If the arc-length methods are combined with a modified' Newton method, an enhancement ... implemented and compared with a numerical relaxation technique.
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
59 (1986) 261-279
CONSISTENT LINEARIZATION FOR PATH FOLLOWING METHODS IN NONLINEAR FE ANALYSIS K.H. SCHWEIZERHOF Institute fiir Baustatik, Universitiit Stuttgart, Stuttgart, Federal Republic of Germany
P. WRIGGERS Institut fiir Baumechanik
und numerische Mechanic, Universitiit Hannover, Federal Republic of Germany
Hannover,
Received 3 March 1986
*
This paper is focussed on path following methods which are derived from consistent linearizations. The linearization procedure leads to some well-known constraint equations--like the constant arc length in the load-displacement space-and to different formulations than those given in the literature. A full Newton scheme for the unknown quantities (displacements and load parameter) can be formulated. A comparison of the derived algorithms with other path following methods is included to show advantages and limits of the methods. Using the linearization technique together with scaling a family of path following methods is introduced. Here, the scaling bypasses physical inconsistencies associated with mixed quantities like displacements and rotations in the global vector of the unknowns. Several possible scaling procedures are derived from a unified formulation. A discussion of these methods by means of numerical examples shows that up to now the choice of the scaling procedure is problem-dependent. If the arc-length methods are combined with a modified’ Newton method, an enhancement of the algorithms is achieved by line search techniques. Here, a simple but efficient line search was implemented and compared with a numerical relaxation technique. Both methods improve the convergence rate considerably.
1. Introduction Within the field of continuation methods [26] the so-called path following methods have found wide interest in engineering for the solution of nonlinear equation systems. Particularly after the ideas developed in [l, 2,161 were cast into computationally efficient schemes [3-51, the field of application has grown substantially. The relationship of path following and turning point methods with those used in constraint optimization theory can be found in [22]. Focussing on stability problems, Riks [6] gives a good overview of the theoretical background for these algorithms with a broad literature review. Therefore, one is referred to [6] for the general aspect, and we want to concentrate further on the specific aspects with which we are concerned. The weak form of a geometrically or/and materially nonlinear structural problem can be written conveniently in matrix form as follows [23]: 00457825/86/$3.50
@ 1986, Elsevier Science Publishers
B.V. (North-Holland)
K.H.
262
Schweizerhof,
G(x, A, q) =
P. Wriggers, Linearization metho& in nonlinear FE analysis
~t(~(x) - AP) = 0 ,
(1)
with the virtual displacement field q, the position vector of the current configuration X, the stress divergence term R(X), the vector of the external forces P, and the propo~ional load factor A. By the fundamental theorem of variational analysis, (1) leads to the equilibrium equations R(x)-AP=O.
(2)
A classical method for solving (2) is Newton’s method, tal iteration scheme
K$) Au(‘)
‘=
-
[@‘))
- up] ,
x(i+i)
=I
which leads naturally to an incremen-
xW + Au(“)
,
(3)
where the tangent stiffness matrix lu, follows directly by consistent linearization of (2). In the classical approach of incremental schemes like (3) the load factor A (load level) is usually kept constant and iterations are performed until a state of equilibrium is reached. It is well known that these methods fail in the case of limit points. The geometrical meaning is that the plane A = constant does not necessarily intersect the load-deflection curve. These problems can be overcome by adding a constraint equation to the standard equilibrium equations (2). The most popular methods [l, 21 keep a specific arc length (s) constant, which is defined by the Euclidean norm of the general displacements and loads: S=:
x’x + A2PtP = const. ,
a: = xtx + A2PtP - s2 = 0.
