OPTIMAL CONTROL METHODS IN SHAPE OPTIMIZATION Dan TIBA INSTITUTE - PowerPoint PPT Presentation

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension OPTIMAL CONTROL METHODS IN SHAPE OPTIMIZATION Dan TIBA INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY ACADEMY OF ROMANIAN SCIENTISTS dan.tiba@imar.ro

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension CONTENTS Introduction 1 Parametrization in Arbitrary Dimension 2 Examples in dimension three 3 Generalized solutions 4 Examples critical case: dimension three 5 Penalization 6 Plate with holes 7 Applications 8

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Introduction A typical example of shape optimization problems has the form: � j ( x , y Ω ( x ) , ∇ y Ω ( x )) dx , Min Ω ∈O Λ − ∆ y Ω = f in Ω , y Ω = 0 on ∂ Ω with other supplementary constraints (on y , Ω , etc.), if necessary. Here, Ω ⊂ D is an (unknown) domain, D is some given bounded Lipschitzian domain, f ∈ L 2 ( D ) , j ( ., ., . ) is a Caratheodory mapping and Λ is either Ω or some fixed subdomain E ⊂ D .

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Introduction Let me also mention that many geometric optimization problems arising in mechanics (for plates, beams, arches, curved rods or shells), are expressed, as well, as optimal control problems by the coefficients, due to the special form of their models. This point is not discussed here. The presentation will discuss in detail two cases: optimization of a plate with holes and a penalization approach to a general shape optimization problem. Boundary observation problems will also be presented if time allows. An essential ingredient in these developments is the new implicit parametrization method that allows an advantageous description of implicitly defined manifolds via iterated Hamiltonian systems. We start with a short description in this respect.

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension We impose the classical independence assumption on some family of C 1 mappings F 1 , F 2 , . . . , F l , in some point x 0 ∈ Ω ⊂ R d , l ≤ d − 1. To fix ideas, we assume F j ( x 0 ) = 0 and D ( F 1 , F 2 , . . . , F l ) in x 0 = ( x 0 1 , x 0 2 , . . . , x 0 D ( x 1 , x 2 , . . . , x l ) � = 0 d ) . This remain valid in a neighborhood V and we introduce the undetermined linear algebraic system with unknowns v ( x ) ∈ R d , x ∈ V : v(x) ·∇ F j ( x ) = 0 , j = 1 , l .

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension We denote by A ( x ) the corresponding l × l nonsingular matrix and the vectors ∇ F 1 ( x ) , . . . , ∇ F l ( x ) are independent, for x ∈ V . We shall use d − l solutions obtained by fixing the last d − l components of the vector v ( x ) ∈ R d to be the rows of the identity matrix in R d − l , multiplied by det A ( x ) . Then, the first l components are uniquely determined, by inverting A ( x ) . In this way, the obtained d − l solutions, denoted by v 1 ( x ) , . . . , v d − l ( x ) ∈ R d are linear independent, for any x ∈ V . Moreover, these vector fields are continuous in V .

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension We introduce now d − l nonlinear systems of first order partial differential equations associated to the vector fields ( v j ( x )) j = 1 , d − l , x ∈ V ⊂ Ω . We use here the order v 1 , v 2 , . . . , v d − l to fix ideas. Moreover, we denote the sequence of independent variables by t 1 , t 2 , . . . , t d − l . These systems have an iterated character in the sense that the solution of one of them is used as initial condition in the next one. Consequently, the independent variables in the "previous" systems enter as parameters in the next system just via the initial conditions.

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension ∂ y 1 ( t 1 ) = v 1 ( y 1 ( t 1 )) , t 1 ∈ I 1 ⊂ R , ∂ t 1 y 1 ( 0 ) = x 0 ; ∂ y 2 ( t 1 , t 2 ) = v 2 ( y 2 ( t 1 , t 2 )) , t 2 ∈ I 2 ( t 1 ) ⊂ R , ∂ t 2 y 2 ( t 1 , 0 ) = y 1 ( t 1 ); . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂ y d − l ( t 1 , t 2 , . . . , t d − l ) = v d − l ( y d − l ( t 1 , t 2 , . . . , t d − l )) , ∂ t d − l t d − l ∈ I d − l ( t 1 , . . . , t d − l − 1 ) , y d − l ( t 1 , . . . , t d − l − 1 , 0 ) = y d − l − 1 ( t 1 , t 2 , . . . , t d − l − 1 ) .

