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Banach Algebras
Machovičová, Tatiana ; Nechvátal, Luděk (oponent) ; Franců, Jan (vedoucí práce)
By Banach algebra we mean Banach space enriched with a multiplication operation. It is a mathematical structure that is used in the non-periodic homogenization of composite materials. The theory of classical homogenization studies materials assuming the periodicity of the structure. For real materials, the assumption of a periodicity is not enough and is replaced by the so-called an abstract hypothesis based on a concept composed mainly of the spectrum of Banach algebra and Sigma convergence. This theory was introduced in 2004.
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Homogenization in Perforated Domains
Rozehnalová, Petra ; Bock, Igor (oponent) ; Rohan, Eduard (oponent) ; Franců, Jan (vedoucí práce)
The numerical solving of mathematical models describing the mechanical behavior of materials with a fine structure (composite materials, finely perforated materials etc.) usually requires huge computational performance. Hence in numerical modeling the original material is replaced by an equivalent homogeneous one. In this work a two-scale convergence based on a periodical unfolding operator is used to find the homogenized material. The operator was for the first time used by J. Casado-Díaz. In this Ph.D. thesis, the operator is defined in a slightly different way which allows us to prove some of its new properties. The unfolding operator for functions defined on a perforated domain is defined analogically and its properties are proved. Finally, this operator is used to find the homogenized solution of a special family of problems with an integral boundary condition; some numerical results are presented.
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Banach Algebras
Machovičová, Tatiana ; Nechvátal, Luděk (oponent) ; Franců, Jan (vedoucí práce)
By Banach algebra we mean Banach space enriched with a multiplication operation. It is a mathematical structure that is used in the non-periodic homogenization of composite materials. The theory of classical homogenization studies materials assuming the periodicity of the structure. For real materials, the assumption of a periodicity is not enough and is replaced by the so-called an abstract hypothesis based on a concept composed mainly of the spectrum of Banach algebra and Sigma convergence. This theory was introduced in 2004.
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|
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Homogenization in Perforated Domains
Rozehnalová, Petra ; Bock, Igor (oponent) ; Rohan, Eduard (oponent) ; Franců, Jan (vedoucí práce)
The numerical solving of mathematical models describing the mechanical behavior of materials with a fine structure (composite materials, finely perforated materials etc.) usually requires huge computational performance. Hence in numerical modeling the original material is replaced by an equivalent homogeneous one. In this work a two-scale convergence based on a periodical unfolding operator is used to find the homogenized material. The operator was for the first time used by J. Casado-Díaz. In this Ph.D. thesis, the operator is defined in a slightly different way which allows us to prove some of its new properties. The unfolding operator for functions defined on a perforated domain is defined analogically and its properties are proved. Finally, this operator is used to find the homogenized solution of a special family of problems with an integral boundary condition; some numerical results are presented.
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