Claudio Verdi | Università degli Studi di Milano - State University of Milan (Italy) (original) (raw)

Papers by Claudio Verdi

Research paper thumbnail of Un generatore automatico di reticolazioni triangolari per domini piani

Research paper thumbnail of Adaptive Solution of Parabolic Free Boundary Problems with Error Control

Free boundary problems:, 2019

We derive a posteriori error estimates in natural energy norms with weights. They are useful in l... more We derive a posteriori error estimates in natural energy norms with weights. They are useful in localizing the error extraction in regions of interest, and form the basis of an adaptive procedure. We use them to study the evolution of a persistent corner singularity and elucidate the issue of critical angle for instantaneous smoothing. 1 Introduction The presence of interfaces, and associated lack of regularity, is responsible for global numerical pollution effects for parabolic free boundary problems. A cure consists of equidistributing discretization errors in adequate norms by means of highly graded meshes and varying time steps. Their construction relies on a posteriori error estimates, which are a fundamental component for the design of reliable and efficient adaptive algorithms for PDEs. These issues have been recently tackled in [6], [7], [8], [9], and are briefly discussed here. We consider for simplicity the classical two-phase Stefan problem for an ideal material with consta...

Research paper thumbnail of On the Convergence of the Approximate Free Boundary for the Parabolic Obstacle Problem

Research paper thumbnail of Claudio Verdi Numerical methods for phase transition problems

Unione Matematica Italiana http://www.bdim.eu/item?id=BUMI\_1998\_8\_1B\_1\_83\_0 L'utilizzo e la sta... more Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_1998_8_1B_1_83_0> L'utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l'utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (

Research paper thumbnail of Gamma-convergence of discrete approximations to interfaces with prescribed mean curvature

Calcolo delle variazioni.-• T-convergence of discrete approximations to interfaces with prescribe... more Calcolo delle variazioni.-• T-convergence of discrete approximations to interfaces with prescribed mean curvature. Nota di GIOVANNI BELLETTINI, MAURIZIO PAOLINI e CLAUDIO VERDI, presentata (*) dal Socio E. DE GIORGI. ABSTRACT.-The numerical approximation of the minimum problem: min<^04), is considered, where A QÛ if (A) = P Q (A) + cos(6)X tt~1 (dA n dû)-Jx. The solution to this problem is a set A cQcR n with pre-A. #. ~. . scribed mean curvature x and contact angle 6 at the intersection of dA with dû. The functional & is first relaxed with a sequence of nonconvex functionals defined in H 1 (Q) which, in turn, are discretized by finite elements. The T-convergence of the discrete functionals to & as well as the compactness of any sequence of discrete absolute minimizers are proven. KEY WORDS: Calculus of variations; Surfaces with prescribed mean curvature; Finite elements; Convergence of discrete approximations. RIASSUNTO.-T-convergenza di approssimazioni discrete di interfacce con curvatura media prescritta. Si studia l'approssimazione numerica del seguente problema di minimo: min^4), ove &(A) = P Q {A) + + Còè{B)DC H~ 1 {dA n dû)-Ix, teso alla ricerca di un insieme A ç Û e R n con curvatura media x e angolo di A ~ contatto 6 all'intersezione di dA con dû. Il funzionale $ viene preliminarmente rilassato mediante una successione di funzionali non convessi definiti in H 1 (Û), che sono successivamente discretizzati con elementi finiti. Si dimostrano la T-convergenza dei funzionali discreti al funzionale tf e la compattezza di qualunque successione di minimi assoluti dei funzionali discreti.

