Length Scale Constraints Research Articles

Optimal design of electro-thermo-mechanical microactuators considering minimum length scale constraints

To ensure the manufacturability of compliant mechanisms, this paper presents a new approach for optimal design of electro-thermo-mechanical microactuators by employing minimum length scale constraints. A sequential coupling method is adopted to carry out finite element analysis for electric-thermal-mechanical coupling multiphysics. The inflection point fields of the solid and void phases are identified by structural indicator functions. The minimum length scales of the solid and void phases are adopted as constraints. The optimization objective is designed to maximize the output displacements of electro-thermo-mechanical microactuators. The validity of the proposed optimal design method is demonstrated through several numerical examples. In the optimal designs obtained by the proposed method, the minimum length scales of the two phases can be properly controlled. The electro-thermo-mechanical microactuator’s output displacements decrease when the allowable value of the minimum length scale is increased. The effects of different mesh discretizations and output spring stiffnesses on the optimized designs are discussed.

Interplay between entanglement and crosslinking in determining mechanical behaviors of polymer networks

ABSTRACT In polymer physics, the concept of entanglement refers to the topological constraints between long polymer chains that are closely packed together. Both theory and experimentation suggest that entanglement has a significant influence on the mechanical properties of polymers. This indicates its promise for materials design across various applications. However, understanding the relationship between entanglement and mechanical properties is complex, especially due to challenges related to length scale constraints and the difficulties of direct experimental observation. This research delves into how the polymer network structure changes when deformed. We specifically examine the relationship between entanglement, crosslinked networks, and their roles in stretching both entangled and unentangled polymer systems. For unentangled polymers, our findings underscore the pivotal role of crosslinking bond strength in determining the system’s overall strength and resistance to deformation. As for entangled polymers, entanglement plays a pivotal role in load bearing during the initial stretching stage, preserving the integrity of the polymer network. As the stretching continues and entanglement diminishes, the responsibility for bearing the load increasingly shifts to the crosslinking network, signifying a critical change in the system’s behavior. We noted a linear correlation between the increase in entanglement and the rise in tensile stress during the initial stretching stage. Conversely, the destruction of the network correlates with a decrease in tensile stress in the later stage. The findings provide vital insights into the complex dynamics between entanglement and crosslinking in the stretching processes of polymer networks, offering valuable guidance for future manipulation and design of polymer materials to achieve desired mechanical properties.

Reaction–diffusion equation driven topology optimization of high-resolution and feature-rich structures using unstructured meshes

Motivated by the need for porous structures in the design of biodegradable implants, as well as for the diverse and competitive designs in architecture, this paper builds upon the recent advances in the single-scale (macroscopic) topology optimization (TO) approach and proposes a level-set method (LSM) for the design of lattice structures. The key idea is to introduce the maximum length-scale constraint, realized by a PDE filter, into the reaction–diffusion equation (RDE)-based LSM which can end up with feature-rich shape, starting from scratch. Then, the proposed TO technique is coupled with distributed computing algorithms based on PETSc, interfaced in the open-source software FreeFEM using unstructured meshes, to deliver both two- and three-dimensional designs with complex-shape geometry. The parallel efficiency is validated by solving large-scale 3D benchmarks with 20 million tetrahedral elements on a cloud-based cluster. Furthermore, we integrate local mesh refinement (h-adaptation scheme) into this workflow, allowing one to reach high-resolution and feature-rich 3D designs on a desktop, paving the way to future investigations, aimed at including the maximum length-scale constraint in large-scale multiphysics topology optimization at a reasonable cost.

An explicit formulation for minimum length scale control in density-based topology optimization

Topology optimization is widely applied in various engineering areas for obtaining innovative designs. However, the optimized results often contain many small features that cannot satisfy the manufacturable requirements. It is important to develop an effective minimum length scale control method which can be easily implemented and control the length scale accurately in topology optimization. In this work, an explicit and general mathematical formulation of the minimum length scale constraint in the density-based topology optimization framework is proposed. Compared to the existing implicit method, such as robust formulation, the proposed method can give accurate length-scale control for arbitrary problems. In addition, the proposed formulation has remarkable advantages in terms of implementation simplicity and parameter insensitivity. Only computing the average density of elements in a small circular region is needed, similar to the typical filtering technique. Aggregation functions are used to gather all the local constraints into a single constraint, and the sensitivity analysis of the constraint function is derived. Some representative numerical examples are presented to verify the effectiveness of the proposed algorithm.

