Optimal shape design as a material distribution problem

Unbiased mechanical cloaks

The distinction between "reinforcement" and "cloaking" has been overlooked in optimization-based design of devices intended to conceal a defect in an elastic medium. In the former, a so-called "cloak" is severely biased toward one or a few specific elastic disturbances, whereas in the latter, an "unbiased cloak" is effective under any elastic disturbance. We propose a two-stage approach for optimization-based design of elastostatic cloaks that targets true, unbiased cloaks. First, we perform load-case optimization to find a finite set of worst-case design loads. Then we perform topology optimization of the cloak microstructure under these worst-case loads using a judicious choice of the objective function, formulated in terms of energy mismatch. Although a small subset of the infinite load cases that the cloak must handle, these highly nonintuitive, worst-case loads lead to designs that approach perfect and unbiased elastostatic cloaking. In demonstration, we consider elastic media composed of spinodal architected materials, which provides an ideal testbed for exploring elastostatic cloaks in media with varying anisotropy and porosity, without sacrificing manufacturability. To numerically verify the universal nature of our cloaks, we compare the elastic response of the medium containing the cloaked defect to that of the undisturbed medium under many random load cases not considered during design. By using digital light processing additive manufacturing to realize the elastic media containing cloaked defects and analyzing their response experimentally using compression testing with digital image correlation, this study provides a physical demonstration of elastostatic cloaking of a three-dimensional defect in a three-dimensional medium.

Design of the multi-material structure using an MMC-SIMP sequential topology optimization method

The present paper introduces a sequential topology optimization method for multi-material structure design. In the proposed method, the moving morphable component (MMC) method and the solid isotropic material with penalization (SIMP) method are performed in sequence to achieve multi-material structure optimization. First, the structural topology is obtained using the MMC method, and then the material layout of the determined structural topology is optimized using the SIMP method. The connection between MMC method and SIMP method is established through components mapping elements, where changes in the material layout within these elements do not affect the external structural topology. The proposed method effectively combines the advantages of two optimization methods, enabling the achievement of an explicit external structural topology and greater freedom in internal multi-material layouts, while remaining easy to implement. The key MATLAB codes for the proposed method are provided in the paper. Finally, the feasibility of the proposed method is verified through some numerical examples.

Inverse design of 3D nanophotonic devices with structural integrity using auxiliary thermal solvers

3D additive manufacturing enables the fabrication of nanophotonic structures with subwavelength features that control light across macroscopic scales. Gradient-based optimization offers an efficient approach to design these complex and non-intuitive structures. However, expanding this methodology from 2D to 3D introduces complexities, such as the need for structural integrity and connectivity. This work introduces a multi-objective optimization method to address these challenges in 3D nanophotonic designs. Our method combines electromagnetic simulations with an auxiliary heat-diffusion solver to ensure continuous material and void connectivity. By modeling material regions as heat sources and boundaries as heat sinks, we optimize the structure to minimize the total temperature, thereby penalizing disconnected regions that cannot dissipate thermal loads. Alongside the optical response, this heat metric becomes part of our objective function. We demonstrate the utility of our algorithm by designing two 3D nanophotonic devices. The first is a focusing element. The second is a waveguide junction, which connects two incoming waveguides for two different wavelengths into two outgoing waveguides, which are rotated by 90° to the incoming waveguides. Our approach offers a design pipeline that generates digital blueprints for fabricable nanophotonic materials, paving the way for practical 3D nanoprinting applications.

Two-Phase Approach for Fast Topology Optimization of Multi-Resonant MEMS Involving Model Order Reduction

In this work, we propose a two-phase approach for a fast topology optimization of multi-resonant MEMSs. The approach minimizes the computation effort required to achieve an optimal design. In the first step, we perform a pre-optimization using bi-directional evolutionary structural optimization (BESO). We found in previous research that BESO can achieve optimal MEMS designs in a significantly lower number of iterations when compared to classical density-based methods. However, we encountered convergence issues with BESO towards the end of the optimization. Therefore, we introduced a second, density-based optimization phase to circumvent this issue. Finally, we introduced model order reduction to reduce the optimization time further. The novel approach is benchmarked with the design task of two common multi-resonant MEMS devices: a linear gyroscope and a micromirror. We show that the two-phase approach can achieve an optimal design within 200 iterations. With the addition of MOR, the computation of the goal function can be further reduced by 50% in our examples.

Inverse Design of Unitary Transmission Matrices in Silicon Photonic Coupled Waveguide Arrays Using a Neural Adjoint Model

The development of low-loss reconfigurable integrated optical devices enables further research into technologies including photonic signal processing, analogue quantum computing, and optical neural networks. Here, we introduce digital patterning of coupled waveguide arrays as a platform capable of implementing unitary matrix operations. Determining the required device geometry for a specific optical output is computationally challenging and requires a robust and versatile inverse design protocol. In this work we present an approach using high speed neural network surrogate-based gradient optimization, capable of predicting patterns of refractive index perturbations based on switching of the ultralow loss chalcogenide phase change material, antimony triselinide (Sb2Se3). Results for a 3 × 3 silicon waveguide array are presented, demonstrating control of both amplitude and phase for each transmission matrix element. Network performance is studied using neural network optimization tools such as data set augmentation and supplementation with random noise, resulting in an average fidelity of 0.94 for unitary matrix targets. Our results show that coupled waveguide arrays with perturbation patterns offer new routes for achieving programmable unitary operators, or Hamiltonians for quantum simulators, with a reduced footprint compared to conventional interferometer-mesh technology.

