• Structural Engineering and Mechanics
• /
• 제85권5호
• /
• pp.621-633
• /
• 2023

The major objective of this paper is to study the receding contact problem between two functional graded layers under a flat indenter. The gravity is assumed negligible, and the shear moduli of both layers are assumed to vary exponentially along the thickness direction. In the absence of body forces, the problem is reduced to a system of Fredholm singular integral equations with the contact pressure and contact size as unknowns via Fourier integral transform, which is transformed into an algebraic one by the Gauss-Chebyshev quadratures and polynomials of both the first and second kinds. Then, an iterative speediest descending algorithm is proposed to numerically solve the system of algebraic equations. Both semi-analytical and finite element method, FEM solutions for the presented problem validate each other. To improve the accuracy of the numerical result of FEM, a graded FEM solution is performed to simulate the FGM mechanical characteristics. The results reveal the potential links between the contact stress/size and the indenter size, the thickness, as well as some other material properties of FGM.

• Structural Engineering and Mechanics
• /
• 제6권4호
• /
• pp.457-466
• /
• 1998

The dynamic response of an elastic torsion shaft of revolution is analysed by the Line-Loaded Integral Equation Method (LLIEM). A "Dynamic Point Ring Couple" (DPRC) is used as a fictitious fundamental load and is distributed in an elastic space along the axis of the shaft outside the shaft occupation. According to the boundary condition, our problem is reduced to a 1-D Fredholm integral equation of the first kind, which is simpler for solving than that of a 2-D singular integral equation of the same kind obtanied by Boundary Element Method (BEM), for steady periodically varied loading. Numerical example of a shaft with quadratic generator under sinusoidal type of torque is given. Formulas for stresses and dangerous frequency are mentioned.

• Journal of Mechanical Science and Technology
• /
• 제14권8호
• /
• pp.845-850
• /
• 2000

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing a Griffith crack under in-plane normal loading within the framework of linear piezoelectricity. The potential theory method and Fourier transforms are used to reduce the problem to the solution of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the field intensity factors are obtained, and the influences of the electric fields for PZT-6B piezoelectric ceramic are discussed.

• Journal of applied mathematics & informatics
• /
• 제5권2호
• /
• pp.293-300
• /
• 1998

The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernels using the Toeplitz matrices. The solution has a computing time requir-ment of O(N2) where 2N+1 is the number of discretization points used. Also the error estimate is computed. Some numerical Exam-ples are computed using the Mathcad package.

• Structural Engineering and Mechanics
• /
• 제34권3호
• /
• pp.377-390
• /
• 2010

This paper studies fracture initiation direction of two parallel non-coplanar cracks of equal length. Using the dislocation pile-up modelling, singular integral equations for two parallel cracks subjected to mixed-mode loading are derived and the crack-tip field including singular and non-singular terms is obtained. The kinking angle is determined by using the maximum hoop stress criterion, or the ${\sigma}_{\theta}$-criterion. Results are presented for simple uniaxial tension and biaxial loading. The biaxiality ratio has a noticeable influence on crack growth direction. For the case of biaxial tension, when neglecting the T-stress the crack branching angle is overestimated for small crack inclination angles relative to the largest applied principal stress direction, and underestimated for large crack inclination angles.

• 대한수학회지
• /
• 제41권6호
• /
• pp.957-976
• /
• 2004

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation $\Omega$$_{m}$(b;$\chi$), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation $\Omega$$_{m}$(b;$\chi$), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation $\Omega$$_{m}$(b;$\chi$).TEX>).

• Ocean Systems Engineering
• /
• 제7권4호
• /
• pp.399-412
• /
• 2017

The Gauss-Legendre integral method is applied to numerically evaluate the Green function and its derivatives in finite water depth. In this method, the singular point of the function in the traditional integral equation can be avoided. Moreover, based on the improved Gauss-Laguerre integral method proposed in the previous research, a new methodology is developed through the Gauss-Legendre integral. Using this new methodology, the Green function with the field and source points near the water surface can be obtained, which is less mentioned in the previous research. The accuracy and efficiency of this new method is investigated. The numerical results using a Gauss-Legendre integral method show good agreements with other numerical results of direct calculations and series form in the far field. Furthermore, the cases with the field and source points near the water surface are also considered. Considering the computational efficiency, the method using the Gauss-Legendre integral proposed in this paper could obtain the accurate numerical results of the Green function and its derivatives in finite water depth and can be adopted in the near field.

• 충청수학회지
• /
• 제33권1호
• /
• pp.57-63
• /
• 2020

We investigate the boundedness of the spectrum of a convex base function for a new function space. The result guarantees the continuity of the Calderón-Zygmund operators on the new space.

• 한국항행학회논문지
• /
• 제7권2호
• /
• pp.180-190
• /
• 2003

실수축 상의 적분 방법에 의한 정확한 closed-form 그린함수를 이용하여 코플래너 도파로의 불연속에 대한 공간영역 full-wave 해석을 하였다. MPIE(Mixed Potential Integral Equation)를 풀기 위한 수치계산 방법으로는 삼각형 요소를 이용한 갤러킨 방법을 사용하였다. 경계면에서 삼각형 요소상의 기저함수로는 선형함수를 사용하였으며, 관측점과 전원점이 일치하는 특이점 근방의 적분 계산을 위해 면적분을 선적분 형태로 바꾸어 피적분 함수의 특이점이 사라지도록 하는 해석적인 방법을 사용하였다. 실수축 적분방법에 의한 그린함수를 이용함으로써 불연속에 대한 정확한 특성을 구하였다.

• Korean Journal of Mathematics
• /
• 제24권3호
• /
• pp.545-565
• /
• 2016

In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: $$T_{\lambda}(f;x)={\large\int_{\Omega}}K_{\lambda}(t,x,f(t))dt,\;x{\in}{\Psi},\;{\lambda}{\in}{\Lambda}$$, where ${\Psi}$ = and ${\Omega}$ = stand for arbitrary closed, semi-closed or open bounded intervals in ${\mathbb{R}}$ or these set notations denote $\mathbb{R}$, and ${\Lambda}$ is a set of non-negative numbers, to the function $f{\in}L_{p,{\omega}}({\Omega})$, where $L_{p,{\omega}}({\Omega})$ denotes the space of all measurable functions f for which $\|{\frac{f}{\omega}}\|^p$ (1 ${\leq}$ p < ${\infty}$) is integrable on ${\Omega}$, and ${\omega}:{\mathbb{R}}{\rightarrow}\mathbb{R}^+$ is a weight function satisfying some conditions.