(4)
(5)
The satisfaction of the introduced constraint is required in every iteration. This does not pose a problem for linear ~nstraints as suggested by Ramm [4] and Keller [S], but may lead to problems in the case of nonlinear constraints [3]. If the constraint is linearized in a consistent manner for a Newton-type scheme (3) [7], the above-mentioned problems vanish as will be shown in the following. The influence on the iteration algorithm is shown for varidus examples including geometrical and material nonlinearities. The characteristics of algorithms using arc-length schemes together with the modified Newton method can be largely improved by a combination with line search schemes, as has been pointed out already by Crisfield [8]. This is also true for a pure Newton method if large load steps have to be performed, since Newton’s method is only locally convergent. The application of two simple line search schemes as well as their limitations in combination with various arc-length methods are discussed.
2. privation
of consistent tangent matrices
In the, following we will restrict ourselves, for simplicity, to proportional loading with non-deformation-dependent loading. For a classification and examples involving arc-length
K.H.
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
263
methods and deformation-dependent loading we refer to [20]. The calculation of loaddeflection curves can be performed best using a continuation method [6], which leads to an incremental scheme, see Fig. 1. Within this scheme we will assume that a load level A”, which represents an equilibrium state, is the starting point for the following load increment. Since due to the constraint equation (5) the loading factor A may vary during the iteration process, the consistent linearized form of the nonlinear equilibrium equation (2) has to be written as Kc’) Au(i)
T
_ p AA(‘)
=
(A”
+
A(‘))p
_ R(x(i))
.
(6)
Constraints for path following algorithms can be arbitrary functions of displacements factors as well, see (4), (5), f(x, A) = 0 . The linearization ,@b Au(‘)
.(i)t
= ;
a(i)= 2
the complete
(7)
of the constraint + a(i)
With the abbreviations
and load
AA(‘)
equation for a Newton scheme yields
= _f(x(i),
for the directional
A(‘))
= _ fci)
.
(8)
derivatives
f(#) + E Au(‘), A(i))l,,o , f(n('), A(')+ E AA(‘)l
system of equations
[ $t,Z.][
t;]‘i’
,
can be written as
= _ [ “(~‘iQ-;;;~;:‘i’,pl
= _ [ $)I
.
(9)
T
To permit different algorithms for the solution of (9) the superscript (k) becomes (i) for a pure Newton method and is set to (0) for a modified The new system matrix A is nonsymmetric for a general constraint like [4,9]. Constraints resulting in symmetric matrices can be found as well
Fig. 1. Notation
and scheme for an incremental
iteration
is introduced, which Newton scheme. (4), as is well known [lo].
in the load-displacement
space.
264
K.H.
Schweiterhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
The Hessian A is by construction nonsingular, see [6], but this advantage has to be paid for by a large bandwidth in addition to its unsymmetric form. Furthermore, it has to be pointed out that within an increment the linearized expressions vfi) , crfi) are not constant in the case of nonlinear constraints. Solutions of the nonsymmetric system of equations can be achieved in various ways as is shown in Section 2.1.
The straightforward approach applies general equation solvers for nonsymmetric matrices in order to avoid all difficulties with nonsingularities. But this method is far too expensive for large equation systems arising in finite element problems. Some special equation solvers [ll] therefore explore the specific nature of the unsymmetries of the matrix in a more efficient way, since they are restricted to one row and one column. If constraints are used which remain constant within an increment [4,5], these equation solvers are numerically as efficient as the corresponding arc-length schemes [ll]. However, in the case of nonlinear constraints an almost complete refactorization has to be performed in each iteration step. Therefore, from a numerical point of view, this type of solution is only competitive for constant constraints, or if full Newton schemes are employed which require a refactorization anyhow. Furthermore, information about the structural stiffness matrix K,, like its determinant or its eigenvalues, can only be calculated by additional special procedures. 2.2. Reduction to a system of equations of original size The reduction to a smaller system of equations is done via elimination of Ah”! This is only = 0 a partitioning of (9) has to be performed as possible using (10) if afi) f0. For CX(~’ suggested in [4,5]: AA(‘)
= _
_.$ (f”’
_ #)*
Au(‘))
_
(W
This leads to:
I
Au(‘)
=: _ (;fi)
_
5 p,
In (11) the original stiffness matrix iK$’ is updated with a matrix of rank one, and the ~ght-hand side shows a modi~cation due to the residual in the constraint. This rank-one update AK”’ produces in general also an unsymmetric coefficient matrix, as in (9). The meaning of these updates is shown in Figs. 2 and 3 for constraints which are constant, respectively variable, within an increment. It is obvious that AK@’changes within the iteration for variable constraints. The inversion of the modified system of equations is performed by applying the Sherman-
K.H.