Two stage multiobjective topology optimization method via SwinUnet with enhanced generalization

Topology optimization is a critical tool for modern structural design, yet existing methods often prioritize single objectives (e.g., compliance minimization) and suffer from prohibitive computational costs, especially in multi-objective scenarios. To address these limitations, this paper introduces a novel two-stage multi-objective topology optimization (MOTO) method that uniquely integrates data-driven learning with physics-informed refinement, and both stages are implemented within nearly identical network frameworks, ensuring simplicity and consistency in execution. Firstly, a MOTO mathematical model based on the constraint programming method that considers competing objectives of compliance, stress distribution, and material usage was constructed. Secondly, a novel neural network that incorporates shifted windows attention mechanism and lightweight modules was developed to enhance feature extraction while maintaining computational efficiency. Finally, the proposed model was trained in two stages: In Stage-1, utilizing adaptive input tensors, the network predicts near-optimal geometries across variable design domains (including rectangular and L-shaped configurations) and diverse boundary conditions in real time, requiring only 1,650 samples per condition. In Stage-2, the near-optimal structures from Stage-1 were physically optimized to achieve optimal performance. Experimental results demonstrate that the method's capability to generate high-accuracy, computationally efficient solutions with robust generalization capabilities. It effectively tackles challenges associated with multi-scale design domains and non-convex geometries, various and even untrained boundary conditions while significantly reducing data dependency, a critical advancement for data-driven topology optimization. The novel approach offers new insights for multi-objective structural design and promotes advancements in structural design practices.

The lightweight design of steel pistons for diesel engines based on thermo-mechanical characteristics

Traditional aluminum-silicon alloy pistons are gradually replaced by steel pistons, which has become the trend of future diesel engine development. However, the efficient design and broad application of steel pistons are limited by the higher density of steel. For this reason, a new lightweight design method for steel pistons in diesel engines was proposed in this paper. The progressive design of the cooling gallery cross-section and the topology optimization of the window region were carried out to meet the strength and deformation requirements. Ultimately, a lightweight steel piston (LSP) was obtained. The results of the study showed that the total mass of LSP could be reduced by 8.86% compared to the original steel piston (OSP). The thermal states of LSP are all superior to those of OSP. The highest temperature, maximum thermal stress, maximum longitudinal thermal deformation, and maximum transverse thermal deformation can be reduced by 0.96 °C, 3.03%, 11.65%, and 8.99%, respectively. LSP has a larger thermo-mechanical coupling stress manifestation compared to OSP. However, the maximum longitudinal and transverse thermo-mechanical coupling deformations can be obtained with a significant reduction of 8.33% and 42.19%, respectively.

Near-Isotropic, Extreme-Stiffness, Continuous 3D Mechanical Metamaterial Sequences Using Implicit Neural Representation

Mechanical metamaterials represent a distinct category of engineered materials characterized by their tailored density distributions to have unique properties. It is challenging to create continuous density distributions to design a smooth mechanical metamaterial sequence in which each metamaterial possesses stiffness close to the theoretical limit in all directions. This study proposes three near-isotropic, extreme-stiffness, and continuous 3D mechanical metamaterial sequences by combining topology optimization and data-driven design. Through innovative structural design, the sequences achieve over 98% of the Hashin-Shtrikman upper bounds in the most unfavorable direction. This performance spans a relative density range of 0.2-1, surpassing previous designs, which fall short at medium and higher densities. Moreover, the metamaterial sequence is innovatively represented by the implicit neural function; thus, it is resolution-free to exhibit continuously varying densities. Experimental validation demonstrates the manufacturability and high stiffness of the three sequences.

Multiscale Concurrent Topology Optimization and Mechanical Property Analysis of Sandwich Structures

Based on the basic theoretical framework of the Bi-directional Evolutionary Structural Optimization method (BESO) and the Solid Isotropic Material with Penalization method (SIMP), this paper presents a multiscale topology optimization method for concurrently optimizing the sandwich structure at the macro level and the core layer at the micro level. The types of optimizations are divided into macro and micro concurrent topology optimization (MM), macro and micro gradient concurrent topology optimization (MMG), and macro and micro layered gradient concurrent topology optimization (MMLG). In order to compare the multiscale optimization method with the traditional macroscopic optimization method, the sandwich simply supported beam is illustrated as a numerical example to demonstrate the functionalities and superiorities of the proposed method. Moreover, several samples are printed through micro-nano 3D printing technology, and then the static three-point bending experiments and the numerical simulations are carried out. The mechanical properties of the optimized structures in terms of deformation modes, load-bearing capacity, and energy absorption characteristics are compared and analyzed in detail. Finally, the multiscale optimization methods are extended to the design of 2D sandwich cantilever beams and 3D sandwich fully clamped beams.

