Optimal Location of Support Points in the Kirchhoff Plate

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.

Appendix

Let us assume that the points P ¹,…,P ^J in are different, i.e., for j≠k, 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 ^j≠X ^k for j≠k, |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 ⁴⊂C 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 u−u ^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 ^ε−χu 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+1−Q|=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.