线源三维多孔介质中的渗流与力学耦合
Coupled Flow and Mechanics in a 3D Porous Media with Line Sources
摘要：
在本文中，我们考虑的准静态，线性Biot模型在三维域$Ω$的质量平衡方程的右手侧集中在一个一维线源$δ_Λ$的数值逼近。该模型在医学背景下是感兴趣的，其中它可以用于对通过血管化组织的流动和变形进行建模。该模型本身是具有挑战性的近似作为线源诱导的压力和通量的解决方案是奇异的。为了克服这一点，我们在这里结合两种方法：（i）一个固定的应力分裂计划解耦的流动和力学方程和（ii）的压力和流量变量的奇异性去除方法。奇异性去除是基于将解分裂成捕获解奇异性的较低正则性项和表示剩余部分的较高正则性项。有了这一点，流动方程就可以重新表述，使它们相对于余项成立。重新制定的系统，然后近似使用固定应力分裂计划。最后，我们通过显示与通过血管化组织的流量相关的测试用例的结果得出结论。这种数值方法被发现是最佳收敛的。
英文摘要：
In this paper, we consider the numerical approximation of the quasi-static, linear Biot model in a 3D domain $Ω$ when the right-hand side of the mass balance equation is concentrated on a 1D line source $δ_Λ$. This model is of interest in the context of medicine, where it can be used to model flow and deformation through vascularized tissue. The model itself is challenging to approximate as the line source induces the pressure and flux solution to be singular. To overcome this, we here combine two methods: (i) a fixed-stress splitting scheme to decouple the flow and mechanics equations and (ii) a singularity removal method for the pressure and flux variables. The singularity removal is based on a splitting of the solution into a lower regularity term capturing the solution singularities, and a higher regularity term denoted the remainder. With this in hand, the flow equations can now be reformulated so that they are posed with respect to the remainder terms. The reformulated system is then approximated using the fixed-stress splitting scheme. Finally, we conclude by showing the results for a test case relevant for flow through vascularized tissue. This numerical method is found to converge optimally.
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