islands are less important than holes, since a hole in a fine region

may result in reduced accuracy, but an island in a coarse region should

only increase or not change the accuracy of the algorithm.

After all mesh changes have been made, the grid geometry is recalcu-

lated, along with some quantities used for vectorization. The solution is

interpolated onto the new grid, and the calculation proceeds. In a typi-

cal problem, an adaptation take as much time as several iterations, but

since adaptation occurs infrequently, the time involved is not significant.

Figure 6.1 shows an example of how elements are refined and unrefined

in two dimensions. The elements marked with the _ are each subdivided

into 4 elements. The elements marked with the 0 are fused back to

form one element. This figure also illustrates the main rule about which

elements can be refined or unrefined. This rule is simply that the refining

or unrefining of an element cannot result in more than one interface node

on the face of any element. For example, the element marked with the x

78

Page 178

Test cases, three dimensions

adaptative grids, 90

50 converging channel, 54

100 double wedge, 54, 90

Test cases, two dimensions

adaptative grids, 88-91

4% circular arc bump, 46-47,52,89-

90

50 converging channel, 43-46,51,88-

89

10% cosine bump, 47, 52

distorted grid, 91

Test functions, 23

166

cell-vertex method, 27

central difference method, 29

galerkin method, 26

Time integration, 38

stability, 38

time step computation, 38

Total enthalpy, definition, 6

Total pressure, 8

Total pressure loss, 8, 125

Trial functions, see interpolation func-

tions

Trilinear Element, 19

geometry of, 20

interpolation functions, 19

jacobian matrix, 21

spatial discretization, 28

Vectorization issues, 144-146

Page 179

Addresses of the editors of the series

"Notes on Numerical Fluid Mechanics":

Prof. Dr. Ernst Heinrich Hirschel (General Editor)

Herzog-Heinrich-Weg 6

D-8011 Zorneding

Federal Republic of Germany

Prof. Dr. Kozo Fujii

High-Speed Aerodynamics Div.

The ISAS

Yoshinodai 3-1-1, Sagamihara

Kanagawa 229

Japan

Prof. Dr. Bram van Leer

Department of Aerospace Engineering

The University of Michigan

Ann Arbor, MI 48109-2140

USA

Prof. Dr. Keith William Morton

Oxford University Computing Laboratory

Numerical Analysis Group

8-11 Keble Road

Oxford OXI 3QD

Great Britain

Prof. Dr. Maurizio Pandolfi

Dipartimento di Ingegneria Aeronautica e Spaziale

Politecnico di Torino

Corso Duca Degli Abruzzi, 24

1-10129 Torino

Italy

Prof. Dr. Arthur Rizzi

FFA Stockholm

Box 11021

S-16111 Bromma 11

Sweden

Dr. Bernard Roux

Institut de Mecanique des Fluides

Laboratoire Associe au C. R. N. S. LA 03

1, Rue Honnorat

F -13003 Marseille

France

Adaptive Finite Element

Solution Algorithm

for the Euler Equations

by Richard A. Shapiro

Page 2

Notes on Numerical Fluid Mechanics (NNFM)

Series Editors: Ernst Heinrich Hirschel, Miinchen

Kozo Fujii, Tokyo

Bram van Leer, Ann Arbor

Keith William Morton, Oxford

Maurizio Pandolfi, Torino

Arthur Rizzi, Stockholm

Bernard Roux, Marseille

(Adresses of the Editors: see last page)

Volume 32

Volume 6 Numerical Methods in Laminar Flame Propagation (N. Peters I J. Warnatz, Eds.)

Volume 7 Proceedings of the Fifth GAMM-Conference on Numerical Methods in Fluid Mechanics

(M. Pandolfi I R. Piva, Eds.)

Volume 8 Vectorization of Computer Programs with Applications of Computational Fluid Dynamics

(W Gentzsch)

Volume 9 Analysis of Laminar Flow over a Backward Facing Step (K. Morgan I J. Periauxl F. Thomasset,

Eds.)

Volume 10 Efficient Solutions of Elliptic Systems (W. Hackbusch, Ed.)

Volume II Advances in Multi-Grid Methods (D. Braess I W Hackbusch I U. Trottenberg, Eds.)

Volume 12 The Efficient Use of Vector Computers with Emphasis on Computational Fluid Dynamics

(W. Schiinauer I W. Gentzsch, Eds.)

