1 Towards Robust Two-Level Methods For Indefinite Systems J. Fish, Y. Qu and A. Suvorov Departments of Civil, Mechanical and Aerospace Engineering Rensselaer Polytechnic Institute Troy, NY 12180 We present a black-box two-level solver for indefinite algebraic linear system of equations arising from the finite element discretization. Numerical experiments show the applicability of the method to 3D Helmholtz equations and shear banding problems with strain softening. 1.0 Introduction Despite the fact that iterative solution techniques are recently gaining recognition among practitioners and finding their way into commercial software arena, the current state of the art in iterative methods remains unsatisfactory in many respects. Users of large production codes such as ANSYS, NASTRAN, ALGOR, SDRC, EMRC and ANSYS often observe many “bad” cases resulting from poorly conditioned or indefinite systems for which their iterative solvers converge prohibitively slow or systematically break. Con- sequently and perhaps rightfully so, a number of commercial software houses, including ABAQUS, refrain from utilizing iterative solver technology primarily because of its lack of robustness. For positive definite well conditioned systems iterative solvers developed into mature technology, and in many cases, far more effective than the direct methods. For such sys- tems the multilevel solution techniques possess an optimal rate of convergence by which computational work required to obtain a fixed accuracy is linearly proportional to the number of unknowns, whereas for indefinite systems not even convergence is guaranteed. Numerical analysis of multilevel methods for indefinite systems shows that convergence is only guaranteed provided that the coarse model is sufficiently fine [1]. For some “bad” cases coarsening factor required might be close to one effectively turning the multilevel solver into a direct method. Indefinite problems arise in many areas of scientific computing. Examples falling into this category are: Helmholtz equations, Galerkin or least squares methods with con- straints, and problems with indefinite constitutive tensor arising as a result of damage/ localization in solids or shocks in fluids. The manuscript is organized as follows. After briefly describing the principles of mul- tilevel methods in Section 2, we present a numerical example which demonstrates the crit- ical role of the prolongation operator and serves as a motivation for developing a family of two-level methods for indefinite systems. Attention is restricted to symmetric systems.
Transcript
itionrrentf largeYStemson-
ludingck
atureh sys-
which thenteed.
nce isbad”tilevel
into con-age/
ul-e crit-ily of
ems.
Towards Robust Two-Level Methods For Indefinite Systems
J. Fish, Y. Qu and A. SuvorovDepartments of Civil, Mechanical and Aerospace Engineering
Rensselaer Polytechnic InstituteTroy, NY 12180
We present a black-box two-level solver for indefinite algebraic linear system ofequations arising from the finite element discretization. Numerical experimentsshow the applicability of the method to 3D Helmholtz equations and shear bandingproblems with strain softening.
1.0 Introduction
Despite the fact that iterative solution techniques are recently gaining recognamong practitioners and finding their way into commercial software arena, the custate of the art in iterative methods remains unsatisfactory in many respects. Users oproduction codes such as ANSYS, NASTRAN, ALGOR, SDRC, EMRC and ANSoften observe many “bad” cases resulting from poorly conditioned or indefinite sysfor which their iterative solvers converge prohibitively slow or systematically break. Csequently and perhaps rightfully so, a number of commercial software houses, incABAQUS, refrain from utilizing iterative solver technology primarily because of its laof robustness.
For positive definite well conditioned systems iterative solvers developed into mtechnology, and in many cases, far more effective than the direct methods. For suctems the multilevel solution techniques possess an optimal rate of convergence bycomputational work required to obtain a fixed accuracy is linearly proportional tonumber of unknowns, whereas for indefinite systems not even convergence is guaraNumerical analysis of multilevel methods for indefinite systems shows that convergeonly guaranteed provided that the coarse model is sufficiently fine [1]. For some “cases coarsening factor required might be close to one effectively turning the mulsolver into a direct method.
Indefinite problems arise in many areas of scientific computing. Examples fallingthis category are: Helmholtz equations, Galerkin or least squares methods withstraints, and problems with indefinite constitutive tensor arising as a result of damlocalization in solids or shocks in fluids.
The manuscript is organized as follows. After briefly describing the principles of mtilevel methods in Section 2, we present a numerical example which demonstrates thical role of the prolongation operator and serves as a motivation for developing a famtwo-level methods for indefinite systems. Attention is restricted to symmetric syst
1
s prior modelectione testion onion of
n or
are
of the
oarse
pro-
e the
es of
maylevel
from
Complex symmetric algebraic systems are transformed into real symmetric systemto the solution (Section 6.1). In Section 4 we conduct a convergence analysis on aproblem and identify key factors affecting the convergence of two-level schemes. S5 details various approaches for constructing efficient prolongators. In Section 6 wvarious two-level schemes on a sequence of examples involving Helmholtz equatbounded domains and shear banding problems with strain softening. A brief discussfuture research directions conclude the manuscript.
2.0 Principles of multilevel methods
Consider a linear or linearized system of equations within a Newton-Raphsorelated scheme
(1)
where is an symmetric sparse matrix. The following notation is adopted in this section. Auxiliary model functions
denoted with subscript . For example, denotes the discrete values
solution in the auxiliary model. We also denote the prolongation operator from the c
to the fine model by :
(2)
The restriction operator from the fine-to-coarse model is conjugated with thelongation operator, i.e.,
(3)
In this section superscripts are reserved to indicate the iteration count. Let b
residual vector in the -th iteration defined as
(4)
where is the current approximation of the solution in the -th iteration.The problem of the coarse model correction consists of finding the stationary valu
the following functional on the subspace :
(5)
where (.,.) denotes the bilinear form defined by
(6)
A direct solution of (5) yields a classical two-level procedure. Alternatively, one introduce an additional auxiliary model for and so forth, leading to a natural multi
sequence. In the present manuscript we will consider a two-level process resulting(5) which yields
Ku f= u Rn∈ f R
n∈K n n×
0 u0 Rm
m n<,∈
Q
Q:Rm
Rn→
QT
QT:R
nR
m→
ri
i
ri
f Kui
–=
ui
i
Rm
1 2⁄( )K ui
Qu0i+( ) u
iQu0
i ) f ui
Qu0i+,( ) stationary u0
i Rm∈⇒–+,
u v,( ) ujvj
j 1=
n
∑= u v Rn∈,
u0
2
vel
the
rse
n by:
given
two-f the
RES
(7)
where is the restriction of the matrix . The resulting classical two-le
algorithm can be viewed as a two-step procedure:a) Coarse model correction
(8)
where is a partial solution obtained after the coarse model correction.b) Smoothing
(9)
where is an inverse of smoothing preconditioner.
