Multiscale Optimization Method Research Articles

Multi-scale simulation and optimization on a thermal management system of intermediate circuit and experiment validation

In the system-level simulations of heat transfer systems, the characteristics of heat exchangers are often considered constant or used in empirical formulas, which affect the accuracy of the system model. Herein a multi-scale simulation method to analyze the heat transfer characteristics of a thermal management system, including an intermediate loop is proposed. The system model is developed based on the heat current method and flow resistance models. By applying COMSOL LiveLink for MATLAB module, a multi-scale model is obtained, which combines three-dimensional numerical models of the heat exchangers with the system model. Compared to the experimental results, the relative errors associated with the outlet temperatures and heat transfer rate are less than 1.1% and 8.2 %, respectively, thus validating the model. Discussions on the effects of the geometric structure of the heat exchangers on the system show that a plate heat exchanger with 12 plates and a plate spacing of 8 mm is better. For achieving systematic maximum heat dissipation for a certain pumping power consumption, a multi-scale optimization method is proposed by coupling the Lagrange function and numerical model. The results show that the total heat transfer rate improves by 5.89 %, from 8314.81 W to 8804.35 W.

Integrated multi-material and multi-scale optimization of compliant structure with embedded movable piezoelectric actuators

Piezoelectric actuators embedded in compliant structures have many potential applications, which call for effective design approaches. Existing optimization driven design methods focus on the piezoelectric actuator location or the macro topology of the host compliant structure while ignoring the possibility to exploit the use of multiple materials and hierarchical material design. In this study, an integrated, density-based multi-material and multi-scale optimization method for compliant structures with embedded movable piezoelectric actuators is proposed. The design variables in the optimization represent the location of the piezoelectric actuators and the layout of the macro/micro host structures. To mitigate the risk of suboptimal designs, the three sets of design variables are simultaneously optimized using gradient-based optimization where the gradients are computed using the adjoint method. A material interpolation model that describes the piezoelectric actuators, multiple lattice materials of the host structure as well as their coating layer is proposed. To illustrate the formulation, several compliant structures embedded with movable piezoelectric actuators are designed such that the displacement at an output port is maximized. Macro and micro volume fractions, and overlapping of piezoelectric actuators are constrained in the optimization.

Concurrent multiscale and multi‐material optimization method for natural vibration design of porous structures

AbstractThis paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aimed at maximizing a specified natural frequency. The multi‐material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization, respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi‐material topology optimization using the generalized solid isotropic material with penalization (GSIMP) method, which is a method to obtain multi‐material topology by implementing the successive binarization. The KS (Kreisselmeier–Steinhauser) function is introduced to solve the repeated natural frequencies issue that lies in the maximization of a specified natural frequency. An area‐constrained multiscale optimization problem is formulated as a distributed‐parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method (i.e., a nonparametric shape/topology optimization method). The effectiveness of the proposed method for maximizing a specified natural frequency is confirmed through 2D and 3D numerical examples, in which the influences of sub‐regions of the macrostructure and anisotropic material of the microstructures are also investigated and the results are discussed.

A multiscale optimization methodology for cyclic elastoplastic performance of beam-pile joint member in pile-supported wharf

A material-member-structure multiscale optimization methodology, which can be utilized for improving the elastoplastic behaviour of the beam-pile joint member in pile-supported wharf under cyclic loading, was developed on the base of the application of high-specific stiffness material. Based on the representative volume element (RVE) method, the parameterized relations between the mesoscale phases (e.g., steel fibers, interfacial transition zone, and matrix) and macroscale homogenized elastoplasticity of ultrahigh-performance concrete (UHPC) reinforced with steel fibers were obtained under periodic boundary conditions. The physical equivalence relation between material scale and member scale was established by utilizing the refined constitutive model with the macroscopic mechanical parameters and exponential decay damage evolution factors. UHPC material property tests and cyclic loading tests of small-scale beam-pile joint specimens were performed respectively, and the results were analyzed and utilized to validate the numerical model and multiscale optimization method. Using the computational platform ABAQUS, the cyclic elastoplastic performances of the novel beam-pile joint members were evaluated numerically by pseudo-static hysteretic loading. As an illustration, a multiscale modelling approach for the pile-supported wharf frame system was presented and employed to assess the seismic performance improvement through the optimization of the beam-pile joint members. The coupled solution of refined member damage and large-scale structural bearing capacity was realized by utilizing the deformation coordination condition and force balance equation, which bridges the member scale and structure scale. This study aims to provide a reference for future design and optimization of the joint members in marine frame structures, to enhance cyclic bearing capacities and reduce reinforcement consumptions.