Topology Optimization: A Review for Structural Designs Under Statics Problems

Topology optimization is a powerful structural design method that determines the optimal configuration by distributing materials efficiently within a given design domain while satisfying specified load, performance, and volume constraints. Unlike size and shape optimization, topology optimization is independent of the initial design, offering a broader design space. This paper provides a systematic review of topology optimization methods, covering two theoretical frameworks: linear elasticity and nonlinear theory. Specifically, the review focuses on sensitivity analysis, optimization criteria, and topology solution smoothing within the context of linear elasticity. In the context of nonlinear theory, the review primarily addresses nonlinear phenomena arising from stress constraints, geometric, material, and contact nonlinearities. The paper concludes by summarizing the current state of the field, identifying limitations in existing methods, and suggesting directions for future research.

Mechanobiological optimization of scaffolds for bone tissue engineering

Synthetic bone graft scaffolds aim to generate new bone tissue and alleviate the limitations of autografts and allografts. To meet that aim, it is essential to have a design approach able to generate scaffold architectures that will promote bone formation. Here, we present a topology-varying design optimization method, the "mixed-topology" approach, that generates new designs from a set of starting structures. This approach was used with objective functions focusing on improving the scaffold's local mechanical microenvironments to mechanobiologically promote bone formation within the scaffold and constraints to ensure manufacturability and achieve desired macroscale properties. The results demonstrate that this approach can successfully generate scaffold designs with improved microenvironments, taking into account different combinations of relevant stimuli and constraints.

Topological optimization of hip spacer reinforcement

The use of an antibiotic-enriched hip spacer represents the optimal treatment for periprosthetic joint infections (PJI). The addition of reinforcement significantly enhances its mechanical properties. Employing the explicit method enables accurate prediction of the mechanical behavior of both the spacer and its reinforcement. Topological optimization of the reinforcement emerges as the most effective strategy to prevent bone demineralization, enhance antibiotic diffusion, and improve spacer resistance. The objective of this study is to conduct topological optimization of a validated numerical model of a reinforced hip spacer and to select, from the obtained topologies, the one that best improves mechanical properties and prevents stress shielding while minimizing volume. The results indicate that an 8 mm thick titanium reinforcement, optimized to 70% of its original volume, proves to be the most effective choice.

Topology Optimization of Periodic Structures Subject to Self-Weight Loading Using a Heuristic Method

This paper deals with the actual and challenging process of the optimal design of topologies of periodic structures taking into account the design-dependent loads. The topology formulation used in this paper minimizes the compliance value of the structure and is subject to a total volume constraint while maintaining a periodic pattern and self-weight load. This combination represents a promising and original contribution to the field of ongoing research, although it is not yet widely recognized. This paper aims to fill this gap by presenting the first results of numerical optimization tests. The redistribution of material within a design domain is governed by the rules of Cellular Automata, a locally oriented optimization tool that can be applied to all types of structural optimization, including topology optimization. The technique has been demonstrated by numerical tests on two- and three-dimensional examples. The calculations were performed for different types of periodic schemes. The optimized structures did not show the checkerboard effect or the presence of residual gray elements in the final topologies. The strategy used in this paper ensures connectivity between periodic subdomains without imposing additional conditions on the algorithm.

Topology optimization of adaptive sandwich plates with magnetorheological core layer for improved vibration attenuation

In this study the optimum topology distribution of the magnetorheological elastomer (MRE) layer in an adaptive sandwich plate is investigated. The adaptive sandwich plate consists of an MR elastomer layer embedded between two thin elastic plates. A finite element model has been first formulated to derive the governing equations of motion. A design optimization methodology incorporating the developed finite element model has been subsequently developed to identify the optimum topology treatment of the MR layer to enhance the vibration control in wide-band frequency range. For this purpose, the dynamic compliance and density of each element are defined as the objective function and design variables in the optimization problem, respectively. The method of the solid isotropic material with penalization (SIMP), is extended for material properties interpolation leading to a new MRE-based penalization (MREP) model. Method of moving asymptotes (MMA) has been subsequently utilized to solve the optimization problem. The developed finite element model and design optimization method are first validated using benchmark problems. The proposed design optimization methodology is then effectively utilized to investigate the optimal topologies of the magnetorheological elastomer (MRE) core layer in MRE-based sandwich plates under various boundary and loading conditions. Results show the effectiveness of the proposed design optimization methodology for topology optimization of MRE-based sandwich panels to mitigate the vibration in wide range of frequencies.