Let be the exact solution of the source problem, then the error resulting fromcoarse model correction (8) can be cast into the following form
(10)
where is an identity matrix and is an inverse of the coa
model preconditioner. Likewise the influence of smoothing on error reduction is give
(11)
Furthermore if we denote
(12)
then the error in the two-level process with pre- and post-smoothing iterations is as:
(13)
In practice, however, the solution increment obtained from a single level cycle is used in the determination of the search direction within the framework oConjugate Gradient (CG) method for positive definite systems and QMR [2] or GM[3] for indefinite systems.
3.0 Motivation and goals
Consider a spectral decomposition of the prolongation operator
(14)
K0uoi Q
Tf Ku
i–( )=
K0 QTKQ= K
ri
f Kui
–=
u0i K0
1– QTr
i=
ui
ui
Qu0i+=
ui
ui 1+ u
iD f Ku
i–( )+=
D
u
ei
u ui
– In CK–( )ei= =
Is ℜsxs∈ C QK01–Q
T=
ei 1+
u ui 1+
– In DK–( )ei= =
M In DK–=
T In CK–=
v
ei 1+ M
vTM
ve
i=
uk 1+ u
k–
Q
Q Φ0α0
Φ1
+= α1 Φ= α
3
d as
ss of
that
ion
igureystemal ele-fteningure 1.ly. We
defi-
ii) - two-es of
pproxi-SSOR to the
It candef-
qual- veryrfor-
roxi-
obustrk tons of
cost-tivesary toe will
where ; ; ; ; ;
; ; and
.
Note that if vanishes, the prolongation operator is optimal for a given an
such the term can be viewed as an error in the optimal prolongation. Without lo
generality we will consider the normalized form of the prolongation operator , such
. Let denote a measure of quality of the prolongat
operator, i.e., if the prolongation operator is optimal for a given .
As a motivation we consider a linearized shear banding problem, illustrated in F1, where the material in the band is softening, and thus giving rise to an indefinite sof algebraic equations. The specimen has been discretized with 8x8x8 hexahedrments totaling 2187 degrees-of-freedom. We assume that a shear band (or a sozone) develops on the diagonal plane of two layers of elements [18] as shown in FigLet and be the stiffnesses inside and outside the shear band, respective
consider three material models having the ratio equal to: (i) 0.1 (positive
nite system with oscillatory coefficients), (ii) - 0.1 (weakly indefinite system), and (i0.5 (strongly indefinite system). The three problems have been analyzed with thelevel method outlined in Section 2 and with prolongators generated by different valu
. For all problems considered the coarse model had 250 degrees-of-freedom, or amately 11% of the modes in the source mesh. The QMR accelerator [2] and the smoother have been employed. Note that for the eigenmodes correspondinglowest eigenvalues in the absolute value comprise the prolongation operator.
Figure 1 shows the iteration count versus the prolongation quality parameter . be seen that for positive definite system with oscillatory coefficients and for weakly ininite system the performance of the two-level method is only mildly sensitive to the ity of prolongation. On the other hand, it is evident that highly indefinite systems aresensitive to the quality of prolongation operator making it a key factor affecting the pemance of two- or multi- level methods for indefinite systems.
Although this approach is impractical due to the large computational effort in app
mating to the desired accuracy, it still shows that it is possible to construct a rmultilevel solver for indefinite problems. Furthermore, it is used in the present womotivate the efficient construction of local approximations to the eigenvector colum
as discussed in the sequel. As an alternative, we will examine the feasibility ofefficient utilization of normal equations. For highly indefinite systems for which posiand negative eigenvalues are of the same order of magnitude, it might be necesapply a two- or multi- level scheme to normal equations. For such ‘hard’ cases w
Φ0 ℜ∈nxm
KΦ0 Φ0= λ0
Φ1 ℜnx n m–( )∈ KΦ1 Φ1λ1
= α0 ℜ∈mxm
α1 ℜ n m–( )xm∈ 0diag λ1
0 λ20 … λm
0, , ,(= λ1diag λ1
1 λ22 … λn m–
0, , ,(=
λ i0 λj
1 1 i m 1 j n m–≤ ≤,≤ ≤∀<
α1m
Φ1α1
Q
Q 2 1= ε α12 α0
2⁄=
ε 0= m
Eband E
Eband E⁄
ε
ε 0=
ε
Φ0
Φ0
4
ided
sys-
a lin-
e
e
transform the indefinite source system into a positive definite one, , (prov
that is not singular), where is defined as .
Figure 1: Iteration count versus prolongation quality parameter for various indefinitetems
4.0 Convergence studies
In this section we study the rate of convergence of two-level methods applied toear system of equations:
(15)
arising from either the source system, , or normal equations, .
4.1 The prolongation operator
We will consider the following model problem: , , ar
both diagonal matrices, i.e., , for . We assume that
and define by choosing arbitrary m columns from to express the error in th
K2x f=
K x Rn∈ u Kx=
0
100
200
300
400
500
600
700
Eband
E
σ
ε
/ EEband/ EEband
/ EEband
= - 0.5= - 0.1= 0.1
Eband
E
Num
ber
of It
erat
ions
Prolongation Quality Parameter10
010-3 10-2 10-1
Kρx f=
ρ 1= ρ 2=
α0 ℜmxm∈ α1 α1 ℜmxm∈=
αkmi
0= i j, 0 1,= k m≠ 2m n<
Φ1 ℜ∈nxm
Φ1
5
igen-
r
it is
n hastemsltingtion
r nor-
value
to thes. Foration
prolongation operator as . Furthermore, let be the corresponding e
values of .
We will seek to enhance the quality of the initial or tentative prolongation operatoby means of smoothing denoted as
(16)
where is a prolongation smoother, represents the number of times
applied, and is termed as the enhanced prolongator.
