An optimal shape design problem for plates
Abstract
We consider an optimal shape design problem for the plate equation, where the variable thickness of the plate is the design function. This problem can be formulated as a control in the coefficient PDEconstrained optimal control problem with additional control and state constraints. The state constraints are treated with a MoreauYosida regularization of a dual problem. Variational discretization is employed for discrete approximation of the optimal control problem. For discretization of the state in the mixed formulation we compare the standard continuous piecewise linear ansatz with a piecewise constant one based on the lowestorder RaviartThomas mixed finite element. We derive bounds for the discretization and regularization errors and also address the coupling of the regularization parameter and finite element grid size. The numerical solution of the optimal control problem is realized with a semismooth Newton algorithm. Numerical examples show the performance of the method.
 Key words.

elliptic optimal control problem, optimal shape design, pointwise state constraints, MoreauYosida regularization, error estimates.
 Mathematics Subject Classification (2010).

49J20, 65N12, 65N30.
1 Introduction
This work is devoted to the numerical analysis and solution of an optimal control problem for a plate with variable thickness. The state equation
can be used to model the relation between the (small) deflection and the thickness of a (thin) plate under the force of a transverse load . The domain represents the unloaded plate’s midplane and we assume its boundary to be simply supported, i.e.,
Invoking a pointwise lower bound on the state and pointwise almost everywhere box constraints on the control , and minimizing the volume of the plate given by the cost functional
lead to a control in coefficients problem, which can also be viewed as an optimal shape design problem. In [Sprekels and Tiba 1999] this optimization problem is analyzed using a transformation to a dual problem.
Building upon this duality, it is our aim to solve the control problem with a finite element approximation that is suitable with regard to the necessary optimality conditions. To this end we compare variational discretization of the control problem (cf. [Hinze 2005]) based on either the lowestorder RaviartThomas mixed finite element or piecewise linear continuous finite elements for the discretization of the Poisson equation. The pointwise state constraints, which are responsible for the low regularity of the Lagrange multiplier, are treated with the help of MoreauYosida regularization (cf. [Hintermüller and Kunisch 2006]). The numerical solution to the control problem is computed via a pathfollowing algorithm that simultaneously refines the mesh and follows the homotopy generated by the regularization parameter. The resulting subproblems are solved by a semismooth Newton method.
To the best of the authors’ knowledge this is the first contribution to numerical analysis of a “control in the coefficients” problem for biharmonic equations including state constraints. The mathematical techniques applied in the numerical analysis of the regularized control problem are related to the relaxation of state constraints as proposed in [Hintermüller and Hinze 2009] and to [Deckelnick, Günther and Hinze 2009], where the RaviartThomas mixed finite element was employed in the context of gradient constraints.
The present work is organized as follows. In Section 2 the optimal control problem and its dual problem are introduced. The regularization of the dual problem is investigated in Section 3. Section 4 deals with the discretization of the regularized problems and with the related error bounds. Finally, in Section 5 the original control problem is solved with a Newtontype pathfollowing method. Numerical examples are presented which validate our analytical findings.
2 The optimization problems
Let , , be a bounded domain with smooth boundary . The Dirichlet problem for the Poisson equation
(2.1) 
admits for every a unique solution satisfying
(2.2) 
In order to define the control problems considered in this paper we introduce the admissible sets for controls and states according to
where and are positive real constants. For a given we consider the following optimal control problems (cf. [Sprekels and Tiba 1999], problems and ):
() 
subject to
(2.3)  
(2.4)  
(2.5)  
denoted as the primal problem , representing the physical control problem motivated in the introduction, and secondly, with the datum induced by
the dual problem
() 
subject to
(2.6)  
(2.7)  
which is analytically and numerically advantageous, in that it is convex and contains two coupled second order equations instead of the fourth order equation (2.3). It will therefore serve as a basis for our analysis in the remaining sections. For every the system (2.3)–(2.5) has a unique weak solution with . Due to we have , and there is a strong solution to the dual system (2.6)–(2.7).