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
265
x
“A -u
-”
(a) Fig. 2. Incremental iterative within the increment.
(b)
scheme with: (a) constant
Fig. 3. Iteration
constraint
within the increment:
(b) variable
constraint
on normal planes.
Morrison formula [ 121
=
Jp-1
T
-
K(k)-1
_
pxz’ .
(12)
With
there follows K(k)
T
I
‘,
a(l)
pu(iP
I= -I
KcqlD&ow
T
a(i)
+ pTK(k)-lv(i)
T
*
(14)
The solution of the modified equation system (11) is obtained with the aid of (14). Introducing Au’ = Kr’-p,
Au”
= _ K$)-‘G(i)
,
(15)
the solution of (1 _) may be written as
(16)
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266
P. Wriggers, Linearization methods in nonlinear FE analysis
The iterative displacement vector consists now of two parts, where AU” stems from the out-of-balance forces, whereas the other part Au’ is a multiple of a solution vector due to the externally applied load. Finally, inserting (1.5) into (10) leads to a formula for the increment of the load parameter AA”‘,
(17) Remarks (i) Equation (16) confirms the assumptions made in [3,4] by leading to the split of the iterative displacements proposed there. (ii) Equation (17) does no longer have the restriction for LY’~), which might become zero as well. 2.3. Discussion The advantages of the latter type of solution are clearly visible. (i) Kg’ is symmetric and has the original bandwidth. Thus solution strategies for equation solvers remain unaffected by arc-length procedures. (ii) In a modified Newton strategy the stiffness matrix has to be factorized only once, even when nonlinear constraints are used. (iii) Information about KF’ is present as usual: determinant, number of sign changes of diagonal terms, etc. (iv) modifications of the determinant of the stiffness matrix are easily controlled by det
KC“) + T
-.I!+ pan &‘)
1 +
r
a(‘)
Au’v”’
(18)
As major disadvantage it has to be mentioned that, contrary to the nonsymmetric matrix A, the stiffness matrix Kr may become singular, which would cause problems in the solution of the equation system. Numerical experiences of the authors with many examples revealed that this is no problem in numerical applications. KT is almost never singular, at least not in the numerical simulations on a computer.
3. Algorithms Among the infinite number of choices of possible arc-length methods we want to restrict ourselves first to the well-known strategies. The formulas developed in Section 2 are used to show the consequences of the choice of particular constraints on the solution algorithm. Furthermore, differences with the methods proposed in [3,4] are explained. The well-known arc-length methods of Crisfield (31 and Ramm [4] are based on a two-step solution strategy. In the first step-the predictor step-the load is scaled back such that the length of the incremental load-displacement vector keeps a prescribed value. In the succeeding corrector steps the two proposals differ due to the choice of the constraint.
K.H. Schweizerhof, P. Wriggers, Linearization methods in nonlinear FE analysis
3.1.
267
Predictor step
Constraint
Algorithm Step 1.
K(O)A,($) = T
(“A
+
A’*‘)P - G(“x) = A(*+,
xh) = Au(*) . Step 2. X(*)fX(*) + p2A(*). Step 3. x(l) = ,(*),b)
= Aa’
9
A(‘) = &)A(*).
Here “‘A,mx denote an equilibrium state, A(*) is an arbitrary initial value, /3 is a scaling factor which is discussed in Section 4, and s is a given constant value. 3.2. Corrector steps 3.2.1.
