• 대한수학회논문집
• /
• 제34권1호
• /
• pp.155-167
• /
• 2019

In this article we are concerned with some non-local problems of Kirchhoff type with Dirichlet boundary condition in Orlicz-Sobolev spaces. A result of the existence of infinitely many solutions is established using variational methods and Ricceri's critical points principle modified by Bonanno.

• 대한수학회지
• /
• 제56권5호
• /
• pp.1419-1439
• /
• 2019

In this paper, we consider a class of noncooperative fourth-order elliptic systems involving nonlocal terms and critical growth in a bounded domain. With the help of Limit Index Theory due to Li [32] combined with the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity. Our results significantly complement and improve some recent results on the existence of solutions for fourth-order elliptic equations and Kirchhoff type problems with critical growth.

• 대한수학회보
• /
• 제54권3호
• /
• pp.1023-1036
• /
• 2017

In this paper, we study the existence of positive solutions to the p-Kirchhoff elliptic equation involving general subcritical growth $(a+{\lambda}{\int_{\mathbb{R}^N}{\mid}{\nabla}u{\mid}^pdx+{\lambda}b{\int_{\mathbb{R}^N}{\mid}u{\mid}^pdx)(-{\Delta}_pu+b{\mid}u{\mid}^{p-2}u)=h(u)$, in ${\mathbb{R}}^N$, where a, b > 0, ${\lambda}$ is a parameter and the nonlinearity h(s) satisfies the weaker conditions than the ones in our known literature. We also consider the asymptotics of solutions with respect to the parameter ${\lambda}$.

• 대한수학회논문집
• /
• 제36권2호
• /
• pp.247-256
• /
• 2021

Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: $$(\mathcal{P}_{\lambda})\;\{\begin{array}{lll}{\Delta}^2u-(a{\int}_{\Omega}{\mid}{\nabla}u{\mid}^2dx+b){\Delta}u+cu=f(x){\mid}u{\mid}^{-{\gamma}}-{\lambda}{\mid}u{\mid}^{p-2}u&&\text{ in }{\Omega},\\{\Delta}u=u=0&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω is a smooth bounded domain of ℝⁿ (n ≥ 5), ∆² is the biharmonic operator, and ∇u denotes the spatial gradient of u and 0 < γ < 1, λ > 0, 0 < p ≤ 2^# and a, b, c are three positive constants with a + b > 0 and f belongs to a given Lebesgue space.

• 한국소음진동공학회:학술대회논문집
• /
• 한국소음진동공학회 2004년도 추계학술대회논문집
• /
• pp.916-919
• /
• 2004

The monopole theory has long been used to model air-pumped effect from the elastic cavities in car tire. This approach models the change of an air as a piston moving backward and forward on a spring and equates local air movements exactly with the volume changes of the system. Thus, the monopole theory has a restricted domain of applicability due to the usual assumption of a small amplitude acoustic wave equation and acoustic monopole theory. This paper describes an approach to predict the air-pumping noise of a car ave with CFD/Kirchhoff integral method. The type groove is simply modeled as piston-cavity-sliding door geometry and with the aid of CFD technique flow properties in the groove of rolling car tyre are acquired. And these unsteady flow data are used as a air-pumping source in the next Cm calculation of full tyre-road geometry. Acoustic far field is predicted from Kirchhoff integral method by using unsteady flow data in space and time, which is provided by the CFD calculation of full tyre-road domain. This approach can cover the non-linearity of acoustic monopole theory with the aid of using Non-linear governing equation in CFD calculation. The method proposed in this paper is applied to the prediction of air-pumping noise of modeled car tyre and the predicted results are qualitatively compared with the experimental data.

• 한국정보통신학회논문지
• /
• 제26권11호
• /
• pp.1586-1591
• /
• 2022

Kirchhoff-Love 판 (KLP) 방정식은 특정 외력이 얇은 막에 끼치는 변형을 기술하는 잘 알려진 이론이다. 한편, frequency 도메인에서 진동하는 판을 해석하는 것은 주요 진동 주파수와 고유함수들을 구하는 것과 판의 진동을 예측하는데 중요하다. 다양한 모드 분석 방법들 중 dynamic mode decomposition (DMD)는 효율적인 data 기반 방법이다. 이 논문에서 우리는 DMD를 기반으로 sine 유형 외력의 영향력 안에 있는 KLP의 모드 분석을 수행한다. 우리는 먼저 유한차분법을 사용하여 이산적으로 표현된 시계열 형식의 KLP 해를 구한다. 720,00개의 FDM으로 생성된 해중에서, 오직 500개의 해만을 DMD의 구현을 위해 선택한다. 우리는 결과적으로 얻어진 DMD-mode를 보고한다. 또한, DMD를 통하여 KLP의 해를 예측하는 효율적인 방법을 소개한다.

