Multi-material Topology Optimization Method Research Articles

Design of the multiphase material structures with mass, stiffness, stress, and dynamic criteria via a modified ordered SIMP topology optimization

This paper, a multi-material topology optimization method (MMTO) is presented for two-dimensional structures using modified ordered solid isotropic material with penalization (SIMP) under multiple constraints such as mass, stiffness, frequency, and stress. The element density is obtained by a density filtering and the Heaviside approximation, which are combined to avoid gray elements and numerical issues of stress. Effective improvement of multiphase material topology optimization was performed including the criteria for stiffness, stress, and free vibration. P-norm stress aggregation is employed to calculate the maximum von Mises stress and sensitivity of von Mises stress is formulated by the adjoint method. The natural frequency and stress constraints are solved simultaneously to enhance the stability and reduce the stress concentration of the optimized structures, which are not much covered in the literature. The multi-mode phenomenon in the eigenfrequency problem is treated by the simply proposed technique. Results indicate that the multi-material designs with multi-constraints can be illustrated and optimized effectively. The optimized topologies, for stress minimization, indicate that the maximum stress can be significantly reduced compared with compliance minimization. The maximum von Mises stress of the topological results considering the maximum stiffness with natural frequency and mass constraints is higher than that maximum stiffness with only mass constraints, which indicates that the optimized structure created considering the eigenvalues is safer. The proposed approach can obtain a reasonable result that effectively controls the stress level and reduces the stress concentration at the high stress regions of multi-phase material structures with multi-constraints. Benchmark numerical examples are presented to confirm the effectiveness of the presented method.

Design of Viscoelastic Damped Plate Structures with a Multi-Material Topology Optimization Method

A viscoelastic damping layer attached to the plate structure can control vibrations but increases mass. Involving multiple viscoelastic materials, the contribution of different materials needs to be evaluated and allocated. In the paper, a topology optimization model is proposed to find distributions of different viscoelastic materials adopting a multi-material level set method, for maximizing single-order or multi-order modal loss factors. Limited by material volume fraction, the variational principle is applied to establish the velocity field, and the problem is solved by a gradient-based method. Numerical examples are performed to illustrate that loss factors of composite plate structures can be maximized with optimal distributions obtained from the multi-material level set method.

Probabilistic-Ellipsoid Hybrid Reliability Multi-Material Topology Optimization Method Based on Stress Constraint

Probabilistic-Ellipsoid Hybrid Reliability Multi-Material Topology Optimization Method Based on Stress Constraint

Microstructure design of porous viscoelastic composites with prescribed relaxation moduli by multi-material topology optimization

This paper presents a multi-material topology optimization method for designing the microstructures of porous viscoelastic composites with prescribed relaxation moduli. It is assumed that the composite materials are microscopically composed of periodic unit cells. The effective relaxation moduli of the composites are obtained using the homogenization method. The density-based topology optimization is utilized, and a related material interpolation scheme is introduced. The design objective is to determine an optimal material distribution that exhibits the prescribed modulus in both the glassy and rubbery regions. The design sensitivity is derived analytically using the adjoint variable method. A filtering scheme is adopted to impose a minimum length scale for each material phase. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. Various microstructures with prescribed relaxation moduli are presented and discussed. In addition, the bounds on the relaxation moduli of the optimized microstructures are compared with theoretical bounds.

A robust dynamic unified multi-material topology optimization method for functionally graded structures

A robust dynamic unified multi-material topology optimization method for functionally graded structures

Topology optimization of multi-material structures considering a piecewise interface stress constraint

Topology optimization for multi-material structures has become an important and hot research topic due to their great application potential in modern industries. Unfortunately, most existing multi-material topology optimization methods assume that the interface is perfectly bonded, and those that do not, ignore the strength difference between different interfaces, limiting their application in the design of multi-material structures with more than two solid phases. In this study, a multi-material topology optimization method considering a tension/compression-asymmetric piecewise interface stress constraint is proposed to improve the overall interface strengths of the multi-material structure under the SIMP method framework. Firstly, a new interpolation scheme is developed to identify interfaces between two arbitrary materials and classify the interfaces into different pieces based on the two neighbor materials. Then, a novel piecewise interface stress constraint is developed to describe different targeted strengths at multiple material interfaces. In this constraint, the equivalent interfacial stress considering the tension/compression-asymmetric characteristics is defined by tensile stresses and shear stresses at material interfaces. The maximum stresses at material interfaces are measured by the global p-norm aggregation function which makes imposing the constraint easy. After that, the proposed piecewise tension/compression-asymmetric interface stress constraint is introduced into multi-material topology optimization for compliance minimization. The sensitivity analyses are derived by the adjoint method. Several benchmark 2D examples are provided to discuss the influence of some parameters on optimization problems, such as the interface width, strength scaling factor and the filter radii. Other numerical examples include complex 3D problems and problems involving more than three materials. In all examples, the results show that the proposed algorithm with the tension/compression-asymmetric piecewise interface stress constraint can effectively control interface stress levels and improve structural safety.

