An Introduction to Invariant Imbedding

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Invariant imbedding and the reduction of boundary-value problems of thin plate theory to Cauchy formulations. Enpng Sci. Pergamon Press 1. University of Southern California, Los Angeles. Abstract- Using ideas of invariant imbedding. The theory is developed in the context of elementary thin plate theory.


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It is shown that a rectangular plate with three edges clamped and the fourth edge free can be completely described by a system of integro- differential equations subject to initial values. The Betti reciprocity relations are proved in the context of the present theory and some applications of the fundamental solution are considered.

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It is finally shown that the present Cauchy system can generate a variety of novel numerical schemes including those previously de- veloped in references [4] and [5]. Unfortunately boundary value problems generally cannot be solved directly by numerical methods. Thus it is of interest to reformulate the problem in terms of initial-value or Cauchy systems for which many standard numerical procedures are available.

We notice that the reduction of boundary-value problems to initial-value formula- tions occupies at present a meaningful place in several branches of mathematical physics, such as neutron transport , radiative transfer [21 and modem control theory [3]. Invariant imbedding ideas usually underlie these transformations which in effect can also be viewed as alternate mathematical models of the same physical process.

Interest in the reduction of high order elliptic boundary-value problems typically encountered in applications of solid mechanics to Cauchy systems is therefore justified not only from a numerical point of view but also from more fundamental modeling considerations.

In fact, since structural perturbations are at the basis of invariant imbedding, this theory offers a natural, unified device to study the solution of the equa- tions in terms of non-classical imbedding variables such as length, thickness and physi- cal constants.

The usual way to accomplish the reduction of an elliptic boundary-value problem to a Cauchy system is via the use of an approximate operator involving ordinary differ- ential or difference schemes[41, [ However, previous[6] and recent work on the potential equation[71 and the transport equation ] has indicated the possibility of converting elliptic boundary value problems into Cauchy systems without making use of any intermediate steps. In this paper it is shown that a rectangular plate with three edges clamped and the fourth edge free can be completely described by a system of iSupported by the National Institute of Health under Grant No.

AT 1 I- 1j-1 Project No. SUniversity of California at Berkeley, Calif. QThe University of Rochester.

New York, U. JAIN integro-differential equations subject to initial values. The Betti reci- procity relations are proved in the context of the present theory and some applications of the fundamental solution are considered. It is finally shown that the present Cauchy system can generate a variety of novel numerical schemes including those already developed in [4] and [S].

Note that for simplicity we are considering the case where the Poisson ratio of the material is zero.


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  • This restriction is unessential for the present theory. The above problem will be considered as a fundamental problem for the plate in the context of invariant imbedding. We will obtain the solution of this problem via the solu- tion of the following two auxiliary problems. Let v be a parameter such that 0s us 1.

    initocet.cf

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    We will regard u and u as being functions not only of x. To facilitate the use of functions of several variables, the following notation will be introduced. Similar equations can be written for c. Although we shall employ some manipulation with delta functions, it is clear that the results can be easily justified via the theory of distributions [9] under the smoothing conditions usually adopted in the theory of thin elastic plates and generally required for numerical compu- tations. Assuming all necessary conditions for the interchangeability of the order of differentiation, we find that the new functions u3 and u3 satisfy I.

    Using 6. Applying superposition as in 10 we have r4, x,y,u. We will regard these four equations as initial value problems for all a 3 x, x fixed. In order to define more precisely our Cauchy system.

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