Shape Optimization for Poisson Problems: A Computational Approach

• Beatriz Dionizio Gomes USP

DOI: https://doi.org/10.5753/eries.2025.16036

Abstract

This work presents a practical computational implementation for shape optimization applied to the Poisson problem using numerical methods based on finite differences. It demonstrates how to calculate shape derivatives through simplified adjoint methods, with application to thermal diffusion problems. Numerical results in 2D domains show the effectiveness of the approach, with an analysis of the evolution of the optimized geometry and the vector field. The implementation has shown numerical consistency and expected physical behavior.

References

Allaire, G. (2007). Shape Optimization by the Homogenization Method. Springer.

Ciarlet, P. G. (1978). The finite element method for elliptic problems. Studies in Mathematics and its Applications, 4.

Evans, L. C. (2010). Partial Differential Equations. American Mathematical Society, 2 edition.

Laporte, E. and Le Tallec, P. (2004). Numerical methods in sensitivity analysis and shape optimization. Modeling and Simulation in Science, Engineering and Technology.

Larson, M. G. and Bengzon, F. (2013). The Finite Element Method: Theory, Implementation, and Applications. Springer.

Nocedal, J. and Wright, S. J. (2006). Numerical Optimization. Springer, 2 edition.

Poon, C. and Kerrigan, E. C. (2006). A new method for computing the adjoint for optimal control of linear differential-algebraic equations. IEEE Transactions on Automatic Control, 51(6):1026–1031.

Quarteroni, A. (2009). Numerical Models for Differential Problems, volume 2. Springer.

Schmidt, S., Ilic, C., Schulz, V., and Gauger, N. R. (2010). Three-dimensional large-scale aerodynamic shape optimization based on shape calculus. AIAA Journal, 48(6):1214–1227.

Sokolowski, J. and Zolésio, J.-P. (1992). Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag.

Strikwerda, J. C. (2004). Finite Difference Schemes and Partial Differential Equations. SIAM, 2 edition.

Zeidler, E. (1990). Applied functional analysis: Applications to mathematical physics. Applied Mathematical Sciences, 108.