Optimization of clamped plates with discontinuous thickness

To the memory of J.L. Lions, who was an example and an inspiration for us
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Abstract

We consider a new variational method for a clamped plate model and related shape optimization problems. Our approach allows the study of plates with discontinuous thickness.

Introduction

Let Ω⊂RN be a smooth bounded domain. We analyze the fourth-order nonhomogeneous boundary value problemΔ(u3Δy)=finΩ,y=zon∂Ω,∂y∂n=∂z∂non∂Ω.In two space dimensions, , , is a simplified model of a clamped plate, Bendsoe [3], with variable thickness u∈L(Ω)+,0<α⩽u a.e. in Ω, with load f∈L2(Ω) and deflection y∈H2(Ω). The mapping z∈H2(Ω) specifies the boundary conditions. If z=0, one obtains the usual clamped plate conditions. It is known that under these hypotheses the weak solution y of , , exists and is unique in H2(Ω), [7, Chapter 2].

In the next section, we develop an alternative variational approach, based on the use of state constrained optimal control problems. Note that the corresponding dual control problem remains infinite dimensional, but has no constraints. By the Pontryagin maximum principle, we show the equivalence with , , . As basic references for control theory, we quote the books of Lions [6] and Barbu and Precupanu [2]. One advantage of our method is that only second-order operators are necessary and, consequently, in the numerical experiments, first-order finite elements may be used. In the work of Arnăutu et al. [1] a related method, using penalization and approximation of the constraints, is presented together with some numerical examples.

Another advantage of the method introduced in this work is that the new decomposition of , , , provided by the Pontryagin principle, has an orthogonality property. This allows to establish the continuity (in the weak topology of L(Ω)) and the differentiability of the mapping uy, defined by , , . We obtain these results in the last section, in connection with the study of the minimization of the thickness of the plate, under the condition that the deflection remains above a given limit under a prescribed load. Other problems and some bang–bang properties are also studied.

Finally, we mention that in the work of Ignat et al. [5] a similar approach was used in the case of Kirchhoff–Love arches, under Lipschitz assumptions for the parametrization. The corresponding dual control problem is unconstrained and finite dimensional and can be completely integrated, giving an explicit solution to the arch problem.

Section snippets

The control variational approach

We directly formulate some optimal control problems, then we show that they are equivalent with system , , Minh∈L2(Ω)12Ωlh2dxsubject to the state systemΔy=lh+lginΩ,y=zon∂Ωand to the state constraints∂y∂n=∂z∂non∂Ω.Here, g∈H2(Ω)∩H01(Ω) is the solution toΔg=finΩ,g=0on∂Ωand l∈L(Ω)+ is given byl=u−3a.e.inΩ.It is clear that y=z,h=u3(Δz−lg) is an admissible pair for problems , , , . By the coercivity of the cost, due to (2.7), and owing to its strict convexity, we can conclude the existence of a

Thickness optimization

We shall now discuss shape optimization problems associated to , , under the assumption:0⩽α⩽u(x)⩽βa.e.inΩ.As a first step, we analyze the continuity and the differentiability of the mapping ly, defined by , , , . Note that, although (2.7) gives a very simple relation between u and l, no continuity properties are valid in the weak topology of L(Ω), for instance. Reformulations , , , , , , have the advantage to introduce l as the main unknown and to remove the inconvenience caused by (2.7).

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