This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.

本文提出了一种新颖的信任区域方法，用于优化多个昂贵的函数。我们将此方法应用于流体力学中的双目标优化问题，即在密闭容器中的流体中颗粒的最佳混合。三维时间相关的流由洛伦兹力驱动，洛伦兹力是由位于矩形容器下方的振荡永久磁铁产生的。矩形磁体提供了在分析上已知的空间不均匀磁场。磁体振荡产生稳定的平均流量（稳定流），类似于从振荡刚体观察到的平均流量。在优化问题中，流动使无规则分布的无质量粒子平流，以实现均匀分布（目标函数1），同时保持使永磁体移动最小的工作（目标函数2）。对这两个目标函数的一次评估可能需要两个小时以上。因此，为节省计算时间，该方法在信任区域上使用插值模型来查找下降方向。我们表明，即使对于我们显着简化的模型问题，混合模式也会随控制参数而显着变化，这证明了使用改进的优化技术及其进一步发展的合理性。所提出的方法在信任区域上使用插值模型来寻找下降方向。我们表明，即使对于我们显着简化的模型问题，混合模式也会随控制参数而显着变化，这证明了使用改进的优化技术及其进一步发展的合理性。所提出的方法在信任区域上使用插值模型来寻找下降方向。我们表明，即使对于我们显着简化的模型问题，混合模式也会随控制参数而显着变化，这证明了使用改进的优化技术及其进一步发展的合理性。