Advanced Numerical Methods In nonlinear finite-element analysis, solutions are typically sought using Newton iteration, either in classical form or augmented as an arc-length method to b
Trang 1Advanced Numerical Methods
In nonlinear finite-element analysis, solutions are typically sought using Newton iteration, either in classical form or augmented as an arc-length method to bypass critical points in the load-deflection behavior Here, two additional topics of interest are briefly presented
20.1 ITERATIVE TRIANGULARIZATION
OF PERTURBED MATRICES 20.1.1 I NTRODUCTION
In solving large linear systems, it is often attractive to use Cholesky triangularization followed by forward and backward substitutions In computational problems, such as
in the nonlinear finite-element method, solutions are attained incrementally, with the stiffness matrix slightly modified whenever it is updated The goal here is to introduce and demonstrate an iterative method of determining the changes in the triangular factors ensuing from modifying the stiffness matrix A heuristic convergence argu-ment is given, as well as a simple example indicating rapid convergence Apparently, no efficient iterative method for matrix triangularization has previously been established The finite-element method often is applied to problems requiring solution of large linear systems of the form K0γγγγ0=f0, in which the stiffness matrix K0 is positive-definite, symmetric, and may be banded As discussed in a previous chapter, an attractive method of solution is based on Cholesky decomposition (triangularization),
in which K0 =L0L T0 and L0 is lower-triangular, and it is also banded if K0 is banded The decomposition enables an efficient solution process consisting of forward sub-stitution followed by backward subsub-stitution Often, however, the stiffness matrix is updated during the solution process, leading to a slightly different (perturbed) matrix,
K=K0+∆K, in which ∆K is small when compared to K0 For example, this situation may occur in modeling nonlinear problems using an updated Lagrangian scheme and load incrementation Given the fact that triangular factors are available for K0,
it would appear to be attractive to use an iteration scheme for the perturbed matrix
K, in which the initial iterate is L0 The iteration scheme should not involve solving intermediate linear systems except by using current triangular factors A scheme is introduced in the following section and produces, in a simple example, good estimates within a few iterations
The solution of perturbed linear systems has been the subject of many investiga-tions Schemes based on explicit matrix inversion include the Sherman-Morrison-Woodbury formulae (see Golub and Van Loan [1996]) An alternate method is to carry bothersome terms to the right side and iterate For example, the perturbed linear 20
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system can be written as
(20.1) and an iterative-solution procedure, assuming convergence, can be employed as
(20.2) Unfortunately, in a typical nonlinear problem involving incremental loading, espe-cially in systems with decreasing stiffness, it will eventually be necessary to update the triangular factors frequently
20.1.2 N OTATION AND B ACKGROUND
A square matrix is said to be lower-triangular if all super-diagonal entries vanish Similarly, a square matrix is said to be upper-triangular if all subdiagonal entries vanish Consider a nonsingular real matrix A It can be decomposed as
in which diag(A) consists of the diagonal entries of A, with zeroes elsewhere; Al
coincides with A below the diagonal with all other entries set to zero; and Au coincides with A above the diagonal, with all other entries set to zero For later use, we introduce the matrix functions:
(20.4)
Note that: (a) the product of two lower-triangular matrices is also lower-triangular, and (b) the inverse of a nonsingular, lower-triangular matrix is also lower-triangular Likewise, the product of two upper-triangular matrices is upper-triangular, and the inverse of a nonsingular, upper-triangular matrix is upper-triangular In proof of (a), let L(1) and L(2) be two n×n lower-triangular matrices The ij th entry of the product matrix is given by Since L(1) is lower-triangular, vanishes unless
k≤i Similarly, vanishes unless k≥j Clearly, all entries of vanish unless i≥j, which is to say that L(1) L(2) is lower-triangular
In proof of (b), let A denote the inverse of a lower-triangular matrix L We multiply the j th column of A by L and set it equal to the vector ej (e Tj = {0… 1… 0} with unity in the j th position): now,
(20.5)
K0∆γγ=∆f−∆Kγγ0−∆ ∆K γγ,
K0∆γγ( 1) ∆f ∆Kγγ0 ∆ ∆K γγ( )
A=Al+diag( )A +Au
lower( )A =Al+1diag( ), A upper( )A =Au+ diag( ).A
2
1 2
∑k=1,n ik l l( ) ( )1 kj2 l ik( ) 1
l kj( )2 ∑k=1,n ik l l( ) ( )1 kj2
l a
j
11 1
21 1 22 2
31 1 32 2 33 3
1 1 2 2 3 3
0 0 0
1
=
M
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Trang 3Advanced Numerical Methods 259
Forward substitution establishes that a kj= 0, if k<j, and a jj=l−1jj , thus, A=L−1 is
lower-triangular
20.1.3 I TERATION S CHEME
Let K0 denote a symmetric, positive-definite matrix, for which the unique triangular
