scholarly journals Solution a problem for edge crack with a hypersingular governing equation by the mechanical quadrature method

2020 ◽  
Vol 73 (2) ◽  
pp. 44-57
Author(s):  
A.V. Sahakyan
Keyword(s):  
Governing Equation ◽  
Edge Crack ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Mechanical Quadrature Method

A High-Accuracy Mechanical Quadrature Method for Solving the Axisymmetric Poisson's Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 393-406 ◽  
Author(s):  
Hu Li ◽  
Jin Huang
Keyword(s):  
Three Dimensional ◽  
High Accuracy ◽  
Poisson’S Equation ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Poisson's Equation ◽  
Mechanical Quadrature ◽  
Asymptotic Error ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

AbstractIn this article, we consider the numerical solution for Poisson's equation in axisymmetric geometry. When the boundary condition and source term are axisymmetric, the problem reduces to solving Poisson's equation in cylindrical coordinates in the two-dimensional (r,z) region of the original three-dimensional domain S. Hence, the original boundary value problem is reduced to a two-dimensional one. To make use of the Mechanical quadrature method (MQM), it is necessary to calculate a particular solution, which can be subtracted off, so that MQM can be used to solve the resulting Laplace problem, which possesses high accuracy order and low computing complexities. Moreover, the multivariate asymptotic error expansion of MQM accompanied with for all mesh widths hi is got. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least by the splitting extrapolation algorithm (SEA). Meanwhile, a posteriori asymptotic error estimate is derived, which can be used to construct self-adaptive algorithms. The numerical examples support our theoretical analysis.

scholarly journals A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Li Zhang ◽  
Jin Huang ◽  
Yubin Pan ◽  
Xiaoxia Wen
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Richardson Extrapolation ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Singular Kernels ◽  
Mechanical Quadrature Method

In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.

scholarly journals Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hu Li ◽  
Yanying Ma
Keyword(s):  
Integral Equation ◽  
Helmholtz Equation ◽  
Boundary Integral Equation ◽  
Dirichlet Boundary ◽  
Boundary Integral ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Splitting Extrapolation ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote byhmthe mesh width of a curved edgeΓm  (m=1,…,d)of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied withO(hm3)for all mesh widthshmis obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at leastO(hmax⁡5)by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

A Bilayer Model for Incorporating the Coupled Effects of Surface Energy and Microstructure on the Electromechanical Stability of NEMS

2017 ◽  
Vol 17 (04) ◽  
pp. 1771005 ◽  
Author(s):  
Rahman Soroush ◽  
Ali Koochi ◽  
Maryam Keivani ◽  
Mohamadreza Abadyan
Keyword(s):  
Surface Energy ◽  
Governing Equation ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Differential Quadrature Method ◽  
Surface Elasticity ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Stress Theory ◽  
Stability Threshold

Herein, the static and dynamic pull-in instabilities of nanotweezers and nanoswitches fabricated from conductive cylindrical nanowires are studied. A bilayer model based on the Gurtin–Murdoch surface elasticity and the consistent couple stress theory is developed to incorporate the coupled effects of surface energy and microstructure phenomenon. The differential quadrature method is employed to solve the nonlinear governing equation of the system. The effects of various parameters on the static and dynamic pull-in parameters and stability threshold of the nanostructures are demonstrated.

scholarly journals A Symmetric Three-Layer Plate with Two Coaxial Cracks under Pure Bending

2021 ◽  
Vol 11 (6) ◽  
pp. 2859
Author(s):  
Mykhaylo Delyavskyy ◽  
Viktor Opanasovych ◽  
Roman Seliverstov ◽  
Oksana Bilash
Keyword(s):  
Strain State ◽  
Geometrical Parameters ◽  
Bending Moments ◽  
Algebraic Equations ◽  
Optimum Ratio ◽  
Pure Bending ◽  
Mechanical Quadrature ◽  
Linear Algebraic Equations ◽  
Crack Tips ◽  
Mechanical Quadrature Method

The purpose of this research was to investigate the effect of mechanical features and geometrical parameters on the stress–strain state of a cracked layered plate under pure bending (bending moments are uniformly distributed at infinity). The sixth-order bending problem of an infinite, symmetric, three-layer plate with two coaxial through cracks is considered under the assumption of no crack closure. By using complex potentials and methods of the theory of functions of a complex variable, the solution to the problem was obtained in the form of a singular integral equation. It is reduced to the system of linear algebraic equations and solved in a numerical manner by the mechanical quadrature method. The distributions of stresses and bending moments near the crack tips are shown. Numerical results are presented as graphical dependences of the reduced moment intensity factor on various problem parameters. In this particular case, the optimum ratio of layer thicknesses is determined.

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