论文标题
在模型降低中的参数几何处理处理的离散化方法
A discretize-then-map approach for the treatment of parameterized geometries in model order reduction
论文作者
论文摘要
我们提出了一种在基于投影的模型降低中处理参数化几何形状的一般方法。在离线阶段，给定（i）一个参数化域的家族$ \ {ω__：μ\ in \ Mathcal {p} \} \ subset \ subset \ mathbb {r}^d $ where $μ\其中$μ\ in \ in \ mathcal {P}参考域$ω$与参数依赖性域$ω_μ$之间的$φ_μ$，以及（iii）$ω$的有限元三角剖分，我们求助于经验的正交程序，以选择网格元素的子集。在在线阶段，我们首先使用映射来“移动”所选元素的节点，然后使用标准元素的剩余评估例程来评估剩余及其Jacobian。我们讨论了如何设计一种在线效率减少订单模型，并讨论了更标准的“地图-tandiscretize”方法的差异（例如Rozza，Huynh，Patera，Acme，Acme，2007年）；特别是，我们展示了如何离散化 - 然后是映射框架，大大简化了降级模型的实现。我们将方法应用于经过参数化的机翼的二维电位流问题，以及对艾哈迈德机构经过的流动的二维命令。
We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains $\{ Ω_μ: μ\in \mathcal{P} \} \subset \mathbb{R}^D$ where $μ\in \mathcal{P} \subset \mathbb{R}^P$ denotes a vector of parameters, (ii) a parameterized mapping $Φ_μ$ between a reference domain $Ω$ and the parameter-dependent domain $Ω_μ$, and (iii) a finite element triangulation of $Ω$, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to "move" the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard "map-then-discretize" approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.
