arXiv subjects
Publications and source records attributed to Dan Tiba.
We study optimal design problems involving variational inequalities with unilateral conditions in the domain and pointwise boundary observation. We use regularizing and penalization tehniques in the setting of the Hamiltonian approach to shape/topology optimization problems. Numerical examples are also included.
We indicate a new approach to the optimization of the clamped plates with holes. It is based on the use of Hamiltonian systems and the penalization of the performance index. The alternative technique employing the penalization of the state system, cannot be applied in this case due to the (two) Dirichlet boundary conditions. We also include numerical tests exhibiting both shape and topological modifications, both creating and closing holes.
We consider the steady Navier-Stokes system with mixed boundary conditions, in subdomains of a holdall domain. We study, via the penalization method, its approximation properties. Error estimates, obtained using the extension operator, other evaluations and the uniqueness of the solution, when the viscosity may be arbitrarily small in certain subdomains, are also discussed. Numerical tests, including topological optimization applications, are presented. A general convergence result for the approximation of this type of geometric inverse problems and of the associated optimal control problems, is investigated in the last part of the paper.
We discuss first order optimality conditions for geometric optimization problems with Neumann boundary conditions and boundary observation. The methods we develop here are applicable to large classes of state systems or cost functionals. Our approach is based on the implicit parametrization theorem and the use of Hamiltonian systems. It establishes equivalence with a constrained optimal control problem and uses Lagrange multipliers under a new simple constraint qualification. In this setting, general functional variations are performed, that combine topological and boundary variations in a natural way.
We present first a brief review of the existing literature on shape optimization, stressing the recent use of Hamiltonian systems in topology optimization. In the second section, we collect some preliminaries on the implicit parametrization theorem, especially in dimension two, which is a case of interest in shape optimization. The formulation of the problem is also discussed. The approximation via penalization and its differentiability properties are analyzed in Section 3. Next, we investigate the discretization process in Section 4. The last section is devoted to numerical experiments.
The recent approach based on Hamiltonian systems and the implicit parametri\-za\-tion theorem, provides a general fixed domain approximation method in shape optimization problems, using optimal control theory. In previous works, we have examined Dirichlet boundary conditions with distributed or boundary observation. Here, we discuss the case of Neumann boundary conditions, with a combined cost functional, including both distributed and boundary observation. Extensions to nonlinear state systems are possible. This new technique allows simultaneous boundary and topological variations and we also report numerical experiments confirming the theoretical results.
We investigate a fixed domain approach in shape optimization, using a regularization of the Heaviside function both in the cost functional and in the state system. We consider the compliance minimization problem in linear elasticity, a well known application in this area of research. The optimal design problem is approached by an optimal control problem defined in a prescribed domain including all the admissible unknown domains. This approximating optimization problem has good differentiability properties and a gradient algorithm can be applied. Moreover, the paper also includes several numerical experiments that demonstrate the descent of the obtained cost values and show the topological and the boundary variations of the computed domains. The proposed approximation technique is new and can be applied to state systems given by various boundary value problems.
We consider general shape optimization problems governed by Dirichlet boundary value problems. The proposed approach may be extended to other boundary conditions as well. It is based on a recent representation result for implicitly defined manifolds, due to the authors, and it is formulated as an optimal control problem. The discretized approximating problem is introduced and we give an explicit construction of the associated discrete gradient. Some numerical examples are also indicated.
We consider a simply supported plate with constant thickness, defined on an unknown multiply connected domain. We optimize its shape according to some given performance functional. Our method is of fixed domain type, easy to be implemented, based on a fictitious domain approach and the control variational method. The algorithm that we introduce is of gradient type and performs simultaneous topological and boundary variations. Numerical experiments are also included and show its efficiency.
We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the usual implicit functions solution. We also discuss the critical case of the implicit functions theorem, define the notion of generalized solution and prove existence and properties. Examples are also indicated. The applications concern necessary conditions and algorithms in nonconvex optimization problems and their perturbations.