Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method

Abstract

Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poisson's Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the Merriman-Bence-Osher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poisson's ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well.

Current and future trends in topology optimization for additive manufacturing

“…Reconciled level set method (RLSM) is employed for topology optimization of NPR (Negative Poisson's Ratio) metamaterials. The RLSM was first introduced by Merriman, Bence, and Osher [151,152] for modeling multiphase flow and was later applied to multi-material topology optimization of smart energy harvesters [153] and metamaterials [137]. RLSM retains the features of CLSM in multi-material representation and the convenience in specifying arbitrary design velocities on each level set function.…”

“…AM now provides a robust approach to fabricating multi-material components, regardless of the complexity of the interface distribution. A few topology optimization results have been realized into real products through multi-material AM [137,145,156]. On the other hand, it is still an issue to improve the numerical analysis accuracy around the interface areas [157] and reflect the actual behavior of printed materials [158,159].…”

Computational Design and Additive Manufacturing of Conformal Metasurfaces by Combining Topology Optimization With Riemann Mapping Theorem

“…Topology optimization is an iterative process starting from an initial level-set function. In this paper, a Finite Element Analysis (FEA) is carried out each iteration under the plane stress assumption, and the effective properties are calculated using the strain energy method [30,31]. Then, the design velocity field is constructed following the steepest descent method [23,28], and the Hamilton-Jacobi equation is solved using the upwind finite difference approach [24,28].…”

“…38. More details about constructing the design velocity field and implementing the negative Poisson's ratio problem can be found in [31], where results have been numerically verified and experimentally validated.…”

Multi-material 3D printed mechanical metamaterials: Rational design of elastic properties through spatial distribution of hard and soft phases

“…The research into both latter rational design approaches has just started, as the multimaterial 3D printing (¼ additive manufacturing) techniques required for achieving complex spatial distributions of mechanical properties and combining that with complex geometries are just emerging. A few recent studies on 3D lattices with high Poisson's ratio properties, 11 topology optimization of multi-material mechanical metamaterials with negative Poisson's ratio 12 or multifunctionality, 13 and controlling instabilities, 14 are examples of the applications of dual-phase materials 15,16 for achieving new ranges of properties and new types of functionalities in mechanical metamaterials.…”