spectral/hp element method · 2017-05-04 · spectral/hp element method 1 numerical mathematics and...

Numerical Methods for Linear StabilityBeyond Matlab

• Spectral/hp element methods

• Nektar++ code

• Linearised Stability Analysis

Solution Techniques

Spectral/hp methods

finite difference

ij

uδ =

h

uδ = p

spectralmethods

finite element/finite volume

uδ =

h

Spectral/hp element method

1

NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION is a series designed to provide texts and monographs for graduate students and researchers on a wide range of mathematical topics at the interface of computational science and numerical analysis.

George Em Karniadakis and Spencer Sherwin

Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computationaldomains has historically been much more limited. More recently the need to findaccurate solutions to the viscous flow equations around complex configurationshas led to the development of high-order discretisation procedures on unstruc-tured meshes, which are also recognised as more efficient for solution of time-dependent oscillatory solutions over long time periods.

Here Karniadakis and Sherwin present a much-updated and expanded version of their successful first edition covering the recent and significant progress inmulti-domain spectral methods at both the fundamental and application level.Containing over 50% new material, including discontinuous Galerkin methods,non-tensorial nodal spectral element methods in simplex domains, and stabilisa-tion and filtering techniques, this text aims to introduce a wider audience to theuse of spectral/hp element methods with particular emphasis on their applicationto unstructured meshes. It provides a detailed explanation of the key conceptsunderlying the methods along with practical examples of their derivation andapplication, and is aimed at students, academics and practitioners in computa-tional fluid mechanics, applied and numerical mathematics, computationalmechanics, aerospace and mechanical engineering and climate/ocean modelling.

ALSO PUBLISHED BY OXFORD UNIVERSITY PRESS

Ch. Schwab

Peter Davidson

Antonín Novotny and Ivan Straskraba

Ivo Babuska and Theofanis Strouboulis

NUMERICAL MATHEMATICSAND SCIENTIFIC COMPUTATION

Spectral/hp Element Methods for Computational Fluid

DynamicsSECOND EDITION

GEORGE EM KARNIADAKIS and SPENCER SHERWIN

OXFORD SCIENCE PUBLIC ATIONS

Spectral/hp E

lemen

t M

ethods for C

ompu

tational

Fluid D

ynam

ics

2

SecondEdition

KA

RN

IAD

AK

IS

AN

D S

HE

RW

IN

ˇ

KARNIADAKIS FILM C2B 28/4/05 12:01 pm

p-type exponential accuracy

h-type geometric flexibility

Work ! C(h)p(Dim+1)

Error ! K(u, p) hp

Spectral/hp element methods Sec. 4: Spectral/hp elements in 2D

!pq("1,"2) = #p("1) #q("2)

#p("1)

"1

"2

#q("2)

p

q

p

qa

a

a a

!pq("1,"2) = hp("1) hq("2)

hp("1)

"1

"2

hq("2)

p

q

p

q

Figure 17: Construction of a two-dimensional expansion basis from the tensor product of twoone-dimensional expansions of order P = 4. A modal expansion (top) and a nodalexpansion (bottom) are shown.

expansion. Since a large part of the e!ciency of the quadrilateral expansion (particu-larly at larger polynomial orders) arises from the tensor product construction, we wouldlike to use a similar procedure to construct expansions within the triangular domains.Therefore, to extend the tensor product expansion to simplex regions such as a trianglewe need to generalise the tensor product expansion concept, which can be achieved byusing a collapsed coordinate system.

4.1.3 Collapsed coordinate system

In this section we will focus on 2D expansions defined on the standard triangle Tst,defined as

Tst = (!1 , !2)|! 1 " !1 , !2 ; !1 + !2 " 0 .

In the quadrilateral expansions discussed in section 4.1.1 we generated a multidi-mensional expansion by forming a tensor product of one-dimensional expansions basedon a Cartesian coordinate system. The one-dimensional expansion was defined betweenconstant limits and therefore an implicit assumption of the tensor extension was that thecoordinates in the two-dimensional region were bounded between constant limits. How-ever this is not the case in the standard triangular region as the bounds of the Cartesiancoordinates (!1, !2) are dependent upon each other.

4-2


!pq("1,"2) = #p("1) #q("2)

#p("1)

"1

"2

#q("2)

p

q

p

qa

a

a a

!pq("1,"2) = hp("1) hq("2)

hp("1)

"1

"2

hq("2)

p

q

p

q





Tst = (!1 , !2)|! 1 " !1 , !2 ; !1 + !2 " 0 .


