libmesh 实现流程

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libmesh 实现流程

标签： libmesh poisson 有限元

2014-09-19 22:56 371人阅读评论(0)收藏举报

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开源 libmesh（8）

介绍poisson方程的libmesh实现过程：

Poisson Equation $- \Delta u = f$

Weak Form $(F(u),v) = \int _{\Omega} \nabla u \cdot \nabla v dV + \int_{\partial \Omega} \nabla u \cdot \textbf{n} dS$

FEM Approximation $(F(u^h),v^h) = \int _{\Omega^h} \nabla u^h \cdot \nabla v^h dV = \sum_{e=1}^{Ne} \int_{\Omega^e} \nabla u^h \cdot \nabla v^h dV^e$

Basic Function/Shape Function $v^h|_{\Omega^e} = \sum_{i=1}^N v_i \phi_i$

Approximation in Element Level $\int_{\Omega^e}v^h f dV = v_i \sum_{i=1}^N f \phi_i \\ \int_{\Omega^e} \nabla u \nabla v dV = v_i \sum_{i=1}^N \nabla \phi_j \nabla \phi_i u_j$

Matrix Notation $K^e U^e = F^e$

Local To Global Element Map $F^e = \int_{\Omega_e } f \phi_i dV = \int_{\hat \Omega_e} f(x(\xi)) \phi_i |J| d \xi \\K^e = \int_{\Omega_e } \phi_j \cdot \phi_i dV = \int_{\hat \Omega_e} \hat \nabla\phi_j \hat \nabla\phi_i |J| d \xi \\where, \hat \nabla = J^{-1} \nabla$

Integrals Numerical Quadrature with Nq points at \xi_q with weight value w_q $F^e = \int_{\hat \Omega_e} f(x(\xi)) \phi_i |J| d \xi \approx \sum_{q=1}^N f(\xi_q) \phi(\xi_q) J(\xi_q) w_q\\K^e = \int_{\hat \Omega_e} \hat \nabla\phi_j \hat \nabla\phi_i |J| d \xi \approx \sum_{q=1}^N \phi _j(\xi_q) \phi _i(\xi_q) J(\xi_q) w_q$

Libmesh provides the following variables at each Quadrature Point JxW -> J(\xi_q) w_q phi[i][q] -> \phi_i{\xi_q) dphi[i][q] -> \nabla \phi_i(\xi_q) xyz[q] -> x(\xi_q) 至此，给出了poisson方程完整的libmesh分析流程。参考libmesh slides.

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libmesh 实现 流程

libmesh 实现 流程

libmesh 实现流程

libmesh 实现流程