An efficient strategy to implement local porosity constraints in the multiscale design of solids with parameterized biomimetic microstructures

In previous works, the authors introduced a multiscale optimization method to maximize the stiffness of elastic solids with biomimetic cancellous microstructures described by a finite set of parameters. Although effective, the procedure is computationally expensive when solving large-scale problems using per-element non-linear constraints to impose local bounds on the solid volume fraction. This work improves the computational performance of the method by exploring two strategies to completely dispense with nonlinear local constraints: to bound the microparameters so the microsctructures are always within the solid fraction of trabecular bone, and to map the microparameters onto an auxiliary set of parameters that are linearly bounded. As a side effect, the design spaces are reduced. Such reductions are assessed in terms of the bulk and shear moduli and elastic symmetries, which are compared to those of natural bone. Performances of the two strategies are assessed by solving a series of benchmark problems and studying the stiffness of a hip prosthesis. The strategy based on the isoparametric mapping achieves the best results, performing up to 2000 times faster while marginally reducing the design space. Thus, the isoparametric mapping approach makes the multiscale design method a suitable tool for solving large-scale problems of practical interest.

On concurrent multiscale topology optimization for porous structures under hygro‐thermo‐elastic multiphysics with considering evaporation

AbstractLightweight polymeric and natural composite materials are extensively used in modern structures, especially with the demands for environmentally friendly products as well as lowering energy consumption. Furthermore, a high performance‐to‐weight ratio can be attained by utilizing porous composites. However, hygral and thermally induced loads are limiting the robustness of polymeric and natural composite materials, therefore; in this research, concurrent multiscale multiphysics topology optimization is used to design lightweight porous composite structures that have resilience toward mechanical as well as hygral and thermal loads. By establishing two independent representations of the design problem, that is, macro and microscale domains, a concurrent topology optimization framework is implemented, and the effective properties of the microscale (i.e., elastic, thermal conductivity, moisture diffusivity tensors, and hygral as well as thermal expansion coefficients) are calculated and used as the hygro‐thermo‐elastic properties of the macroscale using in‐house MATLAB codes. For hygral physics, moisture transport, as well as evaporation, are simultaneously considered in this study. A sensitivity analysis was conducted on the multiphysics concurrent optimization scheme in order to account for the coupling of macro and microstructure, as well as hygro‐thermo‐elastic physics. Multiple numerical cases were examined, which included different loading and boundary conditions, as well as various spatial configurations. The results showed attaining a high stiffness‐to‐weight ratio for the multiscale optimized porous structure compared to the single‐scale solid structure. Furthermore, a study was conducted on multiple microstructure subsystems to examine the impact of microstructure systems on macrostructure dependence. By combining several microstructures into a single macro design domain, design flexibility was enhanced and the performance‐to‐weight ratio was improved. The study was expanded to include the evaluation of hygro‐thermo‐elastic multiscale multiphysics with an evaporation problem, which was demonstrated through several numerical examples. The introduced formulations showed a successful application of the concurrent multiscale optimization formulations and good coupling on the macro and microscale. Also, the formulations demonstrated a strong influence between the macro and the microscale of the design problem for the topology optimization methods. The successful application of the concurrent multiscale optimization method in this research highlights its potential for designing more efficient and effective structures in the future.

Concurrent multiscale optimization method for natural vibration design of porous structures

This paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aiming at maximizing the vibration eigenvalue. The multi-material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi-material topology optimization using the GSIMP method. The KS function is introduced to solve the repeated-root problem that lies in the maximization of vibrational eigenvalues. The area-constrained multiscale optimization problem is formulated as a distributed parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method. The effectiveness of the proposed method for maximizing vibration eigenvalues is confirmed through numerical examples.

End-to-end pattern optimization technology for 3D shape measurement.

The nonlinear errors caused by the gamma effect seriously affect the measurement accuracy of phase-measuring profilometry systems. We present a novel robust end-to-end pattern optimization technology for phase-height mapping. This method generates standard sinusoidal patterns on the reference plane on a camera image instead of on a projection image by optimizing the projection image, which can avoid phase errors due to phase conversion from projector to camera. The pixel set mapping is achieved by chessboard calibration; then the projection image is optimized with the multiscale optimization method and the dislocation optimization method based on proportional integral derivative control. The experiments show that the proposed method can effectively avoid the influence of nonlinear effect and achieve high-quality three-dimensional measurement.

