Dan Tiba

• PhD
• Senior Researcher at Institute of Mathematics

We examine optimal design problems governed by elliptic variational inequalities with unilateral conditions in the domain. We employ functional variations of the geometry that combine shape and topology optimization. Differentiability properties of the regularized/penalized problems are proved and gradient methods are used in the numerical experime...

We consider the steady Navier–Stokes system with combined Dirichlet and convective boundary conditions, in subdomains of a bounded holdall domain. We study approximation properties via the penalization method, including error estimates obtained using the extension operator. The uniqueness of the penalized solution is also proved. These results play...

This is a survey paper devoted to fixed domain methods in optimal design problems, based on optimal control methods. It discusses penalization of the state system and/or of the cost functional via Hamiltonian equations and implicit parametrizations. A special emphasis is on the treatment of various boundary conditions and some new procedures are al...

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 subdo...

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...

"t has been recently shown that the limit cycle situation is not valid for Hamiltonian systems in dimension two, under appropriate conditions. The applications concern global parametrizations of closed curves in the plane and optimal design problems. Here, we discuss a partial extension of this result, for certain Hamiltonian-type systems in higher...

The recent implicit parametrization theorem, based on simple Hamiltonian systems, allows the description of domains and their boundaries and, consequently, it provides a general fixed domain approximation method in shape optimization problems, using optimal control theory. Here, we discuss topology and shape optimization in the difficult case of Ne...

We review several applications of the implicit parametrization theorem in optimization. In nonlinear programming, we discuss both new forms, with less multipliers, of the known optimality conditions, and new algorithms of global type. For optimal control problems, we analyze the case of mixed equality constraints and indicate an algorithm, while in...

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 Neu...

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 t...

In this article, we discuss some approximation methods for optimal design problems governed by evolution equations of parabolic type. The two investigated approaches are of fixed domain type. We also formulate supplementary questions and problems, related to this subject

This volume is dedicated to the 70th anniversary of Vasile Dr˘agan, an internationally recognized researcher in the area of system and control theory. The impressive scientific results of Vasile Dragan are the output of a life-time sustained work and his special talent for Mathematics. This is a story of ”A Beautiful Mind”. On this occasion, his co...

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 approximati...

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 pr...

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 approximati...

We review some results on optimal control problems with both state and control constraints, or general mixed constraints, including certain recent developments. In the setting of state constrained control problems, we consider an approximation technique involving variational inequalities. The constraints may be automatically satisfied in this proce...

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...

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...

We discuss, in arbitrary dimension, certain Hamiltonian type systems and prove existence, uniqueness and regularity properties, under the independence condition. We also investigate the critical case, define a class of generalized solutions and prove existence and basic properties. Relevant examples and counterexamples are also indicated. The appli...

We describe several results from the literature concerning approximation procedures for variational boundary value problems, via duality techniques. Applications in shape optimization are also indicated. Some properties are quite unexpected and this is an argument that the present duality approach may be of interest in a large class of problems.

We discuss an algorithm for the solution of variational inequalities associated with simply supported plates in contact with a rigid obstacle. Our approach has a fixed domain character, uses just linear equations and approximates both the solution and the corresponding coincidence set. Numerical examples are also provided.

In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one and two phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We pr...

We use fictitious domain method with penalization for the Stokes equation in order to obtain approximate solutions in a fixed larger domain including the domain occupied by the structure. The coefficients of the fluid problem, excepting the penalizing term, are independent of the deformation of the structure. It is easy to check the inf-sup conditi...

We discuss a discretization approach for the p- Laplacian equation and a variational inequality associated to fourth order elliptic operators, via a meshless approach based on duality theory.

We discuss recent constructive parametrizations approaches for implicit systems, via systems of ordinary differential equations. We also present the notion of generalized solution, in the critical case and indicate some numerical examples in dimension two and three, using MatLab. In shape optimizations problems, using this method, we introduce gene...

