Independent Continuous Mapping Research Articles

Layout optimization of pore microstructure in fluid-saturated porous media using Galerkin decoupling technology and independent continuous mapping method (GDT-ICM)

A layout optimal design method combining Galerkin decoupling technology and independent continuous mapping method (GDT-ICM) is proposed to rearrange the layout of pore microstructure. Firstly, the Galerkin decoupling technology (GDT) is employed to solve the Brinkman equation for fluid flow in fluid-saturated porous media, which integrates the Stokes equation and Darcy's law. Within this framework a self-programmable element balance equation is developed. Secondly, a permeability interpolation function is defined for each element, incorporating the permeability of the fluid and pore microstructure. This function distinguishes strictly between the fluid domain and the pore domain. Pore microstructures with arbitrary shape including permeability information are established in an Euler mesh by introducing the Carman-Kozeny equation, shape functions, the Joukowski mapping and filter functions (the modified arctan mapping function and the ‘inpolygon’ function) based on the ICM method. Thirdly, a layout optimization model aiming at minimizing energy dissipation is established to accomplish the optimal distribution of pore microstructures. A sensitivity analysis is calculated with respect to design variables and the optimal model is solved by a genetic algorithm. Numerical results demonstrate that the proposed method is effective and feasible in 2D-space. The maximum velocity within the fluid field is reduced by 45%, and the issue of localized high pressure occurring around the pore is solved. This paper provides guidance for solving the Brinkman coupled equation and the layout optimization of the microstructure in porous media.

Concurrent optimization method of principal stress orientation interpolated continuous fiber angle (PSO-CFAO) and structural topology

Continuous fiber-reinforced polymers (CFRPs) have been widely applied in aerospace and other fields due to their excellent mechanical properties, which highly depends on the material distribution and fiber orientations. The designability of CFRP structures and fiber distributions provides an opportunity for achieving better physical properties through optimization. However, local optima and the dependence of initial fiber angle variables make the concurrent optimization of fiber orientation and topology a challenging problem. In this paper, the principal stress orientation interpolated continuous fiber angle optimization (PSO-CFAO) method combined with the independent continuous mapping (ICM) method is proposed to realize the design of CFRP structures with a clear macroscopic topology and microscopic fiber distribution. A sigmoid function is applied to interpolate the fiber angle variables by the principal stress orientation. The fiber angle variables are modified and a continuous fiber design is obtained during the iteration process, which reduces the possibility of a local optimum. Several examples are provided to prove the effectiveness and stability of the proposed method, and the expected results are acquired for different initial fiber angles, material parameters and mesh densities. The proposed method provides guidance for the design of CFRP structures and the planning of fiber laying paths.

Topology Optimization of Minimum Compliance for Continuous Fiber-Reinforced Polymer

Continuous fiber-reinforced polymer (CFRP) has been paid more attention to due to its advantages of high strength, high stiffness, and lightweight. However, it is challenging for topology optimization to achieve CRRP structure design. A topology optimization (TO) model with minimizing structural compliance under volume fraction constraint is established based on the independent continuous mapping (ICM) method. The continuous fiber angle optimization (CFAO) method solves the fiber orientation. Meanwhile, the local optimal solution problem caused by the CFAO method is greatly avoided by the direction of principal stress. The effectiveness of the method is verified by classic numerical examples. This paper has a significant reference for the TO design of CFRP structure.

The Fatigue topology optimization of orthotropic materials

Combining with the constitutive relation of orthotropic materials, a fatigue topology optimization model considering orthotropic materials is established based on the Independent Continuous Mapping (ICM) method. It uses the minimum structural mass as the optimization objective, the element fatigue life as the constraint condition, and the criterion method. The fatigue life filtration function, mass filtration function, and stiffness filtration function are introduced to identify whether the material is retained. The effectuality of the method is verified by several numerical examples, and the influence of different filter function coefficients on the results is explored. This paper has some theoretical significance for considering the fatigue properties of orthotropic material components in the conceptual design stage.

A systematic design of multifunctional lattice structures with energy absorption and phononic bandgap by topology and parameter optimization

ABSTRACT Lattice structure can realize excellent multifunctional characteristics because of its huge design space, and the cellular configuration directly affects the lattice structural performance and lightweight. A novel energy-absorbing multifunctional lattice structure with phononic bandgap is presented by topology and parameter optimization in this paper. First, the two-dimensional (2D) cellular configuration is lightweight designed by using independent continuous mapping (ICM) topology optimization method. The 2D cell is reconstructed by geometric parameters and rotated into a three-dimensional (3D) cell by using chiral shape to achieve bandgap. Subsequently, the surrogated model with energy absorption as the object and first-order natural frequency as the constraint is established to optimize a parametric 3D cell based on the Response Surface Methodology (RSM). Finally, the lattice structures are assembled with dodecagonal staggered arrangements to avoid the deformation interference among the adjacent cells. In addition, the lattice structural energy absorption and bandgap characteristics are analyzed and discussed. Compared to Kelvin lattice structure, the optimal lattice structure shows significant improvement in energy absorption efficiency. Besides, the proposed design also performs well in damping characteristics of the high-frequency and wide-bandgap. The lattice structural optimization design framework has great meaning to achieve the equipment structural lightweight and multi-function in the aerospace field.

