Two-scale Optimization Research Articles

A framework for constrained large-scale multi-objective white-box problems based on two-scale optimization through decision transfer

Most existing constrained multi-objective evolutionary algorithms (CMOEAs) are not so efficient when handling constrained large-scale multi-objective problems (CLSMOPs). To overcome white-box CLSMOPs with definitive objective functions, a two-scale optimization framework based on decision transfer, which integrates dimensionality reduction of large-scale decision variables and constraint handling technology, is proposed. The Lagrange multiplier is first used to construct the two-scale optimization model, which bridges original large-scale decision space of variables and small-scale (2-scale) decision space of objective-constraint parameter. The decision transfer algorithm is then designed to switch between large-scale original decision space and small-scale parametric decision space, while achieving the maximum dimensionality reduction. Finally, the two-scale evolution strategy is proposed for the alternative optimizations in the two decision spaces, which emphasize objectives and constraints, respectively. In summary, the optimization in the large-scale space pushes the population to unconstrained Pareto front (PF), the optimization in the small-scale space helps the population cross the infeasible areas to approach constrained PF, and the offspring generation by Lagrange multiplier is beneficial to both objectives and constraints. Eight representative and state-of-the-art CMOEAs have been embedded into the CLDTEA framework to demonstrate its effectiveness through comparative experiments on CLSMOPs with equalityandinequalityconstraints and 1000 decision variables. Experimental results show that CLDTEA can significantly improve the performance of these basic CMOEAs.

Two-scale minimum compliance optimization of spatially graded plates with variable-height parallelogram microstructures

This work investigates the two-scale design optimization of graded plates with variable-height parallelogram microstructures based on asymptotic homogenization. First, updated unit cell problems, with an emphasis on rotational degrees of freedom, are proposed to evaluate shell-structured microstructures, so that a pointwise effective stiffness calculation can be efficiently conducted with shell elements, which is otherwise computationally prohibitive with solid elements. Then, a two-scale optimization framework seeking minimum compliance is established, where the design variables controlling both in-plane parallelogram shape and out-of-plane stiffener height are introduced to enlarge the design space. In addition, mapping functions are integrated into the optimization formulation and concurrently optimized, enabling the production of well-defined dehomogenized plates. Finally, numerical examples are presented to validate the effectiveness and correctness of the proposed method by comparing the deflections of the graded plates with those of the homogenized plates.

Buckling optimization of additively manufactured cellular structures using numerical homogenization based on beam models

Interest in components with detailed structures increased with the progress in advanced manufacturing techniques. Parts with lattice elements can provide improved global buckling stability compared to solid structures of the same weight. However, thin features are prone to local buckling. We present a two-scale optimization approach that simultaneously improves the local and global stability of parametrized graded lattice structures. Elastic properties and local buckling behavior are upscaled via homogenization based on geometric exact beam theory. To reduce computational effort, we construct a worst-case model for the homogenized buckling load factor, which acts as a safeguard against local buckling. We briefly discuss advantages and limitations by means of numerical examples.

A sequential global programming approach for two-scale optimization of homogenized multiphysics problems with application to Biot porous media

We present a new approach and an algorithm for solving two-scale material optimization problems to optimize the behaviour of a fluid-saturated porous medium in a given domain. While the state problem is governed by the Biot model describing the fluid–structure interaction in homogenized poroelastic structures, the approach is widely applicable to multiphysics problems involving several macroscopic fields in which homogenization provides the relationship between the microconfigurations and the macroscopic mathematical model. The optimization variables describe the local microstructure design by virtue of the pore shape which determines the effective medium properties, namely the material coefficients, computed by the homogenization method. The numerical optimization strategy involves (a) precomputing a database of the material coefficients associated with the geometric parameters and (b) applying the sequential global programming (SGP) method for solving the problem of macroscopically optimized distribution of material coefficients. Although there are similarities to the free material optimization (FMO) approach, only effective material coefficients are considered admissible, for which a well-defined set of corresponding configurable microstructures exists. Due to the flexibility of the SGP approach, different types of microstructures with fully independent parametrizations can easily be handled. The efficiency of the concept is demonstrated by a series of numerical experiments that show that the SGP method can simultaneously handle multiple types of microstructures with nontrivial parametrizations using a considerably low and stable number of state problems to be solved.

