Constrained Versions of the Free Material Design Methods and Their Applications in 3D Printing

Tomasz Denis Lewiński , Sławomir Adam Czarnecki , Radosław Tomasz Czubacki , Tomasz Mariusz Łukasiak , Paweł Wawruch


The paper deals with the free material design and its constrained versions constructed by imposing: (a) cubic symmetry (cubic material design, CMD), (b) isotropy with: (b1) independent bulk and shear moduli (isotropic material design, IMD), and (b2) fixed Poisson’s ratio (Young’s modulus design, YMD). In the latter case the Young modulus is the only design variable. The moduli are viewed as non-negative, thus allowing for the appearance of void domains within the design domain. The paper shows that all these methods (CMD, IMD, YMD) reduce to two mutually dual problems: − the stress-based minimization problem in which the integrand is equal to a norm of the test stress field. The norm ρ(·) involved reflects the type of the constraints imposed; − the displacement-based problem in which the virtual work is subject to maximization over the adjoint displacement fields associated with strains, the dual norm ρ∗(·) of which is bounded almost everywhere. Upon solving the former problem and finding theminimizer one can determine the optimal moduli; they assume non-zero values within the material domain and they vanish outside this domain, thus allowing for cutting out the final shape from the initial design domain. Therefore, the material design methods discussed determine simultaneously: the topology, the optimal shape and elastic material characteristics. The holes appear there where the minimizer vanishes. The YMD method has been made suitable for 3D printing. By using the inverse homogenization method the isotropic and nonhomogeneous YMD designs are replaced by equivalent discrete structures of hexagonal microstructure of varying cross sections of the ligaments. The appropriate numerical codes are prepared to make it possible to produce these fibrous structures by additive manufacturing. The produced prototypes are characterized by high stiffness with respect to the given load.
Author Tomasz Denis Lewiński (FCE / ICE)
Tomasz Denis Lewiński,,
- The Institute of Civil Engineering
, Sławomir Adam Czarnecki (FCE / ICE)
Sławomir Adam Czarnecki,,
- The Institute of Civil Engineering
, Radosław Tomasz Czubacki (FCE / ICE)
Radosław Tomasz Czubacki,,
- The Institute of Civil Engineering
, Tomasz Mariusz Łukasiak (FCE / ICE)
Tomasz Mariusz Łukasiak,,
- The Institute of Civil Engineering
, Paweł Wawruch (FCE)
Paweł Wawruch,,
- Faculty of Civil Engineering
Publication size in sheets0.75
Book Schumacher A., Vietor T., Fiebig S., Bletzinger K., Maut K. (eds.): Advances in Structural and Multidisciplinary Optimization :Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimization (WCSMO12), 2018, Springer, ISBN 978-3-319-67987-7, [978-3-319-67988-4], 2115 p., DOI:10.1007/978-3-319-67988-4
Keywords in EnglishFree material design, Compliance minimization, 3D printing
Languageen angielski
10.1007_978-3-319-67988-4_99.pdf 6.53 MB
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