COMPUTATION OF DISPLACEMENTS FOR NONLINEAR ELASTIC BEAM MODELS USING MONOTONE ITERATIONS

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.


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 have- u q(x,u) _= f(x u) 0 < x < 6, (I I) EI 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 appli- cable 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 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 inverse- positivity, a property of boundary-value problems, rather than of equations, see [2,3].In [3] we applied monotone methods to general inverse-positive problems, 122 P. KORMAN including (I.I) (1.2).In this paper, we present some further results, and 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 theorem 4. Theorems and 2 are essentially known, and are illustrated here 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 con- vergence 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].
(ii) f is continuous, increasing in u for < u < .
Then the problem (2.1) has a C 4 solution u(x), and < u < @.Moreover, starting with or we get two monotone sequences of Picard iterations.Theorem 2. For the problem (2.1) starting with some continuous function Uo(X) define a sequence of approximations {Un(X) by the formula where G(x,{) is the Green's function for u with the boundary conditions O, which is given by the formula (see [5]) Assume the following for 0 x .
NONLINEAR ELASTIC BEAM MODELS USING MONOTONE ITERATIONS 123 (ii) f(x,u) is continuous, decreasing in u for u 0 < u < u I, and there is a constant fo > O, such that If(x,u) f(x,v)I ( foIU-Vl if u 0 u, v < u 4 (iii) fo 31T6 < I.
Then the problem (2.1) has a solution u(x) C 4, lU-UnlCo 0 as n (uniform convergence), and moreover Solution is unique in the order interval [Uo,Ul].
To prove uniqueness it suffices to write the equation for the difference of two solutions w(x), multiply it by w and integrate by parts twice.
0 Assume now that f(x,O) 0 and B y O. Then (2.1) possesses a trivial solution u O. Next, we discuss existence of a nontrivial solution.Theorem 3. Lef f(x,u) be continuous function increasing in u for 0 < u < (R), 0 < x < 2, f(x,u) > 0 for u > O, and (uniformly in x).
Then the problem (2.1) has a positive solution, provided m, B, y, a > O.
Notice, o(X) > 0 for o < x < .Now it is easy to check that o(X) and Mo(X) are sub and supersolutions, provided ,p are sufficiently small and M is 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 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 solu- tion 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).
Definition.Solution u(x) of Lu f(x,u) with f continuous and increasing in u, xR n, is called stable if for any > 0 there exist sub and super- solutions (x) and @(x) with 0 < lu @ICO + lu ICO , such that u @.
(If this condition holds only with @ u ( u) we say that solution is stable from below (above)).Solution is called unstable if in the conditions above <u.
It is easy to see that the concepts of stability and instability are mutually exclusive for an isolated solution.(Our concept of stability corresponds to the 'strong stability' in H. Matano [7], and it implies stability for elliptic equations of second order, see Proposition 4.2 in [7].) must be unstable if it exists, and hence not computable by monotone iterations.
Proof.If u is a solution, then v u satisfies v v p (-P)u p.
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 one- si ded perturbations.
After fixing z we see that T$ 0 is a supersolution for our problem, provided , , B, y, 6 are sufficiently small.Take 0 for a subsolution, and apply the theorem I.
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) as defined in the theorem 3 is given by 4 0 Our next result provides a simple error estimate.Proposition 3.For the problem (2.1) assume conditions of either theorem Let f(x,u) be Lipshitz in u uni- formly in 0 < x 2, with Lipshitz constant fo for u (u 0 < u < u l) in case of theorem I(2).In case of theorem assume additionally that Gof0 < I. as in the theorem 2. (Notice that for the theorem we get convergence of both monotone sequences to the same solution.) Remark I.In the conditions of the theorem assume that f f(u), f(O) > O.
Then we can start with a subsolution u 0 O, compute u x2(x-) (2.12) which explains the fast convergence that we encountered in our experiments (=2).On the other hand, we see from (2.12) that convergence is slower, in general, for larger , and it cannot be guaranteed beyond a certain value of This is no surprise, since e.g. the problem u u 2 + l, u(O) u'(O) u() u'() 0 has no solution for sufficiently large, as can be seen by rescaling x {, and then applying the theorem 4 in [3].Remark 2. Finally, we mention that all our results hold for other physi- cally important boundary conditions for Lu u since they only depended on the positivity of the Green's function.Next we list two such problems with corresponding Green's functions (see [5]), whose positivity is easy to check.that our Green's function in (?.3) is "small".One measure of its smallness is given in (2.5).Another one is derived by an elementary computation" 3 max G(x,.)G(/2, /2) Table I.Accuracy lO -6" ,-subsolution, -supersolution; number of iterations refers to the larger one in case of two monotone sequences.
)-T (p-l) the problem (2.10) has no positive solution" If u denotes the solution obtained by monotone iterations (u by writing u-u n =k Z =n (Uk+l uk)' and estimating each term