论文标题
不连续解决方案的3D-1D耦合问题的PDE受限优化方法
A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions
论文作者
论文摘要
得出和讨论了在接口处与不连续解的耦合3D-1D问题的数值方法。这扩展了有关仅考虑连续解决方案的主题的先前工作。得益于正确定义的函数空间，从原始的完全3D问题获得了良好的3D-1D问题，然后通过PDE受限的优化重新重新加密找到解决方案。这是一个域分解策略，其中引入了未知的接口变量，并具有适当定义的成本功能，表达在满足接口条件下的错误，受到子域中的本构方程的约束。由于在各个子域中使用独立的离散化，因此，由于几何复杂性，所得的离散问题在几何复杂性方面具有牢固性。可以使用不同尺寸的网格，而不会影响离散线性系统的条件，这是考虑公式的特殊方面。基于使用基于梯度的求解器并产生准备并行实施的方法，进一步提出了有效的分辨率策略。关于已知分析解决方案问题的数值实验显示了该方法的准确性，并提出了两个有关更复杂配置的示例，以解决该方法对实际问题的适用性。
A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient resolution strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.
