The periodic unfolding method in domains with holes

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with non homogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding

Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions

We consider the Stokes system in a thin porous medium Ωε of thickness ε which is perforated by periodically distributed solid cylinders of size ε. On the boundary of the cylinders we prescribe non-homogeneous slip boundary conditions depending on a parameter γ. The aim is to give the asymptotic behavior of the velocity and the pressure of the fluid as ε goes to zero. Using an adaptation of the unfolding method, we give, following the values of γ, different limit systems.Junta de AndalucíaMinisterio de Economía y Competitividad (MINECO). Españ

Multiscale analysis and simulation of a signalling process with surface diffusion

We present and analyze a model for cell signaling processes in biological tissues. The model includes diffusion and nonlinear reactions on the cell surfaces and both inter- and intracellular signaling. Using techniques from the theory of two-scale convergence as well the unfolding method, we show convergence of the solutions to the model to solutions of a two-scale macroscopic problem. We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model. We report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on macroscopic concentrations

Homogenization via unfolding in periodic layer with contact

In this work we consider the elasticity problem for two domains separated by a heterogeneous layer. The layer has an $\varepsilon-$periodic structure, $\varepsilon\ll1$, including a multiple micro-contact between the structural components. The components are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Tresca-friction conditions. In order to obtain preliminary estimates modification of the Korn inequality for the $\varepsilon-$dependent periodic layer is performed. An asymptotic analysis with respect to $\varepsilon \to 0$ is provided and the limit problem is obtained, which consists of the elasticity problem together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior.Comment: 20 pages, 1 figur

Homogenization in Perforated Domains

Numerické řešení matematických modelů popisujících chování materiálů s jemnou strukturou (kompozitní materiály, jemně perforované materiály, atp.) obvykle vyžaduje velký výpočetní výkon. Proto se při numerickém modelování původní materiál nahrazuje ekvivalentním materiálem homogenním. V této práci je k nalezení homogenizovaného materiálu použita dvojškálová konvergence založena na tzv. rozvinovacím operátoru (anglicky unfolding operator). Tento operátor poprvé použil J. Casado-Díaz. V disertační práci je operátor definován jiným způsobem, než jak uvádí původní autor. To dovoluje pro něj dokázat některé nové vlastnosti. Analogicky je definován operátor pro funkce definované na perforovaných oblastech a jsou dokázány jeho vlastnosti. Na závěr je rozvinovací operátor použit k nalezení homogenizovaného řešení speciální skupiny diferenciálních problémů s integrální okrajovou podmínkou. Odvozené homogenizované řešení je ilustrováno na numerických experimentech.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.