4-2

hp finite element - hierarchicalSpectral element method - nodal

Why Spectral/hp elements methods?

φ = U [r + a2/r] cos θ

Mesh A Mesh B

Mesh C Mesh D

Mesh E Mesh F

~Work

Erro

r

Mesh C

P=1

P=2

Mesh A

Degrees of freedom

Erro

r

Mesh A

P=1

P=2

Mesh C

BooksOverview: What is it?

S.J. Sherwin, R. M. KirbyChris Cantwell, Gabrielle Rocco

Nektar++ is an open source software library currently being developed and designed to provide a bridge to the community – to provide a toolbox of data structures and algorithms which

implement the spectral/hp element method, a high-order numerical method yielding fast error convergence. It is

implemented as a C++ object-oriented toolkit which allows developers to implement spectral element solvers for a variety

of different engineering problems.

For more information, go to: www.nektar.info

Libraries to operator mapping

i

vj

j

l

0

∂ΦHi∂x

∂ΦHj∂x

+ λΦHi ΦHj

uj dx =

ΦHi f∗dx

f [i]

C

D

!1

!2

A B

CD

u!(!1, !2) =P1!

p=0

P2!

q=0

upq"pq(!1, !2),xi = !i("1, "2) =

p=P1!

p=0

q=P2!

q=0

xipq#pq("1, "2)

x1

x2

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

u!(x1, x2) =Nel!

e=1

"

P1!

p=0

P2!

q=0

uepq"

epq(x1, x2)

#

(6)

(7)

1

LocalRegionsSpatialDomainsStdRegions

MultiRegions

=

Φif∗dx

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2)

A

B

i

vj

j

l

0

∂ΦHi∂x

∂ΦHj∂x

+ λΦHi ΦHj

uj dx =

ΦHi f∗dx

f [i]

C

D

!1

!2

A B

CD

u!(!1, !2) =P1!

p=0

P2!

q=0

upq"pq(!1, !2),xi = !i("1, "2) =

p=P1!

p=0

q=P2!

q=0

xipq#pq("1, "2)

x1

x2

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

u!(x1, x2) =Nel!

e=1

"

P1!

p=0

P2!

q=0

uepq"

epq(x1, x2)

#

(6)

(7)

1


MultiRegions

=

Φif∗dx =

nel

e

p

Ωe

φp(x)f∗dx

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

(6)

1

A

B


i

vj

j

l

0

∂ΦHi∂x

∂ΦHj∂x

+ λΦHi ΦHj

uj dx =

ΦHi f∗dx

f [i]

C

D

!1 (1)

!2 (2)

(3)

1

!1 (1)

!2 (2)

(3)

1

A B

CD

Ch. 3 Multi-dimensional Formulation 105

where

u!p =

P!

q=0

uq#hq

#!(!)

"""""p

.

This is very significant when calculating non-linear terms such as the advection operator inthe Navier-Stokes equation. For example, to determine the value of the non-linear product

u!(!)#u!

#!(!)

at a point !i we have:

u!(!i)#u!

#!(!i) =

#P!

p=0

uphp(!i)

$#P!

q=0

uq#hq

#!(!i)

$

=

#P!

p=0

uphp(!i)

$#P!

q=0

u!qhq(!i)

$

.

Since hp(!i) = $pi and hq(!i) = $qi then

u!(!i)#u!

#!(!i) = uiu

!i.

Finally, we can represent our nonlinear product in terms of an expansion of Lagrange poly-nomials as

u!(!)#u!

#!(!) !

P!

p=0

upu!php(!).

We note however that if u!(!) is a polynomial of order P then the non-linear product

u!(!)#u!

#" (!) is a polynomial of order (2P " 1) and so it cannot be exactly represented by

the Lagrange polynomial expansion of order P . At the nodal points the coe!cient upu!p will

be identical to the value of u!(!p)#u!

#" (!p). Nevertheless, projecting the non-linear terms toa lower polynomial order in this fashion can lead to aliasing errors as discussed in section1.4.1.2.

Although this example is in one-dimension, the same properties apply in multiple di-mensions provided the expansion can be represented by a tensor product of Lagrange poly-nomials. Using the collapsed Cartesian coordinates systems described in section 2.2.1 itis possible to represent any polynomial expansion as a tensor product of one-dimensionalLagrange polynomials.