Volume 13 Proceedings of the Sixth GAMM-Conference on Numerical Methods in Fluid Mechanics

(D. Rues/W Kordulla, Eds.) (out of print)

Volume 14 Finite Approximations in Fluid Mechanics (E. H. Hirscbel, Ed.)

Volume 15 Direct and Large Eddy Simulation of Turbulence (U. Schumann I R. Friedrich, Eds.)

Volume 16 Numerical Techniques in Continuum Mechanics (W Hackbusch I K. Witsch, Eds.)

Volume 17 Research in Numerical Fluid Dynamics (P. Wesseling, Ed.)

Volume 18 Numerical Simulation of Compressible Navier-Stokes Flows (M. O. Bristeau I R. Glowinski I

J. Periauxl H. Viviand, Eds.)

Volume 19 Three-Dimensional Turbulent Boundary Layers - Calculations and Experiments (B. van den

Bergl D. A. Humphreys I E. Krause I 1. P. F. Lindhout)

Volume 20 Proceedings of the Seventh GAMM-Conference on Numerical Methods in Fluid Mechanics

(M. Deville, Ed.)

Volume 21 Panel Methods in Fluid Mechanics with Emphasis on Aerodynamics (1. Ballmann I R. Eppler I

W. Hackbusch, Eds.)

Volume 22 Numerical Simulation oftbe Transonic DFVLR-F5 Wing Experiment (W. Kordulla, Ed.)

Volume 23 Robust Multi-Grid Methods (W Hackbusch, Ed.)

Volume 24 Nonlinear Hyperbolic Equations - Theory, Computation Methods, and Application (J. Ballmannl

R. Jeltsch, Eds.)

Volume 25 Finite Approximation in Fluid Mechanics II (E. H. Hirschel, Ed.)

Volume 26 Numerical Solution of Compressible Euler Flows (A. Dervieux I B. van Leer I J. Periauxl

A. Rizzi, Eds.)

Volume 27 Numerical Simulation of Oscillatory Convection in Low-Pr Fluids (B. Roux, Ed.)

Volume 28 Vortical Solutions of the Conical Euler Equations (K. G. Powell)

Volume 29 Proceedings of the Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics

(P. Wesseling, Ed.)

Volume 30 Numerical Treatment of the Navier-Stokes Equations (w. Hackbusch I R. Rannacher, Eds.)

Volume 31 Parallel Algorithms for Partial Differential Equations (w. Hackbusch, Ed.)

Volume 32 Adaptive Finite Element Solution Algorithm for the Euler Equations (R. A. Shapiro)

Page 89

A second approach sometimes taken is grid regeneration. In this

method, either the entire computational grid or some smaller portion of

it is regenerated when adaptation occurs. The grid point distribution is

determined by some error indicator based on the solution on the previous

grid. This method is described in detail in articles by Peraire, et al.

[53, 54]. This retains the grid alignment advantages while reducing grid

skewness problems. The main disadvantages are that the cost of grid

regeneration is usually fairly high, the interpolation of the solution from

the old grid to the new grid can be difficult, and the method (so far) has

been limited to triangular and tetrahedral elements.

The third approach is grid enrichment. The survey article by Berger

[4] presents a good overview of grid enrichment methods. In this ap-

proach, additional nodes and elements are inserted into the grid. There

are two main subclasses of these embedded mesh methods: those in

which the embedded mesh is not aligned with the initial mesh [5J and

those in which the mesh is aligned with the initial mesh [20,40, 72, 73J.

Under the latter subclass of embedded mesh methods, one can further

distinguish between methods that result in interface regions [21, 52, 63J

and those that have no interfaces [48, 58J. In this thesis, the grids are

aligned with the initial mesh and have interfaces which require special

treatment.

The next section of this chapter will present an explanation of the

procedure used to determine where and how to adapt. Then the inter-

face treatment will be discussed, and some properties of these interfaces

examined. Some examples demonstrating the utility of adaptation will

be presented, and the computational savings produced by adaptation

will be examined.

