Cristian Barbarosie - Academia.edu (original) (raw)

Papers by Cristian Barbarosie

Mathematical Methods in The Applied Sciences, 2005

We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary... more We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary number of isotropic materials, in the heat conduction framework. The component materials and their proportions are given through the Young measure associated to the sequence of coefficient functions. Upper and lower bounds inequalities are deduced in terms of algebraic relations between this Young measure and the eigenvalues of the H-limit matrix. The proofs employ arguments of compensated compactness and fine properties of Young measures. When restricted to the periodic case, we recover known bounds.

Journal of Computational Physics, Mar 1, 2017

Mechanics of Advanced Materials and Structures, Jun 1, 2012

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic m... more This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

Networks and Heterogeneous Media, 2014

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic m... more This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

Structural and Multidisciplinary Optimization, Dec 1, 1997

We begin by explaining briefly why some shape/topology optimization problems need to be relaxed t... more We begin by explaining briefly why some shape/topology optimization problems need to be relaxed through homogenization. In Section 2 we present, from a mechanical viewpoint, the formula for the homogenized coefficients for a periodic infinitesimal perforation, and then briefly discuss the locally periodic ones (Section 3). Sections 4-6 describe a program which minimizes a certain functional over the set of model holes, and then its integration into a larger program, intended to treat topology and shape optimization problems. Numerical results are presented.

Computing, Dec 1, 1995

Reducing the Wrapping Effect. When solving ODEs by interval methods, the main difficulty is reduc... more Reducing the Wrapping Effect. When solving ODEs by interval methods, the main difficulty is reducing the wrapping effect. Various solutions have been put forward, all of which are applicable for narrow initial intervals or to particular classes of equations only. This paper describes an algorithm which, instead of intervals, uses a larger family of sets. The algorithm exhibits a very small wrapping effect and applies to any type of equation and initial region. For the time being it handles only two-dimensional equations.

arXiv (Cornell University), Sep 14, 2022

This paper discusses properties of periodic functions, focusing on (systems of) partial different... more This paper discusses properties of periodic functions, focusing on (systems of) partial differential equations with periodicity boundary conditions, called cellular problems. These cellular problems arise naturally from the assymptotic study of PDEs with rapidly oscillating coefficients; this study is called homogenization theory. We believe the present paper may shed a new light on well-known concepts, for instance by showing hidden links between Green's formula, the div-curl lemma and Donati's representation theorem. We state and prove three extentions of Donati's Theorem addapted to the periodic framework that, beyond their own importance, are essential for understanding the variational formulations of cellular problems in strain, in stress and in displacement, see [C. Barbarosie, A.M. Toader, Stress formulation and duality approach in periodic homogenization, arXiv, 2022]. Section 4 of the present paper presents a self-contained study of properties of traces of a function and their relations with periodicity properties of that function.

Structural and Multidisciplinary Optimization, Apr 23, 2009

In the present paper we deduce formulae for the shape and topological derivatives for elliptic pr... more In the present paper we deduce formulae for the shape and topological derivatives for elliptic problems in unbounded domains subject to periodicity conditions. Note that the known formulae of shape and topological derivatives for elliptic problems in bounded domains do not apply to the periodic framework. We consider a general notion of periodicity, alowing for an arbitrary parallelipiped as periodicity cell. Our calculations are useful for optimizing periodic composite materials by gradient type methods using the topological derivative jointly with the shape derivative for periodic problems. Important particular cases of functionals to optimize are presented. A numerical algorithm for optimizing periodic composites using the topological and shape derivatives is the subject of a second paper [5].

Mathematische Nachrichten, Oct 1, 2000

World Scientific Publishing Company eBooks, Sep 1, 2000

De Gruyter eBooks, Aug 7, 2017

Application of the Adjoint Method has proven successful in Shape Optimization and Topology Optimi... more Application of the Adjoint Method has proven successful in Shape Optimization and Topology Optimization. In the present paper the Adjoint Method is applied to the optimization of eigenvalues and eigenmodes (eigenvectors). The general case of an arbitrary cost function depending on the first n eigenvalues and eigenmodes is detailed. The direct problem does not involve a bilinear form and a linear form as usual in other applications. However, it is possible to follow the spirit of the method and deduce n adjoint problems and obtain n adjoint states, where n is the number of eigenmodes taken into account for optimization. An optimization algorithm based on the derivative of the cost function is developed. This derivative depends on the derivatives of the eigenmodes and the Adjoint Method allows one to express it in terms of the the adjoint states and of the solutions of the direct eigenvalue problem. The formulas hold for the case when the eigenvalues are simple. A section is dedicated to discussions on the case when there are multiple eigenvalues. The same procedures are applied to optimization of microstructures, modeled by Bloch waves. The results obtained hold for general functionals depending on the eigenvalues and on the eigenmodes of the microstructure. However, the wave vector ⃗ k is a more delicate case of optimization parameter. The derivative of a general functional with respect to ⃗ k is obtained, which has interesting implications in band-gap maximization problems.

