Topology optimization on two-dimensional manifolds

Abstract

This paper implements topology optimization on two-dimensional manifolds. In this paper, the material interpolation is implemented on a material parameter in the partial differential equation used to describe a physical field, when this physical field is defined on a two-dimensional manifold; the material density is used to formulate a mixed boundary condition of the physical field and implement the penalization between two different types of boundary conditions, when this physical field is defined on a threedimensional domain with its boundary conditions defined on the two-dimensional manifold corresponding a surface or an interface of this three-dimensional domain. Based on the homeomorphic property of two-dimensional manifolds, typical two-dimensional manifolds, e.g., sphere, torus, Möbius strip and Klein bottle, are included in the numerical tests, which are provided for the problems on fluidic mechanics, heat transfer and electromagnetics.

Fiber bundle topology optimization for surface flows

“…To control the smoothness of the implicit 2-manifold and ensure the well-poseness of the solution, a surface-PDE filter sketched in Fig. 2 is imposed on the design variable of the implicit 2-manifold [50]:…”

“…Here, the threshold projection is used to remove the gray regions and control the minimum length scale in the derived pattern. The surface-PDE filter for the design variable of the pattern is implemented by solving the following surface-PDE [50]:…”

“…With regard to interfacial patterns, researches have been implemented for stiffness and multi-material structures [25][26][27][28][29][30][31], layouts of shell structures [32][33][34][35][36][37][38], electrode patterns of electroosmosis [21], fluid-structure and fluid-particle interaction [39][40][41], energy absorption [42], cohesion [43], actuation [44] and wettability control [45][46][47], etc. ; topology optimization approaches implemented on 2-manifolds have also been developed with applications in elasticity, wettability control, heat transfer and electromagnetics [48][49][50]; and the fiber bundle topology optimization approach has been developed for wettability control at fluid/solid interfaces [47]; recently, topology optimization of surface flows has extended the design space of fluidic structures onto the 2-manifolds [8].…”

Topology optimization of surface flows

“…It can be derived by sequentially implementing surface-PDE filter and threshold projection operations on a design variable, which is independent of time and defined on the 2-manifold Γ and valued continuously in [0, 1]. Inspired by the PDE filter developed in [77], the surface-PDE filter for the design variable is implemented by solving the following PDE defined on the 2-manifold [66]:…”

“…With regard to material interfaces, related investigations have been implemented for stiffness and multi-material structures [44][45][46][47][48][49][50], layouts of shell structures [51][52][53][54][55][56], electrode patterns of electroosmosis [40], fluid-structure and fluid-particle interaction [57][58][59], energy absorption [60], cohesion [61], actuation [62] and wettability control [63][64][65], etc. ; a topology optimization approach implemented on 2-manifolds has also been generally developed with applications in the areas of wettability control, heat transfer and electromagnetics [66].…”