• Korean Journal of Mathematics
• /
• v.16 no.1
• /
• pp.41-49
• /
• 2008

We show the existence of a negative solution for the system of the following nonlinear wave equations with critical growth, under Dirichlet boundary condition and periodic condition $$u_{tt}-u_{xx}=au+b{\upsilon}+\frac{2{\alpha}}{{\alpha}+{\beta}}u_+^{\alpha-1}{\upsilon}_+^{\beta}+s{\phi}_{00}+f,\\{\upsilon}_{tt}-{\upsilon}_{xx}=cu+d{\upsilon}+\frac{2{\alpha}}{{\alpha}+{\beta}}u_+^{\alpha}{\upsilon}_+^{{\beta}-1}+t{\phi}_{00}+g,$$ where ${\alpha},{\beta}>1$ are real constants, $u_+={\max}\{u,0\},\;s,\;t{\in}R,\;{\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}$ of the wave operator and f, g are ${\pi}$-periodic, even in x and t and bounded functions.

• Korean Journal of Mathematics
• /
• v.16 no.3
• /
• pp.389-402
• /
• 2008

We show the existence of at least two solutions for a class of systems of the critical growth nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition. We first show that the system has a positive solution under suitable conditions, and next show that the system has another solution under the same conditions by the linking arguments.

• Journal of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.609-622
• /
• 1997

We investigate the existence of solutions of the nonlinear beam equation under the Dirichlet boundary condition on the interval $-\frac{2}{\pi}, \frac{2}{\pi}$ and periodic condition on the varible t, $Lu + bu^+ -au^- = f(x, t)$, when the jumping nonlinearity crosses the first positive eigenvalue.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.28 no.12
• /
• pp.1950-1957
• /
• 2004

Analysis for structures composed of materials containing regularly spaced in-homogeneities is usually executed by using averaged material properties. In order to evaluate the effective properties, a unit cell is defined and loaded somehow, and its response is investigated. The imposed loading, however, should accord to the status of unit cells immersed in the macroscopic structure to secure the accuracy of the properties. In this study, mathematical description for the periodicity of the displacement field is derived and its direct implementation into FE models of unit cell is attempted. Conventional finite element code needs no modification, and only the boundary of unit cell should be constrained in a way that the periodicity is preserved. The proposed method is applicable to skew arrayed in-homogeneity problems. Homogenized in-plane elastic properties are evaluated for a few representative cases and the accuracy is examined.

• Transactions of Materials Processing
• /
• v.32 no.1
• /
• pp.28-34
• /
• 2023

In this research, a representative volume element (RVE)-based FE Model is presented to estimate the mechanical properties of additively manufactured continuous fiber-reinforced composites with different fiber orientations. To construct the model, an ABAQUS Python script has been implemented to produce matrix and fiber in the desired orientations at the RVE. A script has also been developed to apply the periodic boundary conditions to the RVE. Experimental tests were conducted to validate the numerical models. Tensile specimens with the fiber directions aligned in the 0, 45, and 90 degrees to the loading direction were manufactured using a continuous fiber 3D printer and tensile tests were performed in the three directions. Tensile tests were also simulated using the RVE models. The predicted Young's moduli compared well with the measurements: the Young's modulus prediction accuracy values were 83.73, 97.70, and 92.92 percent for the specimens in the 0, 45, and 90 degrees, respectively. The proposed method with periodic boundary conditions precisely evaluated the elastic properties of additively manufactured continuous fiber-reinforced composites with complex microstructures.

• Korean Journal of Mathematics
• /
• v.16 no.4
• /
• pp.553-564
• /
• 2008

We have a concern with the existence of solutions (${\xi},{\eta}$) for perturbations of the parabolic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}{\xi}_t=-L{\xi}+{\mu}g(3{\xi}+{\eta})-s{\phi}_1-h_1(x,t)\;in\;{\Omega}{\times}(0,2{\pi}),\\{\eta}_t=-L{\eta}+{\nu}g(3{\xi}+{\eta})-s{\phi}_1-h_2(x,t)\;in\;{\Omega}{\times}(0,2{\pi})\end{array}.$$ We prove the uniqueness theorem when the nonlinearity does not cross eigenvalues. We also investigate multiple solutions (${\xi}(x,t),\;{\eta}(x,t)$) for perturbations of the parabolic system with Dirichlet boundary condition when the nonlinearity f' is bounded and $f^{\prime}(-{\infty})<{\lambda}_1,{\lambda}_n<(3{\mu}+{\nu})f^{\prime}(+{\infty})<{\lambda}_{n+1}$.

• Proceedings of the KIEE Conference
• /
• 1999.07a
• /
• pp.76-78
• /
• 1999

This paper presents Galerkin- and Upwind-finite element analyses solution in the travelling magnetic filed problem. The travelling magnetic field problem is subject to convective- diffusion equation. Therefore, the solution derived from Galerkin-FEM with linear interpolation function may oscillate between the adjacent nodes. A simple model with Derichlet, Noumann and periodic boundary condition respectively, have been analyzed to investigate stabilities of solutions. It is concluded that the solution of Galerkin-FEM may oscillate according to boundary condition and element type, but that of Upwind-FEM is stable regardless boundary condition.

• Honam Mathematical Journal
• /
• v.31 no.1
• /
• pp.75-85
• /
• 2009

We show the existence of the nontrivial periodic solution for a class of the system of the nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition by critical point theory and linking arguments. We investigate the geometry of the sublevel sets of the corresponding functional of the system, the topology of the sublevel sets and linking construction between two sublevel sets. Since the functional is strongly indefinite, we use the linking theorem for the strongly indefinite functional and the notion of the suitable version of the Palais-Smale condition.

• 한국전산유체공학회:학술대회논문집
• /
• 2003.10a
• /
• pp.286-288
• /
• 2003

The physical model considered here is a horizontal layer of fluid heated below and cooled above with a periodic array of evenly spaced square cylinders placed at the center of the layer, whose aspect ratio here varies from unity to six. Periodic boundary condition is employed along the horizontal direction to allow for lateral freedom for the convection cells. Two-dimensional solution for unsteady natural convection is obtained using an accurate and efficient Chebyshev spectral multi-domain methodology for a given Rayleigh numbers of $10^6$

• Communications of the Korean Mathematical Society
• /
• v.14 no.2
• /
• pp.337-352
• /
• 1999

In this paper we find a geometric condition for the weaker principle of spatial averaging (PSA) for a class of polyhedral domains. Let \ulcorner be a polyhedron in R\ulcorner, n$\leq$3. If all dihedral angles of \ulcorner are submultiples of $\pi$, then there exists a parallelopiped \ulcorner generated by n linearily independent vectors {\ulcorner}\ulcorner in R\ulcorner containing \ulcorner so that solutions of $\Delta$u+λu=0 in \ulcorner with either the boundary condition u=0 or ∂u/∂n=0 are expressed by linear combinations of those of $\Delta$u+λn=0 in \ulcorner with periodic boundary condition. Moreover, if {\ulcorner}\ulcorner satisfies rational condition, we guarantee the weaker PSA for the domain \ulcorner.