Two-scale optimization of graded lattice structures respecting buckling on micro- and macroscale

Interest in components with detailed structures increased with the progress in advanced manufacturing techniques in recent years. Parts with graded lattice elements can provide interesting mechanical, thermal, and acoustic properties compared to parts where only coarse features are included. One of these improvements is better global buckling resistance of the component. However, thin features are prone to local buckling. Normally, analyses with high-computational effort are conducted on high-resolution finite element meshes to optimize parts with good global and local stability. Until recently, works focused only on either global or local buckling behavior. We use two-scale optimization based on asymptotic homogenization of elastic properties and local buckling behavior to reduce the effort of full-scale analyses. For this, we present an approach for concurrent local and global buckling optimization of parameterized graded lattice structures. It is based on a worst-case model for the homogenized buckling load factor, which acts as a safeguard against pure local buckling. Cross-modes residing on both scales are not detected. We support our theory with numerical examples and validations on dehomogenized designs, which show the capabilities of our method, and discuss the advantages and limitations of the worst-case model.

Additive manufacturing of cellular structures: Multiscale simulation and optimization

This paper introduces a multiscale and multi-purpose simulation framework to investigate selective beam melting processes for metallic cellular structures along the complete process chain, up to the topology optimization of components made of cellular materials. Process simulation methods on various length scales are introduced to analyze the relation of process strategies and resulting properties of the cellular materials, as e. g. warpage, residual stresses, material grain structure and surface roughness. Numerical homogenization methods are applied to investigate the influence of the grain structure and the topology of the cellular materials on their mechanical properties. Based on these process-dependent mechanical properties, a two-scale optimization of components made of cellular material is performed, aiming in particular on optimizing the buckling resistance of the structures. The interfaces between the different simulation tools are specified and illustrate the flexibility of the numerical framework. It is shown that consistent simulations of additive manufacturing of cellular materials – from the manufacturing process to the optimized component – provide important insights into process–structure interactions and enable tailored additive manufacturing processes.

Two-scale optimal management of urban runoff by linking LIDs and landscape configuration

Landscape composition and configuration of urban patches impact greatly on hydrological process, of which low impact development practices (LIDs) have been incorporated worldwide to management stormwater runoff quantity- and quality-related issues. Although optimization design for LIDs has been studied extensively to improve stormwater management at catchment scale, a further study of LIDs-based landscape configuration at site scale has not been fully explored. In this study, a two-scale optimization framework was constructed to integrate spatial layout and landscape configuration of LIDs using SWMM, NSGA-II and GIS techniques. The spatial optimization at catchment scale showed that area of sink landscape almost doubled as the distance to outlets decreased. A further site-scale optimization by a new landscape metric, in terms of CRSS (configuration ratio between patches of runoff sources and sink), was conduct to achieve LIDs-based landscape configuration optimization. Runoff flow and pollution were significantly correlated with CRSS in power function, quantifying CRSS thresholds of 0.57–0.69 as the most adaptive strategies for stormwater management. Notably, the CRSS thresholds were higher for ammonia nitrogen (NH3-N) and suspended solid (SS) pollution, especially targeting source patches of roofs. The optimization in this study could improve the reduction of runoff peak flow of heavy rain, peak concentration of chemical oxygen demand and SS, and initial scouring of NH3-N.