Iteration on normal planes, [l, 4,131
The constraint
equation
is illustrated
f= t(k) . (t - t(k)) i 0 ) f =
Equations
in Fig. 4 and takes the form
t@) = t(x@), p, Ack’) ,
(x(k) - “x)‘(x - xck’)+ p’( Ack) - “A)( A - ACk’).
(20) (21)
(17) and (21) lead to an expression for the load factor: f (9 + tx(k) _ mx) ,@
AA=-
@A(k) + (x(k) _ ‘“r) Au’ ’
Fig. 4. Iteration
on spherical hyperplanes.
(22)
268
K. H. Schweizerhof.
P. Wriggers, Linearization methods in nonlinear FE analysis
For k = 1 the iteration proceeds on constant normal (hyper) planes, whereas for iteration is performed on updated normal planes [4].
k = i
the
Remarks
(i) The residualf”’ is equal to zero within the suggested procedure, linear and therefore exactly satisfied in every iteration. (ii) The arc length does not remain constant within the corrector 3.2.2.
because the constraint is steps.
Iteration on spherical hyperplanes
Cons train t f=
The length increment.
q/(x - “x)‘(x -
3) + p’( A - mA)2 .
of the incremental
load-displacement
(23)
is forced to remain
constant
within an
Load factor Ah = _ f (i)( f (I’ + s) + (xci) - ‘%)t AU” . pzA(i) + (x(i) _ mX)i Au’
(24)
Remarks
(i) The residual f @I(f (i) + s) is not equal to zero throughout the iteration process unless the solution has converged. (ii) The constraint is only iteratively satisfied. (iii) The updated normal plane method (22) with k = i, which was derived from geometrical considerations [4], approximates (24) by updating the linear constraint condition. 3.3. Discussion The difference in comparison to Crisfield’s original method [3] has to be seen as follows. Crisfield’s method requires the satisfaction of the constraint in every iteration step. This leads to a quadratic expression for AA with two possible roots. Problems arise in the choice of the proper root and when the roots become complex. Nevertheless, the explicit satisfaction of (23) ensures that the solution remains in the domain of attraction. However, numerical experience will clarify if there is a need for this more elaborate scheme. As an improvement we suggest the use of the consistently linearized version whenever the roots in Crisfields method become complex. Algorithm (continued)
Iteration Step 4.
loop:
Au” = -K, (k)-lw’))
.
K.H.
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
269
Step 5.
Ah”’ = - (see (22) or (24)) . Step 6. Au(‘)
= AA”’
Au’
+ Au”
Step 7. A(‘+‘)
= A(‘) + AA”’
,
xG+l)
= .(i)
+ Au(‘)
.
Convergence?
The well-known single-displacement control [9], multiple-displacement control [ 141, and load control method can be easily obtained within the framework of the algorithm shown above by introducing the appropriate constraint equation. 4. Scaling The constraints used in the various algorithms do often mix quantities with different rotations, and loads. These inconsistencies can be physical meanings as displacements, bypassed with a proper scaling of the different quantities. A complete scaling was suggested in [15] using f=~(x-mx)‘Vs(x-“x)+~2(h-mA)2-s=0,
(25)
with a diagonal matrix V, as scaling matrix:
(26) with scaling factors Vi and p. By an arbitrary choice of the scaling factors a large family of “arc-length” methods can be created. This family contains the classical load control as well as the single-displacement control method. On the background of examples with various characteristics numerical experiences with the scaling of the arc-length method can be summarized as follows. (i) Variation of p: p % 1, V, = I. Better convergence for stiffening structures because the arc length is shortened according to the load term too, see also [25]. Reduced convergence rates for “soft” structures, reduced adaptivity of the method in the “soft” domain. (ii) Control via displacement terms: udis = 1, u,,,~= /3 = 0. Convergence as with singledisplacement control, but better adaptivity. The unscaled schemes with the neglection of the load terms as originally proposed in [3,4] are essentially equivalent to this type of control, because the displacement terms mostly dominate.
270
K.H.
