An optimal shape design problem for plates

An optimal shape design problem for plates


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 PDE-constrained optimal control problem with additional control and state constraints. The state constraints are treated with a Moreau-Yosida 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 lowest-order Raviart-Thomas 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, Moreau-Yosida 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 lowest-order Raviart-Thomas 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 Moreau-Yosida regularization (cf. [Hintermüller and Kunisch 2006]). The numerical solution to the control problem is computed via a path-following 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 Raviart-Thomas 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 Newton-type path-following 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


admits for every  a unique solution  satisfying


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


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


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:


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


are satisfied.

The variational inequality (2.9) can be written as a projection formula


where  is the orthogonal projection onto the real interval . For later use, we note that (2.10) is equivalent to


It follows from (2.12) that the support of the measure  is concentrated in the state-active set . In particular , see [Casas 1986]. Furthermore, from Theorems 4 and 5 in [Casas 1986] we deduce  for all  and .

3 Moreau-Yosida regularized problem

To relax the state constraints, we introduce the Moreau-Yosida 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


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


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


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


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 lowest-order Raviart-Thomas element. To begin with we recall the mixed formulation of the Dirichlet-problem for the Poisson equation, i.e., for , and  there holds


where . For a given right-hand 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 quasi-uniform, 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])


as well as, if , the pointwise estimate (see [Gastaldi and Nochetto 1989, Cor. 5.5])


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 infinite-dimensional optimization problem similar to  since the control  is not discretized. It admits a unique solution , which is characterized by the optimality system


Condition (4.6) is equivalent to the projection formula


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


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 .

We test (4.6) with  and with the help of the adjoint equation (4.5) and the definition of  we get

with a constant  independent of  and . We then have

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 


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


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 min-function

Now let  and similar to the proof of [Hintermüller and Hinze 2009, Theorem 3.5] one has for