Topology optimization (TopOpt) is a mathematical-driven design procedure to realize optimal material architectures. This procedure is often used to automate the design of devices involving flow through porous media, such as micro-fluidic devices. TopOpt offers material layouts that control the flow of fluids through porous materials, providing desired functionalities. Many prior studies in this application area have used Darcy equations for primal analysis and the minimum power theorem (MPT) to drive the optimization problem. But both these choices (Darcy equations and MPT) are restrictive and not valid for general working conditions of modern devices. Being simple and linear, Darcy equations are often used to model flow of fluids through porous media. However, two inherent assumptions of the Darcy model are: the viscosity of a fluid is a constant, and inertial effects are negligible. There is irrefutable experimental evidence that viscosity of a fluid, especially organic liquids, depends on the pressure. Given the typical small pore-sizes, inertial effects are dominant in micro-fluidic devices. Next, MPT is not a general principle and is not valid for (nonlinear) models that relax the assumptions of the Darcy model. This paper aims to overcome the mentioned deficiencies by presenting a general strategy for using TopOpt. First, we will consider nonlinear models that take into account the pressure-dependent viscosity and inertial effects, and study the effect of these nonlinearities on the optimal material layouts under TopOpt. Second, we will explore the rate of mechanical dissipation, valid even for nonlinear models, as an alternative for the objective function. Third, we will present analytical solutions of optimal designs for canonical problems; these solutions not only possess research and pedagogical values but also facilitate verification of computer implementations.
翻译:地形优化( TopOpt) 是一个数学驱动的设计程序, 以实现最佳材料结构。 此程序通常用于使设备设计自动化, 包括通过微流质设备等多孔媒体流动的装置。 TopOpt 提供了控制流体通过多孔材料流动的材料布局, 提供了想要的功能 。 许多先前在应用领域的研究都使用达西方程式进行原始分析, 以及最小电荷定理( MPT) 来驱动优化问题。 但这两种选择( 达西方程式和 MPT) 都具有限制性, 并且不适用于现代设备的一般工作环境条件。 简单和线性, 达西方程式往往被用来通过多孔媒体模拟流体流体的流动。 然而, 达西模式的两个固有假设是: 流的粘度是恒定的, 惯性效果是用于原始分析的, 并且通过我们提出的标准, 将这种模型用于模拟的优化操作, 将用来解释。