Robust topology optimization of negative Poisson’s ratio metamaterials under material uncertainty
Introduction
Topology optimization (TO) seeks to find the best material layout under the given boundary conditions by optimizing the objective function under the specific design constraints. It is an iterative computational method. Since the introduction of TO by Bendsøe and Kikuchi [1], there has been extensive research into this field and various methods have been developed which can be grouped into two categories, optimality criteria methods and heuristic or intuitive methods. First group includes techniques such as the homogenization method [1], [2], the level set-based method [3], [4], [5], [6], [7], [8], and the solid isotropic material with penalization (SIMP) method [2], [9], [10], [11], [12]. Heuristic methods are computer-aided optimization (CAO) [13], soft kill method [14], bidirectional evolutionary structural optimization (BESO) [15], [16], [17] and others. Recently, TO has gained great popularity and being applied to various areas like aerospace [18], fluid structure interaction problems [19], multiscale nonlinear structures [20], and so on including metamaterial’s microstructure design, e.g. [21], [22], [23], [24], [25], [26], [27].
Metamaterials are synthetic composite materials that show unique properties that are not found in natural materials [28]. This unusual behavior of materials is due to the periodic arrangement of microstructure rather than their chemical makeup. These materials are now used in a variety of engineering applications, including stealth fighters, aerospace applications, medical devices, and many others. Various types of metamaterials have recently been created, depending on the needs like mechanical metamaterial [29], [30], [31], [32], acoustic metamaterials [33], and pentamode metamaterials [34].
The emphasis of this paper will be on metamaterials with the negative Poisson’s ratio (NPR), also known as auxetics [29], [35], [36], [37], [38], [39], [40], [41]. They have a counterintuitive behavior that extends laterally when under uniaxial stress and contract transversely when under uniaxial compression. This behavior finds application in many areas [42] such as, to improve crack resistance [43], vibration control [44], [45], [46], increase fracture durability [47], [48], improve impact resistance [49], provide sound absorption capability [50] and many others.
Since the first introduction of auxetic metamaterial by Lakes [51], much research has been done in their modeling, architecture, and manufacturing [52], [53], [54], [55], [56], [57], [58], [59], [60]. Microstructural deformation is responsible for auxetic behavior. So the design of microstructure is critical in designing structural metamaterial.
Sigmund [61] designed the microstructure using TO for the first time using an inverse homogenization approach. This was followed by different works using various other methods of TO like density based approach [62], [63], the level set-method [64], the parametric level set-method [65], BESO [66], and others. The level set methods are advantageous while handling drastic topology changes and are versatile in considering the different objective functions and mechanical models, including the design of auxetic metamaterials [65] whereas the SIMP method is the most widely utilized method for its conceptual simplicity and easy implementation.
The majority of auxetic research has been conducted under deterministic assumptions, which assume the material properties of the composite’s constituent materials to be uniform and unchanged. However, the deterministic assumption can lead to a non-optimal design due to various factors such as manufacturing tolerance, load variance, aging, and inhomogeneity of material properties. The optimized solution may fail to meet the desired performance criteria, or even the design may be unfeasible in such situations. So it is necessary to consider uncertain parameters in the optimization process to improve the robustness and reliability of the system quantitatively.
Optimization formulation considering uncertainties has been studied by arranging them in two major groups : reliability-based topology optimization (RBTO) [67], [68] and robust topology optimization (RTO) [69], [70], [71], [72]. The RBTO aims for the design that reduces the stated likelihood of failure (i.e., less than an acceptable small value) and thus avoids the catastrophic conditions [73], [74]. The principle of RTO theory attempts to minimize the mean and variance of the objective function by achieving an optimum structure such that the target output of the structure is less vulnerable to unknown disruption.
In this study, a RTO approach for the design of NPR metamaterials under the material properties uncertainty is investigated. Within the context of SIMP based TO simulations of auxetic metamaterials, the contributions are as follows:
- •
Uncertainty in material properties (modulus of elasticity and base material Poisson’s ratio) are incorporated by considering them as interval variables.
