Multiscale Modeling and Simulation· 2026Q1
Approximation in an Optimal Design Problem Governed by the Heat Equation
- 0citations
- Q1SCImago
- 2026year
Short summary
This paper introduces a novel approximation method for a two-material optimal design problem involving heat diffusion, demonstrating the existence of optimal designs for a perturbed problem and constructing them using a level set method.
AI-generated from the title and abstract; the full text is not read.
Key points
- Develops an approximation method for a two-material optimal design problem governed by the heat equation.
- Introduces a perimeter constraint and Dirichlet energy perturbation to ensure the existence of optimal designs.
- Utilizes a level set function and nonlinear diffusion-based method to construct optimal two-material domains.
- Demonstrates convergence of the approximated minimum value to the elliptic case, clarifying long-time behavior.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. This paper studies a two-material optimal design problem for the time-averaged duality pairing between a (possibly time-dependent) heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In general, such optimal designs are not guaranteed to exist, and geometric constraints such as the perimeter are required. As an approximation of the problem with an additional perimeter constraint, a material representation based on a level set function, together with a perturbation of the Dirichlet energy, is employed. It is then shown that optimal level set functions exist for the perturbation problem, and the corresponding minimum value converges to that of the elliptic case, thereby elucidating the long-time behavior. Furthermore, two-material domains satisfying this property are also constructed via the nonlinear diffusion-based level set method. In particular, the asymptotic behavior with respect to the perturbation parameter is clarified, and the validity of the approximation is established.
The authors' abstract, as published at the source. Multiscale Modeling and Simulation, 2026 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Civil and Structural Engineering
Civil and Structural EngineeringEngineering