Incorporating additive manufacturing constraints into magneto-structural topology optimization

Abstract The sequential using of topology optimization and AM (additive manufacturing) can generate and fabricate superior-performance yet lightweight magneto-structural components. In this paper, a magneto-structural topology optimization method is proposed considering AM constraints to improve the design manufacturability of optimized designs. The design problem is formulated with the weighted combination of magnetic compliance and mechanical compliance as a single objective subjected to two types of manufacturing constraints and the volume fraction constraint. Two important AM constraints are in a sequential manner incorporated into the topology optimization model, in which the overhang angle constraint is followed by the maximum length scale constraint. Corresponding sensitivities of the objective and constraints are derived, and the optimization problem is solved by the method of moving asymptotes. Two numerical examples and a practical conceptual design of linear motor rotor structure are systematically investigated to demonstrate the effectiveness of the proposed method.

Inverse Design of Photonic Devices with Strict Foundry Fabrication Constraints

We introduce a new method for inverse design of nanophotonic devices which guarantees that resulting designs satisfy strict length scale constraints - including minimum width and spacing constraints required by commercial semiconductor foundries. The method adopts several concepts from machine learning to transform the problem of topology optimization with strict length scale constraints to an unconstrained stochastic gradient optimization problem. Specifically, we introduce a conditional generator for feasible designs and adopt a straight-through estimator for backpropagation of gradients to a latent design. We demonstrate the performance and reliability of our method by designing several common integrated photonic components.

Topology Optimization of High Resolution Lattice Structures Using a PDE-filter

Lattice-like structures can be observed in many natural systems and industrial applications. Motivated by the need of such porous structure in the design engineering, the present work uses a single scale approach and proposes a level set-based topology optimization method for the lattice designing. The key idea is to introduce a maximum length-scale constraint, realized by a PDE-filter, into the reaction-diffusion equation based LSM. The proposed TO technique can be applied in the unstructured and/or adaptive mesh-based framework, and it can be coupled with the distributed computing technique to achieve high resolution designs. A numerical test case is presented to demonstrate the proposed methodology.

Evidence theory-based reliability optimization for cross-scale topological structures with global stress, local displacement, and micro-manufacturing constraints

An uncertainty-oriented cross-scale topology optimization model with global stress reliability constraint, local displacement constraint, and micro-manufacturing control based on evidence theory is presented. The model is oriented to two-dimensional porous material structure, which concurrently designs the material distribution of both the macrostructure and the cell microstructure. During the optimization process, the homogenization method is used to solve the equivalent elastic modulus of the cell microstructure, which is then endowed to the macro-elements for subsequent analysis. The local stress constraints are converted to a global constraint by P-norm to reduce the computational consumption. Considering the uncertainty factors, the evidence theory is utilized to process the uncertainty parameters and evaluate the reliability of the structural strength performance. Minimum length-scale constraint is imposed on the cell microstructure by a density projection method for better manufacturability. Three numerical examples are presented to illustrate the availability of the proposed model.

Two-scale topology optimisation of cellular materials under mixed boundary conditions

This work proposes a theoretical/numerical framework for the topology optimisation of anisotropic architected cellular materials at different scales. In particular, the topological variable (i.e., the pseudo-density field) is defined at both the scale of the representative volume element (i.e., the unit cell) of the material and at the macroscopic scale of the structure. The two-scale topology optimisation problem is formulated in the most general sense, i.e., by considering non-zero Neumann-Dirichlet boundary conditions. The proposed method is based on: (a) non-uniform rational basis spline hyper-surfaces to represent the topological variable at each scale, (b) the solid isotropic material with penalisation approach, (c) a general numerical homogenisation scheme based on the strain energy to establish the link between scales. The proposed formulation exploits the properties of non-uniform rational basis spline entities to determine the relationships occurring among the topological variables defined at different scales to correctly state the optimisation problem and to satisfy the hypotheses at the basis of the homogenisation method. In particular, scale separation (a necessary condition to be met in order to apply the homogenisation method) and manufacturing requirements are implicitly ensured by introducing minimum length scale constraints on the topological variables defined at both macroscopic scale and unit cell scale, respectively. Furthermore, the sensitivity of the optimised topology (at each scale) to the applied boundary conditions and to the elastic symmetry group of the representative volume element is investigated by founding new and original results. The effectiveness of the approach is tested on 2D and 3D benchmark problems taken from the literature.

Length scale control schemes for bi‐directional evolutionary structural optimization method

AbstractThis work develops length scale control schemes for bi‐directional evolutionary structural optimization method, enabling an enhanced and flexible control of the method on structural member sizes. Specifically, the maximum length scale control is achieved by constraining local material volumes below a threshold value, which is determined upon the allowed maximum length. The massive per‐element volume constraints are aggregated by the p‐norm global measure. The aggregated volume constraint is augmented to the conventional compliance design objective through a Lagrange multiplier. On the other hand, the minimum length scale control is achieved by a post‐processing modification of local feature according to the skeleton detected from a preliminarily optimized topology. The sensitivity numbers of the local structural features that violate the minimum length scale constraint are compensated during the post‐processing modification. Both 2D and 3D benchmark design results show that the proposed schemes are effective and efficient in controlling both the maximum and minimum structural length scales.

