WCCM V
Fifth World Congress on
Computational Mechanics
July 7–12, 2002, Vienna, Austria
Eds.: H.A. Mang, F.G. Rammerstorfer,
J. Eberhardsteiner
Stabilization Parameters and Local Length Scales in
SUPG and PSPG Formulations
Tayfun Tezduyar
Mechanical Engineering, Rice University – MS 321
6100 Main Street, Houston, Texas 77005, USA
e-mail: [email protected]
Key words: stabilization parameters, local length scales, SUPG formulation, PSPG formulation
Abstract
We highlight how we determine the stabilization parameters and element length scales used in the stabilized finite element formulations in fluid mechanics. These formulations include the interface-tracking
and interface-capturing techniques we developed for computation of flow problems with moving boundaries and interfaces. The stabilized formulations we focus on are the streamline-upwind/Petrov-Galerkin
(SUPG) and pressure-stabilizing/Petrov-Galerkin (PSPG) methods. The stabilization parameters described here are designed for the semi-discrete and space-time formulations of the advection-diffusion
equation and the Navier-Stokes equations of incompressible flows.
Tayfun Tezduyar
1
Introduction
Most finite element techniques and computations reported in recent literature for computational fluid mechanics are based on stabilized formulations. The interface-tracking and interface-capturing techniques
we have developed in recent years (see [1, 2, 3, 4]) for flows with moving boundaries and interfaces are
also based on stabilized formulations. An interface-tracking technique, such as the Deforming-SpatialDomain/Stabilized Space-Time (DSD/SST) formulation [1], requires meshes that “track” the interfaces.
The mesh needs to be updated as the flow evolves. In interface-capturing techniques, such as one designed for two-fluid flows, the computations are based on spatial domains that are typically not moving
or deforming. An interface function, marking the location of the interface, needs to be computed to
“capture” the interface over the non-moving mesh.
The finite element flow techniques we have developed, including the interface-tracking and interfacecapturing techniques, are based on the streamline-upwind/Petrov-Galerkin (SUPG) [5], Galerkin/leastsquares (GLS) [6], and pressure-stabilizing/Petrov-Galerkin (PSPG) [1] formulations. In the interfacecapturing techniques, stabilized semi-discrete formulations are used for both the Navier-Stokes equations
of incompressible flows and the advection equation governing the time-evolution of an interface function
marking the interface location. These stabilization techniques prevent numerical oscillations and other
instabilities in solving problems with advection-dominated flows and when using equal-order interpolation functions for velocity and pressure. In these stabilized formulations, judicious selection of the
stabilization parameter, which is almost always known as “ ”, plays an important role in determining
the accuracy of the formulation. This stabilization parameter involves a measure of the local length scale
(also known as “element length”) and other parameters such as the local Reynolds and Courant numbers.
Various element lengths and s were proposed starting with those in [5] and [7], followed by the one
introduced in [8], and those proposed in the subsequently reported SUPG, GLS and PSPG methods. A
number of s, dependent upon spatial and temporal discretizations, were introduced and tested in [9].
More recently, s which are applicable to higher-order elements were proposed in [10].
Ways to calculate s from the element-level matrices and vectors were first introduced in [11]. These
new definitions are expressed in terms of the ratios of the norms of the relevant matrices or vectors. They
take into account the local length scales, advection field and the element-level Reynolds number. Based
on these definitions, a can be calculated for each element, or even for each element node or degree
of freedom or element equation. Certain variations and complements of these new s were introduced
in [12, 4]. In this paper, we describe the element-matrix-based and element-vector-based s designed for
the semi-discrete and space-time formulations of the advection-diffusion equation and the Navier-Stokes
equations of incompressible flows. We also describe approximate versions of these s that are based on
the local length scales for the advection- and diffusion-dominated limits.