For positive definite systems the concept of smoothing or weighted interpolatiobeen utilized in [6] and [7]. Our studies in [11] indicated that for positive definite syswith efficient two- or multi- level preconditioners the computational savings resufrom prolongation smoothing are often very limited due to minor reduction in iteracount but increased cost associated with prolongation enhancement.
The prolongation smoother can be defined either with respect to the source omal equations. It has the following structure:
(17)
where is a preconditioner of . The exponent can be either one or two. Its
might be different from . For example, we may select to apply an iterative method source system, but to smooth the prolongation with respect to the normal equationthe purpose of convergence studies, we will consider the simplest form of prolongsmoother based on Richardson preconditioner given as
(18)
where is an upper bound of the maximal eigenvalue of in the absolute value.
Consequently, the enhanced prolongation is given by:
(19)
4.2 Auxiliary coarse model stiffness matrix
The auxiliary coarse model stiffness matrix is obtained by restriction:
Φ1α1 λ1 ℜmxm∈
Φ1
Q
Q SpQ=
S ℜnxn∈ p
Q ℜnxm∈
S
S In P 1– Kβ–=
P Kβ βρ
P 1– 1
λβ------In=
λ K
Q In1
λβ------Kβ–
pΦ0α0 Φ1α1+( ) Φi
i 0=
1
∑ Im1
λβ
----- λi( )β
–pαi
= =
6
harac-
-
ing
(20)
Before proceeding with the convergence analysis, we investigate the spectral c
teristics of .
(21)
where . From follows that and we obtain the follow
ing estimate:
(22)
Maximizing with respect to for yields
(23)
where
(24)
We now show that (24) is valid for any tentative prolongation satisfy
or .
(25)
Since and we get
(26)
which is identical to equation (22).
K0 QTK
ρQ Im
1
λβ------ λi( )
β–
2pλi( )
ραi( )
2
i 0=
1
∑= =
K0
K02 Kjj
0 j
max 1λ j
i
λ----
β
–
2p
λji ρ
C0 αi0( )2
C1 αi1( )2
+[ ]
j i,max= =
Ci 1≤ Q 2 1= αji( )2
i 0=
1
∑ 1 j∀≤
K02 1
λji
λ----
β
–
2p
λ ji ρ
j i,max≤
K02 λj
i ρ 1=
K02 ηλ
β≤
η 12pβ---------- 2pβ
2pβ 1+-------------------
2p 1+=
Q ℜnxm∈
Q 2 1= α 2 1=
K0
2 SpQ( )
TK
ρS
pQ( ) 2 α 2
2S
pΦ( )TK
ρS
pΦ( ) 2≤=
KρΦ Φλρ
= SpΦ Φ In
1
λβ------λβ
– p
=
K0
2 In1
λβ
-----λβ–
pλρ
In1
λβ
-----λβ–
p
2
≤ 1λj
i
λ----
β
–
2p
λ ji ρ
j i,max=
7
mpo-
4.3 Auxiliary coarse model iteration matrix
The inverse of the coarse model preconditioner (10) is given as:
(27)
where
(28)
Since all the matrices in (28) are diagonal, is also diagonal with diagonal conents given as
(29)
The coarse model iteration matrix (12) is given by
is a Kronecker delta with respect to the superscripts.
4.4 Two-level iteration matrix
For the purpose of convergence studies we consider a relaxation scheme basedRichardson preconditioner. The corresponding relaxation iteration matrix is given
(33)
Relaxation sweeps can be carried out either with respect to the source system
or normal equations . The number pre- and post- relaxation sweeps is deno
. The resulting two-level iteration matrix is given by
(34)
where is a permutation matrix satisfying ; The block diago
blocks, , are denoted as
(35)
where
(36)
and is a diagonal matrix with diagonal terms given as
The spectral radius of the iteration matrix denoted as is given as:
(38)
The eigenvalues of can be computed from
(39)
which yields
(40)
Substituting (36) into (40) gives:
(41)
where
(42)
and as defined in (37). Since we approximate .
To study the convergence characteristics of the two-level method we consider
cases: (i) , (ii) , and (iii) . Only cases (i) and (iii) sa
isfy the convergence criteria, , provided that:
(43)
(44)
Gk1 1
λk1
λ-----
β
–
2p
=
ρ L( )
ρ L( ) ρ Fi( ) Gk
1, i k,
max=
Fi
detF11
i λ– F12i
F21i F22
i λ–λ2 F11
i F22i+( )λ– F21
i F21 F11i F22 0=+–=
λ i F11i F22
i+
2-----------------------=
F11i F22
i–
2-----------------------
2
F21i F21
i+ F11i F22
i F12i F21
i+ +≤±
ρ Fi( )
εi2 εi Λi Λi+ +
εi2 Gi
0⁄ Λ i Gi1⁄+
------------------------------------------- ≤ i∀
Λi
λi0
λi1
----- ρ
= εiαi
1
αi0
------= Gi0 1
λi0( )
β
λβ
-------------–
2p
=
Gi1 λ i
0 λβ
« Gi0 1=
Λi 0≥ Gi1εi
2– Λi 0< < Λi Gi1εi
2–≤
ρ L( ) 1<
Gi1( )
1–1
εi
Λi
---------+≥ Λ i 0≥ i∀
Gi1( )
1–1
εi
Λ– i
-------------2εi
2
Λi–( )-------------+ +≥ Λi Gi
1εi2–≤ i∀
10
re the
ely:
them (43)
ontext
atrixt withSOR,
is
r, i.e.
t the
tion
me isll. In
e suf-
43)
s slowsrease
del
ro pro-
Multi-ate-h
Equations (43) and (44) describe the quality of the smoother required to ensuconvergence of the two-level scheme for a given tentative prolongation operator.
4.5 Discussion
There are several factors affecting the convergence of the two-level process, nam
(i) The smoother. The ability of the prolongation and the two-level smoothers to reducehigher frequency modes of error is one of the key factors affecting convergence. Fro
and (44) it can be seen that is a necessary condition for convergence. In the c
of Richardson-based preconditioner this condition is satisfied if either the stiffness mis positive definite or both the prolongation and iteration smoothings are carried ourespect to normal equations, i.e., . For stronger preconditioners, such as S
there might be a weaker condition satisfying .
