Duality-Based Algorithm and Numerical Analysis for Optimal Insulation Problems on Non-Smooth Domains

TL;DR

The paper tackles the problem of optimally distributing a fixed amount of insulating material on a boundary portion $Γ_I$ of a polyhedral Lipschitz domain $Ω$, formulated as a convex non-local minimization of the primal energy $I$ and its Fenchel dual $D$. It develops a rigorous duality framework, derives both a posteriori and a priori error identities, and demonstrates convergence under minimal regularity. The discrete analysis pairs a Crouzeix–Raviart primal discretization with a Raviart–Thomas dual discretization and includes a reconstruction of primal solutions from dual ones via an inverse generalized Marini formula, solved efficiently by a primal–dual active-set/semismooth Newton method. Numerical experiments validate the theory, reveal optimal error decay rates, and illustrate adaptive refinement and a practical house-insulation example, highlighting the method’s robustness for non-smooth, non-local optimal insulation problems. Overall, the work provides a complete, rigorously justified computational framework for accurate and efficient simulation of optimal boundary insulation in complex geometries.

Abstract

This article develops a numerical approximation of a convex non-local and non-smooth minimization problem. The physical problem involves determining the optimal distribution, given by $h\colon Γ_I\to [0,+\infty)$, of a given amount $m\in \mathbb{N}$ of insulating material attached to a boundary part $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d \in \mathbb{N}$, subject to conductive heat transfer. To tackle the non-local and non-smooth character of the problem, the article introduces a (Fenchel) duality framework: (a) At the continuous level, using (Fenchel) duality relations, we derive an a posteriori error identity that can handle arbitrary admissible approximations of the primal and dual formulations of the convex non-local and non-smooth minimization problem; (b) At the discrete level, using discrete (Fenchel) duality relations, we derive an a priori error identity that applies to a Crouzeix--Raviart discretization of the primal formulation and a Raviart--Thomas discretization of the dual formulation. The proposed framework leads to error decay rates that are optimal with respect to the specific regularity of a minimizer. In addition, we prove convergence of the numerical approximation under minimal regularity assumptions. Since the discrete dual formulation can be written as a quadratic program, it is solved using a primal-dual active set strategy interpreted as semismooth Newton method. A solution of the discrete primal formulation is reconstructed from the solution of the discrete dual formulation by means of an inverse generalized Marini formula. This is the first such formula for this class of convex non-local and non-smooth minimization problems.

Figures (5)

• Figure 1: A thermally conducting body with fully (left) and partly (right) insulated boundary.
• Figure 2: left: logarithmic plots of $\rho_{\textup{tot},h_k}^2(\Pi_{h_k}^{cr}u,\Pi_{h_k}^{rt}z)=\eta_{\textup{gap},h_k}^2(\Pi_{h_k}^{cr}u,\Pi_{h_k}^{rt}z)$, $k=0,\ldots,5$. We report the expected optimal error decay of order $\mathcal{O}(h_k^2)= \mathcal{O}(N_k)$, $k=1,\ldots, 5$; right: logarithmic plots of $I(\overline{u}_{h_k}^{cr})$, $k=0,\ldots,5$, and $D(z_{h_k}^{rt})$, $k=0,\ldots,5$, where $\overline{u}_{h_k}^{cr}\coloneqq \Pi_{h_k}^{av} u_{h_k}^{cr}\in \mathcal{S}^{1,cr}(\mathcal{T}_{h_k})\cap H^1(\Omega)$, $k=0,\ldots,5$. We report convergence to the true primal/dual energy functional value \ref{['expl1:true_energy']}.
• Figure 3: top row: Setup \ref{['setup1']} (pure insulation); bottom row: Setup \ref{['setup2']} (mixed boundary conditions); left column: logarithmic plots of $\rho_{\textup{tot}}^2(\overline{u}_{h_k}^{cr},z_{h_k}^{rt})=\eta_{\textup{gap}}^2(\overline{u}_{h_k}^{cr},z_{h_k}^{rt})$; right column: logarithmic plots of $I(\overline{u}_{h_k}^{cr})$ and $D(z_{h_k}^{rt})$, where $\overline{u}_{h_k}^{cr}\coloneqq \Pi_{h_k}^{av} u_{h_k}^{cr}\in \mathcal{S}^{1,cr}(\mathcal{T}_{h_k})\cap H^1(\Omega)$; each for $k=0,\ldots,30$, when using adaptive mesh-refinement, and for $k=0,\ldots,6$, when using uniform mesh-refinement.
• Figure 4: The discrete primal solution $u_{h_k}^{cr}\in \mathcal{S}^{1,cr}(\mathcal{T}_{h_k})$ and the adaptively refined triangulation $\mathcal{T}_{h_k}$ pictured at refinement level $k=5$(top row), $k=15$(middle row), and $k=25$(bottom row). The left column corresponds to the test case with purely insulated boundary (cf. Setup \ref{['setup1']}), whereas the right column corresponds to the test case with mixed boundary conditions (cf. Setup \ref{['setup2']}).
• Figure 5: left: surface temperature field $\pi_h u_h^{cr}\in \mathcal{L}^0(\mathcal{S}_h^{\partial})$; right: distribution of the insulating material $\widetilde{h}_{u_h^{cr}} \in \mathcal{L}^0(\mathcal{S}_h^{I})$ (cf. \ref{['def:discrete_distribution']}); each for a uniformly heated home (i.e., $f=1$) with insulating mass $m = \frac{1}{4}|\Gamma_I|$ and uniform outward heat flux (i.e., $g=\frac{1}{5}$) at the windows, doors, and floors. The triangulation $\mathcal{T}_h$ consists of $150,370$ tetrahedral elements and the semi-smooth Newton method (cf. Algorithm \ref{['alg:semismooth_Newton']}) terminates after $8$ iterations (at the exact discrete solution).

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1: strong duality and convex optimality relations
• Remark 3.2: equivalent condition to \ref{['eq:optimality.2']}
• proof : Proof (of Theorem \ref{['thm:duality']}).
• Lemma 4.1: representation of primal-dual gap estimator
• Remark 4.2: interpretation of the components of the primal-dual gap estimator
• proof : Proof (of Lemma \ref{['lem:primal_dual_gap_estimator']}).