Homogenizace rovnice vedení tepla s hysterezí

Homogenization of heat equation with hysteresis

journal article - other

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The contribution delas with heat equaition in the form (c u+W[u])_t=div(a.grad u)=f, where the functional operator W[u] is Prandtl-Ishlinskii hysteresis operator of play type characterized by a distribution function eta. The spatially dependent initial boundary value problem is studied. Proof of existence and uniqueness of the solution is omitted since the proof is a slightly modified proof by Brokate-Sprekels. The homogenization problem for this equation si studied. For eps->0, a sequence of problems of the above type with spatially eps-periodic coefficients c^eps, eta,^eps, a^eps si considered. The coefficients c^star,eta^star and a^star in the homogenized problem are identified and convergence of the corresponding solutions u^eps to u^star is proved.

Příspěvek se zabývá rovnicí vedení tepla ve tvaru (c u+W[u])_t=div(a.grad u)=f, kde functionální operátor W[u] je Prandtlův-Ishlinského hysterézní operátor typu play charakterizováný distribuční functí eta. Je studována prostorově závislá počáteční okrajová úloha. Důkaz existence a jednoznačnosti řešení je vynechán, protože důkaz je lehkou modifikací důkazu Brokate a Sprekelse. Je řešena úloha homogenizace této rovnice. Pro eps->0, uvažujeme posloupnost úloh uvedeného tvaru s prostorově eps-periodickými koeficienty c^eps, eta^eps, a^eps. Koefficienty c^star,eta^star a a^star v homogenizované úloze jsou identifikovány a konvergence příslušných řešení u^eps k u^star je dokázána.

The contribution delas with heat equaition in the form (c u+W[u])_t=div(a.grad u)=f, where the functional operator W[u] is Prandtl-Ishlinskii hysteresis operator of play type characterized by a distribution function eta. The spatially dependent initial boundary value problem is studied. Proof of existence and uniqueness of the solution is omitted since the proof is a slightly modified proof by Brokate-Sprekels. The homogenization problem for this equation si studied. For eps->0, a sequence of problems of the above type with spatially eps-periodic coefficients c^eps, eta,^eps, a^eps si considered. The coefficients c^star,eta^star and a^star in the homogenized problem are identified and convergence of the corresponding solutions u^eps to u^star is proved.

Prandtlův-Ishlinského operátor, homogenizace, rovnice vedení tepla

Prandtl-Ishlinskii operaor, Homogenization, Heat equation

2004

01.01.2003

0378-4754

Mathematics and Computers in Simulation

61

3-5

7