(ii) The prolongation operator. The quality of the tentative prolongation operator, ,
governed by the ratio . In the case of the optimal prolongation operato
, the necessary condition becomes the sufficient condition provided tha
system is non-singular, i.e., .
(iii) Spectral characteristics of the linear system of equations. For positive definite systems
is minimal with respect to exponents and for and equa
(43) represents the sufficient conditions for convergence. When the multilevel scheapplied to normal equations, i.e. , (43) represents sufficient conditions as we
the case of indefinite systems, i.e. , both equations (43) and (44) comprise th
ficiency conditions with (43) for and (44) for . By comparing equations (
and (44) it can be seen that the existence of extreme eigenvalues with opposite signdown the converge of the two-level method or may require stronger smoother (or incthe value of ) to prevent divergence.
(iv) The size of the coarse model. As the size (number of equations) of the coarse mo
approaches the size of the source grid, i.e., , defined in (37) approaches ze
vided that .
5.0 Prolongation operator
The tentative prolongation operator can be constructed using either Geometric grid (GM) method or the aggregation approach [5], [6], [8], [9] which falls into the cgory of Algebraic Multigrid (AMG) methods. While geometric multigrid approac
Gi1 1<
β 2=
Gi1 1<
Q
εi αi1 αi
0⁄=
εi 0→ Gi1 1<
Λ i 0≠
Gi1 β ρ,( ) β ρ β ρ 1= =
ρ 2=
ρ 1=
Λi 0> Λi 0<
p
λi1 λ→ Gi
1
Gi1 1<
11
rega-ource
ce gridainsmber ofource
ctralion: (i)enta-onga-
e the Theargevanta-ff” ele-” and
g the
ting of
t
constructs the prolongation operator from auxiliary coarser grids, the method of aggtion accomplishes the same goal on the basis of the information available in the sgrid. In the present manuscript we focus on the aggregation approach.
In an aggregation scheme the coarse model is directly constructed from the sourby grouping finite elements into either nonoverlapping or overlapping subdomreferred to as aggregates, and then for each aggregate assigning a reduced numodes with an intent of effectively capturing the lower frequency response of the ssystem.
In an attempt to construct an efficient prolongation operator in terms of its specharacteristics, the following key issues are discussed in the remainder of this sectconstruction of the auxiliary aggregated model (Section 4.1), (ii) construction of the ttive prolongation operator (Section 4.2), and (iii) enhancement of the tentative proltion operator (Section 4.3).
5.1 Aggregation algorithm
Prior to describing the technical details of the aggregation algorithm, we introducconcept of “stiff” and “soft” elements which is utilized in the process of aggregation.element is considered “stiff” if the spectral radius of its stiffness matrix is relatively lcompared to other elements and vice versa. It has been shown in [8], [9] that it is adgeous to place the “soft” elements at the interface between the aggregates, and “stiments within the aggregates. This approach is a counterpart of the idea of “weak“strong” nodal connectivity employed in [4].
The maximal eigenvalue of the element stiffness matrix, , estimated usinGerschgorin theorem
(45)
is used to quantify the element stiffness. We consider a finite element mesh consis
elements and nodes. Let be the set of nodes belonging to the elemen
(46)
where subscripts and denote sets of elements and nodes, respectively.
Step 1. Setup.
1.1. For each node select the elements containing this node:
(47)
λmaxe
λmaxe βe≤ βe max
ikij
e
j∑
=
NE NN CN i( ) Ei
CN i( ) Nj:N
jE
i∈ =
E N
Nj
j, 1 NN,[ ]=
BE j( ) Ei:N
jE
i∈ =
12
, i.e.,
ments
(s)-of-
set.
ulti-
igh-
or the
1.2. For each element select the set of neighboring elements
elements containing common nodes:
(48)
Step 2. Start-up aggregation.
2.1. Define the set of elements available for aggregation. These are all the elewhich do not contain nodes with essential boundary conditions or the ‘slave’ nodes:
(49)
where is the set of ‘slave’ nodes, which depends on so called ‘master’ degree
freedom, and nodes with essential boundary conditions. We denote as the initial
Remark 1: We include the slave nodes in the set so that we could deal with m
point constraints in a conventional way. See [9] for details.
2.2. Find the “peripheral” element , i.e., the element with minimal number of nebors:
(50)
where is a number of elements in the set . Element is a starting element faggregation algorithm.
2.3. Setup:
- the current aggregate counter ;
- the set of interface elements , i.e., elements between aggregates.
Step 3. Formation of the current nonoverlapping aggregate.
3.1 An aggregate with zero neighbors is defined as follows:
(51)
3.2 An aggregate with one neighbor, , contains the element and those of its
available neighbors which satisfy the relative stiffness condition:
E i, 1 NE,[ ]= FE i( )
FE i( ) Ek:E
kBE j( ) j CN i( )∈,∈ \E
i=
TE
TE 1 NE,[ ]\ BE j( ) Nj SN∈, =
SN
TE0
SN
Es
s min FE i( )arg=i TE∈
X X Es
i 1=
IE 1 NE,[ ]\TE=
AEi
0( ) Es
=
AEi 1( ) E
s
13
igen-
forma-
bor is
.
) thes the
rface
(52)
where is a Gerschgorin upper bound on -the element stiffness matrix maximal e
value, and is a coarsening parameter. If on the other hand the element stiffness in
tion is not available in the aggregation process, then the aggregate with one neigh
defined as
(53)
Similarly we can define an aggregate with arbitrary number neighbors, denoted as
Remark 2. Numerical experiments have shown that for higher order elements (zero-neighbors version is typically more efficient, whereas for lower order elementone- or two- neighbor aggregation scheme is more appropriate.
Step 4. Update the sets of the interface and available elements.
4.1. Update the set of the interface elements:
(54)
4.2. Update the set of the available elements:
(55)
4.3 Update the set of aggregates:
(56)
Step 5. Find the new starting element.