Research paper thumbnail of Models, experiments, and numerical simulation of isothermal crystallization of polymers

When a quiescent molten polymer is cooled below the equilibrium melting temperature, crystals (sp... more When a quiescent molten polymer is cooled below the equilibrium melting temperature, crystals (spherulites) appear and keep growing as long as the temperature ranges between the melting temperature and the glass transition temperature. The crystallization process depends upon temperature and crystalline microstructure. In particular, the reduction of the free volume and subsequent impingement between crystals influence both nucleation and growth rates of spherulites. Below the glass transition temperature, the polymer consists of crystal and amorphous phases. In the sequel we discuss a mathematical model for a bidimensional isothermal crystallization process which takes into account the nucleation, growth, and impingement of spherulites

Research paper thumbnail of Scalable BDDC Algorithms for Cardiac Electromechanical Coupling

Lecture Notes in Computational Science and Engineering, 2017

Research paper thumbnail of Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates

Asymptotic Analysis, 1995

Research paper thumbnail of Energy error estimates for a linear scheme to approximate nonlinear parabolic problems

ESAIM: Mathematical Modelling and Numerical Analysis, 1987

Energy error estimates for a linear scheme to approximate nonlinear parabolic problems RAIRO-Modé... more Energy error estimates for a linear scheme to approximate nonlinear parabolic problems RAIRO-Modélisation mathématique et analyse numérique, tome 21, n o 4 (1987), p. 655-678. <http://www.numdam.org/item?id=M2AN_1987__21_4_655_0> © AFCET, 1987, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MATHEMATICAL MOOELUMG AND NUMERICAL AHALYS1S MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

Research paper thumbnail of Numerical approximation of the Preisach model for hysteresis

ESAIM: Mathematical Modelling and Numerical Analysis, 1989

Numerical approximation of the Preisach model for hysteresis RAIRO-Modélisation mathématique et a... more Numerical approximation of the Preisach model for hysteresis RAIRO-Modélisation mathématique et analyse numérique, tome 23, n o 2 (1989), p. 335-356. <http://www.numdam.org/item?id=M2AN_1989__23_2_335_0> © AFCET, 1989, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MATHEMATICALMOOEUJHGANOHUMEJUCALAKALYSIS MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

Research paper thumbnail of Adaptive Solution of Phase Change Problems Over Unstructured Tetrahedral Meshes

The IMA Volumes in Mathematics and its Applications, 1999

Research paper thumbnail of Error Control for Phase Change Problems

Solid Mechanics and Its Applications, 1999

Research paper thumbnail of Design and Efficient Implementation of an Adaptive FEM for Parabolic FBPs

Numerical Methods for Free Boundary Problems, 1991

Mesh adaptation is discussed for the two-phase Stefan problem in 2-D. Three local parameters are ... more Mesh adaptation is discussed for the two-phase Stefan problem in 2-D. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature as well as to specify the so-called refined region. An extra parameter is utilized, in the event of mushy regions, to equidistribute L 1-interpolation errors for enthalpy within the mush. Upon failure of certain quality mesh tests, the current mesh is discarded and a new one completely regenerated; consecutive meshes are thus noncompatible. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface for the relation to become hyperbolic. A drastic reduction of spatial degrees of freedom is obtained with these highly graded meshes. A suitable interpolation theory for noncompatible meshes quantifies the error introduced by mesh changes and leads to the mesh selection algorithm. The resulting scheme is stable in various Sobolev norms and convergent with an a priori prescribed rate. Binary search techniques on suitable quadtree structured data are used to reach a quasi-optimal computational complexity in several search operations necessary for both mesh generation and interpolation between noncompatible meshes. Several numerical experiments illustrate the superior performance of this method as well as its efficiency in approximating both solutions and interfaces in maximum norm.

Research paper thumbnail of Convergence of discrete approximations to sets of prescribed mean curvature

Research paper thumbnail of An Adaptive Finite Element Method for Stefan Problems

Proceedings of the Fourth European Conference on Mathematics in Industry, 1991

During recent years, degenerate parabolic equations have attracted the attention of both scientis... more During recent years, degenerate parabolic equations have attracted the attention of both scientists and engineers mainly because of their relevance in modelling industrial processes, [3]. A common feature in dealing with such problems is the intrinsic lack of regularity of solutions across the free boundaries which, in turn, are not known in advance. For the two-phase Stefan problem, for instance, the temperature cannot be better than Lipschitz continuous and the enthalpy typically exhibits a jump discontinuity across the interface. This lack of smoothness makes piecewise linear finite elements, defined on quasi-uniform meshes, perform worse than expected according to the interpolation theory. Methods studied so far are not completely satisfactory in that they do not profit from the fact that singularities occur in a small part of the entire domain of definition, at least whenever the interface is sufficiently smooth. Consequently, a possible remedy is to be found in terms of a suitably designed adaptive algorithm. In fact, one would like to use a finer mesh near singularities in order to equidistribute the errors but still preserve the number of degrees of freedom, and thus the computational complexity. We refer to [1,6] for an account of the state-of-the-art on this topic along with numerous references.

