The skin-core morphology of an injection moulded polymer product is due to the melting-crystalliz... more The skin-core morphology of an injection moulded polymer product is due to the melting-crystallization phase transition. It is thus described in terms of thermal history and overall (nucleation and crystal growth) crystallization kinetic. Two Stefan problems have been analyzed to study the moving boundary appearing during polymer crystallization. Their difference is due to the constitutive equation for the heat flux. An improved Avrami equation is adopted for the overall crystallization kinetic. This accounts for the formation of a mushy region besides liquid and crystal phases. Both models have been efficiently discretized by a stable finite element method based on a semi-explicit finite difference approximation in time. The skin-core structure is well predicted by the Fourier-Stefan model.
ABSTRACT A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed ... more ABSTRACT A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. 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 so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors between consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in {L^p} -based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods on quasi-uniform meshes, namely O({τ^{ - 3/2}}) as opposed to O({τ^{ - 2}}) , to achieve the same global asymptotic accuracy; here τ > 0 is the (uniform) time step. A rate of convergence of essentially O({τ^{1/2}}) is derived in the natural energy spaces provided the total number of mesh changes is restricted to O({τ^{ - 1/2}}) , which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme's efficiency in approximating both solutions and interfaces.
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
Mathematical Models for Phase Change Problems, 1989
This is a report on the adaptive implementation of local mesh refinements for two-phase Stefan pr... more This is a report on the adaptive implementation of local mesh refinements for two-phase Stefan problems. The strategy is based on equidistributing interpolation errors in such a way that a typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas is locally refined in its vicinity for the relation to become hyperbolic. Several numerical tests are performed on the computed solution to extract information about first and second derivatives as well as to predict discrete free boundary locations. The resulting scheme is stable and necessitates less degrees of freedom than previous practical methods on quasi-uniform meshes to achieve the same asymptotic accuracy. Several crucial implementational issues are discussed such as structure and computational complexity of both the mesh generator and the finite element code. Numerical results illustrate the scheme efficiency in approximating both solutions and interfaces.
International Journal for Numerical Methods in Engineering, 1988
An efficient implementation of finite element methods for free boundary parabolic problems in gen... more An efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented. The Stefan problem, the Hele‐Shaw problem and the porous medium equation are included. Backward differences or linearization techniques are used for the time discretization of the problem. The performances of these schemes are discussed with several numerical tests.
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.
In this paper we first review the covering space method with constrained BV functions for solving... more In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, aa soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.
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 ...
In this paper we study embeddings of 3-manifolds with connected boundary in S^3 or, equivalently,... more In this paper we study embeddings of 3-manifolds with connected boundary in S^3 or, equivalently, embeddings of oriented connected closed surfaces in S^3. We develop a complete invariant, the fundamental span, for such embeddings, which generalizes the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived, and they are employed to investigate handlebody knots, bi-knotted surfaces and the chirality of a knot. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of inequivalence of Ishii et al.'s handlebody knots 5_1 and 6_4, and construct an infinite family of pairs of inequivalent bi-knotted surfaces with homeomorphic complements. An interpretation of Fox's invariant in terms of the fundamental span is also discussed; we use it to prove the chirality of 9_42 and 10_71, which are known to be un...
We start to investigate the existence of conformal minimizers for the Dirichlet func-tional in th... more We start to investigate the existence of conformal minimizers for the Dirichlet func-tional in the setting of semicartesian parametrizations, adapting to this context the results in [10] for the Plateau’s problem. The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in [5], in the study of the relaxed area functional for maps from the plane to the plane jumping on a line. 1
Abstract. We investigate two different invariants for the Rubik’s Magic puz-zle that can be used ... more Abstract. We investigate two different invariants for the Rubik’s Magic puz-zle that can be used to prove the unreachability of many spatial configurations, one of these invariants, of topological type, is to our knowledge never been studied before and allows to significantly reduce the number of theoretically constructible shapes. 1.
Abstract. Solutions of the so-called prescribed curvature problem minA⊆Ω PΩ(A) − � A g(x), g bein... more Abstract. Solutions of the so-called prescribed curvature problem minA⊆Ω PΩ(A) − � A g(x), g being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers A ⊂ ⊂ ΩweproveanO(ɛ2 |log ɛ | 2) error estimate (where ɛ stands for the perturbation parameter), and show that this estimate is quasi-optimal. The proof is based on the construction of accurate barriers suggested by formal asymptotics. This analysis is next extended to a finite element discretization of the PDE to prove the same error estimate for discrete minima. 1.
