Fictitious domain methods enable to solve physical problems on unfitted grids, thereby avoiding time-consuming and error prone meshing phases. However, an accurate integration of the weak formulation is still mandatory, leading to the need for efficient quadrature strategies in the elements that are partially located in the physical domain. Various methodologies have been proposed to this end. Among them, the design of element-specific moment-fitting quadrature rules seems promising. However, some of the resulting weights are usually negative and the points located out of the domain which may lead to a lack quadrature stability. In this contribution, we aim to construct quadrature rules whose weights are all positive and points all located in the physical domain. Such rules can be constructed based on a non-negative least square resolution of the moment-fitting equations. The resulting quadrature formulas are compared to empirical quadrature rules and different variants of classical moment-fitting quadrature rules in both 1D and 2D. Benchmarks show that non-negative moment-fitting quadrature rules are robust and efficient although their setup cost can be significantly higher than classical moment-fitting. Finally, their application to linear elastic and small-strain elasto-plastic problems highlight their robustness for engineering applications.

虚拟域方法能够解决未拟合网格上的物理问题，从而避免耗时且容易出错的网格划分阶段。然而，弱公式的精确集成仍然是强制性的，导致需要在部分位于物理域的元素中采用有效的正交策略。为此提出了各种方法。其中，单元特定矩拟合正交规则的设计似乎很有前途。然而，一些由此产生的权重通常是负的，并且点位于域外，这可能导致缺乏正交稳定性。在这个贡献中，我们的目标是构建权重均为正且点均位于物理域中的正交规则。此类规则可以基于矩拟合方程的非负最小二乘分辨率来构建。将得到的正交公式与经验正交规则和一维和二维经典矩拟合正交规则的不同变体进行比较。基准测试表明，非负矩拟合正交规则是稳健且有效的，尽管它们的设置成本可能明显高于经典矩拟合。最后，它们在线性弹性和小应变弹塑性问题中的应用突出了它们在工程应用中的稳健性。基准测试表明，非负矩拟合正交规则是稳健且有效的，尽管它们的设置成本可能明显高于经典矩拟合。最后，它们在线性弹性和小应变弹塑性问题中的应用突出了它们在工程应用中的稳健性。基准测试表明，非负矩拟合正交规则是稳健且有效的，尽管它们的设置成本可能明显高于经典矩拟合。最后，它们在线性弹性和小应变弹塑性问题中的应用突出了它们在工程应用中的稳健性。