约化基、嵌入方法与参数化水平集几何
Reduced Basis, Embedded Methods and Parametrized Levelset Geometry
摘要：
在这一章中，我们将研究几何参数化热交换系统，泊松和流动的基础上斯托克斯，定常和非定常不可压缩的Navier-Stokes和Cahn-Hilliard问题的降阶技术。在嵌入和/或浸入几何框架中采用的全阶有限元法是移位边界（SBM）和切割单元（CutFEM）方法，其应用主要集中在流体中。我们首先介绍Nitsche的方法，SBM/CutFEM和参数化的物理问题，以及高保真近似。我们继续与全阶参数化Nitsche转移边界变分弱制定，降阶建模思想的基础上适当的正交分解Galerkin方法和几何参数化，引用的主要区别和优势，相对于用于经典有限元法的参考域方法，而稳定性问题可能会克服采用极大化富集方法。数值实验验证了效率的介绍''你好，世界''的问题，考虑在几种情况下的一，二，三，四维几何参数化的降阶结果。我们调查执行时间，我们说明运输方法和改进。与未拟合方法和降阶建模相关的重要参考文献列表为[11，8，9，10，7，6，12]。
英文摘要：
In this chapter we examine reduced order techniques for geometrical parametrized heat exchange systems, Poisson, and flows based on Stokes, steady and unsteady incompressible Navier-Stokes and Cahn-Hilliard problems. The full order finite element methods, employed in an embedded and/or immersed geometry framework, are the Shifted Boundary (SBM) and the Cut elements (CutFEM) methodologies, with applications mainly focused in fluids. We start by introducing the Nitsche's method, for both SBM/CutFEM and parametrized physical problems as well as the high fidelity approximation. We continue with the full order parameterized Nitsche shifted boundary variational weak formulation, and the reduced order modeling ideas based on a Proper Orthogonal Decomposition Galerkin method and geometrical parametrization, quoting the main differences and advantages with respect to a reference domain approach used for classical finite element methods, while stability issues may overcome employing supremizer enrichment methodologies. Numerical experiments verify the efficiency of the introduced ``hello world'' problems considering reduced order results in several cases for one, two, three and four dimensional geometrical kind of parametrization. We investigate execution times, and we illustrate transport methods and improvements. A list of important references related to unfitted methods and reduced order modeling are [11, 8, 9, 10, 7, 6, 12].
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