Abstract: In this paper, we first present the THex algorithm that refines a tetrahedral mesh into a hexahedral mesh. Strategies for efficient implementation of the THex algorithm are discussed. Then we present the lowest order weak Galerkin (WG) $ (Q_0,Q_0;RT_{[0]}) $ finite element method for solving the Darcy equation on general hexahedral meshes. This simple solver uses constant pressure unknowns inside hexahedra and on faces but specifies the discrete weak gradients of these basis functions in local Raviart-Thomas $ RT_{[0]} $ spaces. The solver has easy implementation, is locally mass-conservative, and produces continuous normal fluxes, regardless of hexahedral mesh quality. When the mesh is asymptotically parallelopiped, this Darcy solver exhibits optimal order convergence in pressure, velocity, and flux, as demonstrated by numerical results.

Abstract: Aiming at the development of a practically parallelizable algorithm for the simulation of fluid-structure interaction (FSI) problems in the future, in this paper a type of elliptic interface problem with jump coefficients is chosen to begin with and is reformulated to a mixture of elliptic/mixed elliptic interface problem which is analogous to a steady state FSI problem to some extent. A mixture of standard- and mixed finite element method is developed for the reformulation of the elliptic interface problem, and its well-posedness and convergence are studied by proving the inf-sup condition of a total bilinear form. With the body-fitted meshes and a strongly coupled alternating iteration scheme, numerical experiments are carried out for an elliptic interface problem with an immersed interface for different jump ratios, and the obtained numerical results confirm our convergence theorem.

Abstract: In this paper, stabilized continuous finite element methods are analyzed for numerically solving the flux which may be a non $H^1$ solution. Coercivity and error estimates are established. Numerical experiments are performed to illustrate these methods.

Abstract: In the Cartesian grid approach, the immersed boundary method (IBM) is well used to handle the boundary of an object with complicated shape on the Cartesian grid. However, the conventional IBM generates the unphysical pressure oscillations near the boundary because of the pressure jump between inside and outside of the boundary. The IBM considering pressure condition was proposed in order to remove the pressure oscillations by solving the governing equations considering the pressure condition on the boundary. In this method, there are two ways of the handling the pressure on the boundary in the Poisson solver. In this paper, the effect of removing the pressure oscillations by the IBM considering the pressure condition is investigated. And, the influence by the difference in the handling of the pressure on the boundary in the Poisson solver is investigated. In the numerical simulations of incompressible flow around a 2D circular cylinder, the present IBM indicate a greate effect of removing the pressure oscillations. And, it do not occur difference of the result by the difference of the handling the pressure on the boundary in the Poisson solver. Therefore, it is possible to select a method with less computational amount in the Poisson solver without degrading the quality of the result. It is concluded that the present IBM is very promising as improved method in order to remove the pressure oscillations in the conventional IBM.

Abstract: Many fluid flow problems involving turbulence, shocks, and material interfaces create a common issue that we call $k_\infty$ irregularity. The non-linear advection term in the governing equations for all of these problems keep generating higher wave modes as $k$ goes to infinity. In this work, we present an inviscid regularization technique, called observable regularization, for the simulation of two-phase compressible flow. In this technique, we use observable divergence theorem to derive an observable equation for tracking material interface (volume fraction). Using a couple of one-dimensional test cases, first we show that this method preserves pressure equilibrium at material interface, then we compare our results to exact Euler solutions. At the end we demonstrate a two-dimensional simulation of shock-bubble interaction showing good agreement with available experimental data from literature.

Abstract: The kidneys are vital organs that contribute to the maintenance of homeostasis in our body. They fulfill functions such as electrolyte control, blood filtration and initiation of red blood cell production in response to hypoxia, a state characterized by deficiency of oxygen (O2 ) in the renal tissue. A normal human kidney contains between 0.8 to 1.5 million functional units called nephrons. Renal tubules along the nephrons are responsible for the reabsorption of various solutes including sodium ions (Na+ ), a process that requires large amounts of O2. Physiological and pathophysiological variation in Na+ transport can alter O2 consumption, leading to changes in tissue oxygenation. We have developed a one-dimensional (1D) mathematical model for the computation of Na+ reabsorption and corresponding O2 consumption along a nephron, which is parameterized using published data obtained from a rat kidney. Our computations demonstrate that per kidney 8μ-moles of O2 are consumed per minute for the reabsorption of 125μ-moles of Na+ , which is in agreement with previously published results. The model also predicts that changes in arterial pressure adversely impact the efficiency of O2 consumption in the nephron. The model is further being extended to account for anatomically realistic renal vasculature obtained from synchrotron X-ray phase contrast micro computed tomography, with the final goal of determining O2 distribution in the whole kidney.