2
Governing Equations
Let t IRnsd be the spatial fluid mechanics domain with boundary ,t at time t 2 (0; T ), where
the subscript t indicates the time-dependence of the spatial domain. The Navier-Stokes equations of
incompressible flows can be written on t and 8t 2 (0; T ) as
(
@u
+ u r u , f ) , r = 0;
@t
r u = 0;
(1)
(2)
WCCM V, July 7–12, 2002, Vienna, Austria
u
where , and
defined as
f are the density, velocity and the external force, respectively. The stress tensor is
(p; u) = ,pI + 2""(u):
I
(3)
Here p is the pressure, is the identity tensor, = is the viscosity, is the kinematic viscosity, and
1
2
(4)
"(u) is the strain-rate tensor:
" (u) = ((ru) + (ru)T ):
The essential and natural boundary conditions for Eq. (1) are represented as
u = g on (,t)g; n = h on (,t)h;
(5)
where (,t )g and (,t )h are complementary subsets of the boundary ,t , n is the unit normal vector, and
and are given functions. A divergence-free velocity field 0 ( ) is specified as the initial condition.
h
u x
g
If the problem does not involve any moving boundaries or interfaces, the spatial domain does not need to
change with respect to time, and the subscript t can be dropped from t and ,t . This might be the case
even for flows with moving boundaries and interfaces, if in the formulation used the spatial domain is
not defined to be the part of the space occupied by the fluid(s). For example, we can have a fixed spatial
domain, and model the fluid-fluid interfaces by assuming that the domain is occupied by two immiscible
fluids, A and B, with densities A and B and viscosities A and B . In modeling a free-surface problem
where Fluid B is irrelevant, we assign a sufficiently low density to Fluid B. An interface function serves
as a marker identifying Fluid A and B with the definition = f1 for Fluid A and 0 for Fluid Bg. The
interface between the two fluids is approximated to be at = 0:5. In this context, and are defined as
= A + (1 , )B ; = A + (1 , )B :
(6)
The evolution of the interface function , and therefore the motion of the interface, is governed by a
time-dependent advection equation, written on and 8t 2 (0; T ) as
@
+ u r = 0:
@t
3
(7)
Stabilized Formulations and Stabilization Parameters
3.1 Advection-Diffusion Equation
Let us consider over a domain with boundary
equation, written on and 8t 2 (0; T ) as
, the following time-dependent advection-diffusion
@
+ u r , r ( r) = 0;
@t
(8)
where represents the quantity being transported (e.g., temperature, concentration, interface function),
and is the diffusivity. The essential and natural boundary conditions associated with Eq. (8) are represented as
= g on ,g ;
A function 0 (
n r = h on ,h;
x) is specified as the initial condition.
(9)
Tayfun Tezduyar
Let us assume that we have constructed some suitably-defined finite-dimensional trial solution and test
function spaces Sh and Vh . The stabilized finite element formulation of Eq. (8) can then be written as
follows: find h 2 Sh such that 8wh 2 Vh :
Z
Z
@h
h
h
h
h
h
w
+ u r d
+ r w r d
,
wh hh d,
@t
,h
nel Z
X
@h
+ uh rh , r rh d
= 0:
(10)
+
SUPGuh rwh
@t
e
e=1
Here nel is the number of elements, e is the domain for element e, and SUPG is the SUPG stabilization
Z
parameter.
b
R
b
b
Let us use the notation : e (: : :)d
: V to denote the element-level matrix and element-level
R
vector V corresponding to the element-level integration term e (: : :)d
. We now define the following
element-level matrices and vectors:
b
Z
@h
d
@t
e
wh uh rh d
: mV ;
(11)
: cV ;
(12)
rwh rh d
: kV ;
(13)
uh rwh uh rh d
: k~ ;
(14)
m:
Z
c:
e
Z
k:
k~ :
e
Z
e
V
Z
c~ :
wh
h
h rwh @ d
u
@t
e
: ~cV :
(15)
We define the element-level Reynolds and Courant numbers as follows:
kuhk2 kck ;
~k
kk
t kck
;
=
2 kmk
t kkk
=
;
2 kmk
t
kk~k ;
=
2
kmk
Re =
Cru
Cr
Cr~
bk is the norm of matrix b.
where k
SUPG
(16)
(17)
(18)
(19)
The components of element-matrix-based SUPG are defined as follows:
kck ;
kk~k
t kck
=
;
2 k~ck
S1 =
(20)
S2
(21)
S3 = S1
k
c
k
Re = ~ Re:
kkk
(22)
WCCM V, July 7–12, 2002, Vienna, Austria
To construct SUPG from its components we propose the form
SUPG =
1
S1r
1
1 ,r
1
+ r + r
S2 S3
;
(23)
which is based on the inverse of SUPG being defined as the r-norm of the vector with components 1S1 ,
1
1
S2 and S3 . We note that the higher the integer r is, the sharper the switching between S1, S2 and S3
becomes.