- •
A robust objective function is formalized based on the weighted mean and variance of the deterministic objective function.
- •
The proposed RTO formulation is tested for its effectiveness by conducting parametric studies with variation in mesh size, targeted volume fraction, initial density distribution, and their effect on the final optimized design.
The proposed RTO algorithm provides reliable results with significantly reduced variation in the effective negative Poisson’s ratio of the designed metamaterial considering a considerable variation in the input material properties. The proposed algorithm can be easily added to any existing deterministic topology optimization algorithm without much changes as a separate function has been proposed that calls the deterministic output for each uncertain parameter.
This paper is structured as follows: We start with a brief overview of the strain energy-based method for prediction of effective Poisson’s ratio of a unit cell in Section 2. In Section 3, we present the DTO formulation and then formulate the RTO formulation using the variance and the mean of the objective function. The Results of numerical experiments are then presented in Section 4. Finally, in Section 5, we summarize the findings.
Section snippets
Prediction of effective Poisson’s ratio of a unit cell
Considering a unit cell that, when arranged periodically, characterizes the behavior of the metamaterial (see Fig. 1). Let , represent the stress and strain inside the unit cell and , represents the stress and strain in the homogenized metamaterial. The effective elasticity matrix representing the behavior of metamaterial is directly dependent on the architecture of the unit cell and the elastic properties of the base material.
Let represent the elasticity matrix of the
Deterministic topology optimization
The effective properties of a NPR metamaterial are dependent on the architecture of the unit cell. A suitable architecture for NPR until cell could be generated by formulating a TO problem combined with the homogenization process and considering the material properties to be constant. In TO, the objective function subjected to some constraint functions is minimized, and optimal layout of the material inside the design domain is determined. In the case of designing NPR architecture [77], the DTO
Results of numerical experiments
In this section, the proposed RTO formulation for an NPR unit cell design is applied to multiple numerical examples to demonstrate its effectiveness. A basic problem is studied first in Section 4.1 to demonstrate the benefits of the proposed method. A parametric study is carried out in Section 4.2 by varying the mesh size and targeted volume fraction to study their effect on the final optimized design with the proposed formulation. Finally in Section 4.3, the effect of different initial density
Conclusion
This paper proposes a robust topology optimization (RTO) method for the design of metamaterials with negative Poisson’s ratio (NPR), considering uncertainty in the material properties of a linear elastic base material. To incorporate uncertainties into the optimization process, Young’s modulus and Poisson’s ratio of the base material are considered as interval variables. The mean and variance of the deterministic objective function are used to define the robust objective function and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
The authors gratefully acknowledge financial support from the Ministry of Human Resource Development. The third author (Rajib Chowdhury) thanks the funding support from the SERB via file no. CRG/2019/004600 and DRDL via file no. DRDL/24/08P/19/0235/ 43386.
References (96)
- et al.
Generating optimal topologies in structural design using a homogenization method
Comput. Methods Appl. Mech. Engrg.
(1988) - et al.
Structural boundary design via level set and immersed interface methods
J. Comput. Phys.
(2000) - et al.
A level set method for structural topology optimization
Comput. Methods Appl. Mech. Engrg.
(2003) - et al.
Structural optimization using sensitivity analysis and a level-set method
J. Comput. Phys.
(2004) - et al.
Stress-based topology optimization using an isoparametric level set method
Finite Elem. Anal. Des.
(2012) - et al.
Topology optimization for concurrent design of structures with multi-patch microstructures by level sets
Comput. Methods Appl. Mech. Engrg.
(2018) - et al.
A multi-material level set-based topology optimization of flexoelectric composites
Comput. Methods Appl. Mech. Engrg.
(2018) - et al.
Structural topology optimization based on non-local shepard interpolation of density field
Comput. Methods Appl. Mech. Engrg.
(2011) - et al.
Topology optimization of multi-material structures with graded interfaces
Comput. Methods Appl. Mech. Engrg.
