New formulation of the Green element method to maintain its second-order accuracy in 2D/3D |
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| Affiliation: | 1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China;2. Zienkiewicz Centre for Computational Engineering, Swansea University, UK;1. Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Mechanical and Electrical Engineering Institute, Zhengzhou University of Light Industry, Zhengzhou, 450002, Henan, China;2. The 39th Research Institute of China Electronics Technology Group Corporation, Xian 710065, China;3. School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, China;4. Henan Huanghe whirlwind Co.. LTD, Changge, 461500, Henan, China;1. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China;2. Key Laboratory of Advanced Technology for Aerospace Vehicles of Liaoning Province, Dalian University of Technology, Dalian 116024, China;3. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China;1. Institute of Sound and Vibration Research, Hefei University of Technology, Hefei, Anhui, 230009, PR China;2. Department of Civil Engineering, University of Siegen, Siegen, D-57068, Germany;3. CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230027, PR China |
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| Abstract: | The Green element method (GEM) is a powerful technique for solving nonlinear boundary value problems. Derived from the boundary element method (BEM), over the meshes of the finite element method (FEM), the GEM combines the second-order accuracy of the BEM with the efficiency and versatility of the FEM.The high accuracy of the GEM, resulting from the direct representation of normal fluxes as unknowns, comes at the price of very large matrices for problems in 2D and 3D domains. The reason for this is a larger number of inter-element boundaries connected to each internal node, yielding the same number of the normal fluxes to be determined. The currently available technique to avoid this problem approximates the normal fluxes by differentiating the potential estimates within each element. Although this approach produces much smaller matrices, the overall accuracy of the GEM is sacrificed.The first of the two techniques proposed in this work redefines the present approach of approximating fluxes by considering more elements sharing each internal node. Numerical tests on the potential field exp(x+y) show an increase in accuracy by two orders of magnitude.The second approach is a reformulation of the standard GEM in terms of the flux vector, replacing its normal component. The original accuracy of the GEM is preserved while the number of unknowns is reduced as many as ten-times in the case of a mesh consisting of tetrahedrons. The additional benefit of this novel technique is the fact that the entire flux field is a mere by-product of the basic procedure for determining the unspecified boundary values. |
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