• Steel and Composite Structures
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• v.28 no.6
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• pp.655-670
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• 2018

A coupled method, that combines the Ritz method and the finite element (FE) method, is proposed to solve the vibration problem of rectangular thin and thick plates with general boundary conditions. The eigenvalue partial differential equation(s) of the plate is (are) first reduced to a set of eigenvalue ordinary differential equations by the application of the Ritz method. The resulting eigenvalue differential equations are then reduced to an eigenvalue algebraic equation system using the finite element method. The natural boundary conditions of the plate problem including the free edge and free corner boundary conditions are also implemented in a simple and accurate manner. Various boundary conditions including simply supported, clamped and free boundary conditions are considered. Comparisons with existing numerical and analytical solutions show that the proposed mixed method can produce highly accurate results for the problems considered using a small number of Ritz terms and finite elements. The proposed mixed Ritz-FE formulation is also compared with the mixed FE-Ritz formulation which has been recently proposed by the present author and his co-author. It is found that the proposed mixed Ritz-FE formulation is more efficient than the mixed FE-Ritz formulation for free vibration analysis of rectangular plates with Levy-type boundary conditions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.2 no.1
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• pp.27-39
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• 1998

We describe a fast numerical method for non-separable elliptic equations in self-adjoin form on irregular adaptive domains. One of the most successful results in numerical PDE is developing rapid elliptic solvers for separable EPDEs, for example, Fourier transformation methods for Poisson problem on a square, however, it is known that there is no rapid elliptic solvers capable of solving a general nonseparable problems. It is the purpose of this paper to present an iterative solver for linear EPDEs in self-adjoint form. The scheme discussed in this paper solves a given non-separable equation using a sequence of solutions of Poisson equations, therefore, the most important key for such a method is having a good Poison solver. High performance is achieved by using a fast high-order adaptive Poisson solver which requires only about 500 floating point operations per gridpoint in order to obtain machine precision for both the computed solution and its partial derivatives. A few numerical examples have been presented.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.185-190
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• 1989

In [4], J. Leray introduced the notion of partial hyperbolicity to characterize the operators for which the non-characteristic Cauchy problem is solvable in the Geverey class for any data which are holomorphic in a part of variables x"=(x$_{2}$,..,x$_{l}$ ) in the initial hyperplane x$_{1}$=0. A linear partial differential operator is called partially hyperbolic modulo the linear subvarieties S:x"=constant if the equation P$_{m}$(x, .zeta.$_{1}$, .xi.')=0 for .zeta.$_{1}$ has only real roots when .xi.'is real and .xi."=0, where P$_{m}$ is the principal symbol of pp. Limiting to the case of operators with constant coefficients, A. Kaneko proposed a new sharper condition when S is a hyperplane [3]. In this paper, we generalize this condition to the case of general linear subvariety S and show that it is sufficient for the solvability of Cauchy problem for the hyperfunction Cauchy data which contains variables parallel to S as holomorphic parameters.blem for the hyperfunction Cauchy data which contains variables parallel to S as holomorphic parameters.

• Proceedings of the Korean Institute of Navigation and Port Research Conference
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• 2009.10a
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• pp.239-240
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• 2009

Vessels stability problems need to resolve the nonlinear mathematical models of rolling motion. For nonlinear systems subjected to random excitations, there are very few special cases can obtain the exact solutions. In this paper, the specific differential equations of rolling motion for intact ship considering the restoring and damping moment have researched firstly. Then the partial stochastic linearization method is applied to study the response statistics of nonlinear ship rolling motion in beam seas. The ship rolling nonlinear stochastic differential equation is then solved approximately by keeping the equivalent damping coefficient as a parameter and nonlinear response of the ship is determined in the frequency domain by a linear analysis method finally.

• Journal of Navigation and Port Research
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• v.33 no.9
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• pp.629-634
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• 2009

Vessels stability problems need to resolve the nonlinear mathematical models of rolling motion. For nonlinear systems subjected to random excitations, there are very few special cases can obtain the exact solutions. In this paper, the specific differential equations of rolling motion for intact ship considering the restoring and damping moment have researched firstly. Then the partial stochastic linearization method is applied to study the response statistics of nonlinear ship rolling motion in beam seas. The ship rolling nonlinear stochastic differential equation is then solved approximately by keeping the equivalent damping coefficient as a parameter and nonlinear response of the ship is determined in the frequency domain by a linear analysis method finally.