Research paper thumbnail of Convex approximations of functionals with curvature

Analisi numerica.-Convex approximations of functionals with curvature. Nota di GIOVANNI BELLETTIN... more Analisi numerica.-Convex approximations of functionals with curvature. Nota di GIOVANNI BELLETTINI, MAURIZIO PAOLINI e CLAUDIO VERDI, presentata (*) dal Socio E. MAGENES.

Research paper thumbnail of Convergence of discrete approximations to interfaces with prescribed mean curvature

Research paper thumbnail of An axisymmetric analysis of thermal effects during cementation of femoral prostheses

Numerical Methods for Free Boundary Problems, 1991

We analyse an axisymmetric mathematical model which describes the thermal effects during cementat... more We analyse an axisymmetric mathematical model which describes the thermal effects during cementation of a femoral prosthesis. Various numerical simulations illustrate some critical aspects of this implant such as the thermal bone necrosis and the presence of unreacted residual monomer.

Research paper thumbnail of Variational approximation of the geometric motion of fronts

Motion by Mean Curvature and Related Topics

Abstract. A double obstacle problem for a singularly perturbed reaction-diffusion equation is use... more Abstract. A double obstacle problem for a singularly perturbed reaction-diffusion equation is used to approximate curvature-dependent evolving interfaces. The solution exhibits a rapid variation from-1 to 1 within a thin transition layer, and coincides with the obstacle±1 ...

Research paper thumbnail of Hysteresis operators

Lecture Notes in Mathematics, 1994

Research paper thumbnail of Un generatore automatico di reticolazioni triangolari per domini piani

Research paper thumbnail of Adaptive Solution of Parabolic Free Boundary Problems with Error Control

Free boundary problems:, 2019

We derive a posteriori error estimates in natural energy norms with weights. They are useful in l... more We derive a posteriori error estimates in natural energy norms with weights. They are useful in localizing the error extraction in regions of interest, and form the basis of an adaptive procedure. We use them to study the evolution of a persistent corner singularity and elucidate the issue of critical angle for instantaneous smoothing. 1 Introduction The presence of interfaces, and associated lack of regularity, is responsible for global numerical pollution effects for parabolic free boundary problems. A cure consists of equidistributing discretization errors in adequate norms by means of highly graded meshes and varying time steps. Their construction relies on a posteriori error estimates, which are a fundamental component for the design of reliable and efficient adaptive algorithms for PDEs. These issues have been recently tackled in [6], [7], [8], [9], and are briefly discussed here. We consider for simplicity the classical two-phase Stefan problem for an ideal material with consta...

Research paper thumbnail of On the Convergence of the Approximate Free Boundary for the Parabolic Obstacle Problem

Research paper thumbnail of Claudio Verdi Numerical methods for phase transition problems

Unione Matematica Italiana http://www.bdim.eu/item?id=BUMI\_1998\_8\_1B\_1\_83\_0 L'utilizzo e la sta... more Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_1998_8_1B_1_83_0> L'utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l'utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (

Research paper thumbnail of Gamma-convergence of discrete approximations to interfaces with prescribed mean curvature

