Transition from adjoint level set topology to shape optimization for 2D fluid mechanics

Abstract

Most optimization problems in the field of fluid mechanics can be classified as either topology or shape optimization. Although topology and shape have been considered mutually exclusive optimization methods since their inception, it is conceivable that they will find choicest solutions in tandem, with shape optimization refining a solution found by topology. However, linking the topology optimization problem to that of shape is not trivial and, to the authors' knowledge, has yet to be formally attempted. This paper proposes a novel transitional procedure that post-processes 2D adjoint topology solutions, fitting the interface between the solid and fluid topological domains to create a parameterized solution which can be used as either a CAD-compatible representation of the interface or a source for grid generation from which a shape optimization loop can be initialized. The interface to be fit can be extracted from any topological field with distinct fluid and solid domains, meaning that the proposed transition process is independent of the topology approach utilized. To conveniently describe the interface between the solid and fluid topological domains, the topology optimization process employed in this paper is filtered using the level set method. The interface is fit with non-uniform rational B-spline (NURBS) curves through application of sensitivities garnered from the solution of an auxiliary inverse design problem which aims at reducing the difference between signed-distance fields generated about both the NURBS curve being optimized and the section of interface being fit. The geometry defined by the fit NURBS curves is then (optionally) used to build a boundary-fitted grid on which a shape optimization loop is performed. The parameterized result of the topology to shape transition process is compared to that of shape optimization in 2D cases with internal, incompressible fluid flows.

A Review of Topology Optimisation for Fluid-Based Problems

“…Jang and Lee [51] maximise the acoustic transmission loss in a chamber muffler with a constraint on the reverse-flow power dissipation modelled using the Navier-Stokes equations. Koch et al [52] proposed a method for automatic conversion from two-dimensional Ersatz-based level set topology optimisation to NURBS-based shape optimisation. Sato et al [53] applied density-based topology optimisation to the design of no-moving parts fluid valves using a Pareto front exploration method.…”

“…PM is the least used method with only a single paper [79]. Surprisingly, FVM is the second least used method with only 12 papers [21,47,52,80,82,115,129,130,137,144,159,188] or 7%, despite the fact that FVM for many years has been the preferred discretisation method for computational fluid dynamics. This can probably be explained by several factors: topology optimisation originates from solid mechanics where FEM is the preferred method; discrete adjoint approaches are easier using FEM than FVM; stabilised FEM has grown to be a mature and accurate method [193,194].…”

Adjoint variable‐based shape optimization with bounding surface constraints

“…This case was initially shown by Koch et al (2017), 37 in an article dealing with topology optimization in fluid mechanics, without involving the bounding box constraints. The duct geometry is described by B‐spline control points.…”

A Continuous Adjoint Framework for Shape and Topology Optimization and their Synergistic Use

“…To achieve this synergistic optimization, a transition process for 3D geometries (developed in 2D in Ref. 19 and referred to as the TtoST -Topology to Shape Transitionprocess) is employed. The TtoST process is exemplified through a test case in section IV.A.…”