26、工程等几何分析相关技术解析

工程等几何分析相关技术解析

1. 瞬态与动态分析基础

1.1 瞬态分析方程

瞬态分析中，标量场模型的常微分方程为：
(\frac{\partial}{\partial x}(k_x\frac{\partial u}{\partial x}) + \frac{\partial}{\partial y}(k_y\frac{\partial u}{\partial y}) + Tu + Q(x, y) = \rho_p\frac{\partial u}{\partial \tau})
其等效积分形式为：
(\int_{\Gamma_b} u(k_{nn}\frac{\partial u}{\partial n}) d\Gamma - \int_{\Omega}(\frac{\partial u}{\partial x}(k_x\frac{\partial u}{\partial x}) + \frac{\partial u}{\partial y}(k_y\frac{\partial u}{\partial y})) d\Omega - \int_{\Omega} uTu d\Omega - \int_{\Omega} uQ d\Omega - \int_{\Omega} u\rho_c p u d\Omega = 0)
同时，还涉及传导、对流、平流和源矩阵：
- (S_e^{\kappa} = \int_{\Omega_e} \vec{\nabla} N^T \kappa \vec{\nabla} N d\Omega)
- (A_e^v = \int_{\Omega_e} N_e^T m_e v_e B_e d\Omega)
-