Topology Optimization of Strength-Safe Continuum Structures Considering Random Damage

Spacecraft in the aerospace field and military equipment in the military field are at risk of being impacted by external objects, which can cause local damage to the structure. The randomness of local damage is a new challenge for structural design, and it is essential to take random damage into account in the conceptual design phase for the purpose of improving structure’s resistance to external shocks. In this article, a random damaged structure is assumed to have damages of the same size and shape at random locations, and the random damage is considered as multiple damage conditions of the structure. In order to improve the randomness and comprehensiveness of the multiple damage conditions, the stacking strategy is used to generate the distribution of the damage area. Following this strategy, the topology optimization design of the random damaged structure, which is to minimize the weight of the structure with a constraint on the stress of the structure under multiple damage conditions, is formulated based on the independent continuous mapping (ICM) method. The dual sequence quadratic programming (DSQP) algorithm combined with the stress globalization method is adopted to solve the optimization problem. The numerical examples demonstrate the effectiveness and applicability of the proposed method in the topology optimization of strength-safe continuum structures.

Topology Optimization for Hybrid Lattice Compliant Mechanisms with Multiple Microstructures.

Hybrid lattice compliant mechanisms (HLCMs) composed of multiple microstructures have attracted widespread interest due to their superior compliant performance compared to the traditional solid compliant mechanisms. A novel optimization scheme for HLCMs is presented using the independent continuous mapping (ICM) method. Firstly, the effective properties of multiple orthogonal and anisotropic lattice microstructures are obtained by taking advantage of homogenization theory, which are used to bridge the relationship between the macrostructure layout and microstructure recognition. Then, a new parallel topology optimization model for optimizing HLCMs is built via a generalized multi-material, recognizing interpolation scheme with filter functions. In addition, the characterization relationship between independent continuous variables and performance of different elements is established. Sensitivity analysis and linear programming are utilized to solve the optimal model. Lastly, numerical examples with a displacement inverter mechanism and compliant gripper mechanism demonstrate the effectiveness of the proposed method for designing HLCMs with various lattice microstructures. Anisotropic lattice microstructures (ALMs) significantly facilitate the efficient use of constitutive properties of materials. Hence, HLCMs consisting of various ALMs achieve superior compliant performance than counterparts comprising different orthogonal lattice microstructures (OLMs). The presented method offers a reference to optimize HLCMs, as well as promotes the theoretical development and application of the ICM method.

Lightweight topology optimization of graded lattice structures with displacement constraints based on an independent continuous mapping method

Lightweight topology optimization of graded lattice structures with displacement constraints based on an independent continuous mapping method

Design optimization of multifunctional metamaterials with tunable thermal expansion and phononic bandgap

Metamaterials have been extensively investigated owing to their unusual properties. However, the design of multifunctional metamaterials requires further investigation. This study focused on the design of three-dimensional (3D) metamaterials to achieve tunable negative thermal expansion and phononic bandgap properties. First, the independent continuous mapping (ICM) topology optimisation method was applied to create metamaterial microstructures with negative thermal expansion properties based on the multi-scale asymptotic homogenisation theory. Secondly, the conceptual structure from the topology optimisation was reconstructed and parameterized to achieve the desired phononic bandgap widths under negative thermal expansion, using a surrogate model-based optimisation method. Both the negative coefficient of thermal expansion and phononic bandgaps were verified through numerical simulations. The results reveal that, by selecting appropriate parameters, the designed metamaterials can have both a negative coefficient of thermal expansion and a maximum bandgap width ratio. The proposed method provides an important reference for the rational design of multifunctional metamaterials.

Acceleration Design for Continuum Topology Optimization by Using Pix2pix Neural Network

Aiming at speeding up the process of topology optimization and generating a novel high-quality topology configuration, a topology optimization design with lightweight structure based on deep learning is investigated. First, Independent Continuous Mapping (ICM) method is employed to establish the original configurations including the intermediate configurations and the corresponding final configurations, and the original configurations are padded and merged to form the dataset required by the deep learning network. Then the high-dimensional mapping relationship between the intermediate configuration and the final configuration is created by using Pix2pix neural network (Pix2pix NN), which transforms the topology optimization into image-to-image translation problem. Finally, the acceleration algorithm is utilized to accelerate the iteration process by the pre-trained network. Numerical examples show that the coupling method is feasible and efficient in topology optimization design. The method provides a new solution for topology optimization design to shorten the iteration process and broaden the application of ICM method.

