This paper presents a topology optimization approach for surface flows, which can represent the viscous and incompressible fluidic motions at the solid/liquid and liquid/vapor interfaces. The fluidic motions on such material interfaces can be described by the surface Navier-Stokes equations defined on 2-manifolds or two-dimensional manifolds, where the elementary tangential calculus is implemented in terms of exterior differential operators expressed in a Cartesian system. Based on the topology optimization model for fluidic flows with porous medium filling the design domain, an artificial Darcy friction is added to the area force term of the surface Navier-Stokes equations and the physical area forces are penalized to eliminate their existence in the fluidic regions and to avoid the invalidity of the porous medium model. Topology optimization for steady and unsteady surface flows can be implemented by iteratively evolving the impermeability of the porous medium on the 2-manifolds, where the impermeability is interpolated by the material density derived from a design variable. The related partial differential equations are solved by using the surface finite element method. Numerical examples have been provided to demonstrate this topology optimization approach for surface flows, including the boundary velocity driven flows, area force driven flows and convection-diffusion flows.

中文翻译：

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表面流的拓扑优化

本文提出了一种表面流动的拓扑优化方法，该方法可以表示固体/液体和液体/蒸气界面处的粘性和不可压缩流体运动。这种材料界面上的流体运动可以通过定义在 2 流形或二维流形上的表面纳维-斯托克斯方程来描述，其中基本切线微积分是根据笛卡尔系统中表示的外部微分算子实现的。基于多孔介质填充设计域的流体流动拓扑优化模型，在表面 Navier-Stokes 方程的面积力项中添加了人工达西摩擦，并对物理面积力进行了惩罚，以消除它们在流体区域中的存在并避免多孔介质模型的失效。稳态和非稳态表面流的拓扑优化可以通过迭代演化多孔介质在 2 流形上的不渗透性来实现，其中不渗透性由设计变量中的材料密度插值。相关的偏微分方程采用曲面有限元法求解。已经提供了数值例子来演示这种表面流的拓扑优化方法，包括边界速度驱动流、区域力驱动流和对流扩散流。相关的偏微分方程采用曲面有限元法求解。已经提供了数值例子来演示这种表面流的拓扑优化方法，包括边界速度驱动流、区域力驱动流和对流扩散流。相关的偏微分方程采用曲面有限元法求解。已经提供了数值例子来演示这种表面流的拓扑优化方法，包括边界速度驱动流、区域力驱动流和对流扩散流。