3.1.2.1 Two Dimensions Di!erentiation in the Standard Regions, "st

Implementation.note: Numericaldi!erentiation in "st:Quadrilateral andtriangular regions.

Quadrilateral Region

To di#erentiate an expansion within the standard quadrilateral region Q2 of the form:

u!(!1, !2) =P1!

p=0

P2!

q=0

upq"pq(!1, !2),

we first represent the function in terms of Lagrange polynomials so

u!(!1, !2) =Q1"1!

p=0

Q2"1!

q=0

upq hp(!1)hq(!2),

xi = !i("1, "2) =p=P1!

p=0

q=P2!

q=0

xipq#pq("1, "2)

x1

x2

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

u!(x1, x2) =Nel!

e=1

"

P1!

p=0

P2!

q=0

uepq"

epq(x1, x2)

#

(6)

(7)

1


MultiRegions

=

Φif∗dx =

nel

e

p

Ωe

φp(x)f∗dx

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

(6)

1

A

B

=nel

e

p

Ωe

φp(χe(ξ))f∗Jedξ


=nel

e

p

Ωe

φp(χe(ξ))f∗Jedξ

i

vj

j

l

0

∂ΦHi∂x

∂ΦHj∂x

+ λΦHi ΦHj

uj dx =

ΦHi f∗dx

f [i]

C

D

!1 (1)

!2 (2)

(3)

1

!1 (1)

!2 (2)

(3)

1

A B

CD


where

u!p =

P!

q=0

uq#hq

#!(!)

"""""p

.


u!(!)#u!

#!(!)


u!(!i)#u!

#!(!i) =

#P!

p=0

uphp(!i)

$#P!

q=0

uq#hq

#!(!i)

$

=

#P!

p=0

uphp(!i)

$#P!

q=0

u!qhq(!i)

$

.


u!(!i)#u!

#!(!i) = uiu

!i.


u!(!)#u!

#!(!) !

P!

p=0

upu!php(!).


u!(!)#u!










u!(!1, !2) =P1!

p=0

P2!

q=0

upq"pq(!1, !2),


u!(!1, !2) =Q1"1!

p=0

Q2"1!

q=0

upq hp(!1)hq(!2),


!1

!2

x1

x2

xi = fiA(!1)

xi = fiC(!1)

xi = fiD(!2)

xi = fi

B(!2)

!1

!2 "i(!1,!2)

Figure 3.4 A general curved element can be described in terms of a series of parametric functionsfA(!1), fB(!2), fC(!1), and fD(!2). Representing these functions as a discrete expansion we canconstruct an iso-parametric mapping "i(!1, !2) relating the standard region (!1, !2) to the deformedregion (x1, x2).

the hierarchical modal expansion. For example, a quadrilateral domain of the form shownin figure 3.2(b) the mapping can be defined by equation (3.37).

We note that this simply involves the vertex modes of the modified hierarchical expansionbasis within a quadrilateral domain (see section 2.1.1). We could, therefore, have writtenthe expansion as

xi = !i("1, "2) =p=P1!

p=0

q=P2!

q=0

xipq#pq("1, "2) (3.38)

where #pq = $ap ("1)$a

q ("2) and xipq = 0 except for the vertex modes which have a value of

xi00 = xA

i xiP10 = xB

i xiP1P2

= xCi xi

0P2= xD

i .

The construction of a mapping based upon the expansion modes in this form can beextended to include curved sided regions using an isoparametric representation. In thistechnique the geometry is represented with an expansion of the same form and polynomialorder as the unknown variables.

To describe a straight-sided region we needed only to know the values of the vertexlocations. To describe a curved region, however, requires more information. As illustratedin figure 3.4, we typically expect to have a definition of a mapping of the shape of each edgein terms of the local coordinates which we denote as fA

i ("1), fBi ("2), fC

i ("1) and fDi ("2). The

process of defining the mapping functions can be considered as part of the mesh generationprocess, the discussion of which is in section 3.3.3.

Knowing the definition of the edges (or faces in three-dimensions) a mapping for acurvilinear domains can be determined using the isoparametric form of equation (3.38)to include more non-zero expansion coe!cients than simply the vertex contributions. Intwo-dimensions we wish to use the coe!cient along each edge of the element, and in three-dimensions we can use the face coe!cients as well. Along each edge we therefore need toapproximate the shape mapping fi(") if it is not already represented by a polynomial of

x1

x2

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

u!(x1, x2) =Nel!

e=1

"

P1!

p=0

P2!

q=0

uepq"

epq(x1, x2)

#

(6)

(7)

1


MultiRegions

=

Φif∗dx =

nel

e

p

Ωe

φp(x)f∗dx

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

(6)

1

A

B


Nektar++ code

C

D

!1 (1)

!2 (2)

(3)

1

!1 (1)

!2 (2)

(3)

1

A B

CD


where

u!p =

P!

q=0

uq#hq

#!(!)

