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MỘT PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN VỚI HỆ SỐ ALPHA
(αFEM) ĐỂ TÌM NGHIỆM GẦN NHƯ CHÍNH XÁC CHO CƠ HỌC
VẬT RẮN BẰNG CÁCH SỬ DỤNG CÁC PHẦN TỬ TAM GIÁC VÀ TỨ
DIỆN
AN ALPHA FINITE ELEMENT METHOD (αFEM) FOR NEARLY
EXACT SOLUTION TO SOLID MECHANICS USING TRIANGULAR
AND TETRAHEDRAL ELEMENTS
Nguyen Thoi Trung1,3 , Liu Gui Rong1,2, Nguyen Xuan Hung2,3, Ngo Thanh Phong3
1 Center for Advanced Computations in Engineering Science (ACES), Department of
Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1,
Singapore 117576
2Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore,
117576
3Faculty of Mathematics and Informatics, University of Science, VNU-HCMC
Abstract
An alpha finite element method (αFEM) is proposed for
computing nearly exact solution in energy norm for solid mechanics problems
using meshes that can be generated automatically for arbitrarily complicated
domains. Three-node triangular (αFEM-T3) and four-node tetrahedral (αFEM-T4)
elements with a scale factor α are formulated for twodimensional (2D) and
three-dimensional (3D) problems, respectively. The essential idea of the method
thod is the use of a scale factor α ∈ [0,1] to obtain a combined model of the
standard fully compatible model of the FEM and a quasi-equilibrium model of the
node-based smoothed FEM (NS-FEM). This novel combination of the FEM and NS-FEM
makes the best use of the upper bound property of the NS-FEM and the lower bound
property of the standard FEM. Using meshes with the same aspect ratio, a unified
approach has been proposed to obtain a nearly exact solution in strain energy
for linear problems. The proposed elements are also applied to improve the
accuracy of the solution of nonlinear problems of large deformation. Numerical
results for 2D (using αFEM-T3) and 3D (using αFEM-T4) problems confirm that the
present method gives the much more accurate solution comparing to both the
standard FEM and the NS-FEM with the same number of degrees of freedom and
similar computational efforts for both linear and nonlinear problems. Key words:
numerical methods, finite element method (FEM), node-based smoothed finite
element method (NS-FEM), upper bound, lower bound, alpha finite element method
(αFEM)
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