• Honam Mathematical Journal
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• v.40 no.4
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• pp.785-794
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• 2018

In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the origin, and compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple singular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.

• 한국해양학회지
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• v.16 no.2
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• pp.99-107
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• 1981

A partial differential equation for the adjusted sea level, obtained from the long wave equations in shallow water, is reduced to a simpler one by the use of physically reasonable approximations based on the observations. The similar equation for the stream function indicates that shelf waves are generated by the longshore wind stress. This indication is in good agreement with the high correlation between the adjusted sea levels and the longshore wind stress. From the dispersion relationship and the boundary conditions, there exist a countable infinite number of modes which satisfy a first-order wave equations. The adjusted sea level for a given wind stress can easily be calculated by utilizing the convolution and the Fourier transformation. Some detailed solutions are presented here for sinusoidal and exponential wind stress.

• Proceedings of the Korean Geotechical Society Conference
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• 2010.09a
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• pp.808-812
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• 2010

A great deal of attention is focused on coupled Thermo-Hydro-Mechanical (THM) behavior of multiphase porous media in diverse geo-mechanical and geo-environmental areas. This paper presents general governing equations for coupled THM processes in unsaturated porous media. Coupled partial differential equations are derived from 3 mass balances equations (solid, water, and air), energy balance equation, and force equilibrium equation. Finite element code is developed from the Galerkin formulation and time integration of these governing equations for 4 main variables (displacement $\underline{u}$, gas pressure $P_g$, liquid pressure $P_l$), and temperature T). The code is validated with theoretical solutions for linear material with simple boundary conditions.

• Transactions of the Korean Society of Mechanical Engineers
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• v.3 no.3
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• pp.91-97
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• 1979

Boundary layer equations (partial differential equations) can be transformed to ordinary diffential equations with constant coeffieients in terms of similarity transformed to ordinary differential equations with constant coeffieients in terms of similarity transformations in the heat tranfer analysis on the surface of any axiaymmetric boiles. The azimuth functions can not be uniquely determined because of the singular behavior at the stagnation point(X=0.deg.).In spite of the azimuth functions behaving singularly, many of researchers have analyzed the heat transfer problem on a horizontal chlinder or a sphere, supposing the set of solutions( $H_{1}$ & G$_{1}$) of being yieled from the simple differential equation to be unique solution of therazimuth functions. In order to ascertain whether mathematical incompatibility as mentioned above can be admitted in the viewpoint of enginerring or not, condensation heat transfer coefficients on a sphere are computed for all azimuth functions( $H_{1}$ G$_{1}$ & $H_{2}$ G$_{2}$) and comparisons with the experimental result are discussed.

• KIEAE Journal
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• v.16 no.1
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• pp.81-87
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• 2016

Purpose: Heat transfers phenomena are described by the second order partial differential equation and its boundary conditions. In a three-dimensional structure of a building, the heat transfer phenomena generally include more than one material, and thus, become complicate. The analytic solutions are useful to understand heat transfer phenomena, but they can hardly be applied in engineering or design problems. Engineers and designers have generally been forced to use numerical methods providing reliable results. Finite volume methods with the unstructured grid system is only the suitable means of the analysis for the complex and arbitrary domains. Method: To obtain an numerical solution, a discretization method, which approximates the differential equations, and the interpolation methods for temperature and heat flux between two or more materials are required. The discretization methods are applied to small domains in space and time, and these numerical solutions form the descretized equations provide approximated solutions in both space and time. The accuracy of numerical solutions is dependent on the quality of discretizations and size of cells used. The higher accuracy, the higher numerical resources are required. The balance between the accuracy and difficulty of the numerical methods is critical for the success of the numerical analysis. A simple and easy interpolation methods among multiple materials are developed. The linear equations are solved with the BiCGSTAB being a effective matrix solver. Result: This study provides an overview of discretization methods, boundary interface, and matrix solver for the 3-dimensional numerical heat transfer including two materials.