Form the set of “frontal” elements , i.e., available elements neighboring the inteelements
(57)
and select the stiffest new starting element from the set defined as
(58)
AEi 1( ) E
sE
j FE s( ) TE βj µβs≥,∩∈ ∪=
βjj
µ
AEi 1( ) E
s FE s( ) TE∩( )∪=
i AEi
p 3≥
IE IE Ek FE j( ) E
jAE
i∈,∈( ) Ek
AEi∉( )∩ ∪=
TE TE\ Ek FE j( ) E
jAE
i∈,∈( ) AEi∪ =
AE AE Ek FE j( ) E
jAE
i∈,∈( ) AEi∩ ∪=
RE
RE Ek FE j( ) E
j IE∈,∈( ) TE∩ =
RE
s = max βj( )argj :E
jRE∈
14
ss we
o o pos-et :
ggre-achingl and
solu-
If on the other hand the stiffness information is not available in the aggregation procesimply select an arbitrary starting element belonging to .
Step 6. Stopping criteria for nonoverlapping aggregation.
If then stop; else and repeat steps 3-6.
Step 7. Define the element-free aggregates
7.1 Each node in is classified as an element-free aggregate
(59)
7.2 Find the set of nodes which is not contained in one of the aggregates.
(60)
For higher order elements there will be a significant number of nodes belonging tprimarily in the interface region between the aggregates. For linear elements it is alssible that as shown in Figure 2. There are two approaches to deal with the s
(i) collapse and the corresponding elements to one of the neighboring agates as shown in Figure 2. If such collapsing makes the aggregate invalid (attthe node without elements) make a ‘master’ node in the coarse modeclassify it as an element free-aggregate, or
(ii) consider as a ‘slave’ nodes in the coarse model and interpolate thetion in from the adjacent nodes in .
Figure 2: A typical nonoverlapping one-neighbor aggregation model
In the present manuscript the first approach is adopted.
RE
RE ∅= i i 1+=
SN
ANj
Nj
= N∀ jSN∈
VN Nj: 1 NN,[ ]\ N
jCN k( ) E
kAE∈,∈( )\SN( ) =
VN
VN ∅≠ VN
Nj
VN∈
Nj
VN∈
Nj
VN∈N
jAE
E
One-NeighborAggregates A (1)Node to be
collapsed
Interfaceelements
15
regate
ts is
cor-
value)
dual
egates
nction
elas-
areoltz
r onn over
alue
Step 8. Formation of the overlapping aggregates (for overlapping version only)
For each nonoverlapping aggregate define a corresponding overlapping agg with one overlapping layer of elements:
(61)
Similarly, an overlapping aggregate with overlapping layers of elemendefined as
(62)
5.2 Construction of the tentative prolongation
The goal of the aggregation method is to approximate the eigenmodes
responding to the lowest eigenvalues of the source stiffness matrix (in the absolute
by a linear combination of continuous local functions defined over the indivi
aggregates. The following four choices have been considered:
5.2.1 A priori selected functions on nonoverlapping aggregates
By this technique a finite element mesh is decomposed into nonoverlapping aggr
(steps one to seven in Section 4.1). On each aggregate , a low oder polynomial fu
(constant or linear field) is used to approximate the solution (typically for Poisson or
ticity equations with constant coefficients). For problems where eigenfunctions oscillatory, such as in the case of elasticity with oscillatory coefficients or Helmhequation, an analytical solution with either periodic boundary conditions [11] ounbounded domains is used instead. Figure 3(a) illustrates a linear approximationonoverlapping aggregates.
5.2.2 Eigenmodes on nonoverlapping aggregates with Neumann boundary conditions
An alternative to selecting analytical functions on is to conduct a local eigenv
analysis on each aggregate
(63)
AEi
AEi
1( )
AEi 1( ) E
k FE j( ) Ej
AEi∈,∈( ) AE
i∪( ) TE0∩ =
k 1+
AEi k 1+( ) E
k FE j( ) Ej
AEi k( )∈,∈( ) AE
i k( )∪( ) TE0∩ =
Φ0 ℜ∈nxm
C0
AEi
Φ0
AEi
Kiφi λi
diag Ki( )φi
=
16
which
it as
for all
even
sis on
s both
r con-
nt for
eter
rob-
are
ocal
inate
, pres-
ann
local
cting
ed to
with zero Neumann boundary conditions on and to select the eigenmodes for
. In (63) denotes the diagonal of the aggregate stiffness matrix .
The value of controls the effectiveness of the aggregated model. In the lim
, the auxiliary coarse model captures the response of the source system
frequencies and therefore the two-level procedure converges in a single iteration
without smoothing. On the negative side, for large values of , the eigenvalue analy
each aggregate becomes prohibitively expensive and the auxiliary matrix become
large and dense. At the other extreme in the limit as , the prolongation operato
tains rigid body modes only, and thus the auxiliary coarse model becomes inefficie
ill-posed problems. For best performance of the iterative process the value of param
should be in the range of to [8], [9]. The optimal value depends on the p
lem type (3D elasticity, shells, Helmholtz). Typically 6-50 modes satisfying
selected. The Lanczos algorithm with partial orthogonalization [13] is utilized for l
eigenvalue analysis.
The aforementioned approach [8], [9] does not require a priori knowledge of the solu-
tion characteristics nor does it utilizes any information regarding the choice of coord
functions or the nature of the discrete approximation (i.e., rotations, displacements
sures, etc.). As such it falls into the category of ‘black-box’ solvers.
5.2.3 Eigenmodes on overlapping aggregates with Dirichlet boundary conditions
For normal equations, , , it is not trivial to construct a local Neum
problem due to coupling resulting from the product of two global matrices. Instead, a
eigenvalue problem with Dirichlet boundary conditions can be constructed by extra
appropriate information from the global matrix . This approach can be also appli
the source system.
For each overlapping aggregate we conduct a local eigenvalue analysis
(64)
AE
λi γ≤ diag Ki( ) K
i
γ
γ maxi
λ i( )→
γ
γ 0→
γ
10 1– 10 3–
λi γ≤
Kρx f= ρ 2=
K2
AEi
Kiφi λi
diag Ki( )φi
=
17
atrix
tiffer
on for
fy theregatehould
con-
oblem.
modeligen-
itions
value
s on
and select eigenmodes for which ; is a block within the global stiffness m
corresponding to the aggregate . Typically since the Dirichlet problem is s
than the corresponding Neumann problem. Figure 3(b) shows a typical approximati
the 1D problem on overlapping aggregates.