"""""p

.


u!(!)#u!

#!(!)


u!(!i)#u!

#!(!i) =

#P!

p=0

uphp(!i)

$#P!

q=0

uq#hq

#!(!i)

$

=

#P!

p=0

uphp(!i)

$#P!

q=0

u!qhq(!i)

$

.


u!(!i)#u!

#!(!i) = uiu

!i.


u!(!)#u!

#!(!) !

P!

p=0

upu!php(!).


u!(!)#u!










u!(!1, !2) =P1!

p=0

P2!

q=0

upq"pq(!1, !2),


u!(!1, !2) =Q1"1!

p=0

Q2"1!

q=0

upq hp(!1)hq(!2),


!1

!2

x1

x2

xi = fiA(!1)

xi = fiC(!1)

xi = fiD(!2)

xi = fi

B(!2)

!1

!2 "i(!1,!2)

Figure 3.4 A general curved element can be described in terms of a series of parametric functionsfA(!1), fB(!2), fC(!1), and fD(!2). Representing these functions as a discrete expansion we canconstruct an iso-parametric mapping "i(!1, !2) relating the standard region (!1, !2) to the deformedregion (x1, x2).

the hierarchical modal expansion. For example, a quadrilateral domain of the form shownin figure 3.2(b) the mapping can be defined by equation (3.37).

We note that this simply involves the vertex modes of the modified hierarchical expansionbasis within a quadrilateral domain (see section 2.1.1). We could, therefore, have writtenthe expansion as

xi = !i("1, "2) =p=P1!

p=0

q=P2!

q=0

xipq#pq("1, "2) (3.38)

where #pq = $ap ("1)$a

q ("2) and xipq = 0 except for the vertex modes which have a value of

xi00 = xA

i xiP10 = xB

i xiP1P2

= xCi xi

0P2= xD

i .

The construction of a mapping based upon the expansion modes in this form can beextended to include curved sided regions using an isoparametric representation. In thistechnique the geometry is represented with an expansion of the same form and polynomialorder as the unknown variables.

To describe a straight-sided region we needed only to know the values of the vertexlocations. To describe a curved region, however, requires more information. As illustratedin figure 3.4, we typically expect to have a definition of a mapping of the shape of each edgein terms of the local coordinates which we denote as fA

i ("1), fBi ("2), fC

i ("1) and fDi ("2). The

process of defining the mapping functions can be considered as part of the mesh generationprocess, the discussion of which is in section 3.3.3.

Knowing the definition of the edges (or faces in three-dimensions) a mapping for acurvilinear domains can be determined using the isoparametric form of equation (3.38)to include more non-zero expansion coe!cients than simply the vertex contributions. Intwo-dimensions we wish to use the coe!cient along each edge of the element, and in three-dimensions we can use the face coe!cients as well. Along each edge we therefore need toapproximate the shape mapping fi(") if it is not already represented by a polynomial of

x1

x2

!1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

u!(x1, x2) =Nel!

e=1

"

P1!

p=0

P2!

q=0

uepq"

epq(x1, x2)

#

(6)

(7)

1


MultiRegions !1 (1)

!2 (2)

x1 (3)

x2 (4)

u!(x1, x2) =P1!

p=0

P2!

q=0

upq"pq(x1, x2) (5)

(6)

1

A

B

StdSegExp

StdTriExp

Std

Regio

ns

libra

ryLocalR

egio

ns

libra

ry

objecs of these classes contain:

SegExp QuadExp

TriExp

HexExp

PrismExp

PyrExp

TetExp

ExpList3D

ExpList

ExpList1D

this library contains:

• linear algebra routines• block-matrix routines• data managers and memory pools• polynomial manipulation routines

several utilities supporting the other libraries

Spati

alD

om

ain

slibra

ry

the geometry of an element• data: ! a standard expansion (a parametrix mapping from a standard

objects of these classes contain:

Geometry

HexGeom

TetGeom

Mult

iRegio

ns

libra

ry

• data: ! a list of local elemental expansions• da for the classes ContExpList iD, ContField iD, DisContField iD:• data: ! the global coe!cients ug