Topology optimization of arbitrary-shape multi-phase structure with structured meshes based on a virtual phase method

In the community of topology optimization, the contradiction between the frontier research with structured meshes and the complex geometry in the real-world problems has existed for a long time. Structured mesh is attractive in convenient application of loads and boundary conditions, and easy implementation of GPU parallel calculation. Therefore, unlike existing researches solving the issue through unstructured elements, this paper presents an easy-to-accomplish virtual phase method (VPM) based on structured meshes, to systematically solve the problems with complex geometries and arbitrary non-design domains. Structural geometry is denoted by the extended regular design domain and void region initially placed at the border of the extended domain. Based on the extended idea of composite material interpolation in multi-phase density-based topology optimization approaches, the extended structural design domains and the pre-defined non-design domains are considered as the optimized phases in the structure, and they are defined by discrete variable fields. Subsequently, the structural optimization is implemented like as the traditional multi-material topology optimization process. Numerical examples accounting for minimum compliance are presented to demonstrate the effectiveness of the proposed method, including multi-phase macrostructural design, the ease with which the method can accommodate passive regions, and how the length scale constraints are used to devise graded porous infill morphology within a given shell. At last, the manufacturability of the optimized designs is discussed to further verify the applicability of the proposed method. Our approach provides those front researches based on structured meshes with an easily accessible access, to be straightforwardly applied for any complex-domain design problem.

Uncertainty‐oriented double‐scale topology optimization with macroreliability limitation and micromanufacturing control

AbstractThis article proposes an uncertainty‐oriented double‐scale topology optimization method considering macroreliability limitation and micromanufacturing control. The procedure combines the homogenization method to solve the equivalent elastic property of the microstructure, the feature distance theory to evaluate reliability, a density projection for manufacturability control, and the solid isotropic material with penalization model to suppress intermediate density. To coupling the micro–macro scale, the equivalent elastic property of the cell microstructure is evaluated by the numerical homogenization method and then endowed to the macroelement for finite element analysis. Uncertainty factors existed in optimization parameters are evaluated by the interval convex set model. By utilizing the interval parameter vertex method and the feature distance theory, the reliability of displacement constraint in the optimization model is evaluated and constrained. In terms of length scale control, the microdensity design variables are projected by a threshold function to obtain a clear microstructure that satisfies the preset minimum length scale constraint. Finally, three numerical examples are presented to illustrate the effect of micromanufacturing control, as well as the necessity of considering parameter uncertainty.

Design Applicable 3D Microfluidic Functional Units Using 2D Topology Optimization with Length Scale Constraints.

Due to the limits of computational time and computer memory, topology optimization problems involving fluidic flow frequently use simplified 2D models. Extruded versions of the 2D optimized results typically comprise the 3D designs to be fabricated. In practice, the depth of the fabricated flow channels is finite; the limited flow depth together with the no-slip condition potentially make the fluidic performance of the 3D model very different from that of the simplified 2D model. This discrepancy significantly limits the usefulness of performing topology optimization involving fluidic flow in 2D—at least if special care is not taken. Inspired by the electric circuit analogy method, we limit the widths of the microchannels in the 2D optimization process. To reduce the difference of fluidic performance between the 2D model and its 3D counterpart, we propose an applicable 2D optimization model, and ensure the manufacturability of the obtained layout, combinations of several morphology-mimicking filters impose maximum or minimum length scales on the solid phase or the fluidic phase. Two typical Lab-on-chip functional units, Tesla valve and fluidic channel splitter, are used to illustrate the validity of the proposed application of length scale control.

An evolutionary design approach to shell-infill structures

Shell-infill structures with internal porous architecture are increasingly used in additive manufacturing because of their advantages such as material saving, printing efficiency, and thermal stress release. The study is built upon the bi-directional evolutionary structural optimization (BESO) method. An evolutionary design approach to shell-infill structures using flexible control of the infill architecture is proposed in this study. Specifically, the general structural configuration with a coating shell is first designed by assuming that the infill consists of a weak material. The delicate fine infill architecture is subsequently designed by extending BESO to consider the maximum length scale constraints of the infill architecture. Shell-infill designs of variant functionally graded infill architectures are additively manufactured using stereolithography and their mechanical performance are tested. Numerical and experimental results indicate that the proposed method is more effective in providing shell-infill designs with optimal stiffness and robust performance compared to conventional designs with uniform lattice infill.