Multi-material topology optimization and additive manufacturing for metamaterials incorporating double negative indexes of Poisson’s ratio and thermal expansion

Advances in the multi-material additive manufacturing promote a new opportunity for developing metamaterials integrating complex functionalities, especially the metamaterial incorporating double negative indexes of Poisson’s ratio and thermal expansion. However, the systematic design and additive manufacturing of such metamaterial are stilling not developed. Here, an improved multi-material topology optimization method, Alternating Active Phase & Objective (AAPO) algorithm, was developed. A new optimization formulation, in which the core contribution was the dynamical switch of the objective functions according to the active phases, was proposed to successfully overcome the convergence oscillation. The optimization formulation and sensitivity analysis based on the numerical homogenization were established. A series of multi-material re-entrant and chiral metamaterials, which exclusively realized the double negative indexes of Poisson’s ratio and thermal expansion, were devised. Especially, the tri-material and chiral metamaterials were originally obtained through the topology optimization. More importantly, metamaterials were well additively manufactured through the multi-material fused deposition modeling. The experimentally tested Poisson’s ratio and thermal expansion suggest a good agreement between the experiments and topology-optimized results. The multi-material design and additive manufacturing developed here provide a general guidance to develop the multi-functional metamaterials.

Improvement of the Phase Section Method for Multi-material Topology Optimization

Improvement of the Phase Section Method for Multi-material Topology Optimization

A novel multi-material topology optimization method for permanent magnet assisted synchronous reluctance motors

PurposeThe purpose of this paper is to develop a multi-material topology optimization method for permanent magnet-assisted synchronous reluctance motors.Design/methodology/approachIn the proposed method, the optimization procedure consists of two steps. In the first step, the entire rotor area was selected for the design region and the distribution of the core and air materials was optimized. In the second step, the design region was limited to the air region of the former solution and the distribution of magnets and cores or magnets and air was optimized.FindingsBecause of the two-step process of the proposed method, the design parameters can be reduced compared to the conventional method. As a result, this study can prevent the solution space from becoming more complex and superior solutions can be founded effectively.Research limitations/implicationsSince limited case study is denoted in this paper, much more case studies, for example, three-dimensional optimization problems, are needed to be discussed.Practical implicationsThe optimal solutions obtained by the proposed method have a smaller magnet volume and higher average torque than that of the conventional method.Originality/valueIn the proposed methods, optimization methodology, which consists of two-steps process, is differed from the conventional method.

Multimaterial Topology Optimization Method Based on Global Search by Combinatorial Optimization and Local Search by Variable Design Region Method

This paper presents a novel multi-material topology optimization method for electric motors. In the proposed method, the search process is divided into two steps. First, combinatorial optimization is conducted, wherein two different types of motors are used. Owing to this, for example, motor shapes that have both the high torque density of permanent magnet motors and the variable flux capability of wound field motors can be obtained. Second, the optimal solution obtained through the combinatorial optimization is used as the initial solution, and multi-material topology optimization with variable design region method is performed. By using the proposed method, we can obtain design solutions that have high performance on complex motors with various materials, such as hybrid field motors and permanent magnet assisted synchronous reluctance motors. To validate the effectiveness, rotors of a hybrid field motor and a permanent magnet assisted synchronous reluctance motor are optimized using the proposed method. From the numerical results, it can be confirmed that the solution obtained by the proposed method has a better performance than that of the conventional multi-material topology optimization.

