Makoto Natori

An application of the integrated penalty method to free boundary problems of laplace equation

Free boundary problems of Laplace equation defined in the annulus in are numerically solved. The original problem is transformed to an optimization problem. The state equation is approximated by an equation with a penalty term which approximates one of boundary conditions on the free boundary. The flux through the free boundary is calculated by the integration of the penalty term introduced above ( Integrated Penalty Method ). This penalized optimization problem is numerically solved by finite difference method. Incomplete Cholesky decomposition combined with the conjugate gradient method is used to solve systems of linear equations. Some numerical examples are given.

Numerical Solution of the Free Surface Drainage Problem of Two Immiscible Fluids by the Boundary-Element Method

A numerical method is presented for solving the unsteady porous flow problem of two immiscible fluids with free surfaces. The velocity potential of each fluid is expressed as the solution of the Laplace equation, which is solved by the boundary-element method. The free surface equation is solved by a Crank-Nicolson type procedure.

Numerical Investigations on Magnetic Shielding Performance of Axisymmetric HTS Plates against Several-Turn-Coil Fields

The magnetic shielding performance of high-Tc superconducting (HTS) plates is numerically investigated. In order to reflect the experimental conditions accurately, a several-turn coil is assumed as the source of the magnetic field. A numerical code for analyzing the time evolution of the shielding current density has been developed and, by use of the code, the shielding performance of HTS plates is investigated. The results of computations show that the spatial distribution of the shielding current density changes drastically with the distance between the plate and the coil. The shielding current density concentrates near the edge of the plate in case of the large distance, whereas it also concentrates just below the coil in case of the sufficiently small distance.

Numerical Method for Magnetic Shielding Analysis of Axisymmetric HTS Plates by Flux Flow Creep Model

The numerical method for analyzing the magnetic shielding effect of high Tc superconducting (HTS) plates is developed on the basis of the flux flow creep model. By taking account of the crystallographic anisotropy of the superconductor, the multiple-thin-layer approximation is employed to modelize the HTS plate. In this case, the shielding current density is governed by the integral-differential equation. When the equation is discretized by using FEM and the θ-method, the resulting equation has a strong nonlinearity due to the flux flow creep model. As the method for solving the nonlinear equation, the Newton-Raphson method is utilized.

Numerical solution of free boundary problem for unsteady slag flow in the hearth

A numerical method for solving a problem in unsteady slag flow in the hearth of a blast furnace is presented. This problem is reduced to a free boundary problem for an elliptic system. The potential problem for a given free boundary is approximated by the penalty method. The derivatives of the potential function on the free boundary is approximated by the integration of the penalty term, and then the subsequent shape of the free boundary is obtained by solving the differential equation for the motion of the free boundary. The finite difference method is used to solve the penalized problem. A numerical example is given.

Numerical simulation on contactless measurement of critical current density in HTS

According to the experiments by Claassen et al., (1991), the odd harmonics of the magnetic field are excited with an increase in the amplitude of the applied ac magnetic field. Especially, the third harmonic is suddenly excited after the amplitude exceeds a certain limit. This onset of the third harmonics indicates that the critical current density is following in all over the HTS. On the basis of the experimental results, Claassen developed the contactless method for measuring the critical current density. The purpose of this study is to reproduce Claassens results by means of the numerical simulation. To this end, the governing equation of the shielding current density has been formulated and the numerical code for solving the initial-boundary-value problem of the equation has been developed. After the shielding current density is evaluated by using the code, the magnetic field generated by the shielding current density is calculated as a function of time. Spectral analysis of the field is performed. The results of computations show that the flux-flow region covers the whole volume of the HTS above a certain limit of the amplitude of the applied magnetic field.

Effect on Spectral Properties by the Splitting Correction Preconditioning for Linear Systems that Arise from Periodic Boundary Problems

In this paper, the spectral properties of the preconditioned systems by the “Splitting Correction (SC)”, proposed by the present authors, are studied and it is conjectured that the degeneracy not the clustering of the eiganvalues plays an important role in the convergence. The SC preconditioner is one of new pre-conditioners based on block factorization for solving linear systems that arise from periodic boundary problems. From the viewpoint of the convergence of residual norm, the behaviors of the residual norm of the conjugate gradient (CG) method preconditioned by the SC and the block incomplete Cholesky (block IC) are very peculiar. Generally, the convergence of the CG method depends on spectral properties, such as the clustering and the degeneracy of the eigenvalues, of the coefficient matrix. Some numerical results suggest that the fast convergence of the SC is due not to the clustering but to the degeneracy of the eigenvalues of the preconditioned coefficient matrix.

Frequency Dependence of Magnetic Shielding Performance in HTS Plates

The magnetic shielding performance of the HTS and the copper plates has been investigated numerically. A numerical code for analyzing the time evolution of the shielding current density has been developed and, by use of the code, the shielding coefficients are calculated for both the HTS and the copper plates. The results of computations show that the magnetic shielding performance of the HTS plates has no dependence on frequency ω of the applied magnetic field. On the other hand, the magnetic shielding performance of the copper plates depends strongly on ω;the shielding effect of the copper plates is enhanced remarkably With the increase of ω.

An Application of the Fuzzy Theory for an III-Posed Problem

Ill-posed problems are usually solved by the transformation to minimization problems. They are ill-conditioned, then additional techniques, i.e. regularizations, are adopted to avoid the oscillation. In practical problems they are usually so complicated that it is not easy to adopt such effective techniques. Here two points should be focused on for practical problems. First, high accuracy is not necessary. Second, engineers who have much experience about the problems know how to deal with them qualitatively. These points suggest the validity of flexible minimizers. In this paper the fuzzy theory is introduced to construct such minimizers and it is applied to an ill-posed shape design problem. Numerical results are satisfactory.

Estimation of Electrical Conductivity of Composite Materials. : II. Numerical Method for Two-Dimensional Tetragonal Texture

A relation between electrical conductivity and effective thickness of grain boundaries is given for a two-dimensional tetragonal texture as a typical pattern of the composite body by a numerical calculation.