An Equilibrated Fluxes Approach to the Certified Descent Algorithm for Shape Optimization Using Conforming Finite Element and Discontinuous Galerkin Discretizations

Abstract

The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed.

Appendix A: Weak Imposition of the Essential Boundary Conditions

We present a formal derivation of the variational formulation of an elliptic problem featuring weakly-imposed Dirichlet boundary conditions. The idea of this approach dates back to the classical paper by Nitsche [50] and has been extensively studied in recent years by several authors (cf. e.g. [23] and references therein). We recall that the solution of a boundary value problem may be interpreted as an optimization problem. Let us introduce the Lagrangian functional associated with the state problem (21) featuring Dirichlet boundary conditions:

$$\begin{aligned} \Lambda (w,\lambda ) = \frac{1}{2} \int _{\mathcal {D}}{\Big (k_\varOmega | \nabla w |^2 + |w|^2 \Big ) d\mathbf {x}} - \int _{\partial {\mathcal {D}}}{\lambda (w-U_D) ds} . \end{aligned}$$

(49)

The solution of the aforementioned boundary value problem is equivalent to the following min-max problem:

$$\begin{aligned} \min _{w \in H^1({\mathcal {D}})} \max _{\lambda \in H^{-\frac{1}{2}}({\mathcal {D}})} \Lambda (w,\lambda ) . \end{aligned}$$

The first-order optimality conditions for (49) read as

$$\begin{aligned} \left\{ \begin{aligned}&\int _{\mathcal {D}}{\Big ( k_\varOmega \nabla w \cdot \nabla \delta w + w \delta w \Big ) d\mathbf {x}} - \int _{\partial {\mathcal {D}}}{\lambda \delta w \ ds} = 0 ,\\&\int _{\partial {\mathcal {D}}}{(w-U_D) \delta \lambda \ ds} = 0 . \end{aligned} \right. \end{aligned}$$

From the second condition, we retrieve the Dirichlet boundary condition on $\partial {\mathcal {D}}$. Integrating by parts the first condition and owing to the strong form of the problem, we obtain

$$\begin{aligned} \int _{\partial {\mathcal {D}}}{(k_\varOmega \nabla w \cdot \mathbf {n} - \lambda ) \delta w \ ds} = 0 . \end{aligned}$$

By plugging $\lambda = k_\varOmega \nabla w \cdot \mathbf {n} \ \text {on} \ \partial {\mathcal {D}}$ into (49) we may now derive the following dual variational problem by seeking $w \in H^1({\mathcal {D}})$ such that $\forall \delta w \in H^1({\mathcal {D}})$

$$\begin{aligned} \begin{aligned} \int _{\mathcal {D}}{\Big ( k_\varOmega \nabla w \cdot \nabla \delta w + w \delta w \Big ) d\mathbf {x}}&- \int _{\partial {\mathcal {D}}}{\Big ( k_\varOmega \nabla w \cdot \mathbf {n} \delta w + w k_\varOmega \nabla \delta w \cdot \mathbf {n} \Big ) ds} \\&\quad = - \int _{\partial {\mathcal {D}}}{U_D k_\varOmega \nabla \delta w \cdot \mathbf {n} \ ds} . \end{aligned} \end{aligned}$$

(50)

We remark that the bilinear form on the left-hand side of (50) is not coercive thus we cannot establish the well-posedness of this problem. To bypass this issue, we consider the following augmented Lagrangian functional and we construct the corresponding dual variational formulation for the problem under analysis:

$$\begin{aligned} \Upsilon (w,\lambda ,\gamma ) = \Lambda (w,\lambda ) + \frac{1}{2} \int _{\partial {\mathcal {D}}}{\gamma (w-U_D)^2 ds} . \end{aligned}$$

Following the same procedure used to derive (50), we seek $w \in H^1({\mathcal {D}})$ such that $\forall \delta w \in H^1({\mathcal {D}})$

$$\begin{aligned} \begin{aligned} \int _{\mathcal {D}}{\Big ( k_\varOmega \nabla w \cdot \nabla \delta w + w \delta w \Big ) d\mathbf {x}} -&\int _{\partial {\mathcal {D}}}{\Big ( k_\varOmega \nabla w \cdot \mathbf {n} \delta w + w k_\varOmega \nabla \delta w \cdot \mathbf {n} \Big ) ds} + \int _{\partial {\mathcal {D}}}{\gamma w \delta w \ ds} \\&\quad = \int _{\partial {\mathcal {D}}}{U_D \Big (\gamma \delta w - k_\varOmega \nabla \delta w \cdot \mathbf {n} \Big ) ds} . \end{aligned} \end{aligned}$$

(51)

It is straightforward to observe that the bilinear form on the left-hand side of (51) is coercive owing a sufficiently large value of $\gamma$ is chosen.