The skin-core morphology of an injection moulded polymer product is due to the melting-crystalliz... more The skin-core morphology of an injection moulded polymer product is due to the melting-crystallization phase transition. It is thus described in terms of thermal history and overall (nucleation and crystal growth) crystallization kinetic. Two Stefan problems have been analyzed to study the moving boundary appearing during polymer crystallization. Their difference is due to the constitutive equation for the heat flux. An improved Avrami equation is adopted for the overall crystallization kinetic. This accounts for the formation of a mushy region besides liquid and crystal phases. Both models have been efficiently discretized by a stable finite element method based on a semi-explicit finite difference approximation in time. The skin-core structure is well predicted by the Fourier-Stefan model.
ABSTRACT A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed ... more ABSTRACT A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. 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 so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors between consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in {L^p} -based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods on quasi-uniform meshes, namely O({τ^{ - 3/2}}) as opposed to O({τ^{ - 2}}) , to achieve the same global asymptotic accuracy; here τ > 0 is the (uniform) time step. A rate of convergence of essentially O({τ^{1/2}}) is derived in the natural energy spaces provided the total number of mesh changes is restricted to O({τ^{ - 1/2}}) , which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme's efficiency in approximating both solutions and interfaces.
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
Mathematical Models for Phase Change Problems, 1989
This is a report on the adaptive implementation of local mesh refinements for two-phase Stefan pr... more This is a report on the adaptive implementation of local mesh refinements for two-phase Stefan problems. The strategy is based on equidistributing interpolation errors in such a way that a typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas is locally refined in its vicinity for the relation to become hyperbolic. Several numerical tests are performed on the computed solution to extract information about first and second derivatives as well as to predict discrete free boundary locations. The resulting scheme is stable and necessitates less degrees of freedom than previous practical methods on quasi-uniform meshes to achieve the same asymptotic accuracy. Several crucial implementational issues are discussed such as structure and computational complexity of both the mesh generator and the finite element code. Numerical results illustrate the scheme efficiency in approximating both solutions and interfaces.
International Journal for Numerical Methods in Engineering, 1988
An efficient implementation of finite element methods for free boundary parabolic problems in gen... more An efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented. The Stefan problem, the Hele‐Shaw problem and the porous medium equation are included. Backward differences or linearization techniques are used for the time discretization of the problem. The performances of these schemes are discussed with several numerical tests.
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.
In this paper we first review the covering space method with constrained BV functions for solving... more In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, aa soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.
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 ...
In this paper we study embeddings of 3-manifolds with connected boundary in S^3 or, equivalently,... more In this paper we study embeddings of 3-manifolds with connected boundary in S^3 or, equivalently, embeddings of oriented connected closed surfaces in S^3. We develop a complete invariant, the fundamental span, for such embeddings, which generalizes the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived, and they are employed to investigate handlebody knots, bi-knotted surfaces and the chirality of a knot. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of inequivalence of Ishii et al.'s handlebody knots 5_1 and 6_4, and construct an infinite family of pairs of inequivalent bi-knotted surfaces with homeomorphic complements. An interpretation of Fox's invariant in terms of the fundamental span is also discussed; we use it to prove the chirality of 9_42 and 10_71, which are known to be un...
We start to investigate the existence of conformal minimizers for the Dirichlet func-tional in th... more We start to investigate the existence of conformal minimizers for the Dirichlet func-tional in the setting of semicartesian parametrizations, adapting to this context the results in [10] for the Plateau’s problem. The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in [5], in the study of the relaxed area functional for maps from the plane to the plane jumping on a line. 1
Abstract. We investigate two different invariants for the Rubik’s Magic puz-zle that can be used ... more Abstract. We investigate two different invariants for the Rubik’s Magic puz-zle that can be used to prove the unreachability of many spatial configurations, one of these invariants, of topological type, is to our knowledge never been studied before and allows to significantly reduce the number of theoretically constructible shapes. 1.
Abstract. Solutions of the so-called prescribed curvature problem minA⊆Ω PΩ(A) − � A g(x), g bein... more Abstract. Solutions of the so-called prescribed curvature problem minA⊆Ω PΩ(A) − � A g(x), g being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers A ⊂ ⊂ ΩweproveanO(ɛ2 |log ɛ | 2) error estimate (where ɛ stands for the perturbation parameter), and show that this estimate is quasi-optimal. The proof is based on the construction of accurate barriers suggested by formal asymptotics. This analysis is next extended to a finite element discretization of the PDE to prove the same error estimate for discrete minima. 1.
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Papers by Maurizio Paolini