Hybrid Fundamental-Solution-Based FEM for Heat Conduction Problems in Orthotropic Materials
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摘要: 采用基于杂交基本解的有限元法（HFS-FEM）对二维正交各向异性材料进行热传导分析. 单元域内和单元边界上的温度分布由两个温度场独立描述. 采用基本解的线性组合来近似单元内部温度场，采用标准一维线单元形函数来定义网线温度场. 利用修正变分泛函和散度定理导得相应的有限元列式，通过2个算例与ABAQUS结果对比，验证了该方法具有有效性. 数值结果表明，该方法在单元形状极度扭曲情形下仍能保持良好的精度，这是区别于传统有限元法的显著特点.Abstract: A heat conduction analysis of two-dimensional orthotropic materials was carried out by the hybrid fundamental-solution-based finite element method (HFS-FEM). Temperature distributions within the element domain and on the element boundary were independently described by two temperature fields. A linear combination of fundamental solutions was utilized to approximate the intra-element temperature field while standard one-dimensional shape functions were employed to define the frame temperature field. By virtue of the modified variational functional and divergence theorem, the resultant finite element formulation was derived. The effectiveness of the proposed method was verified by comparing two numerical examples with ABAQUS result. The numerical results demonstrate that the proposed method can still keep excellent accuracy even when the element shape degenerates to a situation of extreme distortion. This is one of marked features which differs from conventional finite element methods.
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图 3 正方形区域、边界条件和有限元网格
Figure 3. Square domain, boundary conditions and finite element mesh
图 5 温度u和热流分量
${{{q}}_{{x}}}$ 的相对误差Figure 5. Relative errors of temperature
${{u}}$ and heat flux component${{{q}}_{{x}}}$ 
图 7 三角陀螺域，边界条件及有限元网格
Figure 7. Trigonometric gyroscopic domain, boundary conditions and finite element mesh
图 8 三角陀螺域温度云图
Figure 8. Cloud maps of temperature in the trigonometric gyroscopic domain
图 9 三角陀螺域热流分量qx云图
Figure 9. Cloud maps of heat flux component
${{{q}}_x}$ in the trigonometric gyroscopic domain
图 10 三角陀螺域热流分量qy云图
Figure 10. Cloud maps of heat flux component
${{{q}}_y}$ in trigonometric gyroscopic domain表 1 不同网格畸变下选定5个点的温度结果
Table 1. Results of temperatures at selected five points under different mesh distortions
坐标 $\gamma $=0 $\gamma $=0.1 $\gamma $=0.3 $\gamma $=0.5 $\gamma $=0.7 $\gamma $=0.9 (0.05,0.025) 5.0778 5.0771 5.0788 5.0809 5.0801 5.0596 (0.025,0.025) 2.5386 2.5478 2.5508 2.5525 2.5523 2.5487 (0.035,0.075) 3.6361 3.6393 3.6332 3.6439 3.6471 3.6268 (0.075,0.075) 7.6275 7.6205 7.6280 7.6216 7.6275 7.6356 (0.015,0.045) 1.5415 1.5408 1.5401 1.5430 1.5472 1.5509 
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