Solving of Partial Differential Equations by Numerical Manifold Method with Partially Overlapping Covers

Governing equations of 1D high-order numerical manifold method with partially overlapping covers have been deduced by the general method of weighted residuals. By using the proposed method, a highly oscillatory differential equation has been solved. In addition, a posteriori error method is adopted for evaluating the accuracy of the algorithm. Meanwhile, several factors affect the accuracy are also discussed. The results indicate that accuracy of solution increase with the decrease of overlapping ratio and the order of cover function. When higher order cover function such as 6th is used, higher accuracy will matched even for second derivative of unknowns. This paper attempts to enrich the theory and practical field of the NMM.

Download Full-text

Numerical manifold method based on the method of weighted residuals

Computational Mechanics ◽

10.1007/s00466-004-0636-3 ◽

2004 ◽

Vol 35 (6) ◽

pp. 470-480 ◽

Cited By ~ 44

Author(s):

S. Li ◽

Y. Cheng ◽

Y.-F. Wu

Keyword(s):

Numerical Manifold Method ◽

Weighted Residuals ◽

Method Of Weighted Residuals ◽

Numerical Manifold

Download Full-text

Notice of Retraction: Development local cover function of numerical manifold method based upon natural coordinates system

2010 International Conference on Computer Application and System Modeling (ICCASM 2010) ◽

10.1109/iccasm.2010.5620202 ◽

2010 ◽

Author(s):

Zhang Xuan-bing ◽

Zhang Hong

Keyword(s):

Numerical Manifold Method ◽

Natural Coordinates ◽

Numerical Manifold ◽

Cover Function

Download Full-text

Study on Application of Numerical Manifold Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.182-183.1194 ◽

2012 ◽

Vol 182-183 ◽

pp. 1194-1199

Author(s):

Lei Zheng ◽

Zheng Zhong Shen

Keyword(s):

Continuum Mechanics ◽

Discontinuous Deformation Analysis ◽

Deformation Analysis ◽

Numerical Manifold Method ◽

Analysis Method ◽

Continuous Problems ◽

Discontinuous Deformation ◽

Numerical Manifold ◽

Cover Function ◽

The Continuum

Numerical manifold method (NMM) is based on the blocky theory which absorbed the advantages of finite element method based on the continuum mechanics, discontinuous deformation analysis method based on the non-continuum mechanics and analytic method. According to the finite covering technology of modern manifold analysis method, uniform solving format on continuous-non-continuous problems is established by taking the continuous and discontinuous cover function which can be used for the numerical simulation of large deformation and continuous-discontinuous deformation. The calculation program is worked out based on the method above and it is applied to the actual project.

Download Full-text

Three-Dimensional Numerical Manifold Method Based on Hexahedron Element with Full First-Order Cover Function and its Application in Underground Excavation

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.340-341.365 ◽

2007 ◽

Vol 340-341 ◽

pp. 365-370 ◽

Cited By ~ 2

Author(s):

Y.H. Zhang ◽

Qian Sheng ◽

Y.M. Cheng

Keyword(s):

Three Dimensional ◽

Shear Stiffness ◽

Surface Force ◽

Inertia Force ◽

Numerical Manifold Method ◽

Underground Excavation ◽

Equilibrium Equations ◽

First Order ◽

Numerical Manifold ◽

Cover Function

In this paper, three-dimensional(3D) Numerical Manifold Method (NNM) based on hexahedron element cover with full first-order cover function is proposed and the shape function of C8 isoparametric element in FEM is used as the cover weight function. All sub-matrices in equilibrium equations, including stiffness matrix, initial stress matrix, point force matrix, surface force matrix, body force matrix, inertia force matrix, contact matrix and friction matrix, are derived. Different with 2D contact, the direction of shear stiffness and friction force can not be easily defined in 3D contact. A new iterative method based on vector theory to detect the contact direction is developed. The application of 3D NMM in underground excavation is also presented and show good agreement with real engineering.

Download Full-text

Three-Dimensional Numerical Manifold Method Based on Hexahedron Element with Full First-Order Cover Function and its Application in Underground Excavation

Engineering Plasticity and Its Applications - Key Engineering Materials ◽

10.4028/0-87849-433-2.365 ◽

2007 ◽

pp. 365-370

Author(s):

Y.H. Zhang ◽

Qian Sheng ◽

Y.M. Cheng

Keyword(s):

Three Dimensional ◽

Numerical Manifold Method ◽

Underground Excavation ◽

First Order ◽

Numerical Manifold ◽

Cover Function

Download Full-text

EXPANSION TO HIGH-ORDER COVER FUNCTION AND IMPROVEMENT OF THE STRESS ACCURACY IN NUMERICAL MANIFOLD METHOD

Journal of Mechanical Engineering ◽

10.3901/jme.2000.09.020 ◽

2000 ◽

Vol 36 (09) ◽

pp. 20 ◽

Cited By ~ 1

Author(s):

Yongchang Cai

Keyword(s):

High Order ◽

Numerical Manifold Method ◽

Numerical Manifold ◽

Cover Function

Download Full-text

Direct Simulation of Low-Re Flow around a Square Cylinder by Numerical Manifold Method for Navier-Stokes Equations

Journal of Applied Mathematics ◽

10.1155/2012/465972 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Cited By ~ 3

Author(s):

Zhengrong Zhang ◽

Xiangwei Zhang

Keyword(s):

Viscous Flow ◽

Stokes Equations ◽

Navier Stokes ◽

Numerical Manifold Method ◽

Square Cylinder ◽

Navier Stokes Equations ◽

Weighted Residuals ◽

Numerical Tests ◽

Incompressible Viscous Flow ◽

Numerical Manifold

Numerical manifold method was applied to directly solve Navier-Stokes (N-S) equations for incompressible viscous flow in this paper, and numerical manifold schemes for N-S equations coupled velocity and pressure were derived based on Galerkin weighted residuals method as well. Mixed cover with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. As an application, mixed cover 4-node rectangular manifold element has been used to simulate the incompressible viscous flow around a square cylinder in a channel. Numerical tests illustrate that NMM is an effective and high-order accurate numerical method for incompressible viscous flow N-S equations.

Download Full-text