Multiscale topology optimization of 3D structures: A micro-architectured materials database assisted strategy

This article presents a datadriven multiscale optimization method for solving 3D problems under given boundary conditions. The topology is represented by an extended level-set function that allows the nucleation of new holes throughout the process making it less sensitive to the initial guess. The two scales involved are the structural (macro) scale and the material (micro) scale. For the macro scale, the stiffness is maximized while reducing the weight of the structure; while for the micro scale, the homogenized elasticity tensor is prescribed and the weight is also minimized. In order to decrease the significant computational cost of multi-scale optimization, a datadriven approach is proposed. The method consists in two main steps: (1) an offline step to build a database (catalog of optimal micro-architectured materials indexed by their effective elasticity) for a wide range of desired elasticity tensors, and (2) an online step aiming to optimize the 3D structure at the macroscale. The precomputed catalog is thus interrogated in order to find the best micro-architectured material within the macroscale optimization. This method performs a two scales weight minimization and leads to a significantly light structure design. The optimization formulation is first introduced. Then a few 3D benchmark tests are given for the multi-scale optimization using micro-architectured materials.

Multi-View Multi-Scale Optimization of Feature Representation for EEG Classification Improvement.

Effectively extracting common space pattern (CSP) features from motor imagery (MI) EEG signals is often highly dependent on the filter band selection. At the same time, optimizing the EEG channel combinations is another key issue that substantially affects the SMR feature representations. Although numerous algorithms have been developed to find channels that record important characteristics of MI, most of them select channels in a cumbersome way with low computational efficiency, thereby limiting the practicality of MI-based BCI systems. In this study, we propose the multi-scale optimization (MSO) of spatial patterns, optimizing filter bands over multiple channel sets within CSPs to further improve the performance of MI-based BCI. Specifically, several channel subsets are first heuristically predefined, and then raw EEG data specific to each of these subsets bandpass-filtered at the overlap between a set of filter bands. Further, instead of solving learning problems for each channel subset independently, we propose a multi-view learning based sparse optimization to jointly extract robust CSP features with L2,1 -norm regularization, aiming to capture the shared salient information across multiple related spatial patterns for enhanced classification performance. A support vector machine (SVM) classifier is then trained on these optimized EEG features for accurate recognition of MI tasks. Experimental results on three public EEG datasets validate the effectiveness of MSO compared to several other competing methods and their variants. These superior experimental results demonstrate that the proposed MSO method has promising potential in MI-based BCIs.

An efficient multiscale optimization method for conformal lattice materials

This article presents a multiscale optimization method to solve size distribution problem of conformal lattice materials (CLMs). Full-scale analyses are not needed during optimization process, so this method is much more efficient compared with other optimization methods for CLMs, especially when the number of microbars is considerably large. On microscope, inherent heterogeneity of CLM makes it hard to introduce homogenization method to scale down the problem. For substitute, a new method derived from extended multiscale finite element method (EMsFEM) is used to calculate the effective stiffness of the microlattice structures, and two improvements are given to increase the accuracy and extend its application to CLM. On macroscope, based on gradient-based topology optimization method, a multiscale optimization algorithm is raised for a minimum compliance design under volume constraint. The diameters of microbars are set to be design variables. Sensitive analyses based on EMsFEM are carried out, so the effective calculating process can be seamlessly integrated in the optimization algorithm. Furthermore, considering the discontinuity of bars laid on elements’ edges, a post-processing method is proposed to determine the diameters of these bars. This optimization method is validated by two mechanical experiments on specimens fabricated by 3D printing, and its efficiency is tested by comparing with the optimization method with full-scale FE analyses. The results of both mechanical experiments and finite element simulations show that the optimized structures do have better mechanical properties, exposing the material redistribution tendency during optimizing process.

