This work describes a newly developed, arbitrarily high-order Cartesian-grid method for reconstructing material interfaces from a volume fraction field. The method begins by identifying all of the grid cells in the volume fraction field that are intersected by the interface and need to be approximated by the reconstruction scheme. Finite-differences are used to calculate the gradient of the volume fraction field and provide an estimate of the surface normal in all of the interfacial grid cells. Groups of connected grid cells are then identified which all have the same dominant component of the normal vector. This grouping by orientation determines the proper dependent variable to use in the surface reconstruction (e.g. for a 2D curve, this step determines if the surface will be approximated by a function of x or y). A cumulative integral over the surface is constructed and fit using b-splines for two-dimensional problems or tensor-product b-splines for three-dimensional problems. This construction allows for the interface to be recovered through application of the second fundamental theorem of calculus. Fitting the cumulative integral with $\mathcal{N}$th-order b-splines (or tensor-product b-splines) yields an $(\mathcal{N}-1)$th-order convergence rate of the interface shape. Differentiation of the b-spline interface function(s) allows for the high-order approximation of the normal vector and curvature to be obtained directly anywhere along b-spline. Together, the proposed reconstruction technique can achieve arbitrarily high mesh convergence rates. Validation tests are presented with mesh convergence rates ranging from fourth- to tenth-order.

这项工作描述了一种新开发的，任意高阶笛卡尔网格方法，用于从体积分数场重建材料界面。该方法开始于识别体积分数字段中所有与界面相交且需要通过重构方案近似的网格单元。有限差分用于计算体积分数场的梯度，并提供所有界面网格单元中表面法线的估计。然后确定连接的网格单元组，它们均具有法向矢量的相同主要成分。通过方向分组可以确定在曲面重建中使用的适当因变量（例如，对于2D曲线，此步骤确定曲面是否将通过x的函数近似或y）。使用b样条求解二维问题或使用张量积b样条求解三维问题，构造并拟合表面上的累积积分。这种构造允许通过应用微积分的第二基本定理来恢复接口。用以下公式拟合累积积分$\mathcal{ñ}$三阶b样条曲线（或张量积b样条曲线）产生 $（\mathcal{ñ}-\mathrm{1个}）$界面形状的三阶收敛速度。b样条曲线接口函数的微分允许直接沿b样条曲线的任意位置获得法向矢量和曲率的高阶近似。一起，所提出的重建技术可以实现任意高的网格收敛率。提出的验证测试的网格收敛速度范围为四到十阶。