arXiv subjects

Marko Vrdoljak

Publications and source records attributed to Marko Vrdoljak.

3 recordsLinked to original sources

arXiv2026

Optimal design with uncertainties: a risk-averse approach

We study a class of stochastic optimal design problems for elliptic partial differential equations in divergence form, where the coefficients represent mixtures of two conducting materials. The objective is to minimize a generalized risk measure of the system response, incorporating uncertainty in the loading through probability distributions. We establish existence of relaxed optimal designs via homogenization theory and derive first-order stationarity conditions satisfied by the optima. Based on these conditions, we develop an optimality criteria algorithm for numerical computations. The stochastic component is treated using a truncated Karhunen--Lo\`eve expansion, allowing evaluation of the value-at-risk (VaR) and conditional value-at-risk (CVaR) contributions arising from the sensitivity analysis and featured in the algorithm. The method is illustrated for an example involving CVaR-based compliance minimization.

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arXiv2026

Hybrid Optimization Techniques for Multi-State Optimal Design Problems

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.

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arXiv2026

Classical Optimal Designs for Stationary Diffusion with Multiple Phases

We study optimal design problems for stationary diffusion involving one or more state equations and mixtures of an arbitrary number of anisotropic materials. Since such problems typically do not admit classical solutions, we adopt a homogenization-based relaxation framework. The objective considered is the maximization of a weighted sum of the energies associated with each state equation, with particular emphasis on identifying cases in which the optimal design is classical, that is, of bang-bang type, composed solely of the original pure materials. Such cases provide valuable benchmarks for numerical methods in optimal design. A simplified optimization problem expressed in terms of local material proportions is analyzed through a dual formulation in terms of fluxes. Using a saddle-point characterization, we establish a complete description of its optimal solutions. The proposed approach is applied in detail to spherically symmetric problems. In the case of a ball, the method yields explicit classical solutions of the homogenization-based relaxation problem.

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