Discrete and Continuous Dynamical Systems-series B, 2020

We propose a method to optimize periodic microstructures for obtaining homogenized materials with... more We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.

arXiv (Cornell University), Nov 13, 2017

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2017

The present paper focuses on solving partial differential equations in domains exhibiting symmetr... more The present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D3 symmetry to reduce the computational expense by performing the "unit strain" periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D2, i.e. orthotropic, square unit cell and nonabelian D3, i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D3 symmetric hexagonal unit cell.

arXiv (Cornell University), Sep 14, 2022

This paper describes several different formulations of the so-called "cellular problem" which is ... more This paper describes several different formulations of the so-called "cellular problem" which is a system of partial differential equations arising in the theory of homogenization, subject to periodicity boundary conditions. Variational formulations of the cellular problem are presented where the main unknown is the displacement, the stress or the strain, as well as several different formulations as minimization problems. Two dual formulations are also presented, one in displacement-stress and another one in strain-stress. The corresponding Lagrangians may be used in numerical optimization algorithms based on alternated directions.

Journal of Applied Mathematics and Mechanics, Jan 16, 2008

A version of Saint‐Venant's principle is stated and proven for a scalar elliptic equation in ... more A version of Saint‐Venant's principle is stated and proven for a scalar elliptic equation in a domain of arbitrary shape, loaded only in a small ball. Some links are pointed out to the bubble method in topology optimization: when a small hole is introduced in a given shape, the difference between the perturbed solution and the unperturbed one satisfies the hypotheses of Saint‐Venant's principle. An important tool is the Poincaré‐Wirtinger inequality for functions defined on a sphere; results from spectral geometry are used to determine the constant therein.

Mathematical Methods in The Applied Sciences, 2001

This paper studies the problem of bounding the e ective conductivity coe cients of mixtures made ... more This paper studies the problem of bounding the e ective conductivity coe cients of mixtures made of several materials, in given proportions. Lower and upper bounds are obtained, and the optimality of these bounds is proven under certain hypotheses. Also, necessary and su cient conditions for a mixture to attain the bounds are described. Some of the results were already known, but we give simpler proofs.

Summary We prove bounds on the homogenized coefficients for general non periodic mixtures of an a... more Summary We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary number of isotropic materials, in the heat conduction framework. The component materials and their proportions are given through the Young measure associated to the sequence of coefficient functions. Upper and lower bounds inequalities are deduced in terms of algebraic relations between this Young measure and the eigenvalues of the H-limit matrix. The proofs employ arguments of compensated compactness and fine properties of Young measures. When restricted to the periodic case, we recover known bounds.

Mathematical Methods in The Applied Sciences, 2005

We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary... more We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary number of isotropic materials, in the heat conduction framework. The component materials and their proportions are given through the Young measure associated to the sequence of coefficient functions. Upper and lower bounds inequalities are deduced in terms of algebraic relations between this Young measure and the eigenvalues of the H-limit matrix. The proofs employ arguments of compensated compactness and fine properties of Young measures. When restricted to the periodic case, we recover known bounds.

Journal of Computational Physics, Mar 1, 2017

Mechanics of Advanced Materials and Structures, Jun 1, 2012

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic m... more This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

Networks and Heterogeneous Media, 2014

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic m... more This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

Structural and Multidisciplinary Optimization, Dec 1, 1997

We begin by explaining briefly why some shape/topology optimization problems need to be relaxed t... more We begin by explaining briefly why some shape/topology optimization problems need to be relaxed through homogenization. In Section 2 we present, from a mechanical viewpoint, the formula for the homogenized coefficients for a periodic infinitesimal perforation, and then briefly discuss the locally periodic ones (Section 3). Sections 4-6 describe a program which minimizes a certain functional over the set of model holes, and then its integration into a larger program, intended to treat topology and shape optimization problems. Numerical results are presented.

Computing, Dec 1, 1995

Reducing the Wrapping Effect. When solving ODEs by interval methods, the main difficulty is reduc... more Reducing the Wrapping Effect. When solving ODEs by interval methods, the main difficulty is reducing the wrapping effect. Various solutions have been put forward, all of which are applicable for narrow initial intervals or to particular classes of equations only. This paper describes an algorithm which, instead of intervals, uses a larger family of sets. The algorithm exhibits a very small wrapping effect and applies to any type of equation and initial region. For the time being it handles only two-dimensional equations.

arXiv (Cornell University), Sep 14, 2022

This paper discusses properties of periodic functions, focusing on (systems of) partial different... more This paper discusses properties of periodic functions, focusing on (systems of) partial differential equations with periodicity boundary conditions, called cellular problems. These cellular problems arise naturally from the assymptotic study of PDEs with rapidly oscillating coefficients; this study is called homogenization theory. We believe the present paper may shed a new light on well-known concepts, for instance by showing hidden links between Green's formula, the div-curl lemma and Donati's representation theorem. We state and prove three extentions of Donati's Theorem addapted to the periodic framework that, beyond their own importance, are essential for understanding the variational formulations of cellular problems in strain, in stress and in displacement, see [C. Barbarosie, A.M. Toader, Stress formulation and duality approach in periodic homogenization, arXiv, 2022]. Section 4 of the present paper presents a self-contained study of properties of traces of a function and their relations with periodicity properties of that function.