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analystysis
(iii) Control by rotations: u,,~ = 1, udis = p = 0. For sufficiently large rotations a similar behaviour as in (ii) can be observed. If the quantities are very small compared to the displacement terms numerical problems arise. (iv) Scaling with an initial displacement vector: ui = (liu~“)‘, /3 = 1 IA”‘, u!‘j f 0. All quantities are controlling the arc length with the same intensity. The adaptivity of ;he scheme is very strong. Often the step size is reduced too much to be numerically efficient. Numerical problems can arise if the scaling quantities EL;‘)are small compared to the dominating terms. Up to now none of the above-mentioned schemes can be recommended as best for all possible applications, but the following suggestions can be given. (1) The load term has to be present if stiffening structures are analyzed. Advisable is p > I in the stiffening part of the load-deflection curve. (2) Scaling as suggested in (iv) can be very helpful in parts of the load-deflection curve with strong slope changes. (3) For a broad range of problems the schemes (i) and (ii) are most suf~ciently effective and inexpensive. In a recent paper [24] a similar scaling procedure using the diagonal terms of Z’$ is proposed to enlarge the adaptivity of the arc-length methods, Unfortunately, the basis of numerical examples presented there is not broad enough to allow for general conclusions.
5. Combination with line search techniques The combination of line search techniques with the arc-length methods can be performed in a straightfo~ard manner. The schemes mentioned herein are somewhat simpler than those suggested in [8] but nevertheless very effective. 5.1.
Classical line search
The optimal step length s in the direction of an iterative displacement vector Au is calculated by the orthogonality condition (27), which suppresses the projection of the residual vector in the direction of Au: Q, =
Au’ G(x”’ + sj Au”‘) = 0,
with the new displacement x(i+l)
=
$1 +
sj
(27)
vector being, see Fig. 5, Au(‘) .
(28) h
Is-Ah
S-AU ~ Fig. 5. Combined
iterations
in the load-displacement
U
space and line search techniques.
K.H. Schweizerhof, P. Wriggers, Linearization methods in nonlinear FE analysis
It has to be pointed out that the load level is changing recognized from (29): G(x”’
+
sj
= -(A(‘) +
Au”‘)
sj
with the step length,
AA”‘)P + R(x(‘)+ sj Au(‘))
.
271
as can be
(29
It is often desirable to sacrifice accuracy in the line search routine in order to restrict the overall computing time. Satisfactory rates of convergence are often obtained with inaccurate line searches [21]. Thus, no further line searches are performed when the criterion
IIA~(~)~G(_$) + sj AU(‘))]] 6 7711h~(~)~G(x(~))l]
(SO)
is fulfilled. The search algorithm starts with s = 1 and uses linear interpolations between two successive energy values Q j. The line search tolerance has to be chosen properly to achieve fast convergence and numerical efficiency. To avoid unnecessary iterations the line search is also stopped if two successive values of sj are close together:
ISj- Sj_11 < 0.5rl(sj + Extrapolations
sj-l)
(31)
’
leading to sj > 1 are not admitted.
5.2. Numerical relaxation Another method to accelerate the convergence behaviour is numerical relaxation. technique can be applied if the load factors are oscillating [4], AA(‘)
. AA”-”
step
1
2
3
on sphere 4
5
6
* step
W (a) Fig. 7. Thin shell: (a) comparison of iterations with various constraints, combination with numerical relaxation: (b) iteration on spherical planes with consistent fo~ulation; combination with line searches (tolerances v = 0.9, 0.7, 0.5).
at (EJ deformation patterns-1 half shell) R/i-=5.0 ,R/t =400, t=6.35mm
Fig. 8. Hinged shell: snap-through/snap-back and current stiffness parameter (CSP).
problem (thick shell). Deformation
patterns,
load-deflection
curve,
274
K.H.
