Hirofumi Notsu

Kanazawa University | Kindai · Faculty of Mathematics and Physics

A new reformulation of a free boundary problem for the Stokes equations, which govern a viscous flow with an overdetermined condition on the free boundary, is proposed. The idea of the method is to transform the governing equations into a boundary value problem with a complex Robin boundary condition that couples the two boundary conditions on the...

A two-step Lagrange–Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC) is presented. First, we show the experimental order of convergence to see the second-order accuracy in time realized by the two-step methods for conservative and non-conservative material derivatives along the trajectory of fluid particl...

A new reformulation of a free boundary problem for the Stokes equations governing a viscous flow with overdetermined condition on the free boundary is proposed. The idea of the method is to transform the governing equations to a boundary value problem with a complex Robin boundary condition coupling the two boundary conditions on the free boundary....

A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of \(L^2\)-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated prob...

In this study, a shape optimization problem for the two-dimensional stationary Navier–Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the us...

We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity...

In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of secon...

A convective boundary condition (CBC) on an outflow boundary for flow problems governed by the Navier–Stokes equations is studied, and existence and uniqueness results of solutions are established. By illustrating other known outflow boundary conditions, such as “do-nothing”, “directional do-nothing” and “total pressure”, the difference of flows is...

Due to computational complexity, fluid flow problems are mostly defined on a bounded domain. Hence, capturing fluid outflow calls for imposing an appropriate condition on the boundary where the said outflow is prescribed. Usually, the Neumann-type boundary condition called do-nothing condition is the go-to description for such outflow phenomenon Ho...

In this study, a shape optimization problem for the two-dimensional stationary Navier--Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the u...

A mass-conservative Lagrange--Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange--Galerkin method, i.e., CFL-free robustness for convection-dominated problems an...

In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of secon...

Fluids exist universally in nature and technology. Among the many types of fluid flows is the well-known vortex shedding, which takes place when a fluid flows past a bluff body. Diverse types of vortices can be found in this flow as the Reynolds number increases. In this study, we reveal that these vortices can be employed for conducting certain ty...

We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity...

Herein, the Karman vortex system is considered to be a large recurrent neural network, and the computational capability is numerically evaluated by emulating nonlinear dynamical systems and the memory capacity. Therefore, the Reynolds number dependence of the Karman vortex system computational performance is revealed and the optimal computational p...

Nonlinear magnetization dynamics excited by spin-transfer effect with feedback current is studied both numerically and analytically. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagn...

Nonlinear magnetization dynamics excited by spin-transfer effect with feedback current is studied both numerically and analytically. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagn...

We introduce a new Lagrange–Galerkin scheme for computing fluid flow through porous media. The method of volume averaging of the velocity and pressure deviation in the pore is employed to derive the macroscopic mass and momentum conservation in the porous medium in which this technique was established by Hsu and Cheng. We derive the Lagrange–Galerk...

We generalize pressure boundary conditions of an \(\varepsilon \)-Stokes problem. Our \(\varepsilon \)-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter \(\varepsilon >0\). For the Dirichlet boundary condition, it is proven in Matsui and Muntean (Adv Math Sci Appl, 27:181–191, 2...

An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution i...

The purposes of this work are to study the $L^{2}$-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically.The stability estimate is obtained thanks to the presence of a nonlinear drag f...

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirich-let and the slip boundary conditions. For the SWEs with a transmiss...

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirichlet and the slip boundary conditions. For the SWEs with a transmissi...

We generalize pressure boundary conditions of an $\varepsilon$-Stokes problem. Our $\varepsilon$-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter $\varepsilon>0$. For the Dirichlet boundary condition, it is proven in K. Matsui and A. Muntean (2018) that the solution for the $\v...

In the field of micromanipulation, an in situ three-axial rotation of a microscale object remains difficult to realize, with rotational resolution and repeatability remaining low. In this paper, we describe the fundamental principle, properties, and experimental results of multi-axial non-contact in situ micromanipulation of an egg cell driven by s...

As the sequel to the proceedings of the International Conference of Continuum Mechanics Focusing on Singularities (CoMFoS15), the proceedings of CoMFoS16 present further advances and new topics in mathematical theory and numerical simulations related to various aspects of continuum mechanics. These include fracture mechanics, shape optimization, mo...

Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated pro...

Two stabilized Lagrange–Galerkin schemes for the Navier–Stokes equations are reviewed. The schemes are based on a combination of the Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solv...

In this paper, we propose new energy dissipative characteristic numerical methods for the approximation of diffusive Oldroyd-B equations that are based either on the finite element or finite difference discretization. We prove energy stability of both schemes and illustrate their behavior on a series of numerical experiments. Using both the diffusi...

Tornadoes are one type of violent flow phenomenon and occur in many places in the world. There are many research methods that aim to reduce the loss of human lives and material damage caused by tornadoes. One effective method is numerical simulation such as that in Ishihara et al. (J. Wind Engng Ind. Aerodyn., vol. 99, 2011, pp. 239–248). The swirl...

A nonlinear stabilized Lagrange-Galerkin scheme for the Oseen-type Peterlin viscoelastic model is presented. Error estimates with the optimal convergence order are proved without any relation between the time increment and the mesh size. The result holds valid for both cases of diffusive and non-diffusive conformation tensors. The scheme is a combi...

This is the second part of our error analysis of the stabilized Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi-Pitkäranta's stabilization method for the conforming linear elements, which leads to an efficient computation with a small number of d...