Structural and Multidisciplinary Optimization, Apr 23, 2009

In the present paper we deduce formulae for the shape and topological derivatives for elliptic pr... more In the present paper we deduce formulae for the shape and topological derivatives for elliptic problems in unbounded domains subject to periodicity conditions. Note that the known formulae of shape and topological derivatives for elliptic problems in bounded domains do not apply to the periodic framework. We consider a general notion of periodicity, alowing for an arbitrary parallelipiped as periodicity cell. Our calculations are useful for optimizing periodic composite materials by gradient type methods using the topological derivative jointly with the shape derivative for periodic problems. Important particular cases of functionals to optimize are presented. A numerical algorithm for optimizing periodic composites using the topological and shape derivatives is the subject of a second paper [5].

Mathematische Nachrichten, Oct 1, 2000

World Scientific Publishing Company eBooks, Sep 1, 2000

De Gruyter eBooks, Aug 7, 2017

Application of the Adjoint Method has proven successful in Shape Optimization and Topology Optimi... more Application of the Adjoint Method has proven successful in Shape Optimization and Topology Optimization. In the present paper the Adjoint Method is applied to the optimization of eigenvalues and eigenmodes (eigenvectors). The general case of an arbitrary cost function depending on the first n eigenvalues and eigenmodes is detailed. The direct problem does not involve a bilinear form and a linear form as usual in other applications. However, it is possible to follow the spirit of the method and deduce n adjoint problems and obtain n adjoint states, where n is the number of eigenmodes taken into account for optimization. An optimization algorithm based on the derivative of the cost function is developed. This derivative depends on the derivatives of the eigenmodes and the Adjoint Method allows one to express it in terms of the the adjoint states and of the solutions of the direct eigenvalue problem. The formulas hold for the case when the eigenvalues are simple. A section is dedicated to discussions on the case when there are multiple eigenvalues. The same procedures are applied to optimization of microstructures, modeled by Bloch waves. The results obtained hold for general functionals depending on the eigenvalues and on the eigenmodes of the microstructure. However, the wave vector ⃗ k is a more delicate case of optimization parameter. The derivative of a general functional with respect to ⃗ k is obtained, which has interesting implications in band-gap maximization problems.

Discrete and Continuous Dynamical Systems-series B, 2020

We propose a method to optimize periodic microstructures for obtaining homogenized materials with... more We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.

arXiv (Cornell University), Nov 13, 2017

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2017

The present paper focuses on solving partial differential equations in domains exhibiting symmetr... more The present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D3 symmetry to reduce the computational expense by performing the "unit strain" periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D2, i.e. orthotropic, square unit cell and nonabelian D3, i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D3 symmetric hexagonal unit cell.

arXiv (Cornell University), Sep 14, 2022

This paper describes several different formulations of the so-called "cellular problem" which is ... more This paper describes several different formulations of the so-called "cellular problem" which is a system of partial differential equations arising in the theory of homogenization, subject to periodicity boundary conditions. Variational formulations of the cellular problem are presented where the main unknown is the displacement, the stress or the strain, as well as several different formulations as minimization problems. Two dual formulations are also presented, one in displacement-stress and another one in strain-stress. The corresponding Lagrangians may be used in numerical optimization algorithms based on alternated directions.

Journal of Applied Mathematics and Mechanics, Jan 16, 2008

A version of Saint‐Venant's principle is stated and proven for a scalar elliptic equation in ... more A version of Saint‐Venant's principle is stated and proven for a scalar elliptic equation in a domain of arbitrary shape, loaded only in a small ball. Some links are pointed out to the bubble method in topology optimization: when a small hole is introduced in a given shape, the difference between the perturbed solution and the unperturbed one satisfies the hypotheses of Saint‐Venant's principle. An important tool is the Poincaré‐Wirtinger inequality for functions defined on a sphere; results from spectral geometry are used to determine the constant therein.

Mathematical Methods in The Applied Sciences, 2001

This paper studies the problem of bounding the e ective conductivity coe cients of mixtures made ... more This paper studies the problem of bounding the e ective conductivity coe cients of mixtures made of several materials, in given proportions. Lower and upper bounds are obtained, and the optimality of these bounds is proven under certain hypotheses. Also, necessary and su cient conditions for a mixture to attain the bounds are described. Some of the results were already known, but we give simpler proofs.

Summary We prove bounds on the homogenized coefficients for general non periodic mixtures of an a... more Summary We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary number of isotropic materials, in the heat conduction framework. The component materials and their proportions are given through the Young measure associated to the sequence of coefficient functions. Upper and lower bounds inequalities are deduced in terms of algebraic relations between this Young measure and the eigenvalues of the H-limit matrix. The proofs employ arguments of compensated compactness and fine properties of Young measures. When restricted to the periodic case, we recover known bounds.