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Optimal Location of Support Points in the Kirchhoff Plate

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Part of the Springer Optimization and Its Applications book series (SOIA,volume 66)

Abstract

The Dirichlet problem for the bi-harmonic equation is considered as the Kirchhoff model of an isotropic elastic plate clamped at its edge. The plate is supported at certain points P 1,…,P J, that is, the deflexion u(x) satisfies the Sobolev point conditions u(P 1)=⋯=u(P J)=0. The optimal location of the support points is discussed such that either the compliance functional or the minimal deflexion functional attains its minimum.

Keywords

  • Support Points
  • Kirchhoff Plate
  • Sobolev Problem
  • Sobolev Condition
  • Thin Three-dimensional Plate

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Notes

  1. 1.

    For this work, Mademoiselle Sophie Germain was awarded with a prize by French Academy of Sciences. Actually, the original manuscript contained a mistake corrected by Poisson [25] in 1829.

  2. 2.

    Two Kirchhoff hypotheses contradict to each other and must be used in a proper order and at an appropriate moment of the analysis. An example of the erroneous usage of the hypotheses leading to a wrong conclusion is explained and discussed in [28] (see also [23, Mistake 4.2.5]).

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Acknowledgements

The work of Giuseppe Buttazzo is part of the project 2008K7Z249 Trasporto ottimo di massa, disuguaglianze geometriche e funzionali e applicazioni financed by the Italian Ministry of Research. The work of Sergey A. Nazarov has been supported by the Russian Foundation of Basic Research, Grant 09-01-00759.

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Appendix

Appendix

Let us assume that the points P 1,…,P J in are different, i.e., for jk, but the additional points \(P^{J+1}_{\varepsilon},\dots,P^{N}_{\varepsilon}\) are situated close to P J which is put at the coordinate origin . In other words,

$$P^j_\varepsilon=\varepsilon X^j,\quad j=J+1,\dots,N,$$

where X jX k for jk, |X j|≤1, and ε>0 is a small parameter. We denote by the set

$$\bigl\{P^1,\dots,P^J,P^{J+1}_\varepsilon,\dots,P^N_\varepsilon\bigr\}$$

and by u ε the solution to the Sobolev problem (10), (8), (2) with the support points listed above, i.e., with replacing . The notation u is kept for the solution of the Sobolev problem with .

By representation (29), (30) of u and estimate (4) for w, we recall the smoothness of the regular part of the Green function G and observe that (compare Lemma 2)

where a is a constant, and a neighborhood of the point which is free of the points P 1,…,P J−1. Hence, the Sobolev theorem on embedding H 4C 2 ensures the pointwise estimates

(60)

Notice that the quantities |a|, b, and do not exceed \(c\|f\|_{L^{2}(\varOmega )}\) with some constant c (cf. (13)).

The function vanishes at all points P 1,…,P J, and, therefore, the difference uu ε can be taken as the test function in the integral identity for u, that is,

$$ \bigl(\varDelta _x u,\varDelta _x\bigl(u-u^\varepsilon\bigr) \bigr)_\varOmega =\bigl(f,u-u^\varepsilon \bigr)_\varOmega .$$
(61)

The function may not belong to , and we multiply it by χ(|lnε|−1|ln|x||), where χ is a smooth cut-off function such that

$$0\le\chi\le1,\qquad\chi(t)=0\quad\mbox{for }t\ge1, \quad\chi (t)=1\quad\mbox{for }t\le1/2.$$

Notice that

$$ \begin{aligned}[c]&\chi\bigl(|\ln\varepsilon|^{-1}\bigl|\ln|x|\bigr|\bigr)=0\quad \mbox{for }|x|\le \varepsilon,\\&\chi\bigl(|\ln\varepsilon|^{-1}\bigl|\ln|x|\bigr|\bigr)=1\quad \mbox{for }|x|>\varepsilon^{1/2},\end{aligned}$$
(62)

and, therefore, the product χu vanishes at the points \(P^{J+1}_{\varepsilon},\dots,P^{N}_{\varepsilon}\) which live in the ε-neighborhood of . Here and further, the argument |lnε|−1|ln|x|| is assumed for χ.

We insert u εχu into the integral identity for u ε and, after a simple transformation, obtain

(63)

where [Δ x ,χ] is the commutator of the Laplace operator and the multiplication operator, i.e.,

$$ [\varDelta _x,\chi]u=2(\nabla_xu)^\top\nabla_x\chi+u\varDelta _x\chi,$$
(64)

while, in view of (62), expression (64) differs from zero in the annulus Θ(ε)={x : ε<|x|<ε 1/2} only.

Applying estimates (60) and differentiating the cut-off function, we observe that

Using a similar but simpler argument and taking the first and third estimates (60) into account yield

Thus, adding (63) to (61) and using the second fundamental inequality, we derive the relation

where c Ω >0, and hence

$$ \bigl\|u-u^\varepsilon\bigr\|_{H^2(\varOmega )}\le c|\ln\varepsilon |^{-1/2}\|f\|_{L^2(\varOmega )}.$$
(65)

The following assertion becomes trivial.

Lemma 4

Under the above restriction on the location of the points P 1,…P J and \(P^{J+1}_{\varepsilon},\dots ,P^{N}_{\varepsilon}\), there holds estimate (65) and the relationship

$$ E(u;f)\le E\bigl(u^\varepsilon;f\bigr)\le E(u;f)+c|\ln \varepsilon|^{-1/2}\|f\|_{L^2(\varOmega )}^2.$$
(66)

A formula of type (31) relates the solutions of the Sobolev problems with supports at P 1,…,P J and P 1,…,P J,P J+1,…,P N. Thus, locating the additional points P J+1,…,P N at a small distance from P 1,…,P J gives a small increment to the energy functional, due to estimate (66). This means that placing new support points near the old ones P 1,…,P J is surely not profitable.

If the point P J+1 stays near a point Q on the boundary ∂ω, i.e., |P J+1Q|=ε≪1, then by virtue of the conditions (2), estimates (60) turn into

where \(b=c\|f\|_{L^{2}(\varOmega )}\), \(d(x)=\operatorname {dist}(x,\partial\omega)\), and is a neighborhood of Q. This modification brings the small factor ε instead of |lnε|−1/2 into inequalities (65) and (66). In other words, putting a new support point near the clamped boundary is not advantageous as well.

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Buttazzo, G., Nazarov, S.A. (2012). Optimal Location of Support Points in the Kirchhoff Plate. In: Buttazzo, G., Frediani, A. (eds) Variational Analysis and Aerospace Engineering: Mathematical Challenges for Aerospace Design. Springer Optimization and Its Applications(), vol 66. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-2435-2_5

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