Design of graded porous bone-like structures via a multi-material topology optimization approach

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Graded porous structures combine robustness of porous structures and high stiffness of bulk designs. This study aims to design optimized graded porous bone-like structures through a novel multi-material topology optimization approach, which generalizes the concept of multiple materials. Namely, each material can have not only distinct material property but also a different level of local porosity, or a combination of both, thus allowing the realization of multiple levels of porosity. With separated density and material/porosity fields, we propose two types of multi-porosity local volume constraints to enable graded porosity considering linear and bi-linear material constitutive relations. Through the proposed framework, single- and multi-material structures can be obtained with a natural transition between the bulk and multiple levels of porous regions. We adopt the Bi-value Coding Parameterization (BCP) scheme combined with the Solid Isotropic Material with Penalization (SIMP) method to interpolate the stored energy functions. Through several examples with multiple porosity levels and various material properties, we demonstrate the effectiveness of the proposed framework with two novel constraints to generate optimized multi-material and multi-porosity structures. We further investigate the interactions among material properties, multiple porosity levels, structural stiffness, and robustness. Compared with conventional bulk designs, the optimized bone-like structures with multi-level graded porosity, although less stiff, are found to be more robust, i.e., their structural stiffness is less influenced by the load variations and material deficiency. The resulting graded porous composite designs showcase the capability of the proposed multi-material formulation to optimize the distributions of not only different types of materials but also multiple levels of porosity.

Original languageEnglish (US)
Pages (from-to)677-698
Number of pages22
JournalStructural and Multidisciplinary Optimization
Issue number2
StatePublished - Aug 2021


  • Bi-linear material
  • Bone-like structure
  • Graded porosity
  • Local volume constraint
  • Multi-material topology optimization

ASJC Scopus subject areas

  • Software
  • Control and Optimization
  • Control and Systems Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design


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