• Structural Engineering and Mechanics
• /
• 제61권4호
• /
• pp.437-440
• /
• 2017

This study investigates the bending of an isotropic thin rectangular plate in finite deformation. Employing hyperelastic material of John's type, a non-classical model which generalizes the famous Kirchhoff's plate equation is obtained. Exact solution for deflection of the plate under sinusoidal loads is obtained. Finally, it is shown that the non-classical plate under consideration can be used as a replacement for Kirchhoff's plate on an elastic foundation.

• 한국소음진동공학회:학술대회논문집
• /
• 한국소음진동공학회 2007년도 춘계학술대회논문집
• /
• pp.1062-1073
• /
• 2007

Acoustic Target Strength (TS) is a major parameter of the active sonar equation, which indicates the ratio of the radiated intensity from the source to the re-radiated intensity by a target. In developing a TS equation, it is assumed that the radiated pressure is known and the re-radiated intensity is unknown. This research provides a TS equation for polygonal plates, which is applicable to near field acoustics. In this research, Helmholtz-Kirchhoff formula is used as the primary equation for solving the re-radiated pressure field; the primary equation contains a surface (double) integral representation. The double integral representation can be reduced to a closed form, which involves only a line (single) integral representation of the boundary of the surface area by applying Stoke's theorem. Use of such line integral representations can reduce the cost of numerical calculation. Also Kirchhoff approximation is used to solve the surface values such as pressure and particle velocity. Finally, a generalized definition of Sonar Cross Section (SCS) that is applicable to near field is suggested. The TS equation for polygonal plates in near field is developed using the three prescribed statements; the redection to line integral representation, Kirchhoff approximation and a generalized definition of SCS. The equation developed in this research is applicable to near field, and therefore, no approximations are allowed except the Kirchhoff approximation. However, examinations with various types of models for reliability show that the equation has good performance in its applications. To analyze a general shape of model, a submarine type model was selected and successfully analyzed.

• 대한수학회보
• /
• 제55권5호
• /
• pp.1441-1462
• /
• 2018

In this paper, we investigate the fractional p&q-Kirchhoff type system $$\{M_1([u]^p_{s,p})(-{\Delta})^s_pu+V_1(x){\mid}u{\mid}^{p-2}u\\{\hfill{10}}={\ell}k^{-1}F_u(x,\;u,\;v)+{\lambda}{\alpha}(x){\mid}u{\mid}^{m-2}u,\;x{\in}{\Omega}\\M_2([u]^q_{s,q})(-{\Delta})^s_qv+V_2(x){\mid}v{\mid}^{q-2}v\\{\hfill{10}}={\ell}k^{-1}F_v(x,u,v)+{\mu}{\alpha}(x){\mid}v{\mid}^{m-2}v,\;x{\in}{\Omega},\\u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}{\subset}{\mathbb{R}}^N$ is an unbounded domain with smooth boundary ${\partial}{\Omega}$, and $0<s<1<p{\leq}q$ and sq < N, ${\lambda},{\mu}>0$, $1<m{\leq}k<p^*_s$, ${\ell}{\in}R$, while $[u]^t_{s,t}$ denotes the Gagliardo semi-norm given in (1.2) below. $V_1(x)$, $V_2(x)$, $a(x):{\mathbb{R}}^N{\rightarrow}(0,\;{\infty})$ are three positive weights, $M_1$, $M_2$ are continuous and positive functions in ${\mathbb{R}}^+$. Using variational methods, we prove existence of infinitely many high-energy solutions for the above system.

• Structural Engineering and Mechanics
• /
• 제19권6호
• /
• pp.619-637
• /
• 2005

The objective of this research was to determine the Vlasov soil parameters for quadratically varying elasticity modulus $E_s$(z) of the compressible soil continuum and discuss the interaction affect between two close plates. Interaction problem carried on for uniformly distributed load carrying plates. Plate region was simulated by Kirchhoff plate theory based (mixed or displacement type) 2D elements and the foundation continuum was simulated by displacement type 2D elements. At the contact region, plate and foundation elements were geometrically coupled with each other. In this study the necessary formulas for the Vlasov parameters were derived when Young's modulus of the soil continuum was varying as a quadratic function of z-coordinate through the depth of the foundation. In the examples, first the elements and the iterative FE algorithm was verified and later the results of quadratic variation of $E_s$(z) were compared with the previous examples in order to discuss the general behavior. As a final example two plates close to each other resting on elastic foundation were handled to see their interaction influences on the Vlasov foundation parameters. Original examples were solved using both mixed and displacement type plate elements in order to confirm the results.