AI-cusor实现二维泊松方程的自适应有限元

简介

Adaptive Finite Element Method for Poisson Equation

Generated on: 2024-11-19 11:08:14

Theoretical Background

Problem Statement

The Poisson equation is a second-order elliptic partial differential equation:

-Δu = f    in Ω

u = 0      on ∂Ω

where Ω is the computational domain, ∂Ω is its boundary, and f is the source term.

Variational Formulation

The variational form of the Poisson equation is:          
Find u ∈ H¹₀(Ω) such that:          
a(u,v) = (f,v)    ∀v ∈ H¹₀(Ω)          
where:          
a(u,v) = ∫_Ω ∇u·∇v dx          
(f,v) = ∫_Ω fv dx

Finite Element Discretization

The domain is triangulated and we use piecewise linear functions (P1 elements):          
• The solution is approximated as: u_h = Σᵢ Uᵢφᵢ          
• Test functions are: v_h = φⱼ          
This leads to the linear system:          
AU = b          
where:          
Aᵢⱼ = ∫_Ω ∇φᵢ·∇φⱼ dx          
bⱼ = ∫_Ω fφⱼ dx    

Adaptive Strategy

The adaptive procedure follows these steps:          

·SOLVE: Compute the finite element solution u_h

·ESTIMATE: Compute error indicators η_T for each element T

·MARK: Select elements for refinement using Dörfler strategy

·REFINE: Refine marked elements while maintaining mesh conformity

Error Estimation

We use residual-based error estimators:          
η_T² = h_T² ‖f + Δu_h‖²_T + Σₑ h_s ‖[∂_n u_h]‖²_s          
where:          
• h_T is the element diameter          
• h_s is the edge length          
• [∂_n u_h] is the jump of normal derivatives across edges

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Numerical Results

Iteration History

Iteration 1

Number of nodes: 200          
Maximum error: 2.21e+00          
Average error: 1.34e+00          
Computation time:          
Assembly: 0.014s          
Solve: 0.000s          
Error Estimate: 0.003s          
Mesh Refinement: 0.103s    

Final Solution and Mesh:

Convergence History:    

Computation Time Statistics:    

Total iterations: 5          
Final number of nodes: 1376

Convergence History

Timing Statistics

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使用总结

1.cursor确实很强大，整个流程基本上不需要会编程，之所以选择python就是因为我本身不太熟悉。全程包括安装，使用，安装库，全部给出指令、代码，只需要按照提示流程来执行；

2.对于简单的小程序而言，cursor非常快速，但是如果修改次数过多，会导致上下文较多，有时候会比较慢，比如我做了一个二维网格可视化的小工具，最后就直接不回答了。小工具的结果也秀一把吧（整个过程花了2个小时）。

3.可以不会编程，但是必须得知道如何实现，比如实现泊松方程的过程非常容易，使用结构化的三角形网格实现，但是在自适应过程中，cursor在局部加密网格中始终错误，最终是一步一步给他说如何局部加密成有效网格，多次迭代后才得到如上的结果。

4.可能使用技巧还有很多没有挖掘，接下来会逐步尝试用cursor实现更加复杂的编程。