Numerical Manifold Method with Endochronic Theory for Elastoplasticity Analysis

Numerical manifold method (NMM) was originally developed based on linear elastic constitutive model. For many problems it is difficult to obtain accurate results without elastoplasticity analysis, and an elastoplasticity version of NMM is needed. In this paper, the incremental endochronic theory is extended into NMM analysis and an endochronic NMM algorithm is proposed for elastoplasticity analysis. It is well known that endochronic theory is one of the widely used elastoplasticity theories which can deal with elastoplasticity problems without a yield surface and loading or unloading judgments. Numerical tests show that the proposed algorithm of endochronic NMM possesses a good accuracy. The proposed algorithm is also applied to analyze a crack problem and a soft clay foundation under traffic loading problem. Results demonstrate the convenience of the endochronic NMM in analyzing elastoplasticity discontinuous problems.

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Numerical Manifold Method for the Forced Vibration of Thin Plates during Bending

The Scientific World JOURNAL ◽

10.1155/2014/520958 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11

Author(s):

Ding Jun ◽

Chen Song ◽

Wen Wei-Bin ◽

Luo Shao-Ming ◽

Huang Xia

Keyword(s):

Dynamic Equation ◽

Degrees Of Freedom ◽

Solution Process ◽

Thin Plates ◽

Numerical Manifold Method ◽

Bending Vibration ◽

Linear Elastic ◽

Spline Basis ◽

Generalized Degrees Of Freedom ◽

Numerical Manifold

A novel numerical manifold method was derived from the cubic B-spline basis function. The new interpolation function is characterized by high-order coordination at the boundary of a manifold element. The linear elastic-dynamic equation used to solve the bending vibration of thin plates was derived according to the principle of minimum instantaneous potential energy. The method for the initialization of the dynamic equation and its solution process were provided. Moreover, the analysis showed that the calculated stiffness matrix exhibited favorable performance. Numerical results showed that the generalized degrees of freedom were significantly fewer and that the calculation accuracy was higher for the manifold method than for the conventional finite element method.

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A numerical integration strategy of meshless numerical manifold method based on physical cover and applications to linear elastic fractures

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2021.09.028 ◽

2022 ◽

Vol 134 ◽

pp. 79-95

Author(s):

Wei Li ◽

Xianbin Yu ◽

Shan Lin ◽

Xin Qu ◽

Xizhen Sun

Keyword(s):

Numerical Integration ◽

Numerical Manifold Method ◽

Linear Elastic ◽

Integration Strategy ◽

Numerical Manifold

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Investigation of the Accuracy of the Numerical Manifold Method on n-Sided Regular Elements for Crack Problems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.157-158.1093 ◽

2012 ◽

Vol 157-158 ◽

pp. 1093-1096

Author(s):

Hui Hua Zhang ◽

Jia Xiang Yan

Keyword(s):

Numerical Manifold Method ◽

Quadrilateral Element ◽

Regular Elements ◽

Linear Elastic ◽

Quadrilateral Elements ◽

Crack Problems ◽

Elastic Crack ◽

Triangular Elements ◽

Numerical Manifold ◽

Better Than

The numerical manifold method (NMM) is a representative among different numerical methods for crack problems. Due to the independence of physical domain and the mathematical cover system, totally regular mathematical elements can be used in the NMM. In the present paper, the NMM is applied to solve 2-D linear elastic crack problems, together with the comparison study on the accuracy of n-sided regular mathematical elements, i.e., the triangular elements (n=3), the quadrilateral elements (n=4) and the hexagonal elements (n=6). Our numerical results show that among different elements, the regular hexagonal element is the best and the quadrilateral element is better than the triangular one.

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A High-Order Numerical Manifold Method for Darcy Flow in Heterogeneous Porous Media

Processes ◽

10.3390/pr6080111 ◽

2018 ◽

Vol 6 (8) ◽

pp. 111 ◽

Cited By ~ 3

Author(s):

Lingfeng Zhou ◽

Yuan Wang ◽

Di Feng

Keyword(s):

Porous Media ◽

Heterogeneous Porous Media ◽

High Order ◽

Darcy Flow ◽

Numerical Manifold Method ◽

Material Interfaces ◽

Darcy Velocity ◽

Cover Systems ◽

Numerical Manifold ◽

Discontinuous Problems

One major challenge in modeling Darcy flow in heterogeneous porous media is simulating the material interfaces accurately. To overcome this defect, the refraction law is fully introduced into the numerical manifold method (NMM) as an a posteriori condition. To achieve a better accuracy of the Darcy velocity and continuous nodal velocity, a high-order weight function with a continuous nodal gradient is adopted. NMM is an advanced method with two independent cover systems, which can easily solve both continuous and discontinuous problems in a unified form. Moreover, a regular mathematical mesh, independent of the physical domain, is used in the NMM model. Compared to the conforming mesh of other numerical methods, it is more efficient and flexible. A number of numerical examples were simulated by the new NMM model, comparing the results with the original NMM model and the analytical solutions. Thereby, it is proven that the proposed method is accurate, efficient, and robust for modeling Darcy flow in heterogeneous porous media, while the refraction law is satisfied rigorously.