A multi-material topology optimization method based on the material-field series-expansion model

A multi-material topology optimization method based on the material-field series-expansion model

Multi-material topology optimization of large-displacement compliant mechanisms considering material-dependent boundary condition

Multi-material compliant mechanisms design enables potential design possibilities by exploiting the advantages of different materials. To satisfy mechanical/thermal impedance matching requirements, a method for multi-material topology optimization of large-displacement compliant mechanisms considering material-dependent boundary condition is presented in this study. In the optimization model, the element stacking method is employed to describe the material distribution and handle material-dependent boundary condition. The maximization of the output displacement of the compliant mechanism is developed as the objective function and the structural volume of each material is the constraint. Fictitious domain approach is applied to circumvent the numerical instabilities in topology optimization problem with geometrical nonlinearities. The method of moving asymptotes is applied to solve the optimization problem. Several numerical examples are presented to demonstrate the validity of the proposed method. The optimal topologies of the compliant mechanisms obtained by the proposed method can satisfy the specified material-dependent boundary condition.

Topology Optimization and Prototype of a Multimaterial-Like Compliant Finger by Varying the Infill Density in 3D Printing.

This study presents a multimaterial topology optimization method for design of multimaterial compliant mechanisms. Traditionally, the objective function in topology optimization for design of structures is to minimize the strain energy (SE). For synthesis of compliant mechanisms, the objective function is usually to maximize the mutual potential energy (MPE). To design an adaptive compliant gripper for grasping size-varied objects, a multicriteria objective function considering both the SE and MPE at two different output ports is proposed in this study. In addition, based on the fact that different infill densities in three-dimensional (3D) printing leads to prototypes with different equivalent mechanical properties, this article proposes that a multimaterial design can be approximated by varying the values of infill densities in different portions of a 3D-printed component, which enables the multimaterial designs to be prototyped using the general low-cost, single-material fused deposition modeling 3D printing machines. The proposed method is used to design and prototype a bi-material compliant finger which is 3D printed using a flexible thermoplastic elastomer with infill densities of 30% and 100%. The experimental results demonstrate that the bi-material finger is a better design in terms of reducing the driving force while increasing the output displacement at the fingertip comparing to the single-material finger design with the same volume and weight. Furthermore, a two-finger gripper with two identical multimaterial-like compliant fingers is prototyped and installed on a six-axis industrial robot. The experimental tests are performed to demonstrate the effectiveness of the presented design.

Topology optimization of multi-material structures with elastoplastic strain hardening model

The paper presents a new multi-material topology optimization method with a novel adjoint sensitivity analysis that can accommodate not only multiple plasticity but also multiple hardening models for individual materials in composite structures. Based on the proposed method, an integrated framework is developed which details the nonlinear finite element analysis, sensitivity analysis and optimization procedure. The proposed method and framework are implemented and illustrated by three numerical examples presented in this paper. An in-depth analysis of the numerical results has revealed the significant impact of the selection of plasticity and hardening model on the results of topology.

A new level set based multi-material topology optimization method using alternating active-phase algorithm

This paper proposes a new level set based multi-material topology optimization method, where a difference-set-based multi-material level set (DS-MMLS) model is developed for topology description and an alternating active-phase algorithm is implemented. Based on the alternating active-phase algorithm, a multi-material topology optimization problem with N + 1 phases is split into N(N + 1)/2 binary-phase topology optimization sub-problems. Compared with the initial multi-material problem, each sub-problem involves fewer design variables and volume constraints. In the DS-MMLS model, N + 1 phases are represented by the sequential difference set of N level set functions. Based on this model, the topological evolution of two active phases can be easily achieved by updating a single level set function in a fixed domain, which contributes a great convenience to the implementation of the alternating active-phase algorithm with level set method. Therefore, the proposed method can be easily extended to topology optimization problems with more material phases. To demonstrate its effectiveness, some 2D and 3D numerical examples with different material phases are presented. The results reveal that the proposed method is effective for multi-material topology optimization problems.

An incremental form interpolation model together with the Smolyak method for multi-material topology optimization

An incremental form interpolation model integrated with the anisotropic Smolyak method and the Non-Uniform Rational B-Spline (NURBS) method is proposed for multiple material topology optimization. Only a single category of design variables is exploited to characterize the material properties for any number of material types and the Smolyak method embedded within the NURBS method is devised to represent the topology density field while keeping the number of polynomials to a minimum, leading to a significant reduction in the number of design variables. An arbitrary number of volume and/or mass constraints can be implemented with the introduction of the characterization function to indicate the existence information for each material. Finally, the effectiveness of the proposed methodology is illustrated by three numerical examples.