We impose the following Slater condition:
(2.8) 
and recall from [Sprekels and Tiba 1999] that for each pair admissible for , the pair is admissible for with the same cost and vice versa. Moreover, there exists a unique solution of problem , and is the unique solution of . We denote the associated state by .
Next we derive optimality conditions characterizing . For this purpose we introduce , the space of regular Borel measures, which equipped with norm
is the dual space of . Arguments similar to those used in [Casas 1986, Theorem 2] yield the
Theorem 2.1.
Let the assumption (2.8) hold. A control with associated state is optimal for the dual problem if and only if there exist and , such that
(2.9)  
(2.10)  
are satisfied.
The variational inequality (2.9) can be written as a projection formula
(2.11) 
where is the orthogonal projection onto the real interval . For later use, we note that (2.10) is equivalent to
(2.12) 
It follows from (2.12) that the support of the measure is concentrated in the stateactive set . In particular , see [Casas 1986]. Furthermore, from Theorems 4 and 5 in [Casas 1986] we deduce for all and .
3 MoreauYosida regularized problem
To relax the state constraints, we introduce the MoreauYosida regularization of problem for a parameter . It reads
() 
subject to
Here we set . It admits a unique solution of with associated state denoted by . Furthermore, there exists a unique which together with and satisfies
(3.1)  
(3.2)  
The term can be regarded as a regularized version of the Lagrange multiplier in Theorem 2.1. There holds and
This inequality for can be argued as follows:
Now we want to show convergence of the parameterized subproblems towards the unregularized problem . We begin with uniform boundedness of primal and dual variables with respect to . While the former is obtained immediately through the control constraints and (2.2), the latter can be shown as follows.
Lemma 3.1.
Let and be the solution to the problem with associated state and multipliers according to the optimality conditions. Then there exists a constant independent of such that
Proof.
To uniformly bound in we test (3.2) with the Slater element . With the help of the adjoint equation (3.1) we get
with a constant independent of . The desired estimate now follows from
With this bound we want to prove the one on the dual state . Let solve
Using (3.1), the embedding of into and the continuous dependence of on , we have that
hence . ∎
Next we need to estimate the violation of the state constraint measured in the maximum norm with the help of techniques developed in [Hintermüller, Schiela and Wollner 2014].
Lemma 3.2.
Let be the solution of problem , the corresponding state. Then for we have for every a constant , independent of , such that
Proof.
We show that Corollary 2.6 of [Hintermüller, Schiela and Wollner 2014] is applicable. To begin with, we note that is uniformly bounded in for every , since is uniformly bounded in . This implies that is uniformly bounded in and that is uniformly bounded w.r.t. , which by Sobolev imbedding theorems holds also in , for and all . Thus Corollary 2.6 in [Hintermüller, Schiela and Wollner 2014] is applicable and delivers our desired bound. ∎
We are now in position to estimate the regularization error.
Theorem 3.3.
Let and be the solutions to and , resp., with corresponding states and . Then for every there exists a constant , independent of , for which it holds that
Proof.
Using as test function in (3.2) and as test function in (2.9) we obtain
Since we have . Moreover . The third addend is treated with the complementarity condition (2.12) for the multiplier :
With the help of Lemma 3.2 we arrive at
with independent constants , . The continuous dependence of the states on the controls and the continuous embedding allow to extend this estimate to and . ∎
4 Finite element discretization
4.1 Mixed piecewise constant versus piecewise linear approximation
In this section we turn to the variational discretization of the regularized control problems, taking into account the structure imposed by the optimality systems, especially the projection formula (2.11) and its discrete counterparts (4.7) and (4.15). The function applied to the product of two state variables is evaluated with little effort if those variables are approximated piecewise constant and yields an implicit piecewise constant discretization of the optimal control. Further, the finite element system of the semismooth Newton method in Section 5, in particular the parts (A.2) and (A.3) involving the projection formula and its generalized derivative, in this situation is easily assembled exactly.
