29 o Col´ oquio Brasileiro de Matem´ atica A New Method for the Inverse Potential Problem Based on the Topological Derivative Concept e Novotny 1 , Alfredo Canelas 2 & Antoine Laurain 3 Antonio Andr´ 1 Laborat´ orio Nacional de Computa¸ c˜ ao Cient´ ıfica, LNCC/MCTI Av. Get´ ulio Vargas 333, 25651-075 Petr´ opolis - RJ, Brasil 2 Instituto de Estructuras y Transporte, Facultad de Ingenier´ ıa, Av. Julio Herrera y Reissig 565, C.P. 11.300, Montevideo, Uruguay 3 Institut f¨ ur Mathematik, Technical University Berlin, Street 36, A-8010 Berlin, Germany IMPA - 31 th July, 2013 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 1 / 34
Outline Motivation 1 Topological Derivative Concept 2 Applications of the Topological Derivative 3 Second-Order Topological Derivative 4 Inverse Potential Problem 5 Problem Formulation Topological Derivative Calculation Numerical Results Conclusions 6 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 2 / 34
Motivation ψ (Ω): shape functional Ω ∈E ψ (Ω) inf Ω: geometrical domain E : set of admissible domains IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 3 / 34
Motivation ψ (Ω): shape functional Ω ∈E ψ (Ω) inf Ω: geometrical domain E : set of admissible domains IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 3 / 34
Topological Derivative Concept Sokolowski & Zochowski, 1999 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 4 / 34
Topological Derivative Concept Sokolowski & Zochowski, 1999 ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) , where Ω ε ( � x ) = Ω \ ω ε ( � x ) and f ( ε ) → 0, when ε → 0. IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 4 / 34
Topological Derivative Concept Sokolowski & Zochowski, 1999 ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) , where Ω ε ( � x ) = Ω \ ω ε ( � x ) and f ( ε ) → 0, when ε → 0. In general, f ( ε ) = | ω ε | . It ψ (Ω ε ( � x )) − ψ (Ω) depends on the boundary T ( � x ) = lim . f ( ε ) condition on ∂ω ε . ε → 0 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 4 / 34
Applications of the Topological Derivative ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions or cracks. The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 5 / 34
Applications of the Topological Derivative ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions or cracks. The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 5 / 34
Applications of the Topological Derivative ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions or cracks. The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 5 / 34
Applications of the Topological Derivative ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions or cracks. The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. Multi-Scale Modeling: optimal design of micro-structures IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 5 / 34
Applications of the Topological Derivative ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions or cracks. The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. Multi-Scale Modeling: optimal design of micro-structures Image Processing: segmentation, restoration, denoising IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 5 / 34
Applications of the Topological Derivative ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + o ( f ( ε )) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions or cracks. The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. Multi-Scale Modeling: optimal design of micro-structures Image Processing: segmentation, restoration, denoising Mechanical Modeling: fracture and damage mechanics IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 5 / 34
Topology Optimization Energy-Based Topological Derivative in Linear Elasticity Figure : unperturbed problem defined in the domain Ω. IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 6 / 34
Topology Optimization � � ψ (Ω) := J Ω ( u ) = 1 σ ( u ) · ∇ u s − q · u , 2 Ω Γ N Find u , such that − div σ ( u ) = 0 in Ω , C ∇ u s σ ( u ) = u = u on Γ D , σ ( u ) n = on Γ N . q � � E ν C = I + 1 − 2 ν I ⊗ I , 1 + ν IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 7 / 34
Topology Optimization Topological Derivative Calculation Figure : perturbed problem defined in the domain Ω ε . IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 8 / 34
Topology Optimization Topological Asymptotic Expansion x )) = ψ (Ω) − πε 3 P σ ( u ( � x )) · ∇ u s ( � x ) + o ( ε 3 ) , ψ (Ω ε ( � � � P = 3 1 − ν 10 I − 1 − 5 ν 1 − 2 ν I ⊗ I 4 7 − 5 ν IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 9 / 34
Topology Optimization A benchmark example in 3 D T = P σ ( u ) · ∇ u s − β . Ψ Ω ( u ) := −J Ω ( u ) + β | Ω | , IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 10 / 34
Topology Optimization (a) iteration 13 (b) iteration 35 (c) iteration 52 (d) iteration 76 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 11 / 34
Topology Optimization (a) top (b) bottom (c) lateral IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 12 / 34
Topology Optimization IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 13 / 34
Second-Order Topological Derivative x ) + f 2 ( ε ) T 2 ( � ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + R ( f 2 ( ε )) , IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 14 / 34
Second-Order Topological Derivative x ) + f 2 ( ε ) T 2 ( � ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + R ( f 2 ( ε )) , where f ( ε ) → 0 and f 2 ( ε ) → 0 with ε → 0, and f 2 ( ε ) R ( f 2 ( ε )) lim f ( ε ) = 0 , lim = 0 . f 2 ( ε ) ε → 0 ε → 0 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 14 / 34
Second-Order Topological Derivative x ) + f 2 ( ε ) T 2 ( � ψ (Ω ε ( � x )) = ψ (Ω) + f ( ε ) T ( � x ) + R ( f 2 ( ε )) , where f ( ε ) → 0 and f 2 ( ε ) → 0 with ε → 0, and f 2 ( ε ) R ( f 2 ( ε )) lim f ( ε ) = 0 , lim = 0 . f 2 ( ε ) ε → 0 ε → 0 (first order) topological derivative ψ (Ω ε ( � x )) − ψ (Ω) T ( � x ) := lim . f ( ε ) ε → 0 IMPA - 31 th July, 2013 A.A. Novotny et al. Inverse Gravimetry Problem 14 / 34
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