(2019) - et al.
Coupled finite-element/topology optimization of continua using the Newton-raphson method
Eur. J. Mech. A Solids
(2021)
A new method of structural shape optimization based on biological growth
Int. J. Fatigue
SKO (soft kill option): The biological way to find an optimum structure topology
Int. J. Fatigue
A simple evolutionary procedure for structural optimization
Comput. Struct.
Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method
Finite Elem. Anal. Des.
Design of materials with extreme thermal expansion using a three-phase topology optimization method
J. Mech. Phys. Solids
A new class of extremal composites
J. Mech. Phys. Solids
A topology optimization method based on the level set method for the design of negative permeability dielectric metamaterials
Comput. Methods Appl. Mech. Engrg.
Design of manufacturable 3D extremal elastic microstructure
Mech. Mater.
Topological shape optimization of microstructural metamaterials using a level set method
Comput. Mater. Sci.
Topology optimization for the design of periodic microstructures composed of electromagnetic materials
Finite Elem. Anal. Des.
Topological design of pentamode lattice metamaterials using a ground structure method
Mater. Des.
Composite materials with Poisson’s ratios close to – 1
J. Mech. Phys. Solids
Architected metamaterials with tailored 3D buckling mechanisms at the microscale
Extrem. Mech. Lett.
Anti-impact behavior of auxetic sandwich structure with braided face sheets and 3D re-entrant cores
Compos. Struct.
Materials with prescribed constitutive parameters: an inverse homogenization problem
Int. J. Solids Struct.
Topology optimization of an acoustic metamaterial with negative bulk modulus using local resonance
Finite Elem. Anal. Des.
Topological design of microstructures of cellular materials for maximum bulk or shear modulus
Comput. Mater. Sci.
Reliability-based design of MEMS mechanisms by topology optimization
Comput. Struct.
Reliability-based topology optimization of geometrically nonlinear structures with loading and material uncertainties
Finite Elem. Anal. Des.
Level-set topology optimization for mechanical metamaterials under hybrid uncertainties
Comput. Methods Appl. Mech. Engrg.
Structural design considering the of load positions using the phase field design method
Finite Elem. Anal. Des.
Robust topology optimization for multi-material structures under interval uncertainty
Appl. Math. Model.
Robust topology optimization for concurrent design of dynamic structures under hybrid uncertainties
Mech. Syst. Signal Process.
General purpose software for efficient uncertainty management of large finite element models
Finite Elem. Anal. Des.
Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method
Comput. Aided Des.
Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction
Comput. Methods Appl. Mech. Engrg.
Material interpolation schemes in topology optimization
Arch. Appl. Mech. (Ingenieur Archiv)
The COC algorithm, part II: topological, geometry and generalized shape optimization
Comput. Methods Appl. Mech. Engrg.
Evolutionary Topology Optimization of Continuum Structures: Methods and Applications
On the benefits of applying topology optimization to structural design of aircraft components
Struct. Multidiscip. Optim.
Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation: TOPOLOGY optimization for stationary FSI PROBLEMS
Internat. J. Numer. Methods Engrg.
Recent advances on topology optimization of multiscale nonlinear structures
Arch. Comput. Methods Eng.
Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity
Struct. Multidiscip. Optim.
Unimode metamaterials exhibiting negative linear compressibility and negative thermal expansion
Smart Mater. Struct.
Systematic design of metamaterials by topology optimization
Negative Poisson’s ratio materials [8]
Science
Metamaterials and negative refractive index
Science
Topological design for mechanical metamaterials using a multiphase level set method
Struct. Multidiscip. Optim.
Cited by (27)
A level set reliability-based topology optimization (LS-RBTO) method considering sensitivity mapping and multi-source interval uncertainties
2024, Computer Methods in Applied Mechanics and EngineeringComplex uncertainty-oriented robust topology optimization for multiple mechanical metamaterials based on double-layer mesh
2024, Computer Methods in Applied Mechanics and Engineering