基于聚合型代数多重网格法的三维直流电法自适应有限元正演

摘要:

在各向异性、起伏地形、真实地质模型电法模拟中，经自适应有限元离散后形成的大型稀疏线性系统存在内存消耗高、求解效率低等缺陷。为此，提出了聚合型代数多重网格（AGMG）法与自适应有限元法的联合算法，在提高正演精度的同时提升计算效率，实现复杂模型三维直流电法大规模正演模拟。对于三维直流电法满足的二阶椭圆边值问题，采用非结构化四面体网格的有限元法离散，并通过自适应策略进行局部加密，再利用AGMG法求解离散形成的大规模稀疏线性方程组。最后，通过复杂地电模型和实际地质模型验证了联合算法的有效性。在千万级自由度的求解中，联合算法比传统迭代法快了20多倍，比代数多重网格法快了近10倍，随着模型复杂度的提高，联合算法的效率优势更加明显。

Abstract:

In simulations of electrical methods for anisotropic， rolling and realistic terrains， large sparse linear systems generated by the adaptive finite element discretization suffer from high memory consumption and low solution efficiency. To address these issues， we propose a combined algorithm that integrates the aggregated algebraic multigrid （AGMG） method with the adaptive finite element method. The combined algorithm enhances the forward modeling accuracy while significantly improving the computational efficiency， enabling large-scale 3D direct current resistivity complex models. For second-order elliptic boundary value problems associated with 3D direct current resistivity， we utilize an unstructured tetrahedral mesh for the finite element discretization. The local refinement is applied through adaptive strategies， and the resulting large-scale sparse linear systems are solved using the AGMG method. Finally， the effectiveness of the combined algorithm is validated through simulations of complex geoelectric models and real geological scenarios. In solving systems with tens of millions of degrees of freedom， the combined algorithm is over 20 times faster than traditional iterative methods and nearly 10 times faster than the algebraic multigrid method. The efficiency advantage of the combined algorithm becomes even more pronounced as the model complexity increases.