On topology optimization with elliptical masks and honeycomb tessellation with explicit length scale constraints

Topology optimization using gradient search with negative and positive elliptical masks and honeycomb tessellation is presented. Through a novel skeletonization algorithm for topologies defined using filled and void hexagonal cells/elements, explicit minimum and maximum length scales are imposed on solid states in the solutions. An analytical example is presented suggesting that for a skeletonized topology, optimal solutions may not always exist for any specified volume fraction, minimum and maximum length scales, and that there may exist implicit interdependence between them. A sequence for length scale (SLS) methodology is proposed wherein solutions are sought by specifying only the minimum and maximum length scales with volume fraction getting determined within a specified range systematically. Through four benchmark problems in small deformation topology optimization, it is demonstrated that solutions by-and-large satisfy the length scale constraints though the latter may get violated at certain local sites. The proposed approach seems promising, noting especially that solutions, if rendered perfectly black and white with minimum length scale explicitly imposed and boundaries smoothened, are quite close in performance compared with the parent topologies. Attaining volume-distributed topologies, wherein members are more or less of the same thickness, may also be possible with the proposed approach.

Level set topology optimization of cooling channels using the Darcy flow model

The level set topology optimization method for 2D and 3D cooling channels, considering convective heat transfer for high Reynolds number flows, is presented in this paper. The Darcy potential flow, which is a low-fidelity linear flow model, is used to simulate the flow using the finite element method. The resulting velocity field is used in a convection-diffusion model to simulate the heat transfer using the finite element method. A linear combination of the pressure drop and the average temperature is considered as the objective function, which is minimized subject to a volume constraint and a maximum length scale constraint. The results show that the pressure drop and the average temperature are conflicting criteria, and the trade-off between the two criteria is investigated. We perform a verification study by comparing the Darcy flow field of the obtained optimum designs with that of a high fidelity turbulence model. The verification study shows that there exists a reasonable agreement between the Darcy and the turbulent flow field for narrow channels. Therefore, by restricting the design space to narrow channels, we optimize the cooling performance and sufficiently capture the turbulent flow physics using the low-fidelity Darcy flow model. Finally, we show an example in 3D where we optimize the cooling channel topology that conforms to the surface of a sphere.

Piecewise length scale control for topology optimization with an irregular design domain

This paper presents a piecewise length scale control method for level set topology optimization. Different from the existing methods, where a unique lower limit or upper limit was applied to the entire design domain, this new method decomposes the topological design into pieces of strip-like components based on the connectivity condition, and then, the lower or upper limit for length scale control could be piecewise and dynamically defined based on each component’s real-time status (such as position, orientation, or dimension). Specifically, a sub-algorithm of structural skeleton identification and segmentation is developed to decompose the structure and its skeleton. Then, a skeleton segment-based length scale control method is developed to achieve the piecewise length scale control effect. In addition, a special type of length scale constrained topology optimization problem that involves an irregular design domain will be addressed, wherein the complex design domain plus the length scale constraint may make the conventional length scale control methods fail to work. Effectiveness of the proposed method will be proved through a few numerical examples.

Minimum length scale constraints in multi-scale topology optimisation for additive manufacturing

ABSTRACTThis paper performs a combined numerical and experimental study to explore the role of minimum length scale constraints in multi-scale topology optimisation. Multi-scale topology optimisation is generally performed without considering the actual unit cell size, while an arbitrary value considerably smaller than the part is selected afterwards. However, this procedure would be problematic if including geometric constraints, e.g. minimum length scale constraints, since geometric constraints cannot be applied without knowing the unit cell dimensions. To address this issue, unit cell size should be defined beforehand, and guidelines will be provided in this work through a thorough numerical exploration, i.e. compliance minimisation multi-scale topology optimisation with different unit cell sizes and a consistent minimum length scale limit will be performed. The numerical results indicate that selecting the unit cell size considerably smaller than the part and larger than the length scale limit would be recommended. Then, experiments are conducted to explore the effect of minimum length scale limit on the stiffness and strength of the multi-scale design. It is observed that increasing the minimum length scale limit would reduce the structural mechanical performance in both aspects.

Maximum length scale requirement in a topology optimisation method based on NURBS hyper-surfaces

This paper deals with a new method for handling manufacturing and geometrical requirements in the framework of a general Topology Optimisation (TO) strategy. In particular, the maximum length scale constraint (MLSC) implementation is addressed in order to obtain multiple load paths or to locally limit the size of the component. The classic formulation of the MLSC is revisited in the framework of a density-based TO algorithm wherein the pseudo-density field is represented through a NURBS hyper-surface. The NURBS hyper-surface properties are exploited to effectively formulate the MLSC. The effectiveness of the proposed approach is proven on a meaningful 3D benchmark.

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.