Calculating the yield limit $Y_c$ (the critical ratio of the yield stress to the driving stress), of a viscoplastic fluid flow is a challenging problem, often needing iteration in the rheological parameters to approach this limit, as well as accurate computations that account properly for the yield stress and potentially adaptive meshing. For particle settling flows, in recent years calculating $Y_c$ has been accomplished analytically for many antiplane shear flow configurations and also computationally for many geometries, under either two dimensional (2D) or axisymmetric flow restrictions. Here we approach the problem of 3D particle settling and how to compute the yield limit directly, i.e. without iteratively changing the rheology to approach the yield limit. The presented approach develops tools from optimization theory, taking advantage of the fact that $Y_c$ is defined via a minimization problem. We recast this minimization in terms of primal and dual variational problems, develop the necessary theory and finally implement a basic but workable algorithm. We benchmark results against accurate axisymmetric flow computations for cylinders and ellipsoids, computed using adaptive meshing. We also make comparisons of accuracy in calculating $Y_c$ on comparable fixed meshes. This demonstrates the feasibility and benefits of directly computing $Y_c$ in multiple dimensions. Lastly, we present some sample computations for complex 3D particle shapes.

计算屈服极限 ${ÿ}_C$（粘弹性流体的屈服应力与驱动应力的临界比）是一个具有挑战性的问题，通常需要流变参数的迭代才能达到此极限，并且需要正确计算出正确的屈服应力和潜在的计算结果自适应网格划分。对于颗粒沉降流，近年来进行了计算${ÿ}_C$在二维（2D）或轴对称流动限制下，已经完成了许多反平面剪切流配置的解析分析，还完成了许多几何形状的计算解析。在这里，我们解决了3D粒子沉降的问题以及如何直接计算屈服极限，即无需反复更改流变特性即可接近屈服极限。提出的方法利用优化理论从优化理论开发工具${ÿ}_C$通过最小化问题定义。我们根据原始和对偶变分问题重铸了这种最小化，发展了必要的理论，最后实现了一个基本但可行的算法。我们将结果与针对使用自适应网格划分的圆柱体和椭球体的精确轴对称流计算进行对比。我们还对计算的准确性进行了比较${ÿ}_C$在可比较的固定网格上。这证明了直接计算的可行性和好处${ÿ}_C$在多个维度上。最后，我们介绍了一些复杂3D粒子形状的示例计算。