Layout and material gradation in topology optimization of functionally graded structures: a global–local approach

By means of continuous topology optimization, this paper discusses the influence of material gradation and layout in the overall stiffness behavior of functionally graded structures. The formulation is associated to symmetry and pattern repetition constraints, including material gradation effects at both global and local levels. For instance, constraints associated with pattern repetition are applied by considering material gradation either on the global structure or locally over the specific pattern. By means of pattern repetition, we recover previous results in the literature which were obtained using homogenization and optimization of cellular materials.     

Mapping method for sensitivity analysis of composite material property

Composite properties are dependent on the microstructure of materials, which is depicted with a base cell. The parameters for representing the microstructure should include the shape parameters of the base cell and those used to describe the distribution of materials in the base cell. The goal of material design optimization is to find appropriate values of these parameters to make the materials have specific properties. Design optimization needs the sensitivity information of the material properties with respect to the shape parameter of the base cell and the material distribution parameters. Moreover, sensitivity calculation is often expensive. Thus, it is very important to develop an efficient sensitivity analysis method. In this paper, a mapping method is proposed for predicting the material properties and computing their sensitivities with respect to the shape parameters of the base cell. Through mapping transformation, solutions to the micro-scale homogenization problem defined on the domain of a base cell can be obtained by solving a homogenization problem defined on an initial given domain. The composite properties and their sensitivities with respect to the shape parameters of the base cell are explicitly expressed in terms of the properties and their sensitivities of a virtual material with respect to the distribution parameters. This virtual material has an initially given base cell domain. Thus re-meshing for discretizing the problem is avoided and computing cost savings are realized. Numerical examples show that the proposed method is accurate and efficient in both the prediction of material properties and sensitivity calculation.     

Design of piezocomposite materials and piezoelectric transducers using topology optimization—Part I

Summary Currently developments of piezocomposite materials and piczoelectric actuators have been based on the use of simple analytical models, test of prototypes, and analysis using the finite element method (FEM), usually limiting the problem to a parametric optimization. By changing the topology of these devices or their components, we may obtain an improvement in their performance characteristics. Based on this idea, this paper discusses the application of topology optimization combined with the homogenization method and FEM for designing piezocomposite materials. The homogenization method allows us to calculate the effective properties of a composite material knowing its unit cell topology. New effective properties that improves the electromechanical efficiency of the piezocomposite material are obtained by designing the piezocomposite unit cell. This method consists of finding the distribution of the material and void phases in a periodic unit cell that optimizes the performance characteristics of the piezocomposite. The optimized solution is obtained using Sequential Linear Programming (SLP). A general homogenization method applied to piczoelectricity was implemented using the finite element method (FEM). This homogenization method has no limitations regarding volume fraction or shape of the composite constituents. The main assumptions are that the unit cell is periodic and that the scale of the composite part is much larger than the microstructure dimensions. Prototypes of the optimized piezocomposites were manufactured and experimental results confirmed the large improvement. Department of Mechanical Engineering and Applied Mechanics Department of Mechanical Engineering and Applied Mechanics     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.     

Design of materials using hybrid cellular automata

This paper presents a new approach for the topological design of materials with extreme properties. The method is based on hybrid cellular automaton (HCA), which is an implicit optimization technique that uses local rules to update design variables iteratively until meeting the described optimality conditions. By means of an energy-based homogenization approach, the effective properties of the considered material are calculated in terms of element mutual energies. By this method, no sensitivity information is required to find the optimal topology for the considered design objectives: bulk modulus, shear modulus, and negative Poisson’s ratio. The proposed method is validated by a series of numerical examples.     

Simultaneous optimization of the material properties and the topology of functionally graded structures

A level set based method is proposed for the simultaneous optimization of the material properties and the topology of functionally graded structures. The objective of the present study is to determine the optimal material properties (via the material volume fractions) and the structural topology to maximize the performance of the structure in a given application. In the proposed method, the volume fraction and the structural boundary are considered as the design variables, with the former being discretized as a scalar field and the latter being implicitly represented by the level set method. To perform simultaneous optimization, the two design variables are integrated into a common objective functional. Sensitivity analysis is conducted to obtain the descent directions. The optimization process is then expressed as the solution to a coupled Hamilton-Jacobi equation and diffusion partial differential equation. Numerical results are provided for the problem of mean compliance optimization in two dimensions.     

Thermomechanically coupled sensitivity analysis and design optimization of functionally graded materials

This paper presents a systematic numerical technique for performing sensitivity analysis of coupled thermomechanical problem of functionally graded materials (FGMs). General formulations are presented based on finite element model by using the direct method and the adjoint method. In the modeling of spatial variances of material properties, the graded finite element method is employed to conduct the heat transfer analysis and structural analysis and their sensitivity analysis. The design variables are the volume fractions of FGMs constituents and structural shape parameters. The design optimization model is then constructed and solved by the sequential linear programming (SLP). Numerical examples are presented to demonstrate the accuracy and the applicability of the present method.     