5.2.4 Mixed prolongation
In [8] we have shown that the coarse model approximation space should satishomogenous differential equation. For elasticity problems this means that each aggshould be able to represent rigid body modes, whereas for Helmholtz equations it s
contain functions of the form , where k is the wavenumber.
The eigenvalue problem with Dirichlet boundary conditions (64) is usually over
strained, and thus in general the eigenfunction does not satisfy homogeneous pr
For this reason we define a mixed approximation scheme by which the coarse approximation space consists of: (i) functions satisfying homogeneous solution or e
functions computed from the eigenvalue problem with Neumann boundary cond
on nonoverlapping aggregates, and (ii) eigenfunctions computed from the eigen
problem with Dirichlet boundary condition on overlapping aggregates.
Figure 3: (a) Linear interpolation on non-overlapping aggregates, (b) Eigenfunctionoverlapping aggregates with Dirichlet boundary conditions
φ λ γ≤ K
AEi γ γ<
eikx
φi
φi
φi
o o o o o o o o o o o o o o
o o o o o o o o o o o o o o
mAE
AE
k
mAE(1) AE
k(1)
(a)
(b)
18
m, orationesulttice,mple,unt,
for asg maynt part
truc-
uid-
onal
pond-
where
the
es-of-
condi-
5.3 Enhanced prolongation
The quality of the initial prolongation operator can be improved by smoothing:
(65)
where is equal to one if smoothing is carried out with respect to the source systetwo if it is applied to normal equations. The goal is to construct an efficient prolongsmoothing process (65) with minor or no additional memory requirements that will rin a sparse prolongation and will significantly reduce the iteration count. In prachowever, it is not trivial to bridge between these contradicting requirements. For exaan efficient prolongation smoother, which may significantly reduce the iteration comight increase the total computational cost since smoothing has to be carried outmany vectors as the number of equations. Furthermore, the prolongation smoothinresult in a non-sparse prolongation making the stiffness restriction process a dominaof the solution cost.
The key to constructing an efficient smoothing process is to exploit the sparsity s
ture of the tentative prolongation and the locality of pollution effects. These two ging principles are employed within the framework of the incomplete SSOR.
5.3.1 The incomplete SSOR prolongation smoother for the source system
Consider the decomposition , where and are the diag
and strict lower part of , respectively. Let be a set of degrees-of-freedom corres
ing to a nonoverlapping aggregate , and be the corresponding set on ,
is the user-defined number of overlapping layers of elements. Let be
prolongation operator corresponding to the set , and be the number of degre
freedom in the aggregates . For each we define an incomplete SSOR pre
tioner, , as follows:
(66)
where
(67)
Q
Q In P 1– Kβ
–( )pQ=
β
Q
Q
K D L LT
+ += D L
K Ns
AEs
Ns
AEs ω( )
ω Qs
ℜnxms⊂
Ns
ms
AEs
Qs
Ps
Ps
D Ls
+( )D 1–D L
sT+( )=
Lkls Lkl k l, N
s⊂
0 otherwise
=
19
ation isucesffnessbe in
rma- weak
ons,
[3].
or-
c-
her-
quenceshear
It can be seen that even if aggregates are nonoverlapping the enhanced prolongoverlapping with controlling the size of the overlap. Increasing the value of redthe number of iterations but increases the cost of prolongation smoothing and stirestriction. Numerical experiments indicate that for optimal performance should the range of one to three.
The incomplete SSOR preconditioner (66), (67) is based on the topological infotion only. An incomplete SSOR preconditioner based on the concept of strong andconnections in the stiffness matrix has been developed in [14].
5.3.2 The incomplete SSOR prolongation smoother for normal equations
An efficient implementation of the SSOR preconditioner for normal equati
, which does not require explicit formation of has been developed inHere we focus on the incomplete version of this algorithm.
Starting from the initial approximation of the prolongation , , the fward Gauss-Seidel sweep is based on succession of relaxation steps of the form
(68)
where is the k-th column of the identity matrix and is a column ve
tor of unknowns. For the vector is chosen so that the k-th component of the
residual, , becomes zero, where is a vector of ones. Ot
wise , which yields:
(69)
Relaxation steps (68), (69) are carried out for all and k for which .
6.0 Numerical Examples and Discussion
Various aggregation schemes described in Section 5 have been applied to a seof examples involving Helmholtz equation on bounded domains and linearized banding problems with strain softening.
ω ω
ω
K2x f= K2
Qs
Qs
Qs
=
Qnews
Qolds
ek δk( )T+=
ek ℜn⊂ In δk ℜms⊂
k Ns⊂ δk
f1T
K2Qnews
–( )Tek 1 ℜ
ms⊂
δk 0=
δk
1fTek KQold
s( )TKek–
Kek( )TKek( )
--------------------------------------------------- k Ns⊂
0 otherwise
=
s k Ns⊂
20
cubes
nd
e the
onsist-idered.
plexnto a, and
6.1 Helmholtz equation on bounded domains
Consider Helmholtz’s equation in the region enclosed between two concentric of length 2 ( ) and 6 ( ). The strong form of the governing equations is given as
(70)
(71)
(72)
where n is a coordinate in a direction normal to and ; a
; r is distance from the center of the cube. Equations (70)-(72) describ
acoustic pressure of a wave in a transmitting media.
Because of symmetry, one-eighth of the domain is discretized. Three meshes cing of 3072, 23,925 and 156,009 4-node linear tetrahedral elements have been consThe coarsest discretization is shown in Figure 4.
