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基于功能梯度材料的高阶理论,提出了基于任意四边形子胞的多尺度温度场计算方法。以二维问题为研究对象,首先采用任意四边形子胞离散求解域,并且采用二次多项式来近似子胞内的温度场。通过采用子胞边界温度代替假设温度函数的系数减少未知量数目,应用Fourier热传导定律,建立单个子胞边界平均热流密度与边界平均温度的关系。将此关系代入边界条件以及子胞间的平均热流平衡方程和平均温度连续条件,消去子胞边界热流密度。然后以子胞边界平均温度作为未知量对方程组进行求解。从而求解出整个物体的温度场分布。本文提出的方法突破了原高阶理论中子胞必须是规则长方形的限制,提高了高阶理论的使用范围。通过与有限元的计算结果进行比较,证明了本文方法计算的有效性。
Based on the high-order theory of functionally graded materials, a multi-scale temperature field calculation method based on arbitrary quadrilateral cells is proposed. Taking the two-dimensional problem as the research object, the quadratic polynomials are first used to discretize the domain of arbitrary quadrilateral daughter cells, and the temperature field in the daughter cells is approximated by quadratic polynomials. By using the Fourier heat conduction law, the relationship between the average heat flux density of a single sub-cell boundary and the average temperature of the boundary is established by using the sub-cell boundary temperature instead of the coefficient of the assumed temperature function to reduce the number of unknowns. Substituting this relationship into the boundary conditions and the average heat flow balance equation between sub-cells and the average temperature continuous condition, the heat flux at the daughter cell boundary is canceled. Then the average temperature of the daughter cell boundary as an unknown amount of equations to solve. Thus solving the temperature distribution of the whole object. The method proposed in this paper breaks through the restriction that the original high-order neutron theory must be a regular rectangle and improves the use of high-order theory. By comparing with the finite element calculation results, the validity of the method in this paper is proved.