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
current stiffness parameter for any control-step size or sign of the load-is not advisable for such structures. The comparison of the influence of the various constraints on the convergence (Fig. 9) shows again the superiority of the iteration on circles. At load step 4 schemes (a) and (b) diverge whereas Crisfield’s method shows signs of convergence, but only in an extremely slow manner, being still far from the convergence limits after 200 iterations. Numerical relaxation did improve the convergence characteristics, but did not prevent the divergence of the Newton schemes. It also did not help in the combination with Crisfield’s method. If line searches are applied all schemes converged for a given tolerance I-/d 0.9. The difference between the various constraints is negligible again for the low tolerances r) but still visible. More extensive documentation can be found in [19]. EXAMPLE
6.3.
Unstiffened plate girder under concentrated loads. As the third example we
chose the limit point problem of a plate girder where material and geometrical nonlinearities are involved. System and data are shown in Fig. 10. The numerical analysis using a modified Newton scheme and iteration on updated planes broke down at load step 7 (Fig. 11). Also a choice of other constraints did not lead to convergence in this critical step. Even halfening the step length did not help. The only way to overcome this point was the use of line searches. Then, as can be seen in Table 1, all iteration schemes converged with the same number of iterations and line searches. For comparison the pure Newton scheme with iteration on updated planes was tested also in this step. As expected, convergence was achieved with only three iterations, but the CPU time used for the full Newton scheme was still larger than for a modified Newton method with 9 iterations and 8 line searches. This difference was even more pronounced if the overall cost of the analysis including I/O and CPU time was compared. Another remarkable observation was that the modified Newton scheme was always
"'
5
10
(J)
15
18
srep
5
IO
15
IL'
18
11
12813.51]
step
(’ >
Fig. 9. Thick shell: (a) comparison of iteration with various constraints in combination with numerical relaxation: (b) iteration on spherical planes with consistent formulation in combination with line searches (tolerances 7 = 0.9. 0.7. 0.5).
275
K.H. Schweizerhof, P. Wriggers, Linearization methodr in nonlinear FE analysis [Axis
of
: Svmmeirv
M/(P.h)=
E
= 2.1
v
= 0.3
13662
lo5
Upper flange
N/mm2
:
=
F
Lower flange : F Web
:F
251
195
N/mm2
Y
= 207 N/mm2
Y
= 215 N/mm2
Y
all in [mm1
Fig. 10. Unstikened plate girder, geometry and finite element model. Web: 16 (4 x 4) bicubic elements (16 nodes). Upper flange: 6 biquadratic elements (9 nodes). Lower flange: 12 truss elements. limit load
1200
1000
%
- CSP
BOO
s
600
400
200
0t -2
Fig. 11. Load-deflection
I
il
DlSPLAdEMENTS M2k (CSP)
1
curve for plate girder, current stiffness parameter
sufficient and did not need any line searches or other additional procedures and following.
(CSP).
in the load steps 8
EXAMPLE 6.4. Lee-frame, large rotation problem, application of full Newton schemes. The Lee-frame [28], for which an analytical solution exists, is an example where instability occurs after large displacements happened. It also shows a clear snap-back behaviour. For an
K.H.
276
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
Fig. 12. Lee-frame: (a) system and deformed shapes for different load parameters: (b) load-deflection curve of Lee-frame analysis (letters in circles denote a point in the solution path corresponding to the deformed shapes in (a)). Table 1 Comparison of various iterations (step length ju] = 3.2) Newton modified
pure
Iteration scheme
schemes and combination
with line searches at step 7
Number of iterations
Number of line searches
CPU (s)
-
-
(21) + (22) (23) + (24)
>20 divergence >20 divergence >20 divergence 9 9
(21) f (22) (23) + (24)
3 3
-
(21) + (22) (23) + (24) (23) + circle [3]
8 8
21.17 21.19 25.72 25.73
economic solution the full Newton-Raphson scheme had to be used, since the application of modified schemes even with line searches did result in very small steps with very high numbers of iterations. A comparison of the three schemes with equal step length revealed (Table 2) that all schemes behave similarly as the step length is of reasonable size (ds = 25). By enlarging the load step (ds = 35) we do not recognize a difference between the Newton iteration on updated planes and on spherical hyperp~anes, both diverging at the crossing of the road-detection curve with the horizontal axis after the same number of iterations in each step before. Crisfield’s method with the explicit satisfaction of the constant arc length however did still converge. After a further enlargement of the step length to ds = 40, Crisfield’s scheme did diverge also at the point cited above. Thus this example, with very large displacements and rotations, clearly revealed the advantages of Crisfield’s scheme at critical points, where it had the largest domain of attraction. Opposite to the observations in the other examples, which were analyzed with the modified Newton scheme, almost no difference could be found between the three schemes in the regions where the solutions converged using full Newton.