Computationally efficient dominant load-based local bone microstructure reconstruction method using topology optimization

The bone microstructure of the human proximal femur is clinically crucial for diagnosing skeletal pathologies, such as osteoporosis and bone metastases. The topology optimization-based bone microstructure method obtains these bone microstructures by converting low-resolution (LR) images into high-resolution images. However, this method is inherently computationally inefficient as it requires numerous finite elements, iterative analyses, and parallel computations. Therefore, this study proposes a novel topology optimization-based localised bone microstructure reconstruction method using the dominant load, which highly affects the selected region of interest (ROI), for efficient resolution enhancement. The load dependency of selected ROIs is quantified with a load dependency score. Then, the localised finite element model is constructed based on the local load estimation. Finally, the selected dominant load is applied as an input for the topology optimization-based bone microstructure reconstruction method. The reconstructed bone microstructure was similar to that of the conventional method. The localised finite element model applied by the dominant load effectively and accurately reconstructed the bone morphology and exhibited high computational efficiency. In conclusion, the dominant load-based approach can be used to construct a reasonable trabecular bone structure for ROI with high computational efficiency. The predictive performance of the proposed method was validated and showed promise for accurate trabecular bone structure prediction without additional radiation exposure.

Radular Tooth Coating in Members of Dendronotidae and Flabellinidae (Nudibranchia, Gastropoda, Mollusca)

Nudibranchs, with their mesmerizing diversity and ecological significance, play crucial roles in marine ecosystems. Central to their feeding prowess is the radula, a chitinous structure with diverse morphologies adapted to prey preferences and feeding strategies. This study focuses on elucidating wear coping mechanisms in radular teeth of carnivorous molluscs, employing Dendronotus lacteus (Dendronotidae) and Flabellina affinis (Flabellinidae) as model species. Both species forage on hydrozoans. Through scanning electron microscopy, confocal laser scanning microscopy, nanoindentation, and energy-dispersive X-ray spectroscopy, the biomechanical and compositional properties of their teeth were analyzed. Notably, tooth coatings, composed of calcium (Ca) or silicon (Si) and high hardness and stiffness compared to the internal tooth structure, with varying mineral contents across tooth regions and ontogenetic zones, were found. The presence of the hard and stiff tooth coatings highlight their role in enhancing wear resistance. The heterogeneities in the autofluorescence patterns related to the distribution of Ca and Si of the coatings. Overall, this study provides into the biomechanical adaptations of nudibranch radular teeth, shedding light on the intricate interplay between tooth structure, elemental composition, and ecological function in marine molluscs.

Computational design of mechanical metamaterials

In the past few years, design of mechanical metamaterials has been empowered by computational tools that have allowed the community to overcome limitations of human intuition. By leveraging efficient optimization algorithms and computational physics models, it is now possible to explore vast design spaces, achieving new material functionalities with unprecedented performance. Here, we present our viewpoint on the state of the art of computational metamaterials design, discussing recent advances in topology optimization and machine learning design with respect to challenges in additive manufacturing.

Design of Locally Resonant Acoustic Metamaterials with Specified Band Gaps Using Multi-Material Topology Optimization

Locally Resonant Acoustic Metamaterials (LRAMs) have significant application potential because they can form subwavelength band gaps. However, most current research does not involve obtaining LRAMs with specified band gaps, even though such LRAMs are significant for practical applications. To address this, we propose a parameterized level-set-based topology optimization method that can use multiple materials to design LRAMs that meet specified frequency constraints. In this method, a simplified band-gap calculation approach based on the homogenization framework is introduced, establishing a restricted subsystem and an unrestricted subsystem to determine band gaps without relying on the Brillouin zone. These subsystems are specifically tailored to model the phenomena involved in band gaps in LRAMs, facilitating the opening of band gaps during optimization. In the multi-material representation model used in this method, each material, except for the matrix material, is depicted using a similar combinatorial formulation of level-set functions. This model reduces direct conversion between materials other than the matrix material, thereby enhancing the band-gap optimization of LRAMs. Two problems are investigated to test the method's ability to use multiple materials to solve band-gap optimization problems with specified frequency constraints. The first involves maximizing the band-gap width while ensuring it encompasses a specified frequency range, and the second focuses on obtaining light LRAMs with a specified band gap. LRAMs with specified band gaps obtained in three-material or four-material numerical examples demonstrate the effectiveness of the proposed method. The method shows great promise for designing metamaterials to attenuate specified frequency spectra as required, such as mechanical vibrations or environmental noise.

A Parametric Design Method for Engraving Patterns on Thin Shells

Designing thin-shell structures that are diverse, lightweight, and physically viable is a challenging task for traditional heuristic methods. To address this challenge, we present a novel parametric design framework for engraving regular, irregular, and customized patterns on thin-shell structures. Our method optimizes pattern parameters such as size and orientation, to ensure structural stiffness while minimizing material consumption. Our method is unique in that it works directly with shapes and patterns represented by functions, and can engrave patterns through simple function operations. By eliminating the need for remeshing in traditional FEM methods, our method is more computationally efficient in optimizing mechanical properties and can significantly increase the diversity of shell structure design. Quantitative evaluation confirms the convergence of the proposed method. We conduct experiments on regular, irregular, and customized patterns and present 3D printed results to demonstrate the effectiveness of our approach.