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A high order numerical manifold method and its application to linear elastic continuous and fracture problems

Science China Technological Sciences ◽

10.1007/s11431-016-9070-8 ◽

2017 ◽

Vol 61 (3) ◽

pp. 346-358 ◽

Cited By ~ 17

Author(s):

YongTao Yang ◽

GuanHua Sun ◽

KeJian Cai ◽

Hong Zheng

Keyword(s):

High Order ◽

Numerical Manifold Method ◽

Linear Elastic ◽

Numerical Manifold

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Analysis of Crack Interaction Problem by the Numerical Manifold Method

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.446-449.797 ◽

2012 ◽

Vol 446-449 ◽

pp. 797-801 ◽

Cited By ~ 3

Author(s):

Hui Hua Zhang

Keyword(s):

Numerical Manifold Method ◽

Crack Interaction ◽

Multiple Crack ◽

Linear Elastic ◽

Cover System ◽

Crack Problems ◽

Elastic Fracture ◽

Numerical Manifold ◽

Physical Boundary ◽

Interaction Problem

Due to the use of mathematical cover system and physical cover system, the numerical manifold method (NMM) is very suitable for discontinuity problems, especially for multiple crack problems. In the NMM, the mathematical cover system is independent of the physical boundary, and in this case, fully regular mathematical elements can be used. In the present paper, the NMM, combined with the rectangular mathematical elements, is applied to solve crack interaction problems in the linear elastic fracture mechanics (LEFM). To verify the present method, a typical numerical example is investigated and the results agree well with the reference solutions.

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SIMULATION OF TOPPLING FAILURE OF ROCK SLOPE BY NUMERICAL MANIFOLD METHOD

International Journal of Computational Methods ◽

10.1142/s0219876210002118 ◽

2010 ◽

Vol 07 (01) ◽

pp. 167-189 ◽

Cited By ~ 40

Author(s):

GUOXIN ZHANG ◽

YAN ZHAO ◽

XIAOCHU PENG

Keyword(s):

Rock Slope ◽

Limit Equilibrium ◽

Failure Process ◽

Numerical Manifold Method ◽

Left Bank ◽

Rock Slopes ◽

Rock Bridges ◽

Toppling Failure ◽

Numerical Manifold ◽

Discontinuous Problems

As one type of rock slope failures, topping failure can be accurately simulated only when several aspects are correctly calculated such as deformation and stress, contacts between blocks, contact stress, movement of blocks, open/close of contacts between blocks, development of failure plane, and crack generation and propagation. Current numerical methods encounter many difficulties in simulating toppling failure, especially for rock slope with lots of rock-bridges. Numerical manifold method (NMM) can deal with these highly discontinuous problems and be used to model the toppling failure of rock slopes. This paper first introduces the fundamental principles, modeling of contacts, calculation of contact force and stress, and modeling of failure in NMM. Then, several case studies are conducted to testify the accuracy and convergence of method; comparisons with method, based on limit equilibrium principle, which was proposed by Goodman and Bray (G–B method) and centrifuge test are conducted. Finally, the topping failure of left bank of one high dam is simulated. Results show that the NMM can be used to correctly calculate the toppling safety factor, simulate the failure process of slope toppling, and accurately model the whole failure process of rock slopes with many rock-bridges.

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Novel Insight into High-Order Numerical Manifold Method Using Complex Fourier Element Shape Functions in Statics and Dynamics

International Journal of Applied Mechanics ◽

10.1142/s1758825119500583 ◽

2019 ◽

Vol 11 (06) ◽

pp. 1950058

Author(s):

M. Malekzadeh ◽

S. Hamzehei-Javaran ◽

S. Shojaee

Keyword(s):

Degrees Of Freedom ◽

Numerical Solutions ◽

High Order ◽

Numerical Manifold Method ◽

Shape Functions ◽

Linear Complex ◽

Statics And Dynamics ◽

Vibration Problems ◽

Numerical Manifold ◽

Discontinuous Problems

In this paper, the high-order numerical manifold method (HONMM) with new complex Fourier shape functions is developed for the simulation of elastostatic and elastodynamic problems. NMM uses two separate covers which give it the ability to analyze continuous and discontinuous problems in a unified way. The new shape functions are derived using constant and linear complex Fourier shape functions. These shape functions are able to satisfy exponential and trigonometric function fields in addition to polynomial ones, unlike classic Lagrange shape functions. Compared to the Lagrange shape functions, the proposed shape functions show much more accurate results with fewer degrees of freedom. The superiority of the proposed method over the conventional HONMM in static analysis is demonstrated through a special beam example. As cases of dynamic analysis, four free and forced vibration problems are illustrated. The results of the HONMM with the use of constant and linear complex Fourier shape functions are compared with the classic HONMM results and available analytical and other numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic HONMM.

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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.

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