• data: ! a mapping array from the local to the global degrees of freedom


• da for the classes ContField iD, DisContField iD:• data: ! information about the boundary conditions

• data: ! the basis !p(!i)

expansion on local element u(xi) =!

p !p(xi)up

• data: ! the coe!cients up

• data: ! the geometry of the element

• data: ! the basis !p(xi)

• data: ! the physical values u(xi)

expansion on a global region u(xi) =!

e

!p !e

p(xi)uep

• data: ! to the local element, which entirely describes the geometry)• data: ! the geometric factors of the transformation (Jacobian, ...)

objects of these

• data: ! the physical values u(!i)• data: ! the coe!cients up

classes contain:expansion on standard element u(!i) =

!p !p(!i)up

Lib

Utilities

libra

ry

DisContField2D DisContField3DDisContField1D

ContField1D ContField2D ContField3D

ContExpList3DContExpList2DContExpList1D

ExpList2D

StdExpansion3D

StdHexExp

StdPyrExp

StdQuadExp

EdgeComp

SegGeom QuadGeom

TriGeom

PrismGeom

PyrGeom

VertexComp

TriFaceComp

Geometry3DGeometry2DGeometry1D

StdExpansion

StdPrismExp

StdTetExp

QuadFaceComp

StdExpansion2DStdExpansion1D

StdSegExp

StdTriExp

Std

Regio

ns

libra

ryLocalR

egio

ns

libra

ry

objecs of these classes contain:

SegExp QuadExp

TriExp

HexExp

PrismExp

PyrExp

TetExp

ExpList3D

ExpList

ExpList1D

this library contains:

• linear algebra routines• block-matrix routines• data managers and memory pools• polynomial manipulation routines

several utilities supporting the other libraries

Spati

alD

om

ain

slibra

ry

the geometry of an element• data: ! a standard expansion (a parametrix mapping from a standard


Geometry

HexGeom

TetGeom

Mult

iRegio

ns

libra

ry

• data: ! a list of local elemental expansions• da for the classes ContExpList iD, ContField iD, DisContField iD:• data: ! the global coe!cients ug

• data: ! a mapping array from the local to the global degrees of freedom


• da for the classes ContField iD, DisContField iD:• data: ! information about the boundary conditions

• data: ! the basis !p(!i)

expansion on local element u(xi) =!

p !p(xi)up

• data: ! the coe!cients up

• data: ! the geometry of the element

• data: ! the basis !p(xi)

• data: ! the physical values u(xi)

expansion on a global region u(xi) =!

e

!p !e

p(xi)uep

• data: ! to the local element, which entirely describes the geometry)• data: ! the geometric factors of the transformation (Jacobian, ...)

objects of these

• data: ! the physical values u(!i)• data: ! the coe!cients up

classes contain:expansion on standard element u(!i) =

!p !p(!i)up

Lib

Utilities

libra

ry

DisContField2D DisContField3DDisContField1D

ContField1D ContField2D ContField3D

ContExpList3DContExpList2DContExpList1D

ExpList2D

StdExpansion3D

StdHexExp

StdPyrExp

StdQuadExp

EdgeComp

SegGeom QuadGeom

TriGeom

PrismGeom

PyrGeom

VertexComp

TriFaceComp

Geometry3DGeometry2DGeometry1D

StdExpansion

StdPrismExp

StdTetExp

QuadFaceComp

StdExpansion2DStdExpansion1D

Linear stability analysis

• Eigenvalues from inverse linearised operator

• Time steppers approach to linearised eigenvalues

• Transient growth of linearised operator

base flow + perturbation

Linearised Navier-Stokes

!tu! = !UN(u!) + L(u!)

keep terms linear in

u!· !U + U · !u

!

non-symmetric

ut = L(U, t)u (∇ · u = 0)

Linear system:

u(x, τ) = A(U, τ)u(x, 0)Integrate in time:

A(U, τ) = exp τ

0L(U, t)dt

Direct inversion of linearised Navier-Stokes system(Ainsworth & Sherwin, CMAME, 1999 / Sherwin & Ainsworth APNUM 2000)

Coupled linearised Navier-Stokes solvers

Consider the linearised Navier-Stokes element matrix denoted as!

"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$ (1)

where vbnd denote the degrees of freedom of the elemental velocities on theboundary of the element, vint denote the degrees of freedom of the elemen-tal velocities on the interior of the element and p is the piecewise continuouspressure. The matrices have the interpretation

A[n,m] = (!!bn, "!!b

m) + (!bn,U ·!!b

m) + (!bn!