Figure 4: Typical finite element mesh and boundary conditions
The resulting discrete linear system of equations, , is symmetric, comand indefinite. It is convenient to transform the complex symmetric linear system ireal symmetric system by replacing each term in the stiffness, , force vector,
Γ0 Γ1
u x( ) k2u x( )+∇2 0,= x Ω∈
n∂∂u
r∂∂ e
ikrr⁄( )
Γ0
= x Γ0 ∂Ω⊂∈
n∂∂u
r∂∂ e
ikrr⁄( )
Γ1
= x Γ1 ∂Ω⊂∈
Γ0 Γ1 Γ0 Γ1∪ ∂Ω=
Γ0 Γ1∩ 0=
u
Kx F=
KAB FA
21
er-
s two
erator:ondi-
con-normal
r nor- as ar and
con-ion ofeu-
ping
ns
the solution vector, , by , and , respectively. The sup
scripts and denote the real and imaginary parts, which can be interpreted adegrees-of-freedom per node.
We consider two approaches of constructing aggregation-based prolongation op(i) eigenfunctions defined on nonoverlapping aggregates with Neumann boundary ctions, (ii) eigenfunctions defined on overlapping aggregates with Dirichlet boundaryditions. The two schemes have been applied to both the source system and the equations. We denote the resulting four methods as: Source-N(eumann), Source-D(irichlet), Normal-N(eumann), Normal-D(irichlet).
For normal equations we employ a dedicated conjugate gradient acceleration fomal systems [3] and Incomplete Cholesky preconditioner for normal equations [3]two-level smoother. For the source system a combination of QMR [2] acceleratoSSOR smoother is adopted.
Preliminary numerical investigation revealed that for all problems and methodssidered a nearly optimal performance has been obtained with the following combinatalgorithmic parameters: (i) the limiting eigenvalue parameter, , for both N
mann and Dirichlet problems, (ii) one-neighbor approach, , for nonoverlap
aggregates, and two layers of element overlaps, , for overlapping aggregates.
Figure 5: CPU/Cycles versus kh for discrete Helmholtz linear system with 1478 equatio
xAKAB
RKAB
I
KABI
K– ABR
FAR
FAI
xAR
x– AI
R I
γ 0.1=
AE 1( )
AE 2( )
10 1000
2
4
6
8
10
12
14
16
64
127
409
67911
24
115
59
5 4 4
35 3830
2622
22
43
122
39
25
22 21 21 19
18
20
34
88
1713
4 4 3
21
kh
Sol
ver
CP
U T
ime
(sec
)
1
Source-NSource-DNormal-NNormal-D
Direct
1478 Equat ions
Cycles
22
erage
ng wasred tostate-
ns
Figures 5-7 show the CPU time and iteration count versus the product of the avelement size and the wavenumber, kh, for the three meshes considered. The product kh hasbeen selected since it represents a measure of solution accuracy [16]. No smoothicarried out for prolongation operator. Results of the four iterative methods are compathe state-of-the-art multifrontal solver [15]. Comparison to other recently developed of-the-art direct solvers [21][22] have not been conducted.
Figure 6: CPU/Cycles versus kh for discrete Helmholtz linear system with 9648 equatio
1 10 1000
50
100
150
200
250
300
350
2780
878 859
12 5
18 19 43
4
140 151
105
47
356
6
114
122
77
67
39
266
11 4 3
87
kh
Sol
ver C
PU
Tim
e (s
ec)
Source-NSource-DNormal-NNormal-D
Direct
9648 Equations
Cycles
23
inter-rical
ectrumighly
sitive
rges
5] and
rega-f CPUgates
ure 6o thed by a
iffness
Figure 7: CPU/Cycles vs kh for discrete Helmholtz linear system with 57586 equations
Even though practitioners dealing with wave propagation problems are primarily ested in the range, , required for solution accuracy [17], we conduct numeexperiments outside the range of the usual interest. Our interest in a much wider spof kh values stems from the fact that not only the analyst may frequently encounter hnonuniform meshes, where the precise definition of h is questionable, but primarily,because our ultimate goal is to develop a generic black-box equation solver for podefinite and indefinite systems.
It can be seen from Figures 5-7 that for the two-level method rapidly conve
for the source system. In the case of the break even point between the one [1
two-level methods considered is approximately 5000 equations. For the aggtion scheme based on nonoverlapping aggregates [8], [9] is more efficient in terms otime, whereas for the aggregation scheme based overlapping aggre
works better. For the use of normal equations cannot be avoided. Figshows that the two-level method with nonoverlapping aggregates is competitive tdirect method at approximately 10,000 equations, and is faster than the direct methofactor of 2-10 in the case of 50,000 unknowns. It is not surprising that for the iterative methods converge in 3-5 iterations, since the eigenvalues of the stmatrix are all negative.
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
31 64
970
76182
538
4
412
223109
222
697
98
384
19195
216
612
23
kh
Sol
ver
CP
U T
ime
(sec
) Source-NSource-DNormal-NNormal-D
Direct
0.1 1 10 100
q
Cycles
kh 1<
kh 1<kh 1<
kh 0.5<
0.5 kh 2<<kh 2 4–( )>
kh 20 40–( )>
24
countlonga-mooth-lemsecauserolon-rima-
eless,ationse of
ed in
be istaling
oftening spec-
Figure 8: Influence of prolongation smoothing on the iterative process
Figure 8 compares the CPU time of the iterative process and the iteration obtained with the enhanced (10 smoothings) and the tentative (no smoothing) protion operators on nonoverlapping aggregates. It can be seen that the prolongation sing considerably reduces the iteration count (factor of up to 14 for the probconsidered), and at a lesser extent the CPU time of the iterative process. This is bthe enhanced prolongation is denser, resulting in increased cost of restriction and pgation. The overall CPU time obtained with the enhanced prolongation is increased prily due to the computational cost associated with prolongation smoothing. Neverthfor problems with multiple right hand sides, the overhead generated from prolongsmoothing and coarse model factorization, might be negligible, and thus the uenhanced prolongation could be advantageous.
Other variants of multilevel methods for Helmholtz equation have been describ[10] and [12].