Sufficient conditions for the existence of solutions in strongly nonlinear boundary value problems of elliptic and parabolic type, including ordinary differential equations with unilateral conditions on the boundary, are derived by means of an abstract scheme for continuous perturbations of accretive operators in Banach spaces. © 2015, Academy of R...

We give an example related to certain extensions of the necessary conditions in the classical calculus of variations.

In this paper we propose a new algorithm for the wellknown elliptic bilateral obstacle problem. Our approach enters the category of fixed domain methods and solves just linear elliptic equations at each iteration. The approximating coincidence set is explicitly computed. In the numerical examples, the algorithm has a fast convergence.

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 th...

We introduce a general local parametrization for the solution of the implicit equation f(x; y; z) = 0 by using Hamiltonian systems. The approach extends previous work of the authors and is valid in the critical case as well.

In the present paper, we use a penalization of the Stokes equation in order to obtain approximate solutions in a larger domain including the domain occupied by the structure. The coefficients of the fluid problem, excepting the penalizing term, are constant and independent of the deformation of the structure, which represents an advantage of this a...

The calculus of variations is an important tool in the study of boundary value problems for differential systems. A development of this approach, called the control variational method, is based on the use of the optimal control theory, especially of the Pontryagin maximum principle. In this presentation, we review the results established in the lit...

We present a fixed-domain approach for the solution of shape optimization problems governed by linear or nonlinear elliptic partial differential state equations with Dirichlet boundary conditions, where shape optimization is facilitated via optimal control of a shape function. The method involves extending the state equation to a larger domain usin...

We discuss a differential equations treatment of the implicit functions problem. Our approach allows a precise and complete description of the solution, of continuity and differentiability properties. The critical case is also considered. The investigation is devoted to dimension two and three, but extensions to higher dimension are possible.

We consider shape optimization problems, where the state is governed by elliptic partial differential equations. Using a regularization technique, unknown shapes are encoded via shape functions, turning the shape optimization into optimal control problems for the unknown functions. The method is studied for elliptic PDEs to be solved in an unknown...

For shape optimization problems associated to stationary Navier-Stokes equations, we introduce the corresponding finite element approximation and we prove convergence results.

This work is a review of results in the approximation of optimal design problems, defined in variable/unknown domains, based on associated optimization problems defined in a fixed ‘hold-all’ domain, including the family of all admissible open sets. The literature in this respect is very rich and we concentrate on three main approaches: penalization...

We consider a mathematical model which describes the equilibrium of an elastic beam in contact with two obstacles. The contact is modeled with a normal compliance type condition in such a way that the penetration is allowed but is limited. We state the variational formulation of the problem and prove an existence and uniqueness result for the weak...

We consider a mathematical model which describes the equilibrium of an elastic beam in contact with two obstacles. The contact is modeled with a normal compliance type condition in such a way that the penetration is allowed but is limited. We state the variational formulation of the problem and prove an existence and uniqueness result for the weak...

We report on some very recent results concerning optimal design problems defined in unknown domains in arbitrary dimension. The state equation is of elliptic type (including the case of stationary Navier-Stokes equations) and various boundary conditions will be considered.

For optimal design problems, defined in domains of class $C$ and in arbitrary space dimension, governed by elliptic equations with boundary conditions of Neumann or mixed type, we introduce the corresponding discretized problems and we prove convergence results. The discretization method is of fixed domain type, in the sense that it is given in the...

The control variational method is a development of the variational approach, based on optimal control theory. In this work, we give an application to a variational inequality arising in mechanics and involving unilateral conditions both in the domain and on the boundary, and we explore the extension of the method to time-dependent problems.

We consider a multivalued equation of the form Ay + F(y) = fin a real Hilbert space, where A is a linear operator and F represents the (Clarke) subdifferential of some function. We prove existence and uniqueness results of the solution by using the control variational method. The main idea in this method is to minimize the energy functional associa...

We discuss shape optimization problems and variational methods for fundamental mechanical structures like beams, plates, arches, curved rods, and shells.

Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also included.