Optimal error estimates of stable and stabilized Lagrange-Galerkin (LG) schemes for natural convection problems are proved under a mild condition on time increment and mesh size. The schemes maintain the common advantages of the LG method, i.e., robustness for convection-dominated problems and symmetry of the coefficient matrix of the system of lin...

Error estimates with optimal convergence orders are proved for a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a combination of Lagrange-Galerkin method and Brezzi-Pitkaranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of...

Error estimates with the optimal convergence order are proved for a pressure-stabilized characteristics finite element scheme for the Oseen equations. The scheme is a combination of Lagrange-Galerkin finite element method and Brezzi-Pitkäranta’s stabilization method. The scheme maintains the advantages of both methods; (i) It is robust for convecti...

We study spring-block systems which are equivalent to the P1-finite element methods for the linear elliptic partial differential equation of second order and for the equations of linear elasticity. Each derived spring-block system is consistent with the original partial differential equation, since it is discretized by P1-FEM. Symmetry and positive...

A new finite difference scheme based on the method of characteristics is presented for convection-diffusion problems. The scheme is of single-step and second order in time, and the matrix of the derived system of linear equations is symmetric. Since it is a finite difference scheme, we can get rid of numerical integration which may cause some insta...

A pressure-stabilized characteristics finite element scheme for the Oseen equations is presented. Stability and convergence results with the optimal error estimates for the velocity and the pressure are proved. The scheme can deal with convection-dominated problems and leads to a symmetric coefficient matrix of the system of linear equations. A che...

A new type of mesh generator is developed by using a self-organized pattern in a reaction–diffusion system. The system is the Gray–Scott model, which creates a spot pattern in a specific parameter region. The spots correspond to nodes of a mesh. The mesh generator has several advantages: the algorithm is simple and processes to improve the mesh, su...

This paper describes an analysis of high frequency electromagnetic fields using the finite element method of Maxwell equations including the displacement current. To solve a large-scale complex symmetric problem on a parallel computer efficiently, a domain decomposition method based on a COCR method is proposed. The COCR method improved convergence...

Two new finite difference schemes based on the method of characteristics are presented for convection-diffusion problems. Both of the schemes are of second order in time, and the matrices of the derived systems of linear equations are symmetric. No numerical integration is required for them. The one is of first order in space and the other is of se...

Simulated by sFlow 1.1 (in-house version) in Sun Yat-sen Univeristy, post-processed by MicroAVS.

To solve large scale 3-D non-stationary incompressible flow problems, an algorithm to perform an Incomplete Balancing Domain Decomposition (IBDD) is constructed in this work; as an improvement of the Balancing Domain Decomposition (BDD) method, it reduce the computation cost of the coarse problem by replacing the exact balancing procedure with an i...

Many attempts were made in the past to investigate numerically the metal-hydrogen interactions at macro-scale but the actual microstructure was generally not introduced into the analyses. The objective of this work is to simulate, on an artificial polycrystal, the effect of the microstructure-induced stress-strain field heterogeneity on the interna...

Numerical solutions for large scale models face the problem of poor convergence and for non-stationary problems the situation becomes worse. To speed up the convergence, an algorithm to perform the Balancing Domain Decomposition (BDD) preconditioning for non-stationary incompressible flow problems is constructed. The problem caused by the presence...

The presence of nonlinear convection terms complicates the solving of the non-stationary Navier-Stokes equations, and this difficulty increases with the Reynolds number. In this work, the characteristic-curve method is employed to handle this term. The method is advantageous as it renders the matrix for linear equations symmetric, thus enabling the...

Hydrogen embrittlement is a well identified cause of material degradation and is the origin of major failure in industry. The deleterious effect of hydrogen uptake in metals and alloys is especially observed on mechanical properties. For most of the metal-hydrogen systems, fracture strength and fracture toughness are decreased and fatigue crack pro...

Hydrogen can affect drastically the mechanical properties of stainless steels by the so called Hydrogen embrittlement. It is crucial to be able to describe the hydrogen content in the material especially at the microscopic scale. The diffusion process is depending on the fugacity of hydrogen on the surface, on the diffusivity and solubility in the...

The hydrogen diffusion problem in materials under repeated load is considered. We focus on the difference of the numerical results using isotropic- and kinematic-hardenings. Three dimensional numerical results are shown to see the difference.

In this paper we present a new single-step characteristic-curve finite element scheme of second order in time for the nonstationary incompressible Navier-Stokes equations. After supplying correction terms in the variational formulation, we prove that the scheme is of second order in time. The convergence rate of the scheme is numerically recognized...

We apply a newly developed characteristic-curve finite element scheme to cavity flow problems. The scheme is useful for large scale computation, because P1/P1 element is employed and the matrix of resulting linear system is symmetric. Numerical results of two- and three-dimensional cavity flow problems are presented. Three types of the Dirichlet bo...

We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity. We reveal that the depth of the potential well...

A level set method based on the piecewise linear finite element approximation of the signed distance function is proposed for several moving boundary problems. As a prototype of our level set method, we consider a level set discretization of the mean curvature flow problem and give an effective algorithm guaranteed by the maximum principle. Two-pha...

A level set method based on the piecewise linear finite element approximation of the signed distance function is proposed for several moving boundary problems. As a prototype of our level set method, we consider a level set discretization of the mean curvature flow problem and give an effective algorithm guaranteed by the maximum principle. Two-pha...