factors are L0 and L T0 If K0 is banded, the maximum width of its rows (the bandwidth)
equals 2b− 1, in which b is the bandwidth of L0 The factors of the perturbed matrix
K can be written as
(20.6)
We can rewrite Equation 20.6 as
(20.7) from which
(20.8) Note that L0
−−−−1
∆L is lower-triangular It follows that
(20.9) The factor of 1/2 in the definition of the lower and upper matrix functions is
motivated by the fact that the diagonal entries of and are the same
Furthermore, for banded matrices, if ∆L and L0 have the same semibandwidth,
banded, with a bandwidth no greater than b Unfortunately, it is not yet clear how
to take advantage of this behavior
An iteration scheme based on Equation 20.9 is introduced as
(20.10)
by using forward and backward substitution In particular,
, where ∆k1 is the first column of ∆K We can now solve for
by solving the system L0 bj= ∆kj
20.1.4 H EURISTIC C ONVERGENCE A RGUMENT
For an approximate convergence argument, we use the similar relation
(20.11)
[K0+∆K]=[L0+∆L L][ T0+∆L T]
[I+L−1 L I][ + L L T −T]=L−1[K + K L] −T,
L−01∆L+∆L L T −0T =L−01∆KL−0T−L−01∆ ∆L L L T −0T
∆L=L (L−1∆KL−T−L−1∆ ∆L L L T −T)
L−01∆L ∆L L T T
0
−
∆L, L−01∆KL−0T−L−01∆ ∆L L L T −0T
lower lower
+
1
1
( ) ( )
L−01 and L−0T
L−01∆K=[L−01∆k1
1
2 0
1
−∆ K − ∆ n]
bj=L− kj
0
1∆
∞
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Trang 4260 Finite Element Analysis: Thermomechanics of Solids
in which (∆A)∞ is the solution (converged iterate) for ∆A Consider the iteration
scheme
(20.12)
Subtraction of two successive iterates and application of matrix-norm inequali-ties furnish
(20.13)
spectral radius (see Dahlquist and Bjork [1974]) An approximate convergence criterion is obtained as
(20.14)
in which λj(∆A) denotes the j th
eigenvalue of the n × n matrix ∆A Clearly, convergence
is expected if the perturbation matrix has a sufficiently small norm Applied to the current
20.1.5 S AMPLE P ROBLEM
Let L0 andK0 be given by
(20.15)
Now suppose that the matrices are perturbed according to
(20.16)
so that
(20.17)
∆A(j+1) =∆K− A−1(∆A) (∞ ∆A) ( )j
2
∆A(j+2)−∆A(j+1)= A−1∆A∞[∆A(j+1)−∆A( )j ]
2 σ(A−1∆A )∝ < 1
j λj(∆A∝)<1 k λk( ),A
2
max j|λj(∆L∞) |<1min k|λk( ) |L
2
2 2
0
=
a
+
a
0
0 2
,
∆K =
+
0 d(2c d)
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Advanced Numerical Methods 261
We are interested in the case in which d/c << 1, for example, d/c = 0.1, ensuring
that the perturbation is small We also use the fact that
(20.18) The correct answer, which should emerge from the iteration scheme, is
(20.19) The initial iterate is found from straightforward manipulation as
(20.20)
The ratio of the norms of the error is
(20.21) Letting , the second iterate is found, after straightforward manip-ulation, as
(20.22)
The relative error is now
(20.23) Clearly, this is a significant improvement over the initial iterate
L01 1
0
− =
−
ac
c
∆L∝=
∆L1
2
c
norm
d
d c
=
∝
∝
%
( )
∆
L
1
1
5
1 2
∆ =d(1+12 d c)
0
1 2
1 8
2
2 2 3 3
=
c
d c
d c
d
c
d c
2
2 1 1 8
80
Trang 6262 Finite Element Analysis: Thermomechanics of Solids
20.2 OZAWA’S METHOD FOR INCOMPRESSIBLE
MATERIALS
In this section, thermal and inertial effects are neglected and the traction is assumed
to be prescribed on the undeformed exterior boundary Using a two-field formulation for an incompressible elastomer leads to an incremental relation, in global form, as follows (see Nicholson, 1995):
If KMM is singular, it can be replaced with K′MM= KMM+ χKMP K TMP, and χ can be
chosen to render K′MM positive-definite (see Zienkiewicz, 1989)
The presence of zeroes on the diagonal poses computational difficulties, which have received considerable attention Here, we discuss a modification of the Ozawa method discussed by Zienkiewicz and Taylor (1989) In particular, Equation 20.24
is replaced with the iteration scheme
(20.25)
in which the superscript j denotes the j th iterate and r a is an acceleration parameter
This scheme converges rapidly for suitable choices of r a
If the assumed pressure fields are discontinuous at the element boundaries, this method can be used at the element level to eliminate pressure variables (see Hughes, 1987) In this event, the global equilibrium equation only involves displacement degrees-of-freedom
For each iteration, it is necessary to solve a linear system Computation can be
expedited using a convenient version of the LU decomposition Let L1 and L2 denote lower-triangular matrices arising in the following Cholesky decompositions:
K′MM= L1L1T I/r a+ K T
MPL1L1T KMP = L2L2T (20.26) Then, a triangularization is attained as
Forward and backward substitution can now be exploited to solve the linear system arising in the incremental finite-element method
K
f 0
MM MP MP T
M
−
=
=
0
d d
d
γγ ψ
′
−
+
f
MP T
a
j
M j a
d d
d d
γγ
1
′
−
= −
−
−
MP T
T
MP
/ρ
1
1 1 1
2
Trang 7Advanced Numerical Methods 263 20.3 EXERCISES
1 Examine the first two iterates for the matrices
2 Verify that the product and inverse of triangular matrices are lower-triangular using
3 Verify the triangularization scheme in the matrix
Use the triangular factors to solve the equation
K
=
+
+
a
f
a
0 0
0 0
2
2 2
2 2 2
f
2
2 2
a
d
A=
−
−
Ab= −
2 1 1