The components of the element-vector-based SUPG are defined as follows:
kc k ;
kk~ k
kc k ;
=
k~c k
SV1 =
V
(24)
V
SV2
V
V
SV3 = SV1
(25)
k
k
c
Re = ~
Re:
kk k
V
(26)
V
With these three components,
(SUPG )V =
1
1
1
r + SV2
r + SV3
r
SV1
, 1
r
:
(27)
Remark 1 The definition of SUPG given by Eq. (27) can be seen as a nonlinear definition because it
depends on the solution. However, in marching from time level n to n + 1 the element vectors can be
evaluated at level n. This might be preferable in some cases, as it spares us from ending up with a
nonlinear semi-discrete equation system.
3.2 Navier-Stokes Equations of Incompressible Flows
Given Eqs. (1)-(2), let us assume that we have some suitably-defined finite-dimensional trial solution and
test function spaces for velocity and pressure: Suh , Vuh , Sph and Vph = Sph . The stabilized finite element
formulation of Eqs. (1)-(2) can then be written as follows: find h 2 Suh and ph 2 Sph such that 8 h 2 Vuh
and q h 2 Vph :
u
w
h
wh @@tu + uh ruh , f d
+ "(wh) : (ph; uh)d
, wh hhd,
,h
Z
nel Z
X
1
+
q h r uh d
+
[SUPGuh rwh + PSPG rq h ] e
e=1 @ uh
h
h
h
h
+ u ru , r (p ; u ) , f d
@t
nel Z
X
+
LSIC r wh r uh d
= 0:
(28)
e
e=1 Z
Z
Z
Here PSPG and LSIC are the PSPG and LSIC (least-squares on incompressibility constraint) stabilization
parameters.
Tayfun Tezduyar
We now define the following element-level matrices and vectors:
: mV ;
(29)
wh (uh ruh )d
: cV ;
(30)
"(wh) : 2""(uh )d
: kV;
(31)
(r wh )ph d
: gV ;
(32)
q h (r uh )d
: gVT ;
(33)
(uh rwh) (uh ruh )d
: k~ V;
(34)
e
Z
k:
e
Z
g:
T
e
Z
:
k~ :
~c :
~ :
e
Z
e
(uh rwh ) : ~cV ;
(35)
: ~V;
(36)
@u
r
qh d
@t
e
: V ;
(37)
rqh (uh ruh )d
: V ;
(38)
rqh rphd
: V ;
(39)
(r wh )(r uh )d
: eV :
(40)
e
Z
h
Z
e
Z
:
e:
d
@ uh
d
@t
e
Z
(uh rwh ) rph d
Z
:
:
h
Z
c:
g
h @u
w
e
@t
Z
m:
e
Z
e
u
Remark 2 In the definition of the element-level matrices listed above, we assume that h appearing in
the advective operator (i.e. in h r h and h r h ) is evaluated at time level n rather than n + 1.
The definition would essentially be the same if we, alternatively, assumed that it is evaluated at time
level n + 1 but nonlinear iteration level i rather than i + 1. Except, in the first option, in the advective
operator we use ( h )n , whereas in the second option we use ( h )in+1 . The second option can be seen as
a nonlinear definition. The first option might be preferable in some cases, as it spares us from another
level of nonlinearity coming from the way is defined. In the definition of the element-level-vectors, we
face the same choices in terms of the evaluation of h in the advective operator.
u
u
u
w
u
u
u
The element-level Reynolds and Courant numbers are defined the same way as they were defined before,
as given by Eqs. (16)-(19). The components of the element-matrix-based SUPG are defined the same way
as they were defined before, as given by Eqs. (20)-(22). SUPG is constructed from its components the
same way as it was constructed before, as give by Eq. (23). The components of the element-vector-based
SUPG are defined the same way as they were defined before, as given by Eqs. (24)-(26). The construction
of (SUPG )V is also the same as it was before, given by Eq. (27).
