This paper addresses optimal design problems governed by multi-state stationary diffusion equations, aiming at the simultaneous optimization of the domain shape and the distribution of two isotropic materials in prescribed proportions. Existence of generalized solutions is established via a hybrid approach combining homogenization-based relaxation in the interior with suitable restrictions on admissible domains. Based on this framework, we propose a numerical method that integrates homogenization and shape optimization. The domain boundary is evolved using a level set method driven by the shape derivative, while the interior material distribution is updated via an optimality criteria algorithm. The approach is demonstrated on a representative example.
Shape optimization refers to the determination of an optimal domain that minimizes a prescribed objective functional, usually defined by the solution of a partial differential equation (PDE) known as the state equation. As such, shape optimization can be considered as a branch of distributed control theory, where the domain acts as a control variable.
A fundamental difficulty in this field lies in the nonexistence of classical optimal shapes. Minimizing sequences of domains tend to develop oscillations, holes, or degeneracies, so that no Lipschitz domain attains the infimum of the cost functional. This nonexistence has profound implications: numerical algorithms may fail to converge or depend strongly on the initial guess. Two principal strategies have emerged to address this: (i) restricting the class of admissible domains to ensure compactness and existence, or (ii) relaxing the problem by enlarging the admissible set where existence can be recovered.
The homogenization theory plays a central role in obtaining this relaxation [32,26], by introducing a notion of generalized designs that ensures the optimization problem remains well-posed and physically meaningful. This, in turn, leads to more stable and efficient numerical computations of relaxed designs [8,2,9].
On the other hand, imposing additional (uniform) regularity assumptions on the class of admissible domains can ensure the existence of solutions for certain shape optimization problems. Examples include the uniform cone condition [16,35], a bounded number of connected components of the complement in two-dimensional cases [39], perimeter constraints [6], uniform continuity conditions [29], the uniform cusp property [18], and uniform boundedness of the density perimeter [13]. Further theoretical insights are provided in [38,23,12,34].
This theoretical development was accompanied by a numerical treatment of the problem using shape calculus, building on the pioneering work of Hadamard [33,35,38,5]. Numerical methods in shape optimization differ mainly in how they represent geometry. Some approaches, such as the level set and phase-field methods, describe shapes implicitly using a scalar field that marks the region occupied by the material. Other approaches represent shapes explicitly through a computational mesh or CAD model, allowing for accurate analysis but requiring frequent remeshing or geometric updates as the shape evolves.
In this paper, we consider the problem of determining not only the optimal distribution of two phases but also the optimal shape of the region they occupy. The problem has been addressed in the literature using alternative approaches. Extensions of the SIMP scheme to multiple phases were proposed in [37,19], considering three-phase mixtures (two materials plus void). In [45,4], the level set method was applied to multi-phase shape optimization, also allowing one of the phases to represent void.
Motivated by the robustness of the homogenization method, our approach combines homogenization-based relaxation within the domain with shape optimization based on the shape derivative of the overall region. For this reason, we refer to it as a hybrid method. Moreover, the optimality criteria method is known to produce accurate approximations within only a few iterations, which facilitates the application of the shape derivative method to the external boundary.
The resulting optimal design can, through penalization techniques, be reduced to a classical design with pure phases, or it can serve as an informed initial guess for the subsequent application of the shape derivative method to the distribution of the two phases within the “optimal” domain. In the numerical example presented in the final section, the penalization step was not implemented, since the optimal design produced by the proposed method turned out to be classical. This is due to the radial symmetry of the problem: the resulting design is known to be optimal for the fixed-domain problem posed on the resulting domain [27].
We now proceed to formulate the specific two-material optimal design problem under consideration.
Let D ⊂ R d , d ≥ 2, be a bounded open fixed set and Ω ⊂ D a Lipschitz domain. The open set Ω is made of two materials with isotropic conductivities 0 < α < β, so its overall conductivity is given by (1)
where χ ∈ L ∞ (Ω; {0, 1}) represents the characteristic function of the less-conductive phase.
For different right-hand sides f 1 , . . . f m ∈ L 2 (D) we denote the corresponding temperatures u := (u 1 , . . . , u m ) ∈ H 1 (Ω; R m ) of the body, as the unique solutions of the following boundary value problems
(2)
Here, the boundary ∂Ω consists of three parts: ∂Ω = Γ 0 ∪Γ 1 ∪Γ 2 , where Γ 0 is relatively open and fixed, and h i ∈ H 1 2 (Γ 0 ) are prescribed. The sets Γ 1 and Γ 2 represent subsets of D on which the Dirichlet and Neumann boundary conditions, respectively, can be applied. Note that simply requiring h i ∈ H 1 2 (Γ 0 ) is insufficie
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