In this work a new approach to generalized Naghdi shell and curved rod models is discussed. The method is based on optimal control theory and has a wide range of applications. Some abstract variants are also indicated.

This work discusses geometric optimization problems governed by stationary Navier-Stokes equations. Optimal domains are proved to exist under the assumption that the family of admissible domains is bounded and satisfies the Lipschitz condition with a uniform constant, and in the absence of the uniqueness property for the state system. Through the p...

Fixed domain methods have well-known advantages in the solution of variable domain problems, but are mainly applied in the case of Dirichlet boundary conditions. This paper examines a way to extend this class of methods to the more difficult case of Neumann boundary conditions.

Error estimates for the approximation of optimization problems involving nonlinear operators were presented. The application to variational inequalities of obstacle types was discussed and the case without state constraints was analyzed. The first order optimality conditions in abstract form was obtained using subdifferential calculus.

This paper recalls the work of D. Pompeiu who introduced the notion of set distance in his thesis published one century ago. The notion was further studied by F. Hausdorff, C. Kuratowski who acknowledged in their books the contribution of Pompeiu and it is frequently called the Hausdorff distance.

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In this note we derive optimality conditions for distributed control problems governed by a Stefan two phases free boundary problem. The main idea behind our approach is to consider a family of smoth approximating problems and afterwards to tend to the limit in the approximate optimality conditions, Our result can be mainly compared with some recen...

We indicate a formulation of optimal shape design problems as boundary control problems, based on some approximate controllability-type results. Numerical examples and a comparison with the standard method are included.

A boundary control for a two-phase Stefan problem is considered. The problem is regularized by utilizing the Yosida approximation and the Friedrichs mollifier. Next it is discretized by finite elements in space and finite differences in time. The solution of these auxiliary problems are shown to be minimizing sequences for the original problem when...

We prove new properties for the linear isotropic elasticity system and for thickness minimization problems. We also present very recent results con- cerning shape optimization problems for three-dimensional curved rods and for shells. The questions discussed in this paper are related to the control variational method and to control into coecients p...

We address the question of improving the stability properties of a numerical approximation for differential equations involving small parameters, both in the elliptic and time-dependent cases. This is mainly discussed in connection with a model for the deformation of elastic curved rods introduced by the authors in a previous paper [C. R. Acad. Sci...

We study optimal design problems for three-dimensional curved rods and for shells under minimal regularity assumptions for the geometry. The results that we establish concern the existence of optimal shapes and the sensitivity analysis. We also compute some numerical examples for the optimization of curved rods. The models used have been investigat...

This paper recalls the work of D. Pompeiu who introduced the notion of set distance in his thesis published one century ago. The notion was further studied by F. Hausdorff, C. Kuratowski who acknowledged in their books the contribution of Pompeiu and it is frequently called the Hausdorff distance. Full Text at Springer, may require registration or...

We show that the asymptotic method can provide an advantageous approach to obtain models of thin elastic bodies under minimal regularity assumptions on the geometry. Our investigation is devoted to clamped curved rods with a nonsmooth line of centroids, and the obtained model is a generalization of results already available in the literature.

We prove new properties for the linear isotropic elasticity system and for thickness minimization problems. We also present very recent results concerning shape optimization problems for three-dimensional curved rods and for shells. The questions discussed in this paper are related to the control variational method and to control into coefficient p...

In this paper we realize a study of various constraint qualification conditions for the existence of Lagrange multipliers for convex minimization problems in general normed vector spaces; it is based on a new formula for the normal cone to the constraint set, on local metric regularity and a metric regularity property on bounded subsets. As a by-pr...

Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W 0 l,p (Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet b...

We consider a new variational method for a clamped plate model and related shape optimization problems. Our approach allows the study of plates with discontinuous thickness.

We indicate a new approach to the deformation of three-dimensional curved rods with variable cross-section. The model consists of a system of nine ordinary differential equations for which we prove existence and uniqueness via the coercivity of the association bilinear form. From the geometrical point of view, we are using the Darboux frame or a ne...