Dynamic topology optimization of 3D-Printed transtibial orthopedic implant using tunable isotropic porous metamaterials

In this paper, we introduce the design and manufacturing process of a transtibial orthopedic implant. We used medical-grade polyurethane polymer resin to fabricate a 3D porous architected implant with tunable isotropy, employing a high-speed printing method known as Continuous Liquid Interface Production (CLIP). Our objective is to enhance the weight-bearing capabilities of the bone structures in the residual limb, thereby circumventing the traditional reliance on a natural bridge. To achieve a custom-made design, we acquire the topology and morphology of the residual limb as well as the bone structure of the tibia and fibula, utilizing computed tomography (CT) and high-resolution 3D scanning. We employed a dynamic topological optimization method, informed by gait cycle data, to effectively reduce the mass of the implant. This approach, which differs from conventional static methods, enables the quantification of variations in applied forces over time. Using the Euler-Lagrange energy approach, we propose the equations of motion for a homologous multibody model with three degrees of freedom. The versatility of the Solid Isotropic Material with Penalization (SIMP) method facilitates the integration of homogenization methods for microscale porous architectures into the optimized domain. The design of these porous architectures is based on a bias-driven tuning symmetry isotropy of a Triply Periodic Minimal Surface (Schwarz Primitive surface). The internal porosity of the structure significantly reduces weight without compromising the isotropic behavior of the implant.

Performance of biological food processing interfaces: Perspectives on the science of mollusc radula

The Mollusca comprises a diverse range of organisms, with the class Gastropoda alone boasting approximately 80 000 extant species. Their adaptability across various habitats is facilitated by the evolution of the radula, a key structure for food acquisition. The radula's composition and mechanical properties, including its chitinous membrane, teeth, and supporting structures, enable efficient food gathering and processing. Through adaptive tooth morphology and composition, an interplay between radular components is facilitated, which results in collective effects to withstand forces encountered during feeding and reduce structural failure, with the broad range of variations reflecting ecological niches. Furthermore, teeth consist of composite materials with sometimes high contents of iron, calcium, or silicon to reduce wear. During interaction with the food, the radula performs complex three-dimensional motions, challenging to document. Here, we provide a review on the morphology, the mechanical properties, the composition, and various other parameters that contribute to radular performance. Due to, e.g., the smallness of these structures, there are, however, limitations to radular research. However, numerical simulations and physical models tested on substrates offer avenues for further understanding radular function and performance during feeding. These studies not only advance our knowledge of molluscan biology and ecology but also provide inspirations for biomimetic design and further advances in materials engineering.

A biophysically constrained brain connectivity model based on stimulation-evoked potentials

BACKGROUND

Single-pulse electrical stimulation (SPES) is an established technique used to map functional effective connectivity networks in treatment-refractory epilepsy patients undergoing intracranial-electroencephalography monitoring. While the connectivity path between stimulation and recording sites has been explored through the integration of structural connectivity, there are substantial gaps, such that new modeling approaches may advance our understanding of connectivity derived from SPES studies.

NEW METHOD

Using intracranial electrophysiology data recorded from a single patient undergoing stereo-electroencephalography (sEEG) evaluation, we employ an automated detection method to identify early response components, C1, from pulse-evoked potentials (PEPs) induced by SPES. C1 components were utilized for a novel topology optimization method, modeling 3D electrical conductivity to infer neural pathways from stimulation sites. Additionally, PEP features were compared with tractography metrics, and model results were analyzed with respect to anatomical features.

RESULTS

The proposed optimization model resolved conductivity paths with low error. Specific electrode contacts displaying high error correlated with anatomical complexities. The C1 component strongly correlated with additional PEP features and displayed stable, weak correlations with tractography measures.

COMPARISON WITH EXISTING METHOD

Existing methods for estimating neural signal pathways are imaging-based and thus rely on anatomical inferences.

CONCLUSIONS

These results demonstrate that informing topology optimization methods with human intracranial SPES data is a feasible method for generating 3D conductivity maps linking electrical pathways with functional neural ensembles. PEP-estimated effective connectivity is correlated with but distinguished from structural connectivity. Modeled conductivity resolves connectivity pathways in the absence of anatomical priors.

Wear patterns of radular teeth in Loligo vulgaris (Cephalopoda; Mollusca) are related to their structure and mechanical properties

Radular teeth have to cope with wear, when interacting with ingesta. In some molluscan taxa, wear-coping mechanisms, related to the incorporation of high contents of iron or silica, have been previously determined. For most species, particularly for those which possess radulae without such incorporations, wear-coping mechanisms are understudied. In the present study, we documented and characterized the wear on radular teeth in the model species Loligo vulgaris (Cephalopoda). By applying a range of methods, the elementary composition and mechanical properties of the teeth were described, to gain insight into mechanisms for coping with abrasion. It was found that the tooth regions that are prone to wear are harder and stiffer. Additionally, the surfaces interacting with the ingesta possessed a thin coating with high contents of silicon, probably reducing abrasion. The here presented data may serve as an example of systematic study of radular wear, in order to understand the relationship between the structure of radular teeth and their properties.