6.2 Shear banding problem
We consider a linearized shear banding problem, illustrated in Figure 1. The cudiscretized with 16x16x16, 24x24x24 and 32x32x32 8-node hexahedral elements toto 14739, 46875 and 107811 degrees-of-freedom. We assume that a shear band (szone) develops on the diagonal plane of two layers of elements [18]. We consider the
0
50
100
150
200
250
300
350
400
450
59
76 70 66
41
344
34
4 3
33
3124 23
21
93
81970
431
6 55
55
9
q
kh
Itera
tive
Pro
cess
CP
U T
ime
(sec
)
9
1 10 100
Normal-N (no smoothing)
Normal-N (10 smoothings)
Source-N (no smoothing)
Source-N (10 smoothings)
25
in the
ted: (i)ondi-
finedpping normal
limit-ce-D/ggre-
same as
r the
illatorywo-
ationswo
over- then
trum of ratios between the stiffness inside and outside the shear band, ,
range of 0.3 and - 0.7.
Three approaches of constructing aggregation-based prolongation have been teseigenfunctions defined on nonoverlapping aggregates with Neumann boundary ctions, (ii) linear fields defined on nonoverlapping aggregates, (iii) eigenfunctions deon overlapping aggregates combined with rigid body modes defined on nonoverlaaggregates. The three schemes have been applied to both the source system andequations. We denote the resulting six methods as: Source-N(eumann), Source-L(inear),Source-D(irichlet)/R(igid)B(ody), Normal-N(eumann), Normal-L(inear), Normal-D(irichlet)/R(igid)B(ody).
The following combination of algorithmic parameters have been considered: the ing eigenvalue parameter, , equal to 0.0001, 0.01, 0.1, 0.3 for Normal-D/RB, SourRB, Source-N, and Normal-N aggregation schemes, respectively. The topology of agated model, the acceleration schemes and two-level smoothers employed are the in Section 6.1.
Figures 9, 10 and 11 show the CPU time and iteration count versus fo
three meshes considered. It can be seen that for positive definite systems with osccoefficients and for weakly indefinite system, , the behavior of the t
level methods as applied to the source system is similar to that of Helmholtz equwith . For the break even point between one [15] and t
level methods is approximately 10,000 equations. The linear interpolation over nonlapping aggregates performs well for positive definite systems, but is less efficientmethods based on selection of eigenfunctions for .
Eband E⁄
γ
Eband E⁄
Eband E⁄ 0.1–>
kh 1< Eband E⁄ 0.1–=
Eband E⁄ 0.1–=
26
ns
ns
Figure 9: CPU/Cycles vs for shear banding problem with 14739 equatio
Figure 10: CPU/Cycles vs for shear banding problem with 46875 equatio
0.8 0.6 0.4 0.2 0 0.2 0.4 0.60
200
400
600
800
1000
1200
106115
341
230
283
383
103111
205
251260
306
5862
255
148
216
260
11 11
158
511012
78
301
Direct SourceLinSourceD/RB SourceN NormalLinNormalD/RB NormalN
EbandE /
Sol
ver
CP
U T
ime
(sec
)
14739 Equations
Cycles
Eband E⁄
0.8 0.6 0.4 0.2 0 0.2 0.4 0.60
1000
2000
3000
4000
5000
6000
7000
Direct SourceLinSourceD/RB SourceN NormalLinNormalD/RB NormalN
EbandE /
Sol
ver
CP
U T
ime
(sec
)
46875 Equations
1213
313
78
301
1481111
296346
564
452
167197
395
173203
419
754
583
6065
295
400
620
525
Cycles
Eband E⁄
27
ns
time0000
on themancefact
, andd with
thatPU
with a
we
ral pur-d bedefi-
t fail.hybridn strat-olon-
Figure 11: CPU/Cycles vs for shear banding problem with 107811 equatio
For highly indefinite systems the direct solver is more efficient in terms of CPU than the iterative schemes applied to normal equations for problems below 10unknowns. Among the three two-level schemes considered the prolongator basedlinear interpolation on nonoverlapping aggregates, had in general the best perforwith few exceptions ( in Figure 11). This can be explained by the
that linear fields represent the Kernel of normal equations with constant coefficientsthe prolongator based on linear interpolation does not involve overhead associatelocal eigenvalue analysis.
The influence of prolongation smoothing on the solver performance was similar toillustrated in Figure 8, i.e, significant reduction in iteration count, minor gains in Ctime of the iterative process and increased total computational cost for problems single right hand side vector.
For utilization of geometric multigrid methods in plasticity with strain hardeningrefer to [19], [20].
7.0 Future work
The manuscript represents the first step towards developing an automated genepose multilevel solver for indefinite systems. It is critical that such a solver shoulrobust. It may use different strategies, such as utilizing normal equations for highly innite problems or the source system for weakly indefinite problems, but it should noThis goal have been partially accomplished. We have demonstrated that such a solver exist, but we have not addressed the issue of how to select an optimal solutioegy. In particular, what is an optimal number of levels, how to construct an optimal pr
0.8 0.6 0.4 0.2 0 0.2 0.4 0.60
0.5
1
1.5
2
2.5
3
3.5x 104
Direct SourceLinSourceD/RB SourceN NormalLinNormalD/RB NormalN
EbandE /
Sol
ver
CP
U T
ime
(sec
)
107811 Equations
1213
567
200256
473486
1009
640
216233
644
571
1479
868
12 12195
626
522419
1398
637
6370
Cycles
x 104
x 104
x 104
x 104
x 104
x 104
Eband E⁄
Eband E⁄ 0.5–=
28
ormalroblem
ding thet hand devel-ms.
licits it isied tof pro-licabil-
mberapira
nd
ian
-
ns-
s,”
gation operator, will the method converge for the source system or should the nequations be used instead and what is an optimal accelerator and smoother for a pat hand? Clearly, the answer to these questions depends on the problem data, inclusparsity, the spectrum of eigenvalues, the problem size, and the number of righsides. For positive definite systems a decision graph-based methodology has beenoped in [9] and we indent to generalize this or a similar framework to indefinite syste
Even though a family of efficient two-level solvers, which does not require an expformation of normal system of equations, has been developed for normal equationevident that these normal solvers are below par with two-level methods directly applthe source system (provided that they converge). Therefore, further improvement olongators, smoothers and accelerators is critical if we are to extend the range of appity of the two- and multi- level methods for indefinite (source) system of equations.
ACKNOWLEDGMENT
This work was supported by the Office of Naval Research through grant nuN00014-97-1-0687. The authors express their sincere appreciation to Dr. Yair Shfrom Los Alamos National Laboratory for his constructive suggestions.
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