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

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.

scholarly journals Bending of Nonconforming Thin Plates Based on the Mixed-Order Manifold Method with Background Cells for Integration

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Xin Qu ◽  
Lijun Su ◽  
Zhijun Liu ◽  
Xingqian Xu ◽  
Fangfang Diao ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Displacement Function ◽  
Thin Plates ◽  
Irregular Domain ◽  
Interpolation Accuracy ◽  
Mixed Order ◽  
Convergence Performance ◽  
Nonconforming Elements ◽  
Numerical Manifold ◽  
Thin Plate Bending

As it is very difficult to construct conforming plate elements and the solutions achieved with conforming elements yield inferior accuracy to those achieved with nonconforming elements on many occasions, nonconforming elements, especially Adini’s element (ACM element), are often recommended for practical usage. However, the convergence, good numerical accuracy, and high computing efficiency of ACM element with irregular physical boundaries cannot be achieved using either the finite element method (FEM) or the numerical manifold method (NMM). The mixed-order NMM with background cells for integration was developed to analyze the bending of nonconforming thin plates with irregular physical boundaries. Regular meshes were selected to improve the convergence performance; background cells were used to improve the integration accuracy without increasing the degrees of freedom, retaining the efficiency as well; the mixed-order local displacement function was taken to improve the interpolation accuracy. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate bending, a new scheme was proposed to deal with irregular domain boundaries. Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. The results indicated that the solutions achieved with the proposed NMM rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM.

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

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.

scholarly journals Numerical Manifold Method with Endochronic Theory for Elastoplasticity Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wei Zeng ◽  
Junjie Li ◽  
Fei Kang
Keyword(s):  
Soft Clay ◽  
Crack Problem ◽  
Numerical Manifold Method ◽  
Traffic Loading ◽  
Endochronic Theory ◽  
Linear Elastic ◽  
Numerical Tests ◽  
Loading Problem ◽  
Numerical Manifold ◽  
Discontinuous Problems

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.

Analysis of Crack Interaction Problem by the Numerical Manifold Method

2012 ◽  
Vol 446-449 ◽  
pp. 797-801 ◽  
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.

On Improved Thin Plate Bending Rectangular Finite Element Based on the Strain Approach

2016 ◽  
Vol 27 ◽  
pp. 76-86 ◽  
Author(s):  
Sifeddine Abderrahmani ◽  
Toufik Maalem ◽  
Djamal Hamadi
Keyword(s):  
Finite Element ◽  
Thin Plate ◽  
Degrees Of Freedom ◽  
Body Motion ◽  
Thin Plates ◽  
Elastic Behavior ◽  
Transverse Displacement ◽  
Plate Bending ◽  
Linear Elastic ◽  
Thin Plate Bending

We propose in this paper the development of a new rectangular finite element for thin plate bending based on the strain approach with linear elastic behavior. An analytical integration is used to evaluate the element stiffness matrix. The present element possesses the three main degrees of freedom (d.o.f) per node, namely, one transverse displacement (w) and two normal rotations about x and y axis respectively (Ɵx, Ɵy). The proposed displacement field represents exactly the rigid body motion and satisfies the compatibility equations. The numerical results converges rapidly to the Kirchhoff solution for thin plates, this makes the present element robust, better suitable for computations, and particularly interesting in modeling this type of structures.

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

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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