Green Quasifunction Method for Bending Problem of Clamped Orthotropic Trapezoidal Thin Plates on Winkler Foundation

2011 ◽  
Vol 138-139 ◽  
pp. 705-708 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Integral Equation ◽  
Homogeneous Boundary Condition ◽  
Thin Plates ◽  
Winkler Foundation ◽  
Element Solution ◽  
Biharmonic Operator ◽  
Boundary Equation ◽  
Governing Differential Equation ◽  
Suitable Form ◽  
Good Agreement

The Green quasifunction method (GQM) is applied to solve the bending problem of clamped orthotropic thin plates with trapezoidal boundary shape on Winkler foundation. Firstly the governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Green’s formula. A Green quasifunction is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with ANSYS finite element solution shows good agreement. The proposed method is a novel and effective mathematical one.

Green Quasifunction Method for Bending Problem of Clamped Orthotropic Thin Plates with Trapezoidal Boundary Shape

2011 ◽  
Vol 117-119 ◽  
pp. 456-459 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Differential Equation ◽  
Mathematical Method ◽  
Integral Equation ◽  
Homogeneous Boundary Condition ◽  
Thin Plates ◽  
Biharmonic Operator ◽  
Boundary Equation ◽  
Boundary Shape ◽  
Governing Differential Equation ◽  
Suitable Form

The Green quasifunction method(GQM) is employed to solve the bending problem of clamped orthotropic thin plates with trapezoidal boundary shape. Firstly the governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Green’s formula. A Green quasifunction is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. A numerical example demonstrates the feasibility and efficiency of the proposed method, and it is a novel mathematical method.

A Quasi-Green Function Method for Free Vibration of Clamped Orthotropic Parallelogram Thin Plates on Winkler Foundation

2013 ◽  
Vol 397-400 ◽  
pp. 431-434
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Free Vibration ◽  
Function Method ◽  
Homogeneous Boundary Condition ◽  
Thin Plates ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Green Function Method ◽  
Suitable Form

The quasi-Green function method (QGFM) is applied to solve the free vibration of clamped orthotropic thin plates with parallelogram boundary shape on Winkler foundation. Firstly the model governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Greens formula. A quasi-Green function is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with ANSYS finite element solution shows a good agreement. The proposed method is a novel and effective mathematical one.

Application of the R-Function Theory for the Bending Problem of Shallow Spherical Shells with a Dodecagon Domain

2012 ◽  
Vol 468-471 ◽  
pp. 8-12 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Theory ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Suitable Form ◽  
Good Agreement

The R-function theory is applied to describe the dodecagon domain of shallow spherical shells on Winkler foundation, and it is also used to construct a quasi-Green’s function. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. A comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

A Quasi-Greens Function Method for the Bending Problem of Simply Supported Polygonal Shallow Spherical Shells on Pasternak Foundation

2013 ◽  
Vol 353-356 ◽  
pp. 3215-3219
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Function Method ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Pasternak Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Simply Supported ◽  
Suitable Form ◽  
Greens Function ◽  
Good Agreement

The quasi-Greens function method (QGFM) is applied to solve the bending problem of simply supported polygonal shallow spherical shells on Pasternak foundation. A quasi-Greens function is established by using the fundamental solution and the boundary equation of the problem. And the function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Greens formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the proposed method.

A Quasi-Green’s Function Method for the Bending Problem of Simply Supported Trapezoidal Shallow Spherical Shells on Winkler Foundation

2012 ◽  
Vol 446-449 ◽  
pp. 3582-3586 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Method ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Green’S Function Method ◽  
Simply Supported

The quasi-Green’s function method (QGFM) is applied to solve the bending problem of simply supported trapezoidal shallow spherical shells on Winkler foundation. A quasi-Green’s function is established by using the fundamental solution and the boundary equation of the problem. And the function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the proposed method.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

scholarly journals The features of the use of the waveguide radiators in smart antenna systems

2018 ◽  
Vol 26 (1) ◽  
pp. 89-92
Author(s):  
V. M. Morozov ◽  
V. I. Magro
Keyword(s):  
Integral Equation ◽  
Antenna Array ◽  
Antenna Arrays ◽  
Smart Antenna ◽  
Integral Equation Method ◽  
Thin Plates ◽  
Coupling Region ◽  
Five Elements ◽  
Antenna Systems ◽  
Selection Of

The features of the use of finite waveguide antenna arrays in the structure of modern smart antenna systems are considered. The paper deals with the problem of diffraction of an electromagnetic wave on a finite waveguide antenna array scanning in the E-plane. Antenna array consists of five radiating elements. The open ends of the waveguides are surrounded by a metal screen. The resonator coupling region was chosen as matching elements. The solution of the problem is carried out by the integral equation method on the basis of the selection of overlapping regions. The problem reduces to solving the Fredholm integral equation of the second kind. An array of infinitely thin plates and that of waveguides with a finite wall thickness are considered. The main regularities for choosing the optimal geometric dimensions of the antenna array are established. Studies were carried out for arrays with a number of elements from five to fifteen. The analysis of edge effects in the final antenna array is carried out. It is shown that the introduction of a resonator region into a five-element lattice makes it possible to expand the sector of the radiation angles and avoid the effect of blinding. It is shown that this statement is valid not only for five-element lattices, but also for arrays with a large number of radiating elements. The radiation patterns are calculated. The  coefficients of mutual coupling in an array with five elements are investigated. General recommendations for choosing optimal sizes of the resonator coupling region of radiators are considered.

scholarly journals Numerical Assessment of the Performance of Elastic Cloaks for Transient Flexural Waves

2020 ◽  
Vol 7 ◽  
Author(s):  
Marco Rossi ◽  
Daniele Veber ◽  
Massimiliano Gei
Keyword(s):  
Finite Element ◽  
Thin Plates ◽  
Winkler Foundation ◽  
Finite Element Code ◽  
Flexural Waves ◽  
Transient Waves ◽  
Mathematical Transformation ◽  
Plane Body ◽  
Numerical Assessment ◽  
Subgrade Modulus

A relevant application of transformation elastodynamics has shown that flexural waves in a Kirchhoff-Love plate can be diverted and channeled to cloak a region of the ambient space. To achieve the goal, an orthotropic meta-structural plate should be employed. However, the corresponding mathematical transformation leads to the presence of an unwanted strong compressive prestress, likely beyond the buckling threshold of the structure, with a set of in-plane body forces to warrant equilibrium. In addition, the plate must possess, at the same time, high bending stiffnesses, but a null twisting rigidity. With the aim of estimating the performance of cloaks modelled with approximate parameters, an in-house finite element code, based on a subparametric technique, is implemented to deal with the cloaking of transient waves in orthotropic thin plates. The tool allows us to explore the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. In addition, the finite element code is extended to investigate a meta-plate interacting with a Winkler foundation, to confirm how the subgrade modulus should transform in the cloak region.

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