The components of the element-matrix-based PSPG are defined as follows:
P1 =
kg k ;
k k
T
(41)
WCCM V, July 7–12, 2002, Vienna, Austria
t kgT k
;
2 k k
P2 =
P3 = P1
(42)
k
k
g
Re =
k k Re:
T
(43)
PSPG is constructed from its components as follows:
PSPG =
1
P1r
1
1 ,r
1
+ r + r
P2 P3
:
(44)
The components of the element-vector-based PSPG are defined as follows:
PV1 = P1 ;
PV2 = PV1
(45)
k k ;
k k
V
(46)
V
PV3 = PV1 Re:
With these components,
(PSPG )V =
1
1
(47)
1
r + PV2
r + PV3
r
PV1
The element-matrix-based LSIC is defined as follows:
LSIC =
, 1
r
:
(48)
kck :
kek
(49)
We define the element-vector-based LSIC as:
(LSIC)V = LSIC:
(50)
Remark 3 We can also calculate a separate for each element node, or degree of freedom, or element equation. In that case, each component of would be calculated separately for each element
node, or degree of freedom, or element equation. For this, we first represent an element matrix
in terms of its row matrices: 1 ; 2; : : :; nex and an element vector V in terms of its subvectors:
( V )1; ( V )2; : : :; ( V)nex . If we want a separate for each element node, then 1; 2; : : :; nex and
( V )1; ( V )2; : : :; ( V)nex would be the row matrices and subvectors corresponding to each element
node, with nex = nen , where nen is the number of element nodes. If we want a separate for each
degree of freedom, then 1 ; 2; : : :; nex and ( V )1; ( V)2 ; : : :; ( V)nex would be the row matrices and
subvectors corresponding to each degree of freedom, with nex = ndof , where ndof is the number of
degrees of freedom. If we want a separate for each element equation, then 1 ; 2; : : :; nex and
( V )1; ( V )2; : : :; ( V)nex would be the row matrices and subvectors corresponding to each element
equation, with nex = nee , where nee is the number of element equations. Based on this, the components
of would be calculated using the norms of these row matrices or subvectors, instead of the element
matrices or vectors. For example, a separate S1 or SV1 for each element node would be calculated by
using the expression
b
b
b
b
b b
b
b
b b
b
b
b
b
b
b
b
kc k
(S1)a = ~a ; a = 1; 2; : : :; nen
kkak
(SV1 )a =
k(c )ak ; a = 1; 2; : : :; n :
en
k(k~ )ak
V
V
b b
b
b b
b
b
b
or
b
(51)
(52)
Tayfun Tezduyar
Remark 4 The concept of calculating a separate for each element node or equation can be extended to
calculating a separate for each global node or equation. This can be accomplished by first representing
a global matrix or vector in terms of its row matrices or subvectors associated with the global nodes or
equations, and then by calculating the components of using the norms of these global row matrices or
subvectors. With this approach, applying the class of stabilization techniques described in this paper to
element-free methods would become more direct.
For the purpose of comparison, we define here also the stabilization parameters that are based on an
earlier definition of the length scale h [8]:
hUGN = 2 kuh k
nen
X
a=1
!,1
juh rNaj
;
(53)
where Na is the interpolation function associated with node a. The stabilization parameters are defined
as follows:
hUGN
;
2kuh k
t
=
;
SUGN1 =
SUGN2
(54)
(55)
2
SUGN3 =
(SUPG)UGN =
2
hUGN
;
4
1
(56)
2
SUGN1
+
(PSPG)UGN = (SUPG)UGN ;
(LSIC )UGN =
Here z is given as follows:
2
( z=
where ReUGN
hUGN
1
1
2
SUGN2
+
1
2
SUGN3
3
2
;
(57)
(58)
kuhk z:
ReUGN
, 1
(59)
ReUGN 3;
ReUGN > 3;
(60)
h
= ku 2khUGN .
Comparisons between the performances of these earlier stabilization parameters and the ones proposed
here can be found in [11]. These comparisons show that, especially for special element geometries, the
performances are similar.
