1,721,067 research outputs found
Numerical approximation of hysteresis problems
No full textHysteresis effects are represented by means of continuous or (multivalued) discontinuous Volterra functionals. Constitutive relations of this type are coupled with parabolic equations and existence results are given for the corresponding weak formulations. Implicit time discretization and finite elements are used for the numerical approximation. Several numerical tests show convergence of the approximate solutions
A mathematical model of the austenite-pearlite transformation in plain carbon steel based on the Scheil's additivity rule
No full textThe austenite-pearlite phase transition in steels occurs over a large range of temperatures and gradually in time. According to the Scheil's additivity rule, in plain steel for any prescribed temperature evolution T(t), at any time t the fraction F(t) of transformed austenite is characterized by the condition ∫t0 dξ τ[T(ξ), F(t)] = 1 where τ(T, F) is a prescribed positive function. An interpretation of this law is here proposed and its properties are studied. At lower temperatures the remaining austenite is partially transformed into martensite; the transformed fraction depends on the temperature but not on time. At these temperatures, the austenite-pearlite and the austenite-martensite transformations are coupled. The austenite-pearlite transformation by continuous cooling of an initially austenitic body is then studied, taking account of recalescence and of heat diffusion. A variational formulation is given and an existence result valid for both quenching and normalization is stated. Finally a stable numerical discretization scheme is proposed
Error estimates for a semi-explicit numerical scheme for Stefan-type problems
No full textA parabolic problem of the following form is considered {Mathematical expression} {Mathematical expression} where a is a positive constant, f is a datum and λ is a maximal monotone graph. This system contains the (weak formulation of the)Stefan problem as a particular case. Here the problem (1), (2) is approximated by coupling (1) with the relaxed equation {Mathematical expression} The problem (1), (3) is then discretized in time by the semi-explicit scheme {Mathematical expression} {Mathematical expression} a finite element space discretization and quadrature formulae are then introduced. Thus at each time-step (5) is replaced by a finite number of independent algebraic equations, which can be solved with respect to the barycentral values of wn; then (4) is reduced to a linear system of algebraic equations having as unknowns the nodal values of θ{symbol}n. Assuming the condition τ/ε≦a, the fully discrete scheme is stable and its solution converges to that of (1), (2). Error estimates are proved. The results of some numerical experiments are discussed; they show that the present method is faster than other classical procedures
Numerical analysis of the multidimensional Stefan problem with supercooling and superheating
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