Approximating the states with piecewise linear, continuous finite elements delivers a more involved variational discretization of the controls, since the projection formula then no longer implies a piecewise polynomial discretization of the control variable, but rather the negative power of the pointwise projection of a piecewise quadratic function. Moreover, the approximate computation of the terms (A.4) and (A.5) introduces an additional error. On the other hand, a piecewise linear ansatz delivers the higher approximation order two for the states, as opposed to an order of at most one for a piecewise constant ansatz. This is supported by the convergence rates w.r.t. the grid size in the error plots in Section 5, and also allows for a better resolution of the control active sets.
In the remainder of this section we give estimates for the overall error in both discrete approaches.
4.2 Variational discretization of with mixed finite elements
Following the above remarks, we use a mixed finite element method based on the lowestorder RaviartThomas element. To begin with we recall the mixed formulation of the Dirichletproblem for the Poisson equation, i.e., for , and there holds
(4.1) 
where . For a given righthand side we represent the solution of this mixed problem by . In particular, with this means .
Let a triangulation of be given, where and be the union of the elements of , with boundary elements allowed to have one curved face. We additionally assume that the triangulation is quasiuniform, i.e., there exists a constant , independent of , such that each is contained in a ball of radius and contains a ball of radius . To define the discrete version of (4.1) let us introduce the spaces
For a given we set to be the solution of
The resulting error satisfies (see [Brezzi and Fortin 1991])
(4.2) 
as well as, if , the pointwise estimate (see [Gastaldi and Nochetto 1989, Cor. 5.5])
(4.3) 
The load induces a discrete datum via .
Remark 4.1.
For our error analysis we require that is bounded uniformly in in both of the considered discretization approaches.
This is satisfied, e.g., if , since then by (4.3) and (4.11), resp. Note that this regularity restriction is not essentially necessary, cf. [Deckelnick and Hinze 2014, Lemma 3.4], which holds analogously for the scalar states in both discrete approaches and allows for , , with smaller powers of in (4.3) and (4.11).
Let . The variational discretization of the regularized control problems reads
() 
subject to
We note that is still an infinitedimensional optimization problem similar to since the control is not discretized. It admits a unique solution , which is characterized by the optimality system
(4.4)  
(4.5)  
(4.6)  
Condition (4.6) is equivalent to the projection formula
(4.7) 
We denote and similarly to the proof of Lemma 3.1 one obtains boundedness uniformly in and :
Lemma 4.2.
Let and be the solution to the problem with state . Then there exists an and a constant independent of and of such that
Proof.
Let be the adjoint state and the Slater element with corresponding discrete state , which is a discrete Slater state for all with some small enough. In fact, with and from (2.8) we obtain from that for .
Aiming for an estimate of the overall error induced by regularization and discretization we apply the approach from [Hintermüller and Hinze 2009] to our problem setting and derive a similar asymptotic ,dependent bound on in , which further allows to couple the regularization parameter efficiently to the grid size parameter. To this end we need to estimate the discretization error for the regularized problems.
Theorem 4.3.
Let and be the solutions of and , resp., with corresponding states and . Then there is an and a  and independent constant , such that for all and all
Proof.
We define the auxiliary variable and test the problems’ variational inequalities with the respective solutions to obtain
In view of Lemma 3.1 we have for sufficiently small that
For the second addend we find
To estimate the finite element error we set (cf. the proof of Lemma 8.3.11 on p. 228 in [Brenner and Scott 2008]) and consider
(4.8) 
Defining we have (cf. [Casas 1985], proof of Theorem 3 with piecewise linear finite elements)
and furthermore, setting and using the definitions of and ,
Inserting these into (4.8), we conclude with that
Finally, we set and rewrite the third addend using the definitions of , , and together with the monotonicity of the minfunction
Now let and similar to the proof of [Hintermüller and Hinze 2009, Theorem 3.5] one has for