Remark 5 The expression for SUGN1 can be written more directly as
SUGN1 =
nen
X
a=1
!,1
juh rNaj
;
(61)
and based on that, the expression for hUGN can be written as
hUGN = 2kuhk SUGN1:
(62)
WCCM V, July 7–12, 2002, Vienna, Austria
A rationale for SUGN1 given by Eq. (61) can be provided based on Remark 3 and Eq. (52). For that, we
apply Eq. (52) to the advection-diffusion equation:
(SV1 )a =
Z
e
Z e
Na uh rh d
uh rNa
uh rh d
:
(63)
Assuming one-point integration, we can write:
(SV1 )a =
jNaj :
h
ju rNaj
(64)
u
Let us define SUGN1 to be the weighted average (weighted with j h rNaj) of these nodal SV1 values:
nen
X
SUGN1 =
a=1
!,1
juh rN
nen
X
aj
a=1
jNaj :
For linear, bilinear and trilinear elements jNaj = Na and therefore
nen
X
SUGN1 =
a=1
!,1
juh rNaj
(65)
Pnen
a=1 jNaj = 1. Consequently:
:
(66)
As a potential alternative or complement to the LSIC stabilization, we propose the DiscontinuityCapturing Directional Dissipation (DCDD) stabilization. In describing the DCDD stabilization, we first
define the unit vectors and :
s
r
h
kuh k ;
s = kuuhk ; r = k r
rkuh k k
and the element-level matrices and vectors cr , k~r , (cr ) , and (k~r ) :
V
Z
cr :
k~r :
e
V
: (cr )V ;
(68)
(r rwh ) (r ruh )d
: (k~r )V :
(69)
e
Z
(67)
wh (r ruh)d
Then the DCDD stabilization is defined as
SDCDD =
nel Z
X
e=1 e
DCDDr wh :
rr , (r s)2ss ruh
d
;
(70)
where the element-matrix-based and element-vector-based DCDD viscosities are:
jr uhj kc~rk ;
kkrk
k(c ) k
) = jr uh j ~r :
k(kr) k
DCDD =
(DCDD
V
V
(71)
(72)
V
An approximate version of the expression given by Eq. (71) can be written as
h
DCDD = jr uh j RGN ;
2
(73)
Tayfun Tezduyar
where
hRGN = 2
nen
X
a=1
jr rNaj
!,1
:
(74)
A different way of determining DCDD can be expressed as
2
DCDD = DCDD kuh k ;
(75)
where
DCDD =
Here
k rkuhk k h
2kUk
kUk
hDCDD
DCDD
U represents a global velocity scale, and h
DCDD
:
(76)
can be calculated by using the expression
hDCDD = 2 ~r ;
kkrk
kc k
(77)
hDCDD = hRGN :
(78)
or the approximation
Combining Eqs. (75) and (76), we obtain
DCDD =
1
2
kuhk 2(h
kUk
DCDD
)2 k rkuh k k :
(79)
It was shown by Mittal [13] that flow computations with the SUPG and PSPG formulations (based
on stabilization parameters very much like those given by Eqs. (53)-(60)) and very high aspect-ratio
elements in the boundary layers might in some cases exhibit convergence problems and inaccuracies.
As a remedy, Mittal [13] proposed to use an element length definition that represents the “minimum
dimension” of an element and gives the minimum edge length in the special case of a rectangular element.
In [4], we proposed to re-define PSPG by modifying the definitions of P3 and PV3 given by Eqs. (43) and
(47). We proposed to accomplish that by using the expressions
P3 = P1
kck ;
kk~rk
PV3 = PV1
kck ;
kk~rk
(80)
or the approximations
hRGN 2
P3 = P1 Re
;
hUGN
hRGN 2
PV3 = PV1 Re
:
hUGN
(81)
In [4], we further stated that these modifications can also be applied to S3 and SV3 given by Eqs. (22)
and (26). Here we write that explicitly:
S3 = S1
kck ;
kk~rk
hRGN 2
S3 = S1 Re
;
hUGN
SV3 = SV1
kck ;
kk~rk
(82)
hRGN 2
SV3 = SV1 Re
:
hUGN
(83)
WCCM V, July 7–12, 2002, Vienna, Austria
However, if we are dealing with just an advection-diffusion equation, rather than the Navier-Stokes equations of incompressible flows, then the definition of the unit vector changes as follows:
r
jhj :
r= kr
rjhj k
(84)
We also propose to re-define SUGN3 given by Eq. (56) as follows:
SUGN3 =
2
hRGN
:
4
(85)
Furthermore, we propose to replace (LSIC )UGN given by Eq. (59) as follows:
2
(LSIC )UGN = (SUPG)UGN kuh k :
(86)
Remark 6 The “element length”s hUGN (given by Eq. (53)) and hRGN (Eq. (74)) can be viewed as the
local length scales corresponding to the advection- and diffusion-dominated limits, respectively.
