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Isogeometric design and optimization of spatiallyvarying, multi‑material 3D printed rod structures

Weeger, Oliver; Yeung, Sai‑Kit; Dunn, Martin L.

2018

Weeger, O., Yeung, S.‑K., & Dunn, M. L. (2018). Isogeometric design and optimization ofspatially varying, multi‑material 3D printed rod structures. Proceedings of the 3rdInternational Conference on Progress in Additive Manufacturing (Pro‑AM 2018), 280‑285.doi:10.25341/D4JG6V

https://hdl.handle.net/10356/88561

https://doi.org/10.25341/D4JG6V

© 2018 Nanyang Technological University. Published by Nanyang Technological University,Singapore.

Downloaded on 12 Dec 2021 22:08:42 SGT

ABSTRACT: With the capability to locally control the material composition of a structure, multi-material and multi-method 3D printing technologies provide a new level of design freedom beyond the realization of complex topologies. However, the precise design and optimization of spatially varying material compositions within a structure is beyond the capabilities of traditional computer-aided design approaches and tools. In this work, we apply the concept of isogeometric design and analysis to efficiently model, simulate and optimize spatially varying material compositions in the context of multi-material additive manufacturing. In particular, we apply this concept to nonlinear 3D beam structures with axially and transversally varying geometric and material parameters, including non-homogeneous, functionally graded and laminate cross-sections. In addition to discretizing the kinematic variables using an isogeometric collocation method, we also parameterize the geometric and material properties of the cross-sections as spline curves, which enables efficient modelling and optimization of axially varying material compositions and cross-section geometries. We demonstrate the applicability of the approach for design optimization of multi-material 3D printed, active rod structures with axially varying material distributions and direct 4D printing of self-assembling, multi-material laminate structures.

KEYWORDS: Isogeometric analysis; Multi-material 3D printing; Spatially variable materials; Functionally graded beams; Design optimization

INTRODUCTION

Additive manufacturing technologies provide potentially transformative capabilities such as the ability to realize complex topologies and freeform geometries (Zegard and Paulino, 2016), monolithically combine multiple materials, micro-structures and meta-materials to tailor spatially varying material properties (Chen et al., 2017; Schumacher et al., 2015) and fabricate structures with advanced functionalities such as stimuli response and self-assembly (Ge et al., 2013; Ding et al., 2017) or tailored deformation behavior and instabilities (Babaee et al., 2013; Liu et al., 2016). However, common computer-aided design and engineering (CAD/CAE) tools only focus on the geometric design of a structure and the description of its outer shell, e.g., in terms of NURBS surface representations or triangulations such as STL (stereolithography) files, and uniform material specifications within the components. This makes it difficult to design and engineer structures with spatially varying material constitutions, which can now be fabricated through additive manu-facturing. These advances in fabrication technology thus call for the development of digital design tools that bridge the gap between computational design, simulation, optimization and fabrication, and enable the exploitation of the new design freedom provided by additive manufacturing. Here, we develop such an approach for slender beam and rod structures, which are an important class of structural components in the context of advanced manufacturing, since they can not only be

ISOGEOMETRIC DESIGN AND OPTIMIZATION OF SPATIALLY VARYING, MULTI-MATERIAL 3D PRINTED ROD STRUCTURES

OLIVER WEEGER, SAI KIT YEUNG SUTD Digital Manufacturing and Design Centre, Singapore University of Technology and Design,

8 Somapah Road, Singapore 48737, Singapore

MARTIN L DUNN College of Engineering and Applied Science, University of Colorado Denver

Campus Box 104, Denver, CO 80217-3364, USA

OLIVER WEEGER, SAI-KIT YEUNG

280

Proc. Of the 3rd Intl. Conf. on Progress in Additive Manufacturing (Pro-AM 2018) Edited by Chee Kai Chua, Wai Yee Yeong, Ming Jen Tan, Erjia Liu and Shu Beng TorCopyright © 2018 by Nanyang Technological UniversityPublished by Nanyang Technological University ISSN: 2424-8967 :: https://doi.org/10.25341/D4JG6V

used as lattice structures with high stiffness-to-weight ratios or metamaterials with tailored mechanical behavior, but also as soft, deformable and compliant structures with advanced functiona-lities. Many attempts to model and simulate 3D beams undergoing large elastic deformations are based on the geometrically exact 3D beam theory, commonly referred to as Cosserat rod theory (Antman, 2005). For modelling of functionally graded and composite cross-sections, extensions and modifications of the theory have been made (Vu-Quoc et al., 1996; Murin and Kutis, 2002; Hodges, 2006; Eroglu, 2016). This work is closely related to (Bîrsan et al., 2012), where constitutive coefficients of functionally graded 3D beams with general cross-section shapes and composite layouts were derived and deformation analysis carried out analytically and via finite element methods (FEM). Here, we use the concept of isogeometric analysis (IGA) (Hughes et al., 2005) for the modelling and nonlinear simulation of compliant beam structures with axially and transversally varying cross-section parameters. In particular, we employ an isogeometric collocation method (Auricchio et al., 2010) for the discretization of the Cosserat rod model (Weeger et al., 2017). We extend this formulation to transversally varying, functionally graded cross-sections and introduce spline para-meterizations of geometric and material cross-section parameters, e.g., radius, laminate layer ratio, Young’s moduli, etc., to describe and optimize their axial variation along the centerline of a beam. We demonstrate the applicability of this concept to multi-material inkjet printing of active rod structures with axially varying material distributions (Weeger et al., 2016) and direct 4D printing of self-assembling, multi-material laminate structures (Ding et al., 2018).

METHODS

Mechanical modelling of spatial rods and rod structures To model 3-dimensional elastic deformations of spatial rods, we use the Cosserat model, which takes into account large deformations and rotations, shear and twist at small strains, and is suitable for modeling both thin and thick rods (Antman, 2005). The configuration of a rod is fully determined by its description as a framed curve, i.e., a rod is represented by its centreline curve

and an orthonormal frame , which describes the evolution of the cross-section orientation, see Figure 1a. With the kinematic model, the translational strain vector and curvature strain vector are computed. Then, a linear elastic constitutive model is formulated, including thermal expansion behavior:

(1)

Here, and represent the translational and rotational stress, are the elastic constitutive matrices, which depend on the geometrical (shape and layering) and material properties (Young’s modulus and Poisson’s ratio) of the cross-section, is the temperature change, and and are thermo-elastic constitutive matrices. The governing equations of the Cosserat rod, i.e., the equilibria of internal forces and moments are then given as:

(2)

For the numerical discretization of the rod model and solution of the governing equations, we use an isogeometric collocation method. It provides an accurate and efficient numerical discretization and enables a seamless integration of the simulation method with the design approach through the

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ACKNOWLEDGEMENTS

The authors acknowledge support from the SUTD Digital Manufacturing and Design (DManD) Centre, supported by the Singapore National Research Foundation.

REFERENCES

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Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V. A., and Pietras, D. (2012). “Deformation analysis of functionally graded beams by the direct approach.” Composites Part B, 43(3), 1315–1328.

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