Analysis of a variable time-step discretization    of the three-dimensional Frémond model   for shape memory alloys

Stefanelli, Ulisse

doi:10.1090/s0025-5718-02-01409-6

Cited by 7 publications

(5 citation statements)

References 16 publications

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“…By (13), it turns out that the subdifferential * is a maximal monotone operator with 0 ∈ * (0). In particular, it follows that…”

Section: Introductionmentioning

confidence: 99%

A dissipative model for hydrogen storage: existence and regularity results

Chiodaroli

2010

Math. Meth. Appl. Sci.

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We prove global existence of a solution to an initial and boundary-value problem for a highly nonlinear PDE system. The problem arises from a thermo-mechanical dissipative model describing hydrogen storage by use of metal hydrides. In order to treat the model from an analytical point of view, we formulate it as a phase transition phenomenon thanks to the introduction of a suitable phase variable. Continuum mechanics laws lead to an evolutionary problem involving three state variables: the temperature, the phase parameter and the pressure. The problem thus consists of three coupled partial differential equations combined with initial and boundary conditions. The existence and regularity of the solutions are here investigated by means of a time discretization-a priori estimates-passage to the limit procedure joined with compactness and monotonicity arguments.

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“…By (13), it turns out that the subdifferential * is a maximal monotone operator with 0 ∈ * (0). In particular, it follows that…”

Section: Introductionmentioning

confidence: 99%

A dissipative model for hydrogen storage: existence and regularity results

Chiodaroli

2010

Math. Meth. Appl. Sci.

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“…Indeed, the argument follows closely the lines of the proof of the continuous dependence estimate (4.1). The additional intricacy related to the fact that the continuous and the discrete solutions do not solve the same equations may be overcome by the same techniques of the two papers [34,35], where indeed the abstract analysis of [28,29] is applied in a similar context. On the other hand, some comment is in order.…”

Section: Error Estimatesmentioning

confidence: 99%

Analysis of a Thermomechanical Model for Shape Memory Alloys

Stefanelli¹

2005

SIAM J. Math. Anal.

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“…Concerning the mathematical analysis of the system (1.1-1.6), we notice that the existence proof (as in the most part of the systems related to Frémond model, see, e.g., Colli, 1995) relies on a suitable time discretization-a priori estimates-passage to the limit procedure. We remind that the investigation on the error control for problems Downloaded by [North Dakota State University] related to Frémond's model, and in general for phase transition problems, has recently received a good deal of interest as the papers Klein and Verdi (2003) and Stefanelli (1999Stefanelli ( , 2000Stefanelli ( , 2002 show. More precisely, we present the system (1.1-1.6) as an abstract Cauchy problem for two coupled evolution equations, and we investigate in detail its fully implicit time discretization.…”

Section: Error Estimates For Phase Transition Model 549mentioning

confidence: 99%

Error Estimates for a Variable Time-Step Discretization of a Phase Transition Model with Hyperbolic Momentum

Segatti

2004

Numerical Functional Analysis and Optimization

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This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the nonstationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initialboundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data.

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Analysis of a variable time-step discretization of the three-dimensional Frémond model for shape memory alloys

Cited by 7 publications

References 16 publications

A dissipative model for hydrogen storage: existence and regularity results

A dissipative model for hydrogen storage: existence and regularity results

Analysis of a Thermomechanical Model for Shape Memory Alloys

Error Estimates for a Variable Time-Step Discretization of a Phase Transition Model with Hyperbolic Momentum

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