4
DSD/SST Finite Element Formulation
In the DSD/SST method, the finite element formulation of the governing equations is written over a
sequence of N space-time slabs Qn , where Qn is the slice of the space-time domain between the time
levels tn and tn+1 . At each time step, the integrations involved in the finite element formulation are
performed over Qn . The space-time finite element interpolation functions are continuous within a spacetime slab, but discontinuous from one space-time slab to another. Typically we use first-order polyno+
mials as interpolation functions. The notation (),
n and ()n denotes the function values at tn as approached from below and above respectively. Each Qn is decomposed into space-time elements Qen ,
where e = 1; 2; : : :; (nel )n . The subscript n used with nel is to account for the general case in which
the number of space-time elements may change from one space-time slab to another. The Dirichlet- and
Neumann-type boundary conditions are enforced over (Pn )g and (Pn )h , the complementary subsets of
the lateral boundary of the space-time slab. The finite element trial function spaces (Suh )n for velocity
and (Sph )n for pressure, and the test function spaces (Vuh )n and (Vph)n = (Sph )n are defined by using, over
Qn , first-order polynomials in both space and time.
u
u 2 (Suh)n and ph 2 (Sph)n such
h
The DSD/SST formulation is written as follows: given ( h ),
n , find
that 8 h 2 (Vuh )n and q h 2 (Vph )n :
w
h
wh @@tu + uh ruh , f h dQ + "(wh) : (ph; uh)dQ
Qn
Qn
Z
Z
Z
,
wh hh dP + qhr uh dQ + (wh)+n (uh)+n , (uh),n d
(Pn )h
Qn
n
Z
(nX
)
el n
i
LSME h h h h h
Ł(q ; w ) Ł(p ; u ) , f h dQ
+
e=1 Qen nel Z
X
+
LSICr whr uh dQ = 0;
e
e=1 Qn
Z
Z
(87)
Tayfun Tezduyar
where
@ wh
Ł (q h ; w h ) = + uh rwh
@t
, r (qh; wh);
(88)
and LSME and LSIC are the stabilization parameters (see [14]). This formulation is applied to all spacetime slabs Q0; Q1; Q2; : : :; QN ,1, starting with ( h ),
0 = 0 . For an earlier, detailed reference on this
stabilized formulation see [1].
