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The tsunami generated by the Great East Japan Earthquake (GEJE) caused serious damage to the coastal areas of the Tohoku district. Numerical simulations are used to predict damage caused by tsunamis. Shallow-water equations are generally used in numerical simulations of tsunami propagation from the open sea to the coast. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite-element techniques based on the previous investigations of Kanayama and Ohtsuka (1978). First, the viscous shallow-water equation system is derived from the Navier-Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. The derived equations have the horizontal viscosity term hereditary from the original Navier-Stokes equations. Next, a numerical finite element scheme is shown. Finally, tsunami simulations of Tohoku-Oki are shown using the above-mentioned approach. Our main concern in this chapter is how to set the boundary condition on the open boundary.

Requirements to compute stationary flow patterns are often encountered. With progress of computer environments and increasing demand of precise analyses, the number of degrees of freedom (DOF) of such a computation has become larger. However, as far as we know, computational codes are rare, which are efficient for large scale, stationary, and nonlinear flow problems. Therefore, we have developed ADVENTURE sFlow [3], which is one of modules included in the ADVENTURE project [1]. ADVENTURE sFlow uses the Newton method as the nonlinear iteration, and to compute the problem at each step of the nonlinear iteration a stabilized finite element method is introduced. Moreover, to reduce the computational costs, an iterative domain decomposition method is applied to stabilized finite element approximations of stationary Navier–Stokes equations, for which Generalized Product-type methods based on Bi-CG (GPBiCG) [6] is used as the iterative solver of the reduced linear system in each step of the nonlinear iteration. A parallel computing method using the Hierarchical Domain Decomposition Method (HDDM) is also introduced. Numerical results show that ADVENTURE sFLow can analyze a stationary flow problem with 10 million DOF.

Multi-component gas transport phenomena in a cathode electrode layer of Polymer Electrolyte Fuel Cells (PEFC) are investigated by using a two-dimensional computational model developed by Yi and Nguyen. In the past research, we encountered a problem that the solution did not converge when the same settings for physical parameters with them were used in our steady simulations. In the present study, we extend our simulation from steady to unsteady simulation to avoid the problem. Also, based on the results, the effect of changing permeability in the simulations is discussed with reporting possible errors regarding the value of permeability shown in the reference at the same time.

Hydrogen is expected as new fuel instead of fossil fuel. It will be used as fuel of a fuel cell for which development is performed actively. Many scientists are studying characteristic features of hydrogen. But it is difficult to experiment the hydrogen dispersion in case of hydrogen leaks, because hydrogen has the properties that energy for ignition is lower, rate of flame propagation in air is faster and quenching distance is shorter than hydrocarbon fuel. Therefore clarifying the hydrogen dispersion with numerical analysis becomes important. Furthermore, hydrogen dispersion under various conditions can be clarified with numerical analysis, which is useful to use hydrogen safely. This paper deals with computer simulation of the hydrogen dispersion by a finite element method. The mathematical model of hydrogen dispersion is governed by the momentum equations, the continuity equation and the hydrogen mass conservation equation. The model presented here is a three-dimensional, incompressible, non-stationary model. This paper describes a finite element method with the stabilization technique for solving Navier-Stokes equations and the advection-diffusion equation for hydrogen concentration like the Boussinesq approximation of thermal convection problems. We use Bercovier-Pironneau elements for the velocity and the pressure, and smaller P1 elements for the concentration of hydrogen. A suitable implicit time difference is also used. Numerical results are shown for a sample model.

An efficient and scalable Balancing Domain Decomposition (BDD) type preconditioner for large scale linear systems arising from 3-dimensional heat transfer problems is presented. The new method improves parallel scalability of BDD by employing an incomplete balancing technique to approximate a coarse space problem and a diagonal scaling to precondition the local fine space problems instead of the Neumann-Neumann preconditioner. It may increase the number of iterations but reduces the computation costs of the precondition process for each iteration. Consequently, total computation time and required memory are expected to be reduced. The convergence estimates may also be independent of the number of subdomains. We have implemented this algorithm on the parallel processors and have succeeded in solving some ill-conditioned large scale heat transfer problems.

We have constructed a parallel algorithm of the Balancing Domain Decomposition (BDD) in structural analysis, and succeeded in solving problems with over 100 million degrees of freedom. The results are released as an open source software, named ADVENTURE_Solid, from the ADVENTURE project. In this study, as an application of the parallel BDD algorithm of ADVENTURE_Solid to other fields, we apply it to a stationary Stokes solver. The effectiveness is shown with the consideration of applications to Navier-Stokes solvers.

Balancing Domain Decomposition (BDD) proposed by J. Mandel is an effective preconditioning technique for reducing the number of iterations of iterative Domain Decomposition Method(DDM). Until now many researches have been done on the implementation of BDD in different areas including elasticity problems and semiconductor simulation. In this paper, we implement BDD in another area that is 3-dimensional(3-D) heat conductive analysis in solid. We construct the BDD preconditioner in parallel based on Hierarchical Domain Decomposition Method (HDDM). We report the comparative performance of BDD with diagonal scaling and original DDM without preconditioning to analyse heat conductive problems with about 2 million degrees of freedom and over 11 million degrees of freedom. With BDD the convergence rates are reduced effectively and become independent of the number of subdomains.

An iterative domain decomposition method is applied to numerical analysis of 3-dimensional (3-D) nonlinear magnetostatic problems taking the magnetic vector potential as an unknown function. The nonlinear simultaneous equations are solved with the Picard iteration or the Newton iteration. In Picard iteration, we compute the reluctivity using H-B curves, and in Newton iteration we compute the reluctivity using $ \u-B $ curves or $ \u-B^2 $ curves. The simultaneous linear equations at each step of the nonlinear iteration are solved by the iterative domain decomposition method. The iterative domain decomposition method is combined with the Conjugate Gradient (CG) procedure, and the Hierarchical Domain Decomposition Method (HDDM) which is adopted for the parallel computing. Numerical results show that the iterative procedure converges, and that the computed magnetic flux density is suitable.