Symmetry Based Material Optimization

Zeyun Shi,Jiong Chen,Yao Jin,Jinkeng Lin,Jin Huang

doi:10.3390/sym13020315

Abstract

Many man-made or natural objects are composed of symmetric parts and possess symmetric physical behavior. Although its shape can exactly follow a symmetry in the designing or modeling stage, its discretized mesh in the analysis stage may be asymmetric because generating a mesh exactly following the symmetry is usually costly. As a consequence, the expected symmetric physical behavior may not be faithfully reproduced due to the asymmetry of the mesh. To solve this problem, we propose to optimize the material parameters of the mesh for static and kinematic symmetry behavior. Specifically, under the situation of static equilibrium, Young’s modulus is properly scaled so that a symmetric force field leads to symmetric displacement. For kinematics, the mass is optimized to reproduce symmetric acceleration under a symmetric force field. To efficiently measure the deviation from symmetry, we formulate a linear operator whose kernel contains all the symmetric vector fields, which helps to characterize the asymmetry error via a simple ℓ2 norm. To make the resulting material suitable for the general situation, the symmetric training force fields are derived from modal analysis in the above kernel space. Results show that our optimized material significantly reduces the asymmetric error on an asymmetric mesh in both static and dynamic simulations.

Highlights

Symmetry is a ubiquitous phenomenon in nature and a very essential rule for artistic design
We find that all the above symmetric vector fields fall in the kernel of a linear operator associated with the symmetry
The deformations before and after optimization are presented when the same downward forces are applied on both ends

Summary

Introduction

Symmetry is a ubiquitous phenomenon in nature and a very essential rule for artistic design. It encodes some invariance of given structures under certain transformations. Even if the shape is in perfect symmetry, e.g., modeled by software, the mesh tessellated from the shape could be asymmetric. The elastic simulation based on the asymmetric mesh may result in asymmetric behavior. To reduce the asymmetric artifacts, increasing the mesh density may help, but the additional computational cost is not affordable in some real-time applications (e.g., game, virtual reality). If the mesh has some asymmetrically distributed badly shaped elements, which brings strong shear locking, we notice that refining the mesh by subdivision cannot reduce the asymmetrical behavior (see Figure 1)

Methods

Results

Conclusion