TU!b

m)

B[n,m] = (!!bn, "!!i

m) + (!bn,U ·!!i

m) + (!bn!

TU!i

m)

BT [n,m] = (!!in, "!!b

m) + (!in,U ·!!b

m) + (!in!

TU!b

m)

C[n,m] = (!!in, "!!i

m) + (!in,U ·!!i

m) + (!in!

TU!i

m)

Dbnd[n,m] = (#m,!!b)

Dint[n,m] = (#m,!!i)

where # is the space of pressures typically at order P " 2 and ! is the velocityvector space of polynomial order P . (To be strictly correct probably have tobreak matrices A,B,C into sub-blocks to use the above definitions).

Note B = BT if just considering the stokes operator and then C is alsosymmetric. Also note A,B and C are block diagonal in the Oseen equationwhen the !TU are zero.

Since C is invertible we can premultiply the elemental equation (1)by thefollowing,

!

"

I 0 "BC!1

0 I DTintC

!1

0 0 I

#

$

%

&

'

!

"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$

(

)

*

(2)

which if we multiply out the matrix equation we obtain

!

"

A"BC!1BT DTbnd "BC!1Dint 0

Dbnd "DTintC

!1BT "DTintC

!1Dint 0BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd "BC!1fint

fp = "DTintC

!1fint

fint

#

$

(3)In the above equation the vint degrees of freedom are decoupled and so we

need to solve for the vbnd, p degrees of freedom. The final step is to perform asecond level of static condensation but where we will lump the mean pressuremode (or a pressure degree of freedom containing a mean component) with thevelocity boundary degrees of freedom. To do we define b = [vbnd, p0] where p0is the mean pressure mode and p to be the remainder of the pressure space. Wenow repartition the top 2# 2 block of matrices in (3) as

+

A B

DT C

, +

b

p

,

=

+

fbnd

fp

,

(4)

1



"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$ (1)


A[n,m] = (!!bn, "!!b

m) + (!bn,U ·!!b

m) + (!bn!

TU!b

m)

B[n,m] = (!!bn, "!!i

m) + (!bn,U ·!!i

m) + (!bn!

TU!i

m)

BT [n,m] = (!!in, "!!b

m) + (!in,U ·!!b

m) + (!in!

TU!b

m)

C[n,m] = (!!in, "!!i

m) + (!in,U ·!!i

m) + (!in!

TU!i

m)






!

"

I 0 "BC!1

0 I DTintC

!1

0 0 I

#

$

%

&

'

!

"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$

(

)

*

(2)


!

"


Dbnd "DTintC

!1BT "DTintC

!1Dint 0BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd "BC!1fint

fp = "DTintC

!1fint

fint

#

$



+

A B

DT C

, +

b

p

,

=

+

fbnd

fp

,

(4)

1



"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$ (1)


A[n,m] = (!!bn, "!!b

m) + (!bn,U ·!!b

m) + (!bn!

TU!b

m)

B[n,m] = (!!bn, "!!i

m) + (!bn,U ·!!i

m) + (!bn!

TU!i

m)

BT [n,m] = (!!in, "!!b

m) + (!in,U ·!!b

m) + (!in!

TU!b

m)

C[n,m] = (!!in, "!!i

m) + (!in,U ·!!i

m) + (!in!

TU!i

m)






!

"

I 0 "BC!1

0 I DTintC

!1

0 0 I

#

$

%

&

'

!

"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$

(

)

*

(2)


!

"


Dbnd "DTintC

!1BT "DTintC

!1Dint 0BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd "BC!1fint

fp = "DTintC

!1fint

fint

#

$



+

A B

DT C

, +

b

p

,

=

+

fbnd

fp

,

(4)

1

=


!pq("1,"2) = #p("1) #q("2)

#p("1)

"1

"2

#q("2)

p

q

p

qa

a

a a

!pq("1,"2) = hp("1) hq("2)

hp("1)

"1

"2

hq("2)

p

q

p

q





Tst = (!1 , !2)|! 1 " !1 , !2 ; !1 + !2 " 0 .


4-2

v = Spectral/hp element methods Sec. 4: Spectral/hp elements in 2D

!pq("1,"2) = #p("1) #q("2)

#p("1)

"1

"2

#q("2)

p

q

p

qa

a

a a

!pq("1,"2) = hp("1) hq("2)

hp("1)

"1

"2

hq("2)

p

q

p

q





Tst = (!1 , !2)|! 1 " !1 , !2 ; !1 + !2 " 0 .


