Otimização de Forma para Problemas de Poisson Abordagem Computacional
Resumo
Este trabalho apresenta uma implementação computacional prática para otimização de forma aplicada ao problema de Poisson, utilizando métodos numéricos baseados em diferenças finitas. Demonstra-se como calcular derivadas de forma via métodos adjuntos simplificados, com aplicação em problemas de difusão térmica. Resultados numéricos em domínios 2D mostram a eficácia da abordagem, com análise da evolução da geometria otimizada e do campo vetorial. A implementação demonstrou consistência numérica e comportamento físico esperado.Referências
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.
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.
Publicado
16/10/2025
Como Citar
GOMES, Beatriz Dionizio.
Otimização de Forma para Problemas de Poisson Abordagem Computacional. In: ESCOLA REGIONAL DE INFORMÁTICA DO ESPÍRITO SANTO (ERI-ES), 10. , 2025, Espírito Santo/ES.
Anais [...].
Porto Alegre: Sociedade Brasileira de Computação,
2025
.
p. 162-165.
DOI: https://doi.org/10.5753/eries.2025.16036.