Elastic properties and multi-scale design of long carbon fiber nonwoven reinforced plane-based isotropic composite

Carbon fiber reinforced composite is considered as a potential substitutes used in the lightweight of automotive engineering. Nevertheless, the high price and complicated manufacturing process remain challenges for the large-scale application of conventional carbon fibers. In this study, a novel plane isotropic composite reinforced by long carbon fiber non-woven is introduced, which overcomes the obstacles in the application of carbon fiber at present. A multi-scale optimization method is proposed to realize the performance optimization and lightweight design of composite components. The linear elastic properties of long carbon fiber non-woven composite (LCFNC) were studied experimentally and analytically. The predictions of homogenization theory were found to be very similar to the experimental results within an interval of volume fraction. Then the relationship between microstructure and material performance was evaluated in detail. The results showed that there is a critical point for the influence of aspect ratio to the material properties. Thirdly, thickness of LCFNC at macro scales and the fiber parameters at micro scales were simultaneously optimized by Non-dominated sorting genetic algorithm-II (NSGA- II) to enforce multi-objective optimization design. Finally, the lightweight design of automobile hood was performed by the LCFNC and the proposed method. The results showed that the weight of automobile hood was reduced by 37%, while exhibiting better stiffness and strength performance compared with the conventional steel.

Multiscale design of artificial bones with biomimetic elastic microstructures

Cancellous bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. The hierarchical architecture makes cancellous bone a prime example of a lightweight natural material that combines strength with toughness. Better understanding the mechanics of cancellous bone is of interest for the diagnosis of bone diseases, the evaluation of the risk of fracture, and for the design of artificial bones and bone scaffolds for tissue engineering. A multiscale optimization method to maximize the stiffness of artificial bones using biomimetic cellular microstructures described by a finite set of geometrical micro-parameters is presented here. The most outstanding characteristics of its implementation are the use of: an interior point optimization algorithm, a precalculated response surface methodology for the evaluation of the elastic tensor of the microstructure as an analytical function of the micro-parameters, and the adjoint method for the computation of the sensitivity of the macroscopic mechanical response to the variation of the micro-parameters. The performance and effectiveness of the tool are evaluated by solving a problem that consists in finding the optimal distribution of the microstructures for a proximal end of a femur subjected to physiological loads. Two strategies for the specification of the solid volume fraction constraints are assessed. The results are compared with data of a computed tomography study of an actual human bone. The model successfully predicts the main features of the spatial arrangement of the trabecular and cortical microstructures of the natural bone.

Anisotropic design and optimization of conformal gradient lattice structures

In this work, we present a novel anisotropic lattice structure design and multi-scale optimization method that can generate conformal gradient lattice structures (CGLS). The goal of optimization is to achieve gradient density, adaptive orientation and variable scale (or periodic) lattice structures with the highest mechanical stiffness. The asymptotic homogenization method is employed for the calculation of the mechanical properties of various lattice structures. And an equation of elastic tensor and relative density of the unit cell is established. The established function above is then considered in the numerical optimization schemes. In the post-processing, we propose a numerical projecting method based on Fourier transform, which can synthesize conformal gradient lattice structure without changing the size and shape of the unit cells. Besides, the algorithm allows us to minimize distortion and prevent defects in the final lattice and keep the lattice structures smooth and continuous. Finally, in comparison with different parameters and methods are performed to demonstrate the superiority of our proposed method. The results show that the optimized anisotropic conformal gradient lattice structures are much stiffer and exhibit better structural robustness and buckling resistance than the uniform and the directly mapped designs.

A fuzzy optimization method for octet-truss lattices

Purpose This paper aims to present a multiscale fuzzy optimization (FO) method to optimize both the density distribution and macrotopology of a uniform octet-truss lattice structure. Design/methodology/approach The design formulae for the strut radii are presented based on the effective mechanical properties obtained from the representative volume element. The proposed basic lattice material is applied in a normalization process to determine the material model with penalization. The solid isotropic material with penalization (SIMP) method is extended to solve the minimum compliance problem using the optimality criteria. The evolutionary deletion process is proposed to delete elements corresponding to thin-strut unit cells and to obtain the optimal macrotopology. Findings Both numerical cases indicate that the FO results significantly improved in structural performance compared with the results of the conventional SIMP. The deleting threshold controls the macrotopology of the graded-density lattice structures with negligible effects on the mechanical properties. Originality/value This paper presents one of the first multiscale optimization methods to optimize both the relative density and macrotopology of uniform octet-truss lattices. The material model and corresponding optimality criteria of octet-truss lattices are proposed and implemented in the optimization.

