# Topology optimization on two-dimensional manifolds

## TL;DR

This paper extends topology optimization techniques to two-dimensional manifolds, enabling design on complex surfaces like spheres and tori for applications in fluid mechanics, heat transfer, and electromagnetics.

## Contribution

It introduces a method for implementing topology optimization on 2D manifolds, including material interpolation and boundary condition penalization on complex surfaces.

## Key findings

- Successful numerical tests on sphere, torus, Möbius strip, Klein bottle
- Applications demonstrated in fluid mechanics, heat transfer, electromagnetics
- Method accommodates complex 2D geometries in optimization

## 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 three-dimensional 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, M\"{o}bius strip and Klein bottle, are included in the numerical tests, which are provided for the problems on fluidic mechanics, heat transfer and electromagnetics.