Abstract

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.

Graphical Abstract

We have considered a pressure-driven problem with axisymmetry and got optimal material layouts using topology optimization by maximizing the total rate of dissipation. The left figure shows unphysical finger-like design patterns when the primal analysis does not enforce explicitly the underlying radial symmetry. The right figure shows that one can avoid such numerical pathologies if the primal analysis invokes axisymmetry conditions.

中文翻译：

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使用拓扑优化的多孔介质流动优化设计

摘要

拓扑优化（TopOpt）是一种数学驱动的设计过程，可实现最佳的材料体系结构。此过程通常用于使涉及流经多孔介质的设备（例如微流体设备）的设计自动化。TopOpt提供的材料布局可控制流体通过多孔材料的流动，从而提供所需的功能。该应用领域中的许多先前研究已将Darcy方程用于原始分析，并使用了最小功率定理（MPT）来驱动优化问题。但是这两种选择（达西方程式和MPT）是限制性的，不适用于现代设备的一般工作条件。由于简单且线性，Darcy方程通常用于对通过多孔介质的流体流动进行建模。但是，Darcy模型的两个固有假设是：流体的粘度为常数，而惯性效应可忽略不计。有无可辩驳的实验证据表明，流体（尤其是有机液体）的粘度取决于压力。给定典型的小孔径，惯性效应在微流控设备中占主导地位。其次，MPT不是通用原理，并且对于放松Darcy模型假设的（非线性）模型无效。本文旨在通过提出使用TopOpt的一般策略来克服上述缺陷。。首先，我们将考虑考虑压力依赖的粘度和惯性效应的非线性模型，并研究这些非线性对TopOpt下最佳材料布局的影响。其次，我们将探索机械耗散率，该耗散率甚至对非线性模型也有效，作为目标函数的替代方法。第三，我们将为典型问题提供最佳设计的解析解决方案。这些解决方案不仅具有研究和教学价值，而且还有助于验证计算机实现。

图形概要

我们考虑了轴对称的压力驱动问题，并通过最大化总耗散率使用拓扑优化获得了最佳的材料布局。左图显示了当原始分析未明确实施基本的径向对称性时的非物理手指状设计模式。右图显示，如果原始分析调用轴对称条件，则可以避免此类数字病态。