Two-Scale Optimization and Generation of Anisotropic Cellular Designs in the Context of Additive Manufacturing

Within the context of gradient-based optimization of cellular structures, we aim to optimize designs in a two-scale setting and generate respective geometric descriptions that are additively manufacturable in powder-bed systems. We analyze representative unit cells for their suitability and present a workflow to generate cellular designs by means of a parameterized unit cell, for which an implicit surface description is described. After performing homogenization-based optimization, we compare the predicted performance of the homogenized model to a full-resolution finite element (FE) analysis. Throughout the whole process, special attention is paid to the consistent integration of manufacturing limitations in terms of horizontal overhang and feature size.We discuss the performance of cellular components with maximized stiffness. Our numerical experiment for a single-load scenario demonstrates that using isotropic micro-structures is not beneficial, even when graded. We further observe that using anisotropic micro-structures leads to improvements only in very specific cases, e.g. with a uniformly distributed load. However, the conclusion may be different for anisotropic cells with non-uniform parameterization and other loading scenarios. Instead of conducting further tedious investigations for stiffness maximization, we finally demonstrate the benefit of cellular designs for a thermo-mechanical example.

Concurrent topology optimization design of stiffener layout and cross-section for thin-walled structures

Stiffened plates or shells are widely used in engineering structures as primary or secondary load-bearing components. How to design the layout and sizes of the stiffeners is of great significance for structural lightweight. In this work, a new topology optimization method for simultaneously optimizing the layout and cross-section topology of the stiffeners is developed to solve this issue. The stiffeners and base plates are modeled by the beam and shell elements, respectively, significantly reducing the computational cost. The Giavotto beam theory, instead of the widely employed Euler or Timoshenko beam theory, is applied to model the stiffeners for considering the warping deformation in evaluating the section stiffness of the beam. A multi-scale topology optimization model is established by simultaneously optimizing the layout of the beam and the topology of the cross-section. The design space is significantly expanded by optimizing these two types of design variables. Several numerical examples are applied to illustrate the validity and effectiveness of the proposed method. The results show that the proposed two-scale optimization approach can generate better designs than the single-scale method.

Size-dependent two-scale topological design for maximizing structural fundamental eigenfrequency

A two-scale concurrent topology optimization method based on the couple stress theory is proposed for maximizing structural fundamental eigenfrequency. Because of the fact that the classical mechanics theory cannot reveal the size effect because of neglecting the influence of microstructure, the theory of couple stress including the microscopic properties of materials can be used to describe the size effect in deformations. On the foundation of the couple stress theory, the two-scale optimization model for finding optimal configurations of macrostructures and their periodic composite material microstructures is built. And the fundamental eigenfrequency of the macrostructure is maximized. The effective macroscopic couple stress constitutive constants of macrostructures are calculated by the representative volume element method. And a modified solid isotropic material with a penalization model is used to effectively avoid the localized mode. The optimization algorithm based on the bidirectional evolutionary structural optimization method is proposed. The optimal results of numerical examples show that the optimal topologies and natural frequencies obtained by the couple stress theory may differ significantly from those obtained by the typical Cauchy theory. It is obvious that couple stress theory can effectively describe the size effect in topology optimization.

A low-parametric rhombic microstructure family for irregular lattices

New fabrication technologies have significantly decreased the cost of fabrication of shapes with highly complex geometric structure. One important application of complex fine-scale geometric structures is to create variable effective elastic material properties in shapes manufactured from a single material. Modification of material properties has a variety of uses, from aerospace applications to soft robotics and prosthetic devices. Due to its scalability and effectiveness, an increasingly common approach to creating spatially varying materials is to partition a shape into cells and use a parametric family of small-scale geometric structures with known effective properties to fill the cells. We propose a new approach to solving this problem for extruded, planar microstructures. Differently from existing methods for two-scale optimization based on regular grids with square periodic cells, which cannot conform to an arbitrary boundary, we introduce cell decompositions consisting of (nearly) rhombic cells. These meshes have far greater flexibility than those with square cells in terms of approximating arbitrary shapes, and, at the same time, have a number of properties simplifying small-scale structure construction. Our main contributions include a new family of 2D cell geometry structures, explicitly parameterized by their effective Young's moduli E , Poisson's ratios v , and rhombic angle α with the geometry parameters expressed directly as smooth spline functions of E, v , and α. This family leads to smooth transitions between the tiles and can handle a broad range of rhombic cell shapes. We introduce a complete material design pipeline based on this microstructure family, composed of an algorithm to generate rhombic tessellation from quadrilateral meshes and an algorithm to synthesize the microstructure geometry. We fabricated a number of models and experimentally demonstrated how our method, in combination with material optimization, can be used to achieve the desired deformation behavior.