Machine Learning Aided Design and Optimization of Thermal Metamaterials

Artificial Intelligence (AI) has advanced material research that were previously intractable, for example, the machine learning (ML) has been able to predict some unprecedented thermal properties. In this review, we first elucidate the methodologies underpinning discriminative and generative models, as well as the paradigm of optimization approaches. Then, we present a series of case studies showcasing the application of machine learning in thermal metamaterial design. Finally, we give a brief discussion on the challenges and opportunities in this fast developing field. In particular, this review provides: (1) Optimization of thermal metamaterials using optimization algorithms to achieve specific target properties. (2) Integration of discriminative models with optimization algorithms to enhance computational efficiency. (3) Generative models for the structural design and optimization of thermal metamaterials.

Finite element-based optimization procedure for an irregular domain with unstructured mesh

At present, structural optimization is a highly demanding area of research in engineering. Engineers aim to minimize material in a body while maintaining its usability and safety at the same time. Developing a user-friendly program to optimize a structure using the finite element method (FEM) is the goal of the current study. With the advent of additive manufacturing, the production of complex-shaped designs is showing promise. A detailed optimization algorithm based on solid isotropic material with penalization (SIMP) is presented in this paper. UnTop2D: An object-oriented Python program with a graphical user interface (GUI) has been developed, which can be applied to structures with both structured and unstructured meshes. The mesh is not required to be topologically ball and can be imported from professional meshing software. Any selected element can be frozen to prevent its removal during optimization, and wall elements can also be frozen for real-world scenarios. The optimized structure can be exported as an Abaqus input file for structural analysis and STL file for 3D printing. This paper presents several examples to demonstrate the effectiveness of the proposed procedure.

Computational discovery of microstructured composites with optimal stiffness-toughness trade-offs

The conflict between stiffness and toughness is a fundamental problem in engineering materials design. However, the systematic discovery of microstructured composites with optimal stiffness-toughness trade-offs has never been demonstrated, hindered by the discrepancies between simulation and reality and the lack of data-efficient exploration of the entire Pareto front. We introduce a generalizable pipeline that integrates physical experiments, numerical simulations, and artificial neural networks to address both challenges. Without any prescribed expert knowledge of material design, our approach implements a nested-loop proposal-validation workflow to bridge the simulation-to-reality gap and find microstructured composites that are stiff and tough with high sample efficiency. Further analysis of Pareto-optimal designs allows us to automatically identify existing toughness enhancement mechanisms, which were previously found through trial and error or biomimicry. On a broader scale, our method provides a blueprint for computational design in various research areas beyond solid mechanics, such as polymer chemistry, fluid dynamics, meteorology, and robotics.

A biophysically constrained brain connectivity model based on stimulation-evoked potentials

Background

Single-pulse electrical stimulation (SPES) is an established technique used to map functional effective connectivity networks in treatment-refractory epilepsy patients undergoing intracranial-electroencephalography monitoring. While the connectivity path between stimulation and recording sites has been explored through the integration of structural connectivity, there are substantial gaps, such that new modeling approaches may advance our understanding of connectivity derived from SPES studies.

New Method

Using intracranial electrophysiology data recorded from a single patient undergoing sEEG evaluation, we employ an automated detection method to identify early response components, C1, from pulse-evoked potentials (PEPs) induced by SPES. C1 components were utilized for a novel topology optimization method, modeling 3D conductivity propagation from stimulation sites. Additionally, PEP features were compared with tractography metrics, and model results were analyzed with respect to anatomical features.

Results

The proposed optimization model resolved conductivity paths with low error. Specific electrode contacts displaying high error correlated with anatomical complexities. The C1 component strongly correlates with additional PEP features and displayed stable, weak correlations with tractography measures.

Comparison with existing methods

Existing methods for estimating conductivity propagation are imaging-based and thus rely on anatomical inferences.

Conclusions

These results demonstrate that informing topology optimization methods with human intracranial SPES data is a feasible method for generating 3D conductivity maps linking electrical pathways with functional neural ensembles. PEP-estimated effective connectivity is correlated with but distinguished from structural connectivity. Modeled conductivity resolves connectivity pathways in the absence of anatomical priors.

Compliance Prediction for Structural Topology Optimization on the Basis of Moment Invariants and a Generalized Regression Neural Network

Topology optimization techniques are essential for manufacturing industries, such as designing fiber-reinforced polymer composites (FRPCs) and structures with outstanding strength-to-weight ratios and light weights. In the SIMP approach, artificial intelligence algorithms are commonly utilized to enhance traditional FEM-based compliance minimization procedures. Based on an effective generalized regression neural network (GRNN), a new deep learning algorithm of compliance prediction for structural topology optimization is proposed. The algorithm learns the structural information using a fourth-order moment invariant analysis of the structural topology obtained from FEA at different iterations of classical topology optimization. A cantilever and a simply supported beam problem are used as ground-truth datasets, and the moment invariants are used as independent variables for input features. By comparing it with the well-known convolutional neural network (CNN) and deep neural network (DNN) models, the proposed GRNN model achieves a high prediction accuracy (R2 > 0.97) and drastically shortens the training and prediction cost. Furthermore, the GRNN algorithm exhibits excellent generalization ability on the prediction performance of the optimized topology with rotations and varied material volume fractions. This algorithm is promising for the replacement of the FEA calculation in the SIMP method, and can be applied to real-time optimization for advanced FRPC structure design.