Geometrically Nonlinear Topology Optimization of Continuum Structures Based on an Independent Continuous Mapping Method

A geometrically nonlinear topology optimization method for continuum structures is proposed based on the independent continuous mapping method. The stress constraint problem is studied due to the importance of structural strength in engineering applications. First, a topology optimization model is established for a lightweight structure with element stress as constraints. Second, the stress globalization method is adopted to convert local stress constraints into strain energy constraints, which overcomes the difficulties caused by local stress constraints, such as model establishment, sensitivity analysis, and massive solution calculations. Third, the sensitivity of the objective function and constraint function is analyzed, and the method of moving asymptotes is employed to solve the optimization model. In addition, the additive hyperelasticity technique is utilized to solve the numerical instability induced by structures undergoing large deformation. Numerical examples are given to validate the feasibility of the proposed method. The method provides a significant reference for geometrically nonlinear optimization design.

Fatigue Topology Optimization Design Based on Distortion Energy Theory and Independent Continuous Mapping Method

Fatigue failure is a common failure mode under the action of cyclic loads in engineering applications, which often occurs with no obvious signal. The maximum structural stress is far below the allowable stress when the structures are damaged. Aiming at the lightweight structure, fatigue topology optimization design is investigated to avoid the occurrence of fatigue failure in the structural conceptual design beforehand. Firstly, the fatigue life is expressed by topology variables and the fatigue life filter function. The continuum fatigue optimization model is established with the independent continuous mapping (ICM) method. Secondly, fatigue life constraints are transformed to distortion energy constraints explicitly by taking advantage of the distortion energy theory. Thirdly, the optimization formulation is solved by the dual sequence quadratic programming (DSQP). And the design scheme of lightweight structure considering the fatigue characteristics is obtained. Finally, numerical examples illustrate the practicality and effectiveness of the fatigue optimization method. This method further expands the theoretical application of the ICM method and provides a novel approach for the fatigue optimization problem.

Fail-Safe Topology Optimization of Continuum Structures with Multiple Constraints Based on ICM Method

Traditional topology optimization methods may lead to a great reduction in the redundancy of the optimized structure due to unexpected material removal at the critical components. The local failure in critical components can instantly cause the overall failure in the structure. More and more scholars have taken the fail-safe design into consideration when conducting topology optimization. A lot of good designs have been obtained in their research, though limited regarding minimizing structural compliance (maximizing stiffness) with given amount of material. In terms of practical engineering applications considering fail-safe design, it is more meaningful to seek for the lightweight structure with enough stiffness to resist various component failures and/or to meet multiple design requirements, than the stiffest structure only. Thus, this paper presents a fail-safe topology optimization model for minimizing structural weight with respect to stress and displacement constraints. The optimization problem is solved by utilizing the independent continuous mapping (ICM) method combined with the dual sequence quadratic programming (DSQP) algorithm. Special treatments are applied to the constraints, including converting local stress constraints into a global structural strain energy constraint and expressing the displacement constraint explicitly with approximations. All of the constraints are nondimensionalized to avoid numerical instability caused by great differences in constraint magnitudes. The optimized results exhibit more complex topological configurations and higher redundancy to resist local failures than the traditional optimization designs. This paper also shows how to find the worst failure region, which can be a good reference for designers in engineering.

Fail-safe topology optimization of continuum structures with fundamental frequency constraints based on the ICM method

It is an important topic to improve the redundancy of optimized configuration to resist the local failure in topology optimization of continuum structures. Such a fail-safe topology optimization problem has been solved effectively in the field of statics. In this paper, the fail-safe topology optimization problem is extended to the field of frequency topology optimization. Based on the independent continuous mapping (ICM) method, the model of fail-safe topology optimization is established with the objective of minimal weight integrating with the discrete condition of topological variables and the constraint of the fundamental frequency. The fail-safe optimization model established above is substituted by a sequence of subproblems in the form of the quadratic program with exact second-order information and solved efficiently by the dual sequence quadratic programming (DSQP) algorithm. The numerical result reveals that the optimized fail-safe structure has more complex configuration and preserved materials than the structure obtained from the traditional frequency topology optimization, which means that the optimized fail-safe structure has higher redundancy. Moreover, the optimized fail-safe structure guarantees that the natural frequency meets the constraint of fundamental frequency when the local failure occurs, which can avoid the structural frequency to be sensitive to local failure. The fail-safe optimization topology model is proved effective and feasible by four numerical examples.