A B-spline multi-parameterization method for multi-material topology optimization of thermoelastic structures

A B-spline multi-parameterization method (MPM) is presented in this paper for topology optimization of thermoelastic structures. As thermoelastic topology optimization belongs to a kind of design-dependent problems that are complicated to deal with, this method is aimed to solve thermoelastic problems with multiple materials by means of B-spline parameterization that integrates together the recursive multiphase material interpolation (RMMI) and the uniform multiphase material interpolation (UMMI) schemes. The commonly used discrete pseudo-density variables related to the finite element model are thus replaced with continuous pseudo-density fields dominated by control parameters in the B-spline space. In this sense, B-spline multi-parameterization is used for the first time to represent multiple pseudo-density fields and multi-material properties including elasticity matrix and thermal stress coefficient. Compared with traditional pseudo-density method, the current method has the advantage of not only attaining a reduction of design variables in number but also achieving a regularized distribution of pseudo-density fields with a clear material layout over the whole structure domain. Numerical results show that the stable convergences are achieved with the avoidance of gray elements, checkerboards, and multi-material overlapping owing to the high continuity of B-spline preserved for multi-material distributions. Besides, it is found that the RMMI scheme distributes less stiff materials around stiff materials, while the UMMI scheme tends to gather less stiff materials together.

Modified element stacking method for multi-material topology optimization with anisotropic materials

This article presents a modified element stacking method for anisotropic multi-material topology optimization. This method can transform standard multi-material topology optimization formulations into a series of equivalent single-material topology optimization ones to overcome the various limitations inherent to conventional techniques. First, typical multi-material topology optimization methods utilize material interpolation schemes that restrict the study to the isotropic domain with a constant Poisson’s ratio, limiting practical applications and solution accuracy. Additionally, in attempts to further extend these classical multi-material topology optimization methods to anisotropic materials, preparing the necessary element information matrices becomes increasingly laborious or even impossible for complex models, preventing the application of sensitivity analysis for robust gradient-based optimization methods. To directly address these limitations, the element interpolation method replaces material interpolation schemes with an element interpolation framework, where each design cell property is determined using a weighted sum of various coincident single-material elements. This paper provides a thorough description of element interpolation and presents several numerical examples demonstrating successful implementation.

Multiphase topology optimization with a single variable using the phase‐field design method

SummaryIn this study, a multimaterial topology optimization method using a single variable is proposed by combining the solid isotropic material with penalization method and the reaction‐diffusion equation. Unlike ordinary multimaterial optimization, which requires several variables depending on the number of material types, this method intends to represent various materials as one variable. The proposed method combines two special functions in the sensitivity analysis of the objective function to converge the design variable into prespecified density values defined for each of the multimaterials. The composition constraint based on a normal distribution function is also introduced to estimate the distribution of each target density value in a single variable. It enables density exchange between multiple materials by increasing or decreasing the amount of a specific material. The proposed method is applied to structural and electromagnetic problems to verify its effectiveness, and its usefulness is also confirmed from the viewpoint of cost and computation time.

ICM method for topology optimization of multimaterial continuum structure with displacement constraint

A new topology optimization method is formulated for lightweight design of multimaterial structures, using the independent continuous mapping (ICM) method to minimize the weight with a prescribed nodal displacement constraint. Two types of independent topological variable are used to identify the presence of elements and select the material for each phase, to realize the interpolations of the element stiffness matrix and total weight. Furthermore, an explicit expression for the optimized formulation is derived, using approximations of the displacement and weight given by first- and second-order Taylor expansions. The optimization problem is thereby transformed into a standard quadratic programming problem that can be solved using a sequential quadratic programming approach. The feasibility and effectiveness of the proposed multimaterial topology optimization method are demonstrated by determining the best load transfer path for four numerical examples. The results reveal that the topologically optimized configuration of the multimaterial structure varies with the material properties, load conditions, and constraint. Firstly, the weight of the optimized multimaterial structure is found to be lower than that composed of a single material. Secondly, under the precondition of a displacement constraint, the weight of the topologically optimized multimaterial structure decreases as the displacement constraint value is increased. Finally, the topologically optimized multimaterial structures differ depending on the elastic modulus of the materials. Besides, the established optimization formulation is more reliable and suitable for use in practical engineering applications with structural performance parameters as constraint.