Learning hyperparameter predictors for similarity-based multidisciplinary topology optimization

Topology optimization (TO) plays a significant role in industry by providing engineers with optimal material distributions based exclusively on the information about the design space and loading conditions. Such approaches are especially important for current multidisciplinary design tasks in industry, where the conflicting criteria often lead to very unintuitive solutions. Despite the progress in integrating manufacturing constraints into TO, one of the main factors restricting the use of TO in practice is the users' limited control of the final material distribution. To address this problem, recently, a universal methodology for enforcing similarity to reference structures in various TO methods by applying scaling of elemental energies was proposed. The method, however, requires an expensive hyperparameter sampling, which involves running multiple TO processes to find the design of a given similarity to a reference structure. In this article, we propose a novel end-to-end approach for similarity-based TO, which integrates a machine learning model to predict the hyperparameters of the method, and provide the engineer, at minimal computational cost, with a design satisfying multidisciplinary criteria expressed by the similarity to a reference. The training set for the model is generated based on an academic linear elastic problem, but the model generalizes well to both nonlinear dynamic crash and industrial-scale TO problems. We show the latter by applying the proposed methodology to a real-world multidisciplinary TO problem of a car hood frame, which demonstrates the usefulness of the approach in industrial settings.

Topological Design of Two-Dimensional Phononic Crystals Based on Genetic Algorithm

Phononic crystals are a kind of artificial acoustic metamaterial whose mass density and elastic modulus are periodically arranged. The precise and efficient design of phononic crystals with specific bandgap characteristics has attracted increasing attention in past decades. In this paper, an improved adaptive genetic algorithm is proposed for the reverse customization of two-dimensional phononic crystals designed to maximize the relative bandwidth at low frequencies. The energy band dispersion relation and transmission loss of the optimal structure are calculated by the finite-element method, and the effective wave-attenuation effect in the bandgap range is verified. This provides a solution for the custom-made design of acoustic metamaterials with excellent low-frequency bandgap sound insulation or other engineering applications.

Optimization of Components with Topology Optimization for Direct Additive Manufacturing by DLMS

This paper presents a novel design methodology that validates and utilizes the results of topology optimization as the final product shape. The proposed methodology aims to streamline the design process by eliminating the need for remodeling and minimizing printing errors through process simulation. It also eliminates the repeated export and import of data between software tools. The study includes a case study involving the steering column housing of a racing car, where Siemens NX Topology Optimization was used for optimization, and verification analysis was conducted using the NX Nastran solver. The final solution was fabricated using AlSi10Mg via direct metal laser sintering on a 3D printer and successfully validated under real conditions. In conclusion, this paper introduces a comprehensive design methodology for the direct utilization of topology optimization, which was validated through a case study, yielding positive results.

Elemental composition and material properties of radular teeth in the heterobranch snail Gastropteron rubrum (Mollusca, Gastropoda, Cephalaspidea) foraging on hard organisms

The molluscan feeding structure is the radula, a chitinous membrane with teeth, which are highly adapted to the food and the substrate to which the food is attached. In Polyplacophora and Patellogastropoda, the handling of hard ingesta can be facilitated by high content of chemical compounds containing Fe or Si in the tooth cusps. Other taxa, however, possess teeth that are less mineralized, even though animals have to avoid structural failure or high wear during feeding as well. Here, we investigated the gastropod Gastropteron rubrum, feeding on hard Foraminifera, diatoms and Porifera. Tooth morphologies and wear were documented by scanning electron microscopy and their mechanical properties were tested by nanoindentation. We determined that gradients of hard- and stiffness run along each tooth, decreasing from cusp to basis. We also found that inner lateral teeth were harder and stiffer than the outer ones. These findings allowed us to propose hypotheses about the radula-ingesta interaction. In search for the origins of the gradients, teeth were visualized using confocal laser scanning microscopy, to determine the degree of tanning, and analyzed with energy-dispersive X-ray spectroscopy, to test the elemental composition. We found that the mechanical gradients did not have their origins in the elemental content, as the teeth did not contain high proportions of metals or other minerals. This indicates that their origin might be the degree of tanning. However, in the tooth surfaces that interact with the ingesta high Si and Ca contents were determined, which is likely an adaptation to reduce wear.

