Topology Optimization of Flow Problems Modeled by the Incompressible Navier-Stokes Equations.

摘要

This work is concerned with topology optimization of incompressible flow problems. While size and shape optimization methods are limited to modifying existing boundaries, topology optimization allows for merging boundaries as well as creating new ones. Since topology optimization methods do not require a good initial guess, they are powerful tools for finding new and non-intuitive designs. The latter is particularly beneficial for flow problems which are typically nonlinear as well as transient. Depending on the complexity of the flow problem, predicting a solution may be challenging. Determining an improved or optimized design for complex flow problems is an even greater challenge as it not only requires a solution to the flow problem for a given design, but also a prediction on how a design change will affect the flow. Fluid topology optimization commonly uses a material interpolation approach for describing the geometry during the optimization process: solid material is modeled via an artificial porosity that penalizes the flow velocities. While this approach works well for simple steady-state problems aiming to minimize the dissipated energy, the current study shows that using the porosity approach may cause issues for more complex problems such as coupled fluid-structure-interaction (FSI) systems, unsteady flow problems or problems aiming to match a target performance. To overcome these issues a geometric boundary description based on level sets is developed. This geometric boundary description is applied to both, a steady-state hydrodynamic lattice Boltzmann formulation and a stabilized finite element formulation of the steady-state Navier-Stokes equations. The enforcement of the no-slip condition along the fluid-solid interface is handled via an immersed boundary technique in case of the lattice Boltzmann method, while the Navier-Stokes formulation uses an extended finite element method (XFEM). Through the research conducted in this work, the spectrum of flow problems that can be solved by topology optimization techniques has been broadened significantly.