An immersed boundary fast integration methodology featured by a consistent weight learning is proposed to accelerate Galerkin meshfree computation. In the proposed approach, the problem domain is embedded in a rectangular spatial domain discretized by regular distributions of meshfree nodes and integration sampling points with virtual integration cells. A trimming operation of the rectangular spatial domain by the physical problem boundary with nodal discretization yields the discrete model for meshfree computation, where the integration sampling points are grouped into the interior sampling points and the near boundary sampling points which form a training set. For the interior sampling points, a natural alignment between the influence domains of meshfree shape functions and virtual integration cells can be easily realized through employing integer support sizes, which ensures a satisfactory accuracy for the basis degree correspondent normal order Gauss integration. The background cells for the domain integration are completely avoided, which greatly simplifies the preprocessing for numerical integration. Meanwhile, a machine learning module is devised to optimize the weights of near boundary integration sampling points. The key step to construct this machine learning module for weight optimization is the selection of the variational integration consistency condition as the loss function, which guarantees the convergence of Galerkin meshfree formulation. The accuracy and efficiency of the proposed weight learning immersed boundary fast integration methodology is thoroughly validated through numerical results.

中文翻译：

---

具有一致权重学习的浸入式边界快速无网格集成方法


提出了一种以一致权重学习为特征的浸入式边界快速积分方法来加速伽辽金无网格计算。在所提出的方法中，问题域嵌入到由无网格节点和具有虚拟积分单元的积分采样点的规则分布离散化的矩形空间域中。通过节点离散化的物理问题边界对矩形空间域进行修剪操作，产生用于无网格计算的离散模型，其中积分采样点被分组为内部采样点和近边界采样点，形成训练集。对于内部采样点，通过采用整数支撑尺寸可以很容易地实现无网格形状函数和虚拟积分单元的影响域之间的自然对齐，这保证了基度对应的正阶高斯积分的令人满意的精度。完全避免了域积分的背景单元，这大大简化了数值积分的预处理。同时，设计了机器学习模块来优化近边界积分采样点的权重。构造权重优化机器学习模块的关键步骤是选择变分积分一致性条件作为损失函数，保证了伽辽金无网格公式的收敛性。通过数值结果彻底验证了所提出的权重学习浸入式边界快速积分方法的准确性和效率。