Multi-scale structural topology optimization of free-layer damping structures with damping composite materials

Damping performance of the free-layer damping structure mainly depends on the damping material layout and its material physical properties. This paper proposes a concurrent topology optimization method for the design of the multi-scale free-layer damping structures with damping composite materials. In order to maximize the structural damping performance, the density-based topology optimization method is adopted to find the optimal layout on both macro and micro scales. For this coupled system, the macrostructure is composed of periodic damping composite materials, a stiff damping material which is designed for maintaining the structural stiffness and a soft damping material which is used for improving structural damping. The homogenized effective complex elastic matrix is obtained by considering the two damping materials layout in the microstructure. Mathematical model is established and sensitivity analysis is deduced. Several numerical examples are presented to demonstrate the effectiveness of the proposed multi-scale optimization method. In-depth discussions are given for the effects of the volume fraction of soft damping material in the micro scale on the design results. The results show that there is an optimal volume ratio of the stiff and soft damping material to make the structure reach optimal vibration performance due to the effective competition of soft and stiff characteristics of different damping materials. The optimal microstructure has relatively great loss moduli and high material loss factor, and it also presents a negative Poisson’s ratio. The structural vibration performance of the optimal multi-scale layer damping structure with damping composite materials is significantly improved.

基于特征值分析的多尺度结构优化设计方法

A multi-scale structure optimization method was proposed based on eigenvlue analysis, to find the macrostructure and microstructure of maximum macro stiffness under the worst load. The constraint that the Euclidian norm of the uncertain load is 1 was introduced, the structural compliance was calculated according to the Rayleigh-Ritz theorem, and the compliance was transformed to a symmetric matrix with the same dimensions as the local load vector. In this way, the compliance minimization problem under the worst load was transformed to the minimum problem of the maximum eigenvalue of the symmetric matrix. Moreover, the worst load case was determined with the eigenvector corresponding to the maximum eigenvalue of the matrix. Several numerical experiments demonstrated the validity of the proposed method, and illustrated the reasonability of the macro topological structure and the micro material distribution. The proposed multi-scale optimization method has virtues of iterative stability and rapid convergence. The update of the density function in the topological optimization was performed based on sensitivity analysis and the method of moving asymptotes (MMA).

Research on the optimal dynamical systems of three-dimensional Navier-Stokes equations based on weighted residual

In this paper, the theory of constructing optimal dynamical systems based on weighted residual presented by Wu & Sha is applied to three-dimensional Navier-Stokes equations, and the optimal dynamical system modeling equations are derived. Then the multiscale global optimization method based on coarse graining analysis is presented, by which a set of approximate global optimal bases is directly obtained from Navier-Stokes equations and the construction of optimal dynamical systems is realized. The optimal bases show good properties, such as showing the physical properties of complex flows and the turbulent vortex structures, being intrinsic to real physical problem and dynamical systems, and having scaling symmetry in mathematics, etc.. In conclusion, using fewer terms of optimal bases will approach the exact solutions of Navier-Stokes equations, and the dynamical systems based on them show the most optimal behavior.

Multiscale Collaborative Optimization of Processing Parameters for Carbon Fiber/Epoxy Laminates Fabricated by High-Speed Automated Fiber Placement

Processing optimization is an important means to inhibit manufacturing defects efficiently. However, processing optimization used by experiments or macroscopic theories in high-speed automated fiber placement (AFP) suffers from some restrictions, because multiscale effect of laying tows and their manufacturing defects could not be considered. In this paper, processing parameters, including compaction force, laying speed, and preheating temperature, are optimized by multiscale collaborative optimization in AFP process. Firstly, rational model between cracks and strain energy is revealed in order that the formative possibility of cracks could be assessed by using strain energy or its density. Following that, an antisequential hierarchical multiscale collaborative optimization method is presented to resolve multiscale effect of structure and mechanical properties for laying tows or cracks in high-speed automated fiber placement process. According to the above method and taking carbon fiber/epoxy tow as an example, multiscale mechanical properties of laying tow under different processing parameters are investigated through simulation, which includes recoverable strain energy (ALLSE) of macroscale, strain energy density (SED) of mesoscale, and interface absorbability and matrix fluidity of microscale. Finally, response surface method (RSM) is used to optimize the processing parameters. Two groups of processing parameters, which have higher desirability, are obtained to achieve the purpose of multiscale collaborative optimization.

An improved multiscale method for life-cycle production optimization

This work proposes a new hierarchical multiscale optimization technique to improve the performance of life-cycle production optimization. This approach represents a combination of two previous multiscale approaches presented in the literature and is theoretically motivated by the concept of refinement indicators. The new algorithm is applied to two example problems, and its performance is compared with other life-cycle production optimization algorithms that have been proposed in the literature including the two hierarchical multiscale optimization methods on which it is based. In a separate paper (Oliveira and Reynolds 2015), the proposed algorithm is successfully applied to a field case.