40.0
35.0
65556
65555
Circle
Circle
65554
55554
LP
55444
circle
Updated planes Consistently linearized
44554
55544
65555
LP 65555 LP 65555 LP 65555
44554
44564
LP
LP
LP
Step 678910
55444
25.0
Step 12345
Updated planes Consistently linearized
Iteration scheme
Step length
45555
45567
45566
6-- ----divergence------
@
@
@
Step 16 17 18 19 20
divergence-----_ BP 55610775554
5 5._____
5 5------divergence------
44444
44444
44444
Step 11 12 13 14 15
Table 2 Lee-frame: comparison of iteration algorithms in combination with full Newton schemes. ds = Ilz+ll; tolerances: out-of-b~ance force: ftol = lo-” ; LP = limit point; BP = bottom point; @ - see Fig. 12
rtol = 10-l;
BP 7545444 BP 7645444 BP 7645444
Step 21 22 23 24 25 26 27
displacements:
278
K.H.
Schweizerhof,
P. Wriggers, Linearization methods in nonlinear FE analysis
Also line searches did not lead to an acceleration in convergence where the solution without line searches did not converge.
or to a solution at load levels
7. Conclusions A consistent derivation for a Newton-type algorithm was given for the solution of a set of nonlinear equations with arbitrary constraint functions. In particular the algorithms for a Newton iteration on spheres was compared to the algorithms developed by Ramm [4] and Crisfield [3]. For a modified Newton strategy, finally, the combination with simple line search techniques was investigated, which ensures global convergence of the Newton scheme. The following results were obtained. (1) Within a modified Newton strategy the iteration on spheres is clearly superior to the iteration on updated planes. The explicit satisfaction of the arc-length constraint might present divergence in some critical situations, but then convergence is also very slow and additional strategies should be used. (2) The Newton strategies for the arc-length constraint can be applied in a straightforward manner and problems like choices of roots or complex roots are avoided. (3) Line searches improve the convergence rates of all algorithms in combination with, modified Newton schemes considerably. For low line search tolerances no significant difference in the convergence characteristics of either scheme could be observed. Numerical relaxation, being computationally very attractive due to its inexpensiveness, did improve the convergence characteristics in many cases, but could not prevent divergence generally. (4) In the combination with full Newton schemes the differences between the algorithms are minor. At critical points Crisfield’s algorithm showed the largest domain of attraction. The application of line searches together with full Newton schemes did not reveal any further improvement in convergence for the examples under consideration.
REMARK
7.1. We have to give further credit to Dr. Riks for a recent paper of his [29] showing also the consistent derivation, which he brought to our attention while we were finishing this study. Acknowledgment
The authors gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG) and want to thank Professor E. Ramm for encouragements and valuable discussions. References [l] E. Riks, The application of Newtons method to the problem of elastic stability, J. Appl. Mech. 39 (1972) 1060-1066. [2] G.A. Wempner, Discrete approximations related to nonlinear theories of solids, Internat. J. Solids and Structures 7 (1971) 1581-1599. [3] M.A. Crisfield, A fast incremental/iterative solution procedure that handles snap through, Comput. & Structures 13 (1981) 55-62.
K. H. Schweizerhof, [4] E. Ramm,
P. Wriggers, Linearization methods in nonlinear FE analysis
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