Natural Element Method for Material and Geometrical Bi-Nonlinear Problems

Based on the Voronoi diagram of some nodes, the natural element method (NEM) constructs the shape functions by the natural neighbor interpolation method, and its shape functions satisfy the Kronecker delta property, which makes it impose essential boundary conditions easily. Based on the geometrical nonlinear relations and material nonlinear constitutive relations, we extend the NEM to material and geometrical bi-nonlinear problems in this paper. Numerical examples show that the NEM is effective, rational and feasible in dealing with problems of both material and geometrical bi-nonlinear.

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C1 Natural Element Method for Couple-Stress Elasticity

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.105-107.370 ◽

2011 ◽

Vol 105-107 ◽

pp. 370-373

Author(s):

Ru Jun Han ◽

Zhi Feng Nie

Keyword(s):

Boundary Conditions ◽

Analytical Solutions ◽

Couple Stress ◽

Shape Functions ◽

Galerkin Scheme ◽

Natural Neighbor ◽

Natural Element Method ◽

Couple Stress Elasticity ◽

Natural Element ◽

Element Method

The shape functions in C1 natural element method (C1 NEM) are built upon the natural neighbor interpolation (NNI), and realize the interpolation to nodal function and nodal gradient values, so that the essential boundary conditions (EBCs) can be imposed directly in a Galerkin scheme for the partial differential equations (PDEs). In the present paper, C1 NEM for couple-stress (CS) elasticity is constructed, and the typical example which has analytical solutions is presented to illustrate the effectiveness of the constructed method.

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Hybrid Natural Element Method for Elastic Large Deformation Problems

International Journal of Applied Mechanics ◽

10.1142/s1758825116500447 ◽

2016 ◽

Vol 08 (04) ◽

pp. 1650044 ◽

Cited By ~ 2

Author(s):

Yongqi Ma ◽

Yankai Zhou ◽

Yi Dong ◽

Wei Feng

Keyword(s):

Large Deformation ◽

Iteration Process ◽

Deformation Analysis ◽

Natural Neighbor ◽

Natural Element Method ◽

Incremental Formulation ◽

Natural Element ◽

Incremental Equations ◽

Large Deformation Problems ◽

Element Method

Elastic large deformation analysis based on the hybrid natural element method (HNEM) is presented in this paper. The natural neighbor interpolation is adopted to construct the shape functions for the HNEM. The incremental formulation of Hellinger–Reissner variational principle is used to derive discrete system of incremental equations under the total Lagrangian formulation. And the Newton-Raphson iteration is applied to solve these incremental equations. Compared with the natural element method (NEM), the HNEM can directly obtain nodal stresses of higher precision, which will bring advantage in the iteration process and improve computational efficiency in solving elastic large deformation problems. Some numerical examples demonstrate the validity of the HNEM for elastic large deformation problems.

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Using Meshless Local Natural Neighbor Interpolation Method to Solve Two-Dimensional Nonlinear Problems

International Journal of Applied Mechanics ◽

10.1142/s1758825116500691 ◽

2016 ◽

Vol 08 (05) ◽

pp. 1650069 ◽

Cited By ~ 9

Author(s):

Q. H. Li ◽

S. S. Chen ◽

X. M. Luo

Keyword(s):

Heat Transfer ◽

Boundary Conditions ◽

Interpolation Method ◽

Nonlinear Problems ◽

Transient Heat Transfer ◽

Integration Scheme ◽

Two Dimensional ◽

Shape Functions ◽

Time Step ◽

Natural Neighbor

Based on the meshless local natural neighbor interpolation method (MLNNI), a novel solution procedure is developed for the analysis of nonlinear steady and transient heat transfer of two-dimensional structures in this paper. Nonlinearities arising from temperature dependence of material properties and nonlinear boundary conditions have been taken into account. The present method is developed based on the natural neighbor interpolation (NNI) for constructing shape functions at scattered nodes. The three-node triangular FEM shape function is employed as the test function, which reduces the orders of integrands involved in domain integrals. Due to the delta function property of the natural neighbor shape functions, there is no need to employ special techniques to enforce the essential boundary conditions. The backward difference method is employed for the time integration scheme in transient analysis and the Newton–Raphson iterative procedure is required at each time step. Three numerical examples with different geometries and boundary conditions are presented at the end to demonstrate the validity and accuracy of the proposed method for the solution of a wide class of nonlinear steady and transient heat transfer problems.

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