4-2

p =



"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$ (1)


A[n,m] = (!!bn, "!!b

m) + (!bn,U ·!!b

m) + (!bn!

TU!b

m)

B[n,m] = (!!bn, "!!i

m) + (!bn,U ·!!i

m) + (!bn!

TU!i

m)

BT [n,m] = (!!in, "!!b

m) + (!in,U ·!!b

m) + (!in!

TU!b

m)

C[n,m] = (!!in, "!!i

m) + (!in,U ·!!i

m) + (!in!

TU!i

m)






!

"

I 0 "BC!1

0 I DTintC

!1

0 0 I

#

$

%

&

'

!

"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$

(

)

*

(2)


!

"


Dbnd "DTintC

!1BT "DTintC

!1Dint 0BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd "BC!1fint

fp = "DTintC

!1fint

fint

#

$



+

A B

DT C

, +

b

p

,

=

+

fbnd

fp

,

(4)

1



"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$ (1)


A[n,m] = (!!bn, "!!b

m) + (!bn,U ·!!b

m) + (!bn!

TU!b

m)

B[n,m] = (!!bn, "!!i

m) + (!bn,U ·!!i

m) + (!bn!

TU!i

m)

BT [n,m] = (!!in, "!!b

m) + (!in,U ·!!b

m) + (!in!

TU!b

m)

C[n,m] = (!!in, "!!i

m) + (!in,U ·!!i

m) + (!in!

TU!i

m)






!

"

I 0 "BC!1

0 I DTintC

!1

0 0 I

#

$

%

&

'

!

"

A DTbnd B

Dbnd 0 DTint

BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd

0fint

#

$

(

)

*

(2)


!

"


Dbnd "DTintC

!1BT "DTintC

!1Dint 0BT Dint C

#

$

!

"

vbnd

pvint

#

$ =

!

"

fbnd "BC!1fint

fp = "DTintC

!1fint

fint

#

$



+

A B

DT C

, +

b

p

,

=

+

fbnd

fp

,

(4)

1

Static condensation:

Smallest eigenvalues of L is largest eigenvalue of L-1


!pq("1,"2) = #p("1) #q("2)

#p("1)

"1

"2

#q("2)

p

q

p

qa

a

a a

!pq("1,"2) = hp("1) hq("2)

hp("1)

"1

"2

hq("2)

p

q

p

q





Tst = (!1 , !2)|! 1 " !1 , !2 ; !1 + !2 " 0 .


4-2

vbnd

vint

Restarted Eigenvalue algorithm (IJNMF 08)ARPACK: http://www.caam.rice.edu/software/ARPACK/

3. Use LAPACK to calculate eigensystem of H

1. Generate a Krylov subspace T

2. QR factorize T and Calculate Hessenberg matrix H:

In practice use a fixed Kmax.If iteration exceeds Kmax, then discard oldest vector , i.e. restart with second oldest vector.

Wake transition in staggered arrangements 9

The action of operator A on the perturbation u

was computed by integrating the

linearized Navier-Stokes equations, using the same schemes that were used to compute

the base flow. Two changes were necessary in the integration algorithm. The first was that

a linear operator (−(U.∇)u−(u

.∇)U) had to be used to calculate the advection terms.

In this operator, the values of U were computed by means of a Fourier interpolation of the

time slices resulting from the base flow calculation. The second change was to replace the

operator ∇ by (∂/∂x, ∂/∂y,−iβ), and compute the three velocity components (u, v, w)

and p on the two-dimensional domains.

The Arnoldi method was used to compute the Floquet multipliers of largest magnitude.

We briefly describe this method here; for a more complete discussion the reader should

refer to Saad (1992). Essentially, this is an orthogonal projection method of a matrix A

on to a k-dimensional Krylov subspace Kk ≡ spanu,Au,A2

u, . . . ,Ak−1u. Given an

orthonormal basis Qk = [v0|v1| . . . |vk−1] of the Krylov subspace Kk, it is possible to

decompose the matrix A in the following way:

AQk = QkHk + hk,k−1vkeH

k−1. (2.10)

Hk is a Hessenberg matrix, whose component on row i and column j is denoted as hi,j ,

vk is a unitary vector orthogonal to the basis Qk and ek−1 is a versor pointing in the

direction of the k− 1 component. Multiplying both sides of eq. (2.10) on the left by QH

k

and using the fact that Qk is orthonormal results in

QH

kAQk = Hk. (2.11)