Fatigue Topology Optimization Based on Global Stress Constraint Method

In topology optimization design for continuous structure, common mechanical properties considered in optimization model focus on structural strength, stiffness and stability. However, fatigue characteristic of structure is easily ignored, which is one of the main causes of engineering failure. Therefore, it is necessary to consider the fatigue characteristic for the optimization design of the structure. In order to investigate the influence of fatigue on structural topology in optimization design, fatigue life filter function is introduced and then the model and solution method of fatigue topology optimization for continuous structures are proposed based on independent continuous mapping method (ICM). The fatigue topological optimization model is established with structure minimum weight as objective and fatigue life as constraint, and the model is solved by global stress constraint method. The results extend the ICM method and this basic theory provides a new idea for fatigue topology optimization.

Lightweight Topology Optimization with Buckling and Frequency Constraints Using the Independent Continuous Mapping Method

This research focuses on the lightweight topology optimization method for structures under the premise of meeting the requirements of stability and vibration characteristics. A new topology optimization model with the constraints of natural frequencies and critical buckling loads and the objective of minimizing the structural volume is established and solved based on the independent continuous mapping method. The eigenvalue equations and composite exponential filter functions are applied to convert the optimization formulation into a continuous, solvable mathematical programming model. In the process of topology optimization, suitable initial values of the filter functions are chosen to avoid local modes, and the dynamic frequency gap constraints are added in the optimal model to prevent mode switches. Furthermore, for the optimal structures with grey elements obtained by the continuous optimization model, the bisection–inverse iteration is applied to search the optimal discrete structures. Finally, a detailed scheme is given for the buckling and frequency topology optimization problem. Numerical examples illustrate that the modelling method of minimizing the economic index with given performance requirements is practical and feasible for multi-performance topology optimization problems.

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.

Topology optimization of continuum structures considering damage based on independent continuous mapping method

Continuum topology optimization usually produces results similar to a skeleton structure. In addition, material utilization in the optimized structure is greatly improved compared with the original structure. On the other hand, the redundancy of the structure is greatly reduced due to the removed material. Partial local failure in the optimized structure makes it more difficult for the entire structure to meet strength/stiffness requirements. Using the independent continuous mapping (ICM) method, with minimal weight as the objective and both stress and displacement as the respective constraints, continuum topology optimization models which also consider damage can be employed. A dual-sequence quadratic programming (DSQP) algorithm was used in this work to solve such topology optimization models. Numerical examples confirmed the effectiveness and feasibility of the models. The results indicated that both good load path and weight reduction can be obtained. In addition, compared with the structure obtained using conventional topology optimization, redundancy is greatly improved, and the strength/stiffness requirements for the structure can be satisfied for each damage scenario. Furthermore, the results indicate that the strength/stiffness of the structure, after topology optimization, is only slightly sensitive to local damage.

基于磨光反演映射的拓扑优化ICM方法

The polish mapping and the filter mapping in the ICM (independent continuous mapping) method for topology optimization were extended, and the composite function was used to coordinate the filter function. Due to the superposed discrete effects of composite functions, the composite function of the power function and the sine function was introduced to identify the presence and absence of the elements. The ICM method was used to establish the topology optimization model for the continuum structure with the minimum weight under the displacement constraints, which was solved with the exact dual algorithm of the quadratic programming. Then based on the dynamic inversion strategy, the rational inversion function was constructed with the optimal threshold to obtain the most strict discrete solution. Moreover, the 2-stage ‘discrete-continuous’ and ‘continuous-discrete’ solution method was established for topology optimization. Moreover, a calculator program was developed and compiled based on the MATLAB according to this new method. The results show that the proposed method has the advantages of higher computational efficiency, less optimal gray values and less structural weight after inversion, and gives a more reasonable structural topology.

Optimal design for the support structure of a cavity mirror in a chemical oxygen iodine laser resonator.

The dynamic characteristics of the cavity mirror support structure strongly influence the quality of the output beam. However, the contradiction between excellent dynamic performance and light weight make the design process challenging. To cope with the problems encountered in the original design of a chemical oxygen iodine laser system, this paper presents a two-dimensional adjustable support structure based on spherical constraints with large specific stiffness in the initial design phase. Subsequently, a two-level optimization strategy containing a macro design and a detailed design is adopted to optimize the initial structure. At the macro design stage, a two-step topology optimization procedure is introduced, in which the scale of the optimization model is dramatically reduced using the independent continuous mapping algorithm to improve the calculation speed in the first step, and the gray elements are eliminated using the bi-directional evolutionary structural optimization method to clearly obtain the optimal topology in the second step. This method is verified to overcome the defect of low efficiency, while still eliminating gray elements. At the detailed design stage, an adaptive surrogate model and the multi-objective design optimization method are employed to seek the best compromise between the lower weight and higher dynamic performance. The results from the application to the example of the cavity mirror support structure show that the mass is reduced by 41.8%, and the dynamic performance requirement is fulfilled.