u
u
Remark 7
Space-Time Extension of the Calculations Described in Section 3
Let us write a DSD/SST formulation that is slightly different than the one given by Eq. (87). We do that
by neglecting the (LSME =) r (2""( h)) term and replacing LSME with SUPG and PSPG :
w
h
wh @@tu + uh ruh , f h dQ + "(wh) : (ph; uh)dQ
Qn
Qn
Z
Z
Z
,
wh hhdP + qhr uhdQ + (wh)+n (uh )+n , (uh),n d
(Pn )h
Qn
n
h
(nX
el )n Z
i
1
@ wh
h
h
h
+
SUPG
+ u rw + PSPGrq Ł(ph ; uh) , f h dQ
@t
e=1 Qen nel Z
X
+
LSICr wh r uh dQ = 0:
e
Q
e=1 n
Z
Z
(89)
For extensions of the calculations based on matrix norms, we define the space-time augmented versions
of the element-level matrices and vectors given by Eqs. (30), (34), and (38):
c
A
h
h @ u + uh ruh dQ
w
@t
Qen
:
k~ A :
A :
Z
Z
Qen
@ wh
+ uh rwh
@t
: (cA )V ;
(90)
h
@@tu + uh ruh dQ : (k~ A)V;
(91)
@ uh
r
qh + uh ruh dQ
@t
Qen
Z
: (A)V :
(92)
The components of element-matrix-based SUPG are defined as follows:
kc k ;
kk~ k
kc k ;
= kk~ rk
S12 =
A
(93)
A
S3
k
A
(94)
S12
where ~ r is the space-time version (i.e. integrated over the space-time element domain Qen ) of the
element-level matrix given by Eq. (69). To construct SUPG from its components we propose the form
SUPG =
1
r
S12
1
1 ,r
+ r
S3
:
(95)
WCCM V, July 7–12, 2002, Vienna, Austria
The components of the element-vector-based SUPG are defined as follows:
k(c ) k ;
k(k~ ) k
kc k :
= ~ rk
kk
SV12 =
A V
(96)
A V
SV3
A
(97)
SV12
From these two components,
(SUPG)V =
1
1
r + SV3
r
SV12
, 1
r
:
(98)
The components of element-matrix-based PSPG are defined as follows:
kg k ;
k k
kc k ;
= kk~ rk
T
P12 =
(99)
A
P3
A
(100)
P12
g
where T is the space-time version of the element-level matrix given by Eq. (33). To construct PSPG from
its components we propose the form
PSPG =
1
1 ,r
1
+ r
P3
r
P12
:
(101)
The components of the element-vector-based PSPG are defined as follows:
kg k
T
V
PV12 =
k( ) k ;
kc k :
= ~ rk
kk
(102)
A V
PV3
A
From these components,
(PSPG)V =
1
1
r + PV3
r
PV12
The element-matrix-based LSIC is defined as
LSIC =
where
(103)
PV12
, 1
r
:
(104)
kc k ;
kek
A
(105)
e is the space-time version of the element-level matrix given by Eq. (40).
The element-vector-based LSIC is defined as
(LSIC)V = LSIC:
(106)
The space-time versions of SUGN1 , SUGN2 , SUGN3 , (SUPG)UGN , (PSPG)UGN , and (LSIC )UGN , given respectively
by Eqs. (54), (55), (85), (57), (58), and (86), are defined as follows:
SUGN12 =
nen @N
X
a
@t
a=1
!,1
+ uh rNa
;
(107)
Tayfun Tezduyar
SUGN3 =
(SUPG )UGN =
h2RGN
;
4
1
2
SUGN13
(108)
+
1
2
SUGN3
, 1
2
;
(PSPG )UGN = (SUPG)UGN ;
(LSIC )UGN = (SUPG)UGN kuh k2 :
(109)
(110)
(111)
Here, nen is the number of nodes for the space-time element, and Na is the space-time interpolation
function associated with node a.
5
Concluding Remarks
We described how we propose to determine the stabilization parameters (“ ”s) and element length scales
used in stabilized finite element formulations of flow problems. These stabilized formulations include the
interface-tracking and interface-capturing techniques we developed for computation of flows with moving boundaries and interfaces. The interface-tracking techniques are based on the Deforming-SpatialDomain/Stabilized Space-Time formulation, where the mesh moves to track the interface. The interfacecapturing techniques, typically used with non-moving meshes, are based on a stabilized semi-discrete
formulation of the Navier-Stokes equations, combined with a stabilized formulation of an advection
equation. The advection equation governs the time-evolution of an interface function marking the interface location. As specific stabilization methods, we focused on the streamline-upwind/Petrov-Galerkin
(SUPG) and pressure-stabilizing/Petrov-Galerkin (PSPG) methods. For the Navier-Stokes equations and
the advection equation, we described the element-matrix-based and element-vector-based s designed
for semi-discrete and space-time formulations. These definitions are expressed in terms of the ratios of
the norms of the relevant matrices or vectors. They take into account the local length scales, advection
field and the element-level Reynolds number. Based on these definitions, a can be calculated for each
element, or even for each element node or degree of freedom or element equation. We also described
certain variations and complements of these new s, including the approximate versions that are based
on the local length scales for the advection- and diffusion-dominated limits.
Acknowledgments
This work was supported by the Army Natick Soldier Center and NASA JSC.
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