Parciální diferenciální rovnice hrají důležitou v inženýrských aplikacích. Často je možné tyto rovnice řešit pouze přibližně, tj. numericky. Z toho důvodu vzniklo množství diskretizačních metod pro řešení těchto rovnic. Uvedená nespojitá Galerkinova metoda se zdá jako velmi obecná metoda pro řešení těchto rovnic, především pak pro hyperbolické systémy. Naším cílem je řešit úlohy aeroakustiky, přičemž šíření akustických vln je popsáno pomocí linearizovaných Eulerových rovnic. A jelikož se jedná o hyperbolický systém, byla vybrána právě nespojitá Galerkinova metoda. Mezi nejdůležitější aspekty této metody patří schopnost pracovat s geometricky složitými oblastmi, možnost dosáhnout metody vysokého řádu a dále lokální charakter toho schématu umožnuje efektivní paralelizaci výpočtu. Nejprve uvedeme nespojitou Galerkinovu metodu v obecném pojetí pro jedno- a dvoudimenzionalní úlohy. Algoritmus následně otestujeme pro řešení rovnice advekce, která byla zvolena jako modelový případ hyperbolické rovnice. Metoda nakonec bude testována na řadě verifikačních úloh, které byly formulovány pro testování metod pro výpočetní aeroakustiku, včetně oveření okrajových podmínek, které, stejně jako v případě teorie proudění tekutin, jsou nedílnou součástí výpočetní aeroakustiky.

This thesis covers the basics in the stochastic homogenization of elliptic partial differential equations, from underlying theory up to numerical ap- proaches. In particular, we introduce and analyze a combination of the Fourier-Galerkin method in the spatial domain with a collocation method in the stochastic domain. The material coefficients are assumed to depend on a finite number of random variables. We present a comparison of the Monte Carlo method with the full tensor grid and sparse grid collocation method for two applications. The first one is the checkerboard problem with continuous random variables, the other considers the material coefficients to be described in terms of an autocorrelation function.

Cette thèse est consacrée à l'étude des équations du Darcy Brinkman Forchheimer (DBF) avec des conditions aux limites non standards. Nous montrons d'abord l'existence de différents type de solutions (faible, forte) correspondant au problème DBF stationnaire dans un domaine simplement connexe avec des conditions portants sur la composante normale du champ de vitesse et la composante tangentielle du tourbillon. Ensuite, nous considérons le système Brinkman Forchheimer (BF) avec des conditions sur la pression dans un domaine non simplement connexe. Nous prouvons que ce problème est bien posé ainsi que l'existence de la solution forte. Nous établissons la régularité de la solution dans les espaces L^p pour p >= 2.L'étude et l'approximation du problème DBF non stationnaire est basée sur une approche pseudo-compressibilité. Une estimation d'erreur d'ordre deux est établie dans le cas o\`u les conditions aux limites sont de types Dirichlet ou Navier.Enfin, une méthode d'éléments finis Galerkin Discontinue est proposée et la convergence établie concernant le problème DBF linéarisé et le système DBF non linéaire avec des conditions aux limites non standard. / This thesis is devoted to Darcy Brinkman Forchheimer (DBF) equations with a non standard boundary conditions. We prove first the existence of different type of solutions (weak, strong) of the stationary DBF problem in a simply connected domain with boundary conditions on the normal component of the velocity field and the tangential component of the vorticity. Next, we consider Brinkman Forchheimer (BF) system with boundary conditions on the pressure in a non simply connected domain. We prove the well-posedness and the existence of a strong solution of this problem. We establish the regularity of the solution in the L^p spaces, for p >= 2.The approximation of the non stationary DBF problem is based on the pseudo-compressibility approach. The second order's error estimate is established in the case where the boundary conditions are of type Dirichlet or Navier. Finally, the finite elements Galerkin Discontinuous method is proposed and the convergence is settled concerning the linearized DBF problem and the non linear DBF system with a non standard boundary conditions.

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The thesis is dedicated to the study and numerical analysis of the non- linear flows in the porous media, using general Forchheimer models. In the numerical analysis, the local discontinuous Galerkin method is chosen. The first part of the paper is dedicated to the derivation of the studied equations based on the physical motivation and summarizing the theory needed for the further analysis. Core of the thesis consists of the introduction of the chosen discretization method and the derivation of the main stability and a priory error estimates, optimal for the linear ansatz functions. At the end we present a couple of numerical experiments to verify the results. 1

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A combination of theoretical, numerical, and experimental methods were utilized to determine that shock waves can form in brain tissue from smooth boundary conditions. The conditions that lead to the formation of shock waves were determined. The implication of this finding was that the high gradients of stress and strain that could occur at the shock wave front could contribute to mechanism of brain injury in blast loading conditions. The approach consisted of three major steps. In the first step, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10 1/s ramp rate and 20 s hold time has been developed. The assumption of quasi-linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C_10, C_01 and C_11) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a linear bulk modulus model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06 s and 14 s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0 1/s) and the results in compression and their extrapolation to tension were compared against previous models. In the next step, a framework for understanding the propagation of stress waves in brain tissue under blast loading was developed. It was shown that tissue nonlinearity and rate dependence are key parameters in predicting the mechanical behavior under such loadings, as they determine whether traveling waves could become steeper and eventually evolve into shock discontinuities. To investigate this phenomenon, the QLV material model developed based on finite compression results mentioned above was extended to blast loading rates, by utilizing the stress data published on finite torsion of brain tissue at high rates (up to 700 1/s). It was shown that development of shock waves is possible inside the head in response to compressive pressure waves from blast explosions. Furthermore, it was argued that injury to the nervous tissue at the microstructural level could be attributed to the high stress and strain gradients with high temporal rates generated at the shock front and this was proposed as a mechanism of injury in brain tissue. In the final step, the phenomenon of shock wave formation and propagation in brain tissue was further studied by developing a one-dimensional model of brain tissue using the Discontinuous Galerkin finite element method. This model is capable of capturing high-gradient waves with higher accuracy than commercial finite element software. The deformation of brain tissue was investigated under displacement input and pressure input boundary conditions relevant to blast over-pressure reported in the literature. It was shown that a continuous wave can become a shock wave as it propagates in the tissue when the initial changes in acceleration are beyond a certain limit. The high spatial gradients of stress and strain at the shock front cause large relative motions at the cellular scale at high temporal rates even when the maximum strains and stresses are relatively low. This gradient-induced local deformation occurs away from the boundary and can therefore contribute to the diffuse nature of blast-induced injuries.   / Mechanical Engineering