基于Laplace调和方程的网格重构算法

针对三角形网格向四边形网格的转化问题,基于调和方程构建模型梯度场,追踪表面流线,实现了参数化网格重构。首先,建立了基于离散Laplace方程的三角网格梯度场理论模型、数据结构模型和稀疏矩阵求解方案;其次,提出了局部坐标变换和参数方程相结合求解流线节点的统一算法,并针对流线跟踪无交点、有多个交点等特殊情况,提出了梯度收敛、最短距离和参数极值等优选策略;最后,通过模型实验验证了算法。结果表明,流线网格具有等参、闭合特点,复杂模型网格划分没有歧义,而且网格质量随网格密度增加而提高。因此,相对于传统几何重构算法,数学方法对网格重构表达具有鲁棒性和唯一性,且应用场景更广泛。

Mesh reconstruction algorithm based on Laplace harmonic equation

Upon the problem of transforming triangular mesh into quadrilateral mesh, gradient field via harmonic equation is created, integral flow is tracked, and parameterized mesh is reconstructed. First, gradient field construction theory, data structure model and solution scheme of sparse matrix are constructed. Then, one uniform algorithm for solving flow line node by integration of local coordinate transformation and parameterized equation is advanced, and schemes like gradient convergence, shortest distance and extreme parameter are optimally occupied upon special cases of no intersection or multiple intersections when tracing flow line. Finally, the algorithm is verified via case studies. The result shows that flow lines are characterized as iso-parameterized and closed, mesh reconstruction of complicate models has no bifurcation, and mesh quality is increased with its density. The result proves that mathematical method has robustness and uniqueness for mesh reconstruction representation compared with traditional geometry method, and it will have more application scenarios.