Calcolo delle variazioni.-• T-convergence of discrete approximations to interfaces with prescribe... more Calcolo delle variazioni.-• T-convergence of discrete approximations to interfaces with prescribed mean curvature. Nota di GIOVANNI BELLETTINI, MAURIZIO PAOLINI e CLAUDIO VERDI, presentata (*) dal Socio E. DE GIORGI. ABSTRACT.-The numerical approximation of the minimum problem: min<^04), is considered, where A QÛ if (A) = P Q (A) + cos(6)X tt~1 (dA n dû)-Jx. The solution to this problem is a set A cQcR n with pre-A. #. ~. . scribed mean curvature x and contact angle 6 at the intersection of dA with dû. The functional & is first relaxed with a sequence of nonconvex functionals defined in H 1 (Q) which, in turn, are discretized by finite elements. The T-convergence of the discrete functionals to & as well as the compactness of any sequence of discrete absolute minimizers are proven. KEY WORDS: Calculus of variations; Surfaces with prescribed mean curvature; Finite elements; Convergence of discrete approximations. RIASSUNTO.-T-convergenza di approssimazioni discrete di interfacce con curvatura media prescritta. Si studia l'approssimazione numerica del seguente problema di minimo: min^4), ove &(A) = P Q {A) + + Còè{B)DC H~ 1 {dA n dû)-Ix, teso alla ricerca di un insieme A ç Û e R n con curvatura media x e angolo di A ~ contatto 6 all'intersezione di dA con dû. Il funzionale $ viene preliminarmente rilassato mediante una successione di funzionali non convessi definiti in H 1 (Û), che sono successivamente discretizzati con elementi finiti. Si dimostrano la T-convergenza dei funzionali discreti al funzionale tf e la compattezza di qualunque successione di minimi assoluti dei funzionali discreti.

Research paper thumbnail of Models, experiments, and numerical simulation of isothermal crystallization of polymers

When a quiescent molten polymer is cooled below the equilibrium melting temperature, crystals (sp... more When a quiescent molten polymer is cooled below the equilibrium melting temperature, crystals (spherulites) appear and keep growing as long as the temperature ranges between the melting temperature and the glass transition temperature. The crystallization process depends upon temperature and crystalline microstructure. In particular, the reduction of the free volume and subsequent impingement between crystals influence both nucleation and growth rates of spherulites. Below the glass transition temperature, the polymer consists of crystal and amorphous phases. In the sequel we discuss a mathematical model for a bidimensional isothermal crystallization process which takes into account the nucleation, growth, and impingement of spherulites

Research paper thumbnail of Scalable BDDC Algorithms for Cardiac Electromechanical Coupling

Lecture Notes in Computational Science and Engineering, 2017

Research paper thumbnail of Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates

Asymptotic Analysis, 1995

Research paper thumbnail of Energy error estimates for a linear scheme to approximate nonlinear parabolic problems

ESAIM: Mathematical Modelling and Numerical Analysis, 1987

Energy error estimates for a linear scheme to approximate nonlinear parabolic problems RAIRO-Modé... more Energy error estimates for a linear scheme to approximate nonlinear parabolic problems RAIRO-Modélisation mathématique et analyse numérique, tome 21, n o 4 (1987), p. 655-678. <http://www.numdam.org/item?id=M2AN_1987__21_4_655_0> © AFCET, 1987, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MATHEMATICAL MOOELUMG AND NUMERICAL AHALYS1S MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

Research paper thumbnail of Numerical approximation of the Preisach model for hysteresis

ESAIM: Mathematical Modelling and Numerical Analysis, 1989

Numerical approximation of the Preisach model for hysteresis RAIRO-Modélisation mathématique et a... more Numerical approximation of the Preisach model for hysteresis RAIRO-Modélisation mathématique et analyse numérique, tome 23, n o 2 (1989), p. 335-356. <http://www.numdam.org/item?id=M2AN_1989__23_2_335_0> © AFCET, 1989, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MATHEMATICALMOOEUJHGANOHUMEJUCALAKALYSIS MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

Research paper thumbnail of Adaptive Solution of Phase Change Problems Over Unstructured Tetrahedral Meshes

The IMA Volumes in Mathematics and its Applications, 1999

Research paper thumbnail of Error Control for Phase Change Problems

Solid Mechanics and Its Applications, 1999

Research paper thumbnail of Design and Efficient Implementation of an Adaptive FEM for Parabolic FBPs

