Proof. With $0 as defined in the theorem 3, choose z > 0 so small that .... COLLATZ, L. Eigenwertp.robleme und ihre numerische behandlum Chelsea New.
Internat. J. Math. & Math. Sci. VOL. II NO. (1988) 121-128
121
COMPUTATION OF DISPLACEMENTS FOR NONLINEAR ELASTIC BEAM MODELS USING MONOTONE ITERATIONS PHILIP KORMAN Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221 (Received September 2, 1986)
ABSTRACT.
We study displacement of a uniform elastic beam subject to various physically important boundary conditions. Using monotone methods, we discuss stability and instability of solutions. We present computations, which suggest efficiency of monotone methods for fourth order boundary value problems.
KEY WORDS AND PHRASES.
Monotone schemes, elastic beam, existence and approxima
tion, stability.
1980 AMS SUBJECT CLASSIFICATION CODES.
34,65.
INTRODUCTION We study the displacement curve u u(x) of a uniform elastic beam of length 6, supporting a distributed load of intensity q(x,u(x)). This load causes the beam to bend from its equilibrium configuration along the x-axis. For small displacements we have1.
u
q(x,u) EI
_=
f(x u)
0 < x < 6,
(I I)
where E is Young’s modulus, is the moment of inertia, see e.g. [1]. That is we study the equation (l.l) with appropriate two-point boundary conditions. We show that the monotone iteration scheme and other monotone methods are applicable and provide an effective computational tool, as well as means of proving existence theorems.
Monotone methods are usually associated with maximum principles. Clearly, there is no weak maximum principle for u f(x), since condition f(x) 0 does not preclude u(x) from having extreme points inside of any interval. However, if we add the boundary conditlons
u(O)
u’(O)
B,
u() :y, -u’()
(l .2)
with =, B, y, O, then condition f(x) 0 does imply u(x) 0 (since the Green’s function in (2.2) is positive). This is an example of inversepositivity, a property of boundary-value problems, rather than of equations, see [2,3]. In [3] we applied monotone methods to general inverse-positive problems,
P. KORMAN
122
(1.2). In this paper, we present some further results, and including (I.I) report on computations ith numerous nonlinearities f(x,u). The main results of this note are the theorem 3, and our discussion of stability leading to the and 2 are essentially known, and are illustrated here
Theorems
theorem 4.
computationally. For a previous application of monotone methods For this model,
see J. Schroder [4], where a rather involved splitting method was used. Most of the results in this paper were stimulated by computations (and the Fast convergence that we encountered). Our results apply to other physically important boundary conditions, see Remark 2, as well as to biharmonic equations in higher dimensions, see [3]. Throughout the paper @ will denote subsolution, -supersolution,
lUIco 2.
max
lu(x)I.
0(x(,
GENERAL RESULTS The following theorem was proved in [3]. Consider the problem (one-dimensional)
Theorem 1.
f(x,u)
u
,
u(o)
u’(o)
Assume the following for 0 < x < (i)
’(0)
.
0 < x
O, such that If(x,u) u0 u, v < u
foIU-Vl
4
(iii)
fo 31T6
< I.
C 4,
Then the problem (2.1) has a solution u(x) (uniform convergence), and moreover
u0 < u2
u4
0 as n
(2.4)
u I.
u3
u
(
lU-Unl Co
[Uo,Ul].
Solution is unique in the order interval Proof. Relations (2.4) easily follow by induction.
To prove convergence
write
Un+l(X)
Un(X)
0
f
and hence
max O 0 and 0 > 0 be the principle eigenfunctlon and eigenProof. Let ;u, u(O) u’(O) u() u’() O, whose existence is value for u guaranteed by the Krein-Rutman theorem (see [3] for details). Let Co(X) and u’(+p) with p > O. u(+p) u’(-p) uu, u(-p) u 0 be the same for u Now it is easy to check that Notice, o(X) > 0 for o < x < o(X) and sufficiently small and M is ,p are provided supersolutions, are and sub Mo(X)
0
.
sufficiently large. This theorem covers in particular the sublinear nonlinearities. For the superlinear case, we state the following conjecture. Conjecture. Assume the conditions of the theorem 3 with (2.6) changed to
P. KORMAN
124
f(x,u)
lim u/O +
u
lim
U
(2.7)
O.
U+
Then the problem (2.1) has a positive solution. We expect this solution to be unstable. We discuss the concept of stability next. For the equations of second order, like Au=f(x,u), it is known that if solution can be computed by the monotone iteration method, then it must be stable. The proof of this result, as well as the definition of stability itself, makes use of t6e maximum principle for parabolic equations, see e.g. [6]. Since we do not know of any maximum principle for u t + Uxxxx we have to generalize the concept of stability. We state it for a general inverse-positive operator L, see e.g. [2,3]; in our case Lu u with the boundary conditions of (2.1). f(x,u) with f continuous and increasing Definition. Solution u(x) of Lu > 0 there exist sub and superis called stable if for any in u, xR < u + lu solutions (x) and @(x) with 0 such that @. lu u) we say that solution is stable (If this condition holds only with @ u ( from below (above)). Solution is called unstable if in the conditions above
n,
ICO ,
@ICO
must be unstable if it exists, and hence not computable by monotone iterations. (-P)u p. vp u satisfies v Proof. If u is a solution, then v Hence v is a supersolution for < l, and a subsolution for > I. On the other hand, it is easy to see that the trivial solution of (2.8) is stable from above (take O, 0; if p is an odd integer take -0’ giving two-sided stability). Not surprisingly, it persists under small onesi ded perturbations. Proposition I. Consider the problem
u
up +
f(x,u)
u(O)
,
u’(O)
Here f is continuous, increasing in u; f(x,u),
,,
B, u()
,
B,
Y, 6
y, u’()
(2.9)
6.
O; f(x,O) (
O.
Then for B, y, 6 sufficiently small, the problem (2.9) has a nonnegative solution. Proof. With $0 as defined in the theorem 3, choose z > 0 so small that for 0 ( x ( (with p > 0 chosen so small that 0(0) > O, 0() < 0). > After fixing z we see that T$ is a supersolution for our problem, provided 0 0 for a subsolution, and apply B, y, 6 are sufficiently small. Take the theorem I.
TPoP
,,
NONLINEAR ELASTIC BEAM MODELS USING MONOTONE ITERATIONS
125
In [3] we discussed some general non-existence results which can be applied to (2.1). In particular, we had the following result which will be illustrated compu ta ti ona y. Proposition 2. Consider the problem (p > l) up +
u
u’(O)
u(O)
u’()
u()
(2.10)
O.
P
x
(O)-T (p-l)
the problem (2.10) has no positive solution" 4 as defined in the theorem 3 is given by 4 0 k 4.7300, see [5, p 146]. k Our next result provides a simple error estimate. Proposition 3. For the problem (2.1) assume conditions of either theorem >
Then for
0
4
or 2
max [ O