Multiscale topology optimization for non-uniform microstructures with hybrid cellular automata

Based on hybrid cellular automata (HCA), we present a two-scale optimization model for heterogeneous structures with non-uniform porous cells at the microscopic scale. The method uses the K-means clustering algorithm to achieve locally nonperiodicity through easily obtained elemental strain energy. This energy is used again for a two-scale topological optimization procedure without sensitivity analysis, avoiding drastically the computational complexity. Both the experimental tests and numerical results illustrate a significant increase in the resulting structural stiffness with locally nonperiodicity, as compared to using uniform periodic cells. The effects of parameters such as clustering number and adopted method versus classical Optimality Criteria (OC) are discussed. Finally, the proposed methodology is extended to 3D two-scale heterogeneous structure design.

Two-scale concurrent topology optimization with multiple micro materials based on principal stress orientation

This paper studies two-scale concurrent topology optimization with multiple micro heterogeneous materials subjected to volume constraints. In previous work on concurrent two-scale optimization, either only one material with optimal microstructure is assumed or multiple micro materials are included but are distributed in prescribed geometrical domains. Here the selection of micro heterogeneous materials is based on the criterion for principal stress orientation in the macro structure. To meet this requirement, an additional constraint, called misplaced material volume constraint, is introduced to constrain the volume fraction of material that is misplaced in macro structure to be less than a small parameter e. This constraint comprises several piecewise smooth penalty functions, each of which is a proper modification of Heaviside function. One advantage of the misplaced material volume constraint is that, without much modification to the original formulation, the optimized macro material is distributed in line with the use criterion and the material microstructures automatically converge to different optimized topologies. Three numerical examples are presented to show the effectiveness of the proposed method.

A Hyperelastic Two-Scale Optimization Model for Shape Matching

We suggest a novel shape matching algorithm for three-dimensional surface meshes of disk or sphere topology. The method is based on the physical theory of nonlinear elasticity and can hence handle large rotations and deformations. Deformation boundary conditions that supplement the underlying equations are usually unknown. Given an initial guess, these are optimized such that the mechanical boundary forces that are responsible for the deformation are of a simple nature. We show a heuristic way to approximate the nonlinear optimization problem by a sequence of convex problems using finite elements. The deformation cost, i.e., the forces, is measured on a coarse scale, while ICP-like matching is done on the fine scale. We demonstrate the plausibility of our algorithm through examples taken from different datasets.

Structure-material integrated design by level sets

This paper proposes a structure-material integrated design method in the framework of level sets. A two-scale optimization is performed, where not only is the structural configuration optimized, but the effective properties of the constituent material are also designed by optimizing the configuration of the periodically-arranged microstructures. In this way, the approach can simultaneously generate optimized material distribution patterns as well as optimized materials. Due to the fact that the level set method produces material-void solutions only, the obtained material is uniformly distributed over the material regions of the structure in a strict sense. Three numerical examples are presented to validate the proposed method. The obtained optimal solutions illustrate that whether to design material cells or to optimize structural configurations to achieve the best structural performance would be problem dependent, and thus further demonstrate the necessity and validity of the integrated design scheme.