Numerical Methods for Free Boundary Problems, 1991

Mesh adaptation is discussed for the two-phase Stefan problem in 2-D. Three local parameters are ... more Mesh adaptation is discussed for the two-phase Stefan problem in 2-D. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature as well as to specify the so-called refined region. An extra parameter is utilized, in the event of mushy regions, to equidistribute L 1-interpolation errors for enthalpy within the mush. Upon failure of certain quality mesh tests, the current mesh is discarded and a new one completely regenerated; consecutive meshes are thus noncompatible. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface for the relation to become hyperbolic. A drastic reduction of spatial degrees of freedom is obtained with these highly graded meshes. A suitable interpolation theory for noncompatible meshes quantifies the error introduced by mesh changes and leads to the mesh selection algorithm. The resulting scheme is stable in various Sobolev norms and convergent with an a priori prescribed rate. Binary search techniques on suitable quadtree structured data are used to reach a quasi-optimal computational complexity in several search operations necessary for both mesh generation and interpolation between noncompatible meshes. Several numerical experiments illustrate the superior performance of this method as well as its efficiency in approximating both solutions and interfaces in maximum norm.

Research paper thumbnail of Convergence of discrete approximations to sets of prescribed mean curvature

Research paper thumbnail of An Adaptive Finite Element Method for Stefan Problems

Proceedings of the Fourth European Conference on Mathematics in Industry, 1991

During recent years, degenerate parabolic equations have attracted the attention of both scientis... more During recent years, degenerate parabolic equations have attracted the attention of both scientists and engineers mainly because of their relevance in modelling industrial processes, [3]. A common feature in dealing with such problems is the intrinsic lack of regularity of solutions across the free boundaries which, in turn, are not known in advance. For the two-phase Stefan problem, for instance, the temperature cannot be better than Lipschitz continuous and the enthalpy typically exhibits a jump discontinuity across the interface. This lack of smoothness makes piecewise linear finite elements, defined on quasi-uniform meshes, perform worse than expected according to the interpolation theory. Methods studied so far are not completely satisfactory in that they do not profit from the fact that singularities occur in a small part of the entire domain of definition, at least whenever the interface is sufficiently smooth. Consequently, a possible remedy is to be found in terms of a suitably designed adaptive algorithm. In fact, one would like to use a finer mesh near singularities in order to equidistribute the errors but still preserve the number of degrees of freedom, and thus the computational complexity. We refer to [1,6] for an account of the state-of-the-art on this topic along with numerous references.

Research paper thumbnail of Convex approximations of functionals with curvature

Analisi numerica.-Convex approximations of functionals with curvature. Nota di GIOVANNI BELLETTIN... more Analisi numerica.-Convex approximations of functionals with curvature. Nota di GIOVANNI BELLETTINI, MAURIZIO PAOLINI e CLAUDIO VERDI, presentata (*) dal Socio E. MAGENES.

Research paper thumbnail of Convergence of discrete approximations to interfaces with prescribed mean curvature

Research paper thumbnail of An axisymmetric analysis of thermal effects during cementation of femoral prostheses

Numerical Methods for Free Boundary Problems, 1991

We analyse an axisymmetric mathematical model which describes the thermal effects during cementat... more We analyse an axisymmetric mathematical model which describes the thermal effects during cementation of a femoral prosthesis. Various numerical simulations illustrate some critical aspects of this implant such as the thermal bone necrosis and the presence of unreacted residual monomer.

Research paper thumbnail of Variational approximation of the geometric motion of fronts

Motion by Mean Curvature and Related Topics

Abstract. A double obstacle problem for a singularly perturbed reaction-diffusion equation is use... more Abstract. A double obstacle problem for a singularly perturbed reaction-diffusion equation is used to approximate curvature-dependent evolving interfaces. The solution exhibits a rapid variation from-1 to 1 within a thin transition layer, and coincides with the obstacle±1 ...

Research paper thumbnail of Hysteresis operators

Lecture Notes in Mathematics, 1994