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension Here, the notations I 1 , I 2 ( t 1 ) , . . . . , I d − l ( t 1 , . . . , t d − l − 1 ) are d − l real intervals, containing 0 in interior and depending, in principle, on the "previous" parameters. Due to their simple structure, we stress that each equation may be interpreted as an ordinary differential system with parameters, although partial differential notations are used. The existence of the solutions y 1 , y 2 , . . . , y d − l follows by the Peano theorem due to the continuity of the vector fields ( v j ) j = 1 , d − l on V .

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension Proposition For every k = 1 , l, j = 1 , d − l, we have F k ( y j ( t 1 , t 2 , . . . , t j )) = 0 , ∀ ( t 1 , t 2 , . . . , t j ) ∈ I 1 × I 2 × . . . × I j . Theorem Under the above assumptions, then the differential system consists of d − l subsystems of dimension d with the uniqueness property in V. Importance of Hamiltonian type structure and references.

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension Theorem a) There are closed intervals I j ⊂ R, 0 ∈ int I j , independent of the parameters, such that I j ⊂ I j ( t 1 , t 2 , . . . , t j − 1 ) , j = 1 , d − l. b) The unique solutions of the differential systems are of class C 1 in any existence point and we have: ∂ y d − l ( t 1 , . . . , t d − l ) = v k ( y d − l ( t 1 , . . . , t d − l )) , k = 1 , d − l. ∂ t k Theorem If F k ∈ C 1 (Ω) , k = 1 , l, with the independence property, and the I j are sufficiently small, j = 1 , d − l, then the mapping y d − l : I 1 × I 2 × . . . × I d − l → R d is regular and one-to-one on its image.

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension The local solution of the initial system is a d − l dimensional manifold around x 0 and y d − l ( t 1 , t 2 , . . . , t d − l ) is a local parametrization of this manifold on I 1 × I 2 × . . . × I d − l . Geometric interpretation: basis in tangent space and integration. Choose the last d − l components of the solutions v j ( x ) ∈ R d as the rows of the identity matrix in R d − l . We obtain : Proposition The last d − l components of y d − l have the form ( t 1 + x 0 l + 1 , t 2 + x 0 l + 2 , . . . , t j + x 0 l + j , x 0 l + j + 1 , . . . , t d − l + x 0 d ) , that is the first l components of y d − l give the unique solution of the implicit system on x 0 + ( I 1 × I 2 × . . . × I d − l ) .

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension Arbitrary Dimension Notice that the propositions are valid just for C 1 and independence assumptions. We also get an evaluation of the existence neighborhood in the implicit function theorem, from Peano’s theorem. By the relation y d − l ( t 1 , t 2 , . . . , t j , 0 , . . . , 0 ) = y j ( t 1 , t 2 , . . . , t j ) we see that y 1 , y 2 , . . . , y d − l have continuous partial derivatives with respect to their arguments. IMPORTANT: the above parametrizations may be more advantageous in applications since we may use maximal solutions, as it will be shown in dimension three. They are not constrained by the function condition...

Introduction Parametrization in Arbitrary Dimension Examples in dimension three Generalized solutions Examples critical case: dimension EXAMPLES IN DIMENSION THREE : SURFACES • The hypothesis ∇ f ( x 0 , y 0 , z 0 ) � = 0 is specified in the form f x ( x 0 , y 0 , z 0 ) � = 0. • We associate to the equation f ( x , y , z ) = 0 , f ( x 0 , y 0 , z 0 ) = 0, two iterated Hamiltonian systems: x ′ = − f y ( x , y , z ) , t ∈ I 1 , y ′ = f x ( x , y , z ) , t ∈ I 1 , z ′ = 0 , t ∈ I 1 , x ( 0 ) = x 0 , y ( 0 ) = y 0 , z ( 0 ) = z 0 ;