It is our aim to present a new treatment for some classical models of arches and for their optimization. In particular, our approach allows us to study nonsmooth arches, while the standard assumptions from the literature require W-3,∞-regularity for the parametric representation. Moreover, by a duality-type argument, the deformation of the arches m...

We discuss some existence results for optimal design problems governed by second order elliptic equations with the homogeneous Neumann boundary conditions or with the interior transmission conditions. We show that our continuity hypotheses for the unknown boundaries yield the compactness of the associated characteristic functions, which, in turn, g...

We investigate general control problems governed by ordinary differential systems involving hysteresis operators. Our main hypotheses are of continuity type, and we discuss existence results, discretization methods, and approximation approaches.

A shell model of generalized Naghdi type is studied which requires only low regularity conditions. It is shown that the corresponding system of linear variational equations (representing a boundary value problem for a linear system of six partial differential equations on the shell) admits a unique solution. The main step in the proof is to show th...

We review recent results established in the literature via the optimal control approach to differential equations, and we show that a systematic study of general variational inequalities associated to fourth-order operators can be performed by similar methods.

We study the Kirchhoff-Love model in the case when the middle curve of the arch has corners. Our approach does not use the Dirichlet principle or the Korn inequality. We propose a variational formulation based on optimal control theory and we obtain explicit formulas for the deformation. © 2000 Académie des sciences/Éditions scientifiques et médica...

We discuss existence theorems for shape optimization and material distribution problems. The conditions that we impose on the unknown sets are continuity of the boundary, respectively a certain measurability hypothesis.

We study the Kirchhoff–Love model in the case when the middle curve of the arch has corners. Our approach does not use the Dirichlet principle or the Korn inequality. We propose a variational formulation based on optimal control theory and we obtain explicit formulas for the deformation.

We discuss existence theorems for shape optimization and material distribution problems. The conditions that we impose on the unknown sets are continuity of the boundary, respectively a certain measurability hypothesis. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

We review several recent results devoted to this topic, and we present a new approach to a variational inequality with unilateral conditions on the boundary, associated with a partially clamped plate.

For a simply supported plate, we consider two optimization problems: the volume minimization and the identification of a coefficient. Via a transformation recently introduced by the authors, we obtain the optimality conditions in a qualified form, and their analysis yields the bang-bang properties for the optimal thickness.

For a simply supported plate, we consider two optimization problems: the volume minimization and the identification of a coefficient. Via a transformation recently introduced by the authors, we obtain the optimality conditions in a qualified form, and their analysis yields the bang-bang properties for the optimal thickness.

We introduce a class of nonlinear transformations called "resizing rules" which associate to optimal shape design problems certain equivalent distributed control problems, while preserving the state of the system. This puts into evidence the duality principle that the class of system states that can be achieved, under a prescribed force, via modifi...

this paper. We restrict ourselves to explain the main ideas in a simplified setting. Let r

In this paper we investigate optimal control problems governed by elliptic variational inequalities of the obstacle type. We show how to obtain optimality conditions for a relaxed problem with or without state constraints. Then we present the optimality system related to the original problem with state constraints, using a generalized derivative.

We discuss the nonconvex optimal shape design problem of minimizing the weight of a loaded beam subject to deflection constraints. We associate to it a convex minimization problem which will play the role of a dual. The algorithm we propose has a global character and iterates between the two optimization problems via a so called "resizing rule". 1...

this paper, we use such an approach to develop numerical approximation methods both for Eq. (1.1) and for various optimization problems (optimal control or optimal shape design) that may be associated to it. In Sect. 2, we study the numerical treatment of the clamped plate via second order elliptic systems. Section 3 is devoted to optimization ques...

Abstract In this paper we investigate some optimal convex control problems, with mixed constraints on the state and the control. We give a general condition which allows to set optimality conditions for non qualified problems (in the Slater sense). Then we give some applications and examples involving generalized bang-bang results. Keywords: Optima...