The eigenvalues λ(k)i

of the Hessenberg matrix Hk are approximations of the eigenvalues

of the matrix A, and the approximate eigenvector of A associated with λ(k)i

, also called

the Ritz eigenvector, can be calculated using the expression

w(k)i

= Qkyk

i, (2.12)

λk(A) ≈ λmax(Hk) wmax ≈ Qk ymax

Timestepper stability analysisTuckerman & Barkley (2000), Barkley & Henderson (1996)

Asymptotic instability if:

If is periodic, are Floquet multipliers

equivalently

where

Note: maxi

µi ⇒ maxi

Re(λi)

N(u) = u ·∇u

Pressure Poisson:

∇2pn+1 =1

∆t∇ · u∗

Navier-Stokes Time-stepping∂tu + N(u) = −∇p + ν∇2u

Navier–Stokes:

Velocity correction scheme (aka stiffly stable):Orszag, Israeli, Deville (90), Karnaidakis Israeli, Orszag (1991), Guermond & Shen (2003)

Helmholtz: ∇2un+1 − α0

ν∆tun+1 = − u∗

ν∆t+

1ν∇pn+1

u ·∇u

ν∇2u−∇p = f∇ · u = 0

un

un+1

n = n + 1

u∗

∇ · u = 0

Advection: u∗ = −J

q=1

αqun−q −∆tJ−1

q=0

βqN(un−q)

Transient growth: optimal perturbationsEnergy of perturbations over time τ

Recall:

G(τ)

G(τ)

Equivalent to eigenvalue problem:

G(τ)

G(τ)

Transient growth: optimal perturbations

⇒ λmax = [SV Dmax(A)]2

H∗H

Direct optimal growth analysis for timesteppers, Barkley, D, Blackburn, H.M, Sherwin, S.J , International Journal for Numerical Methods in Fluids, Vol: 57, pages: 1435-145 , 2008

Direct inversion vs. Time stepping.

Direct invert Time stepping

Linear eigenvalue steady base flow

Yes (quick)might need shifting

Yes (slow)Exp. method finds leading eigenvalue

Adjoint eigenvalue steady base flow

Yes (quick)might need shifting

Yes (slow) Exp. method finds leading eigenvalue

Linear Floquet mode about unsteady base

flow No Yes

Transient growth No Yes

Test cases

Backward facing step:

Cylinder flow:

Channel flow:

-4 -2 0 2 4-1

0

1

A few papersNumerical methods: - Direct optimal growth analysis for timesteppers, Barkley, D, Blackburn, H.M, Sherwin, S.J , International Journal for Numerical Methods in Fluids, Vol: 57, 1435-145 , 2008

Cylinder flow papers: - Secondary instabilities in the flow around two circular cylinders in tandem, Carmo, B.S., Meneghini, J. Sherwin, S.J., J. Fluid Mech. Vol 644,395-431 , 2010- Wake transition in the flow around two circular cylinders in staggered arrangements, Carmo, B.S, Sherwin, S.J, Bearman, P.W, Willden, R.H.J , Journal of Fluid Mechanics, Vol: 597, 1-29 , 2008

Backward facing step transient growth: - Convective instability and transient growth in flow over a backwards-facing step, Blackburn, H.M, Barkley, D, Sherwin, S.J , Journal of Fluid Mechanics, Vol: 603, 271-304 , 2008

Stenotic flow: - Convective instability and transient growth in steady and pulsatile stenotic flows, Blackburn, H.M, Sherwin, S.J, Barkley, D , Journal of Fluid Mechanics, Vol: 607, 267-277 , 2008- Instability modes and transition of pulsatile stenotic flow: pulse-period dependence, Blackburn, H.M, Sherwin, S.J , Journal of Fluid Mechanics, Vol: 573, 57 , 2007- Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows, Sherwin, SJ, Blackburn, HM, , J. Fluid Mechanics, Vol: 533, 297 - 327 , 2005

Low pressure turbine flow: - Transient growth mechanisms of low Reynolds number flow over a low pressure turbine blade, A.S. Sharma, N. Abdessemed, S.J.Sherwin, V. Theofilis, Theo & Comp Fluid Dyn. 25, 19-30 20011 - Linear instability analysis of low pressure turbine flows, N. Abdessemed, S.J. Sherwin, and V. Theofilis, J. Fluid Mech. , 628, 57-83, 2009.

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