The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. We present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method.

位移边界方法（SBM）属于未拟合（或沉入或嵌入）有限元方法的类别，最近被引入用于泊松，线性对流/扩散，斯托克斯，纳维尔-斯托克斯，声学和浅水方程式。通过在替代（近似）计算域上重新构造原始边值问题，SBM避免了对切割单元的积分以及与数值稳定性和矩阵条件有关的相关问题。通过使用泰勒展开修改原始边界条件，可以保持精度。因此该方法的名称，偏移的位置和值边界条件。在本文中，我们通过用活性流体的元素体积分数适当地加权其变化形式，将SBM扩展到具有自由表面运动的不可压缩Navier-Stokes流的模拟。这种方法可以防止时间上的杂散压力振荡，否则，如果总有效流体体积在一个时间步长上突然变化，将会产生杂散压力振荡。实际上，所提出的加权SBM方法会引起较小的质量（即体积）守恒误差，在细分线性有限元插值的情况下，随着网格的细化，误差会二次收敛。我们提出了一套广泛的二维和三维测试，以证明该方法的鲁棒性和准确性。