Optimal design of compliant mechanisms using functionally graded materials

This research applies topology optimization to create feasible functionally graded compliant mechanism designs with the aim of improving structural performance compared to traditional homogeneous compliant mechanism designs. Converged functionally graded designs will also be compared with two-material compliant mechanism designs. Structural performance is assessed with respect to mechanical/geometric advantage and stress distributions. Two design problems are presented – a gripper and a mechanical inverter. A novel modified solid isotropic material with penalization (SIMP) method is introduced for representing local element material properties in functionally graded structures. The method of moving asymptotes (MMA) is used in conjunction with adjoint sensitivity analysis to find the optimal distribution of material properties. Geometric non-linear analysis is used to solve the mechanics problem based on the Neo-Hookean model for hyperelastic materials. Functionally graded materials (FGMs) have material properties that vary based on spatial position. Here, FGMs are implemented using two different resource constraints – one on the mechanism’s volume and the other on the integral of the Young’s modulus distribution throughout the design domain. Tensile tests are performed to obtain the material properties used in the analysis. Results suggest that FGMs can achieve the desired improvements in mechanical/geometric advantage when compared to both homogeneous and two-material mechanisms.     

Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level

The aim of this paper is to present a microstructural topology optimization methodology for the structural-acoustic coupled system. In the structural-acoustic system, the structure is considered to be a thin composite plate composed of periodic uniform microstructures. The discrete design variables are used in the microstructural topology optimization, and the constitutive matrix is interpolated by the power-law scheme at the micro scale. The equivalent macro material properties of the microstructure are computed through the homogenization method. The design objective is to minimize the sound pressure level (SPL) in an interior acoustic medium. The sensitivities of the SPL with respect to design variables are derived. The bi-directional evolutionary structural optimization (BESO) method is extended to solve the structural-acoustic coupled optimization problem to find the optimal material distribution of the microstructure. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented microstructural topology optimization method.     

Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

Viscoelastic material design with negative stiffness components using topology optimization

An application of topology optimization to design viscoelastic composite materials with elastic moduli that soften with frequency is presented. The material is a two-phase composite whose first constituent is isotropic and viscoelastic while the other is an orthotropic material with negative stiffness but stable. A concept for this material based on a lumped parameter model is used. The performance of the topology optimization approach in this context is illustrated using three examples.     

Multilevel optimal design of composite structures including materials with negative Poisson's ratio

Multilevel iterative optimal design procedures, horrowed from the theory of structural optimization by means of homogenization, are used in this paper for the optimal material design of composite material structures. The method is quite general and includes materials with appropriate microstructure, which may lead eventually to phenomenological, overall negative Poisson's ratios. The benefits of optimal structural design gained by this approach, together with the first attempts to explain the taskoriented microstructure of natural structures, are investigated by means of numerical examples, and simulation of, among others, human bones.     

Multi-material topology optimization with multiple volume constraints: a general approach applied to ground structures with material nonlinearity

Multi-material topology optimization is a practical tool that allows for improved structural designs. However, most studies are presented in the context of continuum topology optimization – few studies focus on truss topology optimization. Moreover, most work in this field has been restricted to linear material behavior with limited volume constraint settings for multiple materials. To address these issues, we propose an efficient multi-material topology optimization formulation considering material nonlinearity. The proposed formulation handles an arbitrary number of candidate materials with flexible material properties, features freely specified material layers, and includes a generalized volume constraint setting. To efficiently handle such arbitrary volume constraints, we derive a design update scheme that performs robust updates of the design variables associated with each volume constraint independently. The derivation is based on the separable feature of the dual problem of the convex approximated primal subproblem with respect to the Lagrange multipliers, and thus the update of design variables in each volume constraint only depends on the corresponding Lagrange multiplier. Through examples in 2D and 3D, using combinations of Ogden-based, bilinear, and linear materials, we demonstrate that the proposed multi-material topology optimization framework with the presented update scheme leads to a design tool that not only finds the optimal topology but also selects the proper type and amount of material. The design update scheme is named ZPR (phonetically, zipper), after the initials of the authors’ last names (Zhang-Paulino-Ramos Jr.).     

Topology optimization for microstructural design under stress constraints

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.     

Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials

In this paper, two different methods for modelling of functionally graded material (FGM) beam with continuous spatially varying material properties will be presented and compared, namely the multilayering method and the direct integration method. Both the methods are related to homogenization of spatially varying material properties of real FGM beam and to calculation of the secondary variables of the FGM beams. The multilayering method is based on the laminate theory, which is very often used by modelling of the multilayer composite beams. The direct integration method transform spatial continuous varying material properties to the effective ones by direct integration of derived homogenization rules. In next part of the paper, new multiphysical beam finite element will be presented, which in conjunction with the proposed homogenization methods can be used for very effective analysis of the FGM beam structures. The numerical experiment will be presented concerning the multiphysical (electro–thermal–structural) analysis of the chosen FGM beams with spatial continuous variation of material properties.     

Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty

Aim of this work is the synthesis of auxetic structures using a topology optimization approach for micropolar (or Cosserat) materials. A distributed compliant mechanism design problem is formulated, adopting a SIMP–like model to approximate the constitutive parameters of 2D micropolar bodies. The robustness of the proposed approach is assessed through numerical examples concerning the optimal design of structures that can expand perpendicularly to an applied tensile stress. The influence of the material characteristic length on the optimal layouts is investigated. Depending on the inherent flexural stiffness of micropolar solids, truss–like solutions typical of Cauchy solids are replaced by curved beam–like material distributions. No homogenization technique is implemented, since the proposed design approach applies to elements made of microstructured material with prescribed properties and not to the material itself.     

Topology optimization of hyperbolic metamaterials for an optical hyperlens

This paper presents a level set-based topology optimization method for the microstructural design of an optical hyperlens. The resolution of conventional optics is generally diffraction limited, but the diffraction limit can be overcome by a hyperlens that converts evanescent waves to propagation waves, utilizing a cylindrical geometry to magnify the subwavelength features of imaged objects, which allows such features to be resolved beyond the diffraction limit at the hyperlens output. To support electromagnetic waves that contain information of subwavelength features, the hyperlens must have material properties that include a positive permittivity in the angular direction and a negative permittivity in the radial direction. Here, an energy-based homogenization method is used to obtain the effective permittivity of a metamaterial hyperlens unit cell. A level set-based topology optimization is applied so that the boundaries of the structure are clearly expressed. Moreover, a finite element mesh regeneration scheme is used to precisely capture the boundaries of the metal and dielectric materials of the unit cell that will ultimately form the hyperlens. The optimization algorithm uses the finite element method (FEM) to solve the equilibrium and adjoint equations. Optimum design examples for the design of a hyperlens microstructure are provided to confirm the utility and validity of the presented method.     

Interface study and boundary smoothing on designed composite material microstructures for manufacturing purposes

The manufacturability of composite material microstructures designed by the homogenized topology optimization method has to be considered when bringing the design into reality. In this paper, numerical studies are conducted on multiphase material microstructures that have negative coefficients of thermal expansion, for the purpose of improving manufacturability. Realistic manufacturing factors are considered, including the diffusion interface between two constituent phases and vectorized toolpath design. The effect of the mixture interface between two solid phases is examined by homogenization analysis. An image-processing program is developed to smooth out the boundaries of the topological structure to facilitate toolpath design. Numerical results show that the diffusion interface between two solid material phases reduces the effective negative thermal expansion property of the whole microstructure. The designed material properties are retained and converge after boundary smoothing .     

Multi-phase field topology optimization of polycrystalline microstructure for maximizing heat conductivity

The present study proposes multi-scale topology optimization for polycrystalline microstructures applying a multi-phase field method. The objective function is to maximize the heat compliance of macrostructure and the equality constraint is the material volume of constituents in an alloy consisting of two components with different heat conductivity. Two-scale steady-state heat conduction problem based on a homogenization method is conducted. The Allen-Cahn non-conserved time evolution equation with the additional volume constraint scheme is employed as the optimization strategy for updating the crystal configuration. In the time evolution equation, sensitivities of objective function with respect to phase-field variables are considered to relate topology optimization to the multi-phase field method. It is verified from a series of numerical examples that the proposed method has great potential for the development of material design underlying polycrystalline structure.     

Optimal shape design in biomimetics based on homogenization and adaptivity

Optimal shape design of microstructured materials has recently attracted a great deal of attention in materials science. The shape and the topology of the microstructure have a significant impact on the macroscopic properties. The paper is devoted to the shape optimization of new biomorphic microcellular silicon carbide ceramics produced from natural wood by biotemplating. This is a novel technology in the field of biomimetics which features a material synthesis from biologically grown materials into ceramic composites by fast high-temperature processing. We are interested in finding the best material-and-shape combination in order to achieve the optimal prespecified performance of the composite material. The computation of the effective material properties is carried out using the homogenization method. Adaptive mesh-refinement technique based on the computation of recovered stresses is applied in the microstructure to find the homogenized elasticity coefficients. Numerical results show the reliability of the implemented a posteriori error estimators.