瞬态热传导问题分析的虚边界无网格伽辽金法研究

As an important numerical method, boundary element method (BEM) is often used to analyze the transient heat conduction problem. The virtual boundary least square method (VBLSM) can avoid the singular integral, "Vertex Question" and "Boundary Layer Effect" of BEM. Its numerical precision is high. However, there are some problems for the weighting coefficient of VBLSM, such as artificial value, and how to select. The above problems are solved by virtual boundary meshfree galerkin method (VBMGM) for steady-state heat conduction problem analysis presented by applicant. The weighted coefficient of VBMGM is unique. And the meaning of the weighted coefficient is clear. The equation of VBMGM for steady-state heat conduction problem has the sparse symmetry, the good stability and accuracy. In this project, VBMGM is extended to analyze the transient heat conduction problem with multi domain variable coefficient containing heat source. The time step scheme is chosen according to no oscillation and numerical result can accurately meet the partial differential equation. According to the initial condition, the boundary condition, the continuous condition and the control equation, the integral equation of VBMGM for the transient heat conduction problem is obtained by the galerkin method. Numerical discretization formula of the integral equation is gotten by using the radial basis function to construct the virtual source function and the heat source. At last, the stability and the accuracy of the proposed method are proved by changing the parameters of the numerical discretization formula. This method is not only simple and effective, but also provides a new way to solve the multi-medium transient heat conduction problem in the complex engineering.

边界元法作为一种重要的数值方法，常被用来分析瞬态热传导问题。虚边界无网格最小二乘法能够避免边界元法的奇异积分、边界层效应、角点问题，且数值精度高，但加权系数存在人为取值、如何取值等问题。申请人提出了稳态热传导问题分析的虚边界无网格伽辽金法，解决了上述问题，且加权系数数值唯一、意义明确，方程具有稀疏对称性、稳定性与精确性好。本项目拟推广虚边界无网格伽辽金法，分析多域变系数含热源的瞬态热传导问题。选择没有振荡、数值结果可以精确满足偏微分方程的时间步长方案。根据初始条件、边界条件、连续条件、控制方程，采用伽辽金法，获得瞬态热传导问题分析的虚边界无网格伽辽金法的积分方程式。利用径向基函数构造虚拟源函数、热源，得到积分方程对应的数值离散公式。最后，改变数值离散公式中的参数，证明方法的数值稳定性、精确性。该方法不但简单有效、精度高，而且为复杂工程多介质瞬态热传导问题的求解提供了新的途径。

虚边界无网格伽辽金法作为虚边界元法的推广，能够避免边界元法的奇异积分、边界层效应、角点问题，具有加权系数数值唯一、意义明确，方程具有稀疏对称性，已被用于稳态热传导问题的计算。此项目进一步推广虚边界无网格伽辽金法，根据初始条件、边界条件、连续条件、控制方程，采用伽辽金法，使用向后差分方案，获得二维、三维多域变热传导系数、变热源瞬态热传导问题分析的虚边界无网格伽辽金法，计算多个算例说明了项目的精确性与稳定性。项目方法具有编程简单、精度高的优点，已被用于计算功能梯度材料的热传导问题、裂纹问题、弹性问题、热弹性问题，为复杂工程多介质瞬态热传导问题的求解提供了新的途径。