6.2 Adaptation Procedure

In a typical problem, one starts out with an idea of what the flow

will look like, but one usually does not know exactly where the features

lie. The adaptive approach starts with an initial grid coarse enough to

be inexpensive to compute, yet fine enough so that most of the essential

features can appear. The first step in the adaptive solution procedure is

to compute a solution on this coarse grid. It is usually not necessary to

77

Page 90

allow this solution to converge completely, and in the method describe

here the solution is allowed to evolve "halfway" to convergence, that is,

until the convergence parameter reaches the square root of its converged

value. Then, an adaptation parameter is calculated, and cells are flagged

either for refinement or unrefinement. After the initial flagging, elements

adjacent to the cells marked for division are marked for division. This

flagging of adjacent cells can be continued for any number of passes to

enlarge the adapted region. In the test cases computed, only the adjacent

cells need to be flagged to obtain good results. The flagged cells at the

coarsest level are divided, then those at the next coarsest, and so on.

After all division is done, unrefinement proceeds from the finest level

to the coarsest. After the initial set of refinements and unrefinements

in performed, the mesh is scanned for "holes". In two dimensions, an

element is considered part of a hole if 3 or more of its faces have been

subdivided. In three dimensions, an element is considered a hole if 4

or more of its faces are subdivided. If an element is a hole, then it

is subdivided to eliminate the hole. This can result in the formation

of more holes, so the process is repeated until the grid stops changing.

The analogous "island" of fine cells in the midst of a region of coarse

cells is not treated explicitly as the adjacent element flagging algorithm

prevents the formation of excessively small islands in most cases. In any

case, islands are less important than holes, since a hole in a fine region

may result in reduced accuracy, but an island in a coarse region should

only increase or not change the accuracy of the algorithm.

After all mesh changes have been made, the grid geometry is recalcu-

lated, along with some quantities used for vectorization. The solution is

interpolated onto the new grid, and the calculation proceeds. In a typi-

cal problem, an adaptation take as much time as several iterations, but

since adaptation occurs infrequently, the time involved is not significant.

Figure 6.1 shows an example of how elements are refined and unrefined

in two dimensions. The elements marked with the _ are each subdivided

into 4 elements. The elements marked with the 0 are fused back to

form one element. This figure also illustrates the main rule about which

elements can be refined or unrefined. This rule is simply that the refining

or unrefining of an element cannot result in more than one interface node

on the face of any element. For example, the element marked with the x

78

Page 178

Test cases, three dimensions

adaptative grids, 90

50 converging channel, 54

100 double wedge, 54, 90

Test cases, two dimensions

adaptative grids, 88-91

4% circular arc bump, 46-47,52,89-

90

50 converging channel, 43-46,51,88-

89

10% cosine bump, 47, 52

distorted grid, 91

Test functions, 23

166

cell-vertex method, 27

central difference method, 29

galerkin method, 26

Time integration, 38

stability, 38

time step computation, 38

Total enthalpy, definition, 6

Total pressure, 8

Total pressure loss, 8, 125

Trial functions, see interpolation func-

tions

Trilinear Element, 19

geometry of, 20

interpolation functions, 19

jacobian matrix, 21

spatial discretization, 28

Vectorization issues, 144-146

Page 179

Addresses of the editors of the series

"Notes on Numerical Fluid Mechanics":

Prof. Dr. Ernst Heinrich Hirschel (General Editor)

Herzog-Heinrich-Weg 6

D-8011 Zorneding

Federal Republic of Germany

Prof. Dr. Kozo Fujii

High-Speed Aerodynamics Div.

The ISAS

Yoshinodai 3-1-1, Sagamihara

Kanagawa 229

Japan

Prof. Dr. Bram van Leer

Department of Aerospace Engineering

The University of Michigan

Ann Arbor, MI 48109-2140

USA

Prof. Dr. Keith William Morton

Oxford University Computing Laboratory

Numerical Analysis Group

8-11 Keble Road

Oxford OXI 3QD

Great Britain

Prof. Dr. Maurizio Pandolfi

Dipartimento di Ingegneria Aeronautica e Spaziale

Politecnico di Torino

Corso Duca Degli Abruzzi, 24

1-10129 Torino

Italy

Prof. Dr. Arthur Rizzi

FFA Stockholm

Box 11021

S-16111 Bromma 11

Sweden

Dr. Bernard Roux

Institut de Mecanique des Fluides

Laboratoire Associe au C. R. N. S. LA 03

1, Rue Honnorat

F -13003 Marseille

France