Concurrent multi-scale design optimization of composite frame structures using the Heaviside penalization of discrete material model

This paper deals with the concurrent multi-scale optimization design of frame structure composed of glass or carbon fiber reinforced polymer laminates. In the composite frame structure, the fiber winding angle at the micro-material scale and the geometrical parameter of components of the frame in the macro-structural scale are introduced as the independent variables on the two geometrical scales. Considering manufacturing requirements, discrete fiber winding angles are specified for the micro design variable. The improved Heaviside penalization discrete material optimization interpolation scheme has been applied to achieve the discrete optimization design of the fiber winding angle. An optimization model based on the minimum structural compliance and the specified fiber material volume constraint has been established. The sensitivity information about the two geometrical scales design variables are also deduced considering the characteristics of discrete fiber winding angles. The optimization results of the fiber winding angle or the macro structural topology on the two single geometrical scales, together with the concurrent two-scale optimization, is separately studied and compared in the paper. Numerical examples in the paper show that the concurrent multi-scale optimization can further explore the coupling effect between the macro-structure and micro-material of the composite to achieve an ultra-light design of the composite frame structure. The novel two geometrical scales optimization model provides a new opportunity for the design of composite structure in aerospace and other industries.

Microstructural topology optimization of viscoelastic materials for maximum modal loss factor of macrostructures

The geometric layout and physical properties of a viscoelastic damping material have a significant influence on the damping performance of a passive constrained layer damping (PCLD) structure. This paper presents a two-scale optimization method and aims to find the optimal microstructural configuration of the viscoelastic material (i.e., the optimal effective properties of the material) with maximum modal loss factors of the macrostructures. The modal loss factor is obtained by using the Modal Strain Energy (MSE) method. The material microstructure is assumed to be homogeneous in the macro-scale, i.e., the macrostructure is composed of periodic unit cells (PUC). In the optimization formulation, the relative densities are introduced as the design variables for the material microstructure design, based upon the idea of the Solid Isotropic Material with Penalization (SIMP) method of topology optimization. The modal loss factor of the structure is assigned as the objective function. All the sensitivities of the modal loss factor with respect to the design variables are derived analytically and the optimization problem is solved by Method of Moving Asymptote (MMA) 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 PCLD treatment.

Two-scale optimal design of structures with thermal insulation materials

This paper introduces a two-scale topology optimization approach by integrating optimized structures with the design of their materials. The optimization aims to find a multifunctional structure composed of homogeneous porous material. Driven by the multi-objective functions, macrostructural stiffness and material thermal conductivity, stiff but lightweight structures composed of thermal insulation materials can be achieved through optimizing the topologies of the macrostructures and their material microstructure simultaneously. For such a two-scale optimization problem, the effective properties of materials derived from the homogenization method are applied to the analysis of macrostructure. Meanwhile, the displacement field of the macrostructure under given boundary conditions is used for the sensitivity analysis of the material microstructure. Then, the bi-directional evolutionary structural optimization (BESO) method is employed to iteratively update the macrostructures and material microstructures by ranking elemental sensitivity numbers at the both scales. Finally, some 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm. A variety of optimal macrostructures and their optimal material microstructures are obtained.

Topology optimization of microstructures of cellular materials and composites for macrostructures

This paper introduces a topology optimization algorithm for the optimal design of cellular materials and composites with periodic microstructures so that the resulting macrostructure has the maximum stiffness (or minimum mean compliance). The effective properties of the heterogeneous material are obtained through the homogenization theory, and these properties are integrated into the analysis of the macrostructure. The sensitivity analysis for the material unit cell is established for such a two-scale optimization problem. Then, a bi-directional evolutionary structural optimization (BESO) approach is developed to achieve a clear and optimized topology for the material microstructure. Several numerical examples are presented to validate the proposed optimization algorithm and a variety of anisotropic microstructures of cellular materials and composites are obtained. The various effects on the topological design of the material microstructure are discussed.

Efficient Two-Scale Optimization of Manufacturable Graded Structures

A new method for the optimal design of inhomogeneous meso-structured components is presented. The method is based on a variant of the free material optimization (FMO) method. Instead of coupling a macroscopic model directly with the meso-scale in the framework of a two-scale approach, information from the meso-scale is incorporated into the FMO model by additional constraints. The FMO result, given in terms of optimized tensors, is interpreted by the inverse homogenization approach. Rather than only computing the meso-structure for each “FMO cell” independently, approximate continuity of the meso-structure is enforced. Thus, the final result is a manufacturable structurally graded material. The feasibility of the whole method is demonstrated by means of an academic example.