Influence of topology optimization parameters on the mechanical response of an additively manufactured test structure

Topology Optimization (TO) determines a material distribution within a domain under given conditions and design constraints, and generally generates complex geometries as a result. Complementary to TO, Additive Manufacturing (AM) offers the ability to fabricate complex geometries which may be difficult to manufacture using traditional techniques such as milling. AM has been used in multiple industries including the medical devices area. Hence, TO may be used to create patient-matched devices where the mechanical response is catered to a particular patient. However, during a medical device regulatory 510(k) pathway, demonstrating that worst-cases are known and tested is critical to the review process. Using TO and AM to predict worst-case designs for subsequent performance testing may be challenging and does not appear to have been thoroughly explored. Investigating the effects of TO input parameters when AM is employed may be the first step in determining the feasibility of predicting these worst-cases. In this paper, the effect of selected TO parameters on the resulting mechanical response and geometries of an AM pipe flange structure are investigated. Four different input parameters were chosen in the TO formulation: (1) penalty factor, (2) volume fraction, (3) element size, and (4) density threshold. Topology optimized designs were fabricated using PA2200 polyamide and the mechanical responses (reaction force, stress, and strain) were observed through experiments (universal testing machine and 3D Digital Image Correlation) and in silico environments (finite element analysis). In addition, 3D scanning and mass measurement were performed to inspect the geometric fidelity of the AM structures. A sensitivity analysis is performed to examine the effect of each TO parameters. The sensitivity analysis revealed mechanical responses can have non-monotonic and non-linear relationships between each tested parameter.

Simulation-Based Support Generation for Laser Powder Bed Fusion Processes

The rise of additive manufacturing, particularly laser powder bed fusion, has enabled new degrees of freedom in terms of design and functionality. Notably high-performance industries such as the medical, automotive, and aerospace industries are on the edge of broad industrial application. However, the quality standards required for these industries are not yet entirely met. Process-related temperature-induced residual stresses within the component can lead to warpage and deformations causing rejects. Support structures are a vital counter measure to absorb these residual stresses and ensure the required print quality. While the current industrial standard for support structure generation mainly focuses on geometrical boundary conditions, this study presents a simulation-based approach taking into account residual stresses. The proposed approach determines the stress distribution during the process from a thermomechanical finite element process simulation and uses these results for the allocation of topology-optimized stackable unit cells. To assure a reliable connection to the component while offering easy removability of the support, different connection support structures in the interface area were tested and integrated in the proposed approach. The result is a robust tailored lattice support structure minimizing residual stresses to achieve high component quality, while focusing on cost-efficiency factors such as minimal material usage, easy support removability, and numerical efficiency. Finally, the proposed approach was tested on a demonstrator part printed from AlSi10Mg.

Diatom-inspired stiffness optimization for plates and cellular solids

Diatoms, a class of aquatic autotrophic microorganisms, are characterized by silicified exoskeletons with highly complex architectures. These morphologies have been shaped by the selection pressure that the organisms have been subjected to during their evolutionary history. Two properties which are highly likely to have contributed to the evolutionary success of current diatom species are lightweightness and structural strength. Thousands of diatom species are present in water bodies today, and although each has its unique shell architecture, a strategy that is common across species is the uneven and gradient solid material distribution across their shells. The aim of this study is to present and evaluate two novel structural optimization workflows inspired by material grading strategies in diatoms. The first workflow mimics theAuliscus intermidusdiatoms' surface thickening strategy and generates continuous sheet structures with optimal boundaries and local sheet thickness distributions when applied to plate models subjected to in-plane boundary conditions. The second workflow mimics theTriceratium sp.diatoms' cellular solid grading strategy, and produces 3D cellular solids with optimal boundaries and local parameter distributions. Both methods are evaluated through sample load cases, and prove to be highly efficient in transforming optimization solutions with non-binary relative density distributions into highly performing 3D models.

Topology optimization with local stress constraints and continuously varying load direction and magnitude: towards practical applications

Topology optimization problems typically consider a single load case or a small, discrete number of load cases; however, practical structures are often subjected to infinitely many load cases that may vary in intensity, location and/or direction (e.g. moving/rotating loads or uncertain fixed loads). The variability of these loads significantly influences the stress distribution in a structure and should be considered during the design. We propose a locally stress-constrained topology optimization formulation that considers loads with continuously varying direction to ensure structural integrity under more realistic loading conditions. The problem is solved using an Augmented Lagrangian method, and the continuous range of load directions is incorporated through a series of analytic expressions that enables the computation of the worst-case maximum stress over all possible load directions. Variable load intensity is also handled by controlling the magnitude of load basis vectors used to derive the worst-case load. Several two- and three-dimensional examples demonstrate that topology-optimized designs are extremely sensitive to loads that vary in direction. The designs generated by this formulation are safer, more reliable, and more suitable for real applications, because they consider realistic loading conditions.

High-Efficiency Achromatic Metalens Topologically Optimized in the Visible

Metalens, composed of arrays of nano-posts, is an ultrathin planar optical element used for constructing compact optical systems which can achieve high-performance optical imaging by wavefront modulating. However, the existing achromatic metalenses for circular polarization possess the problem of low focal efficiency, which is caused by the low polarization conversion efficiencies of the nano-posts. This problem hinders the practical application of the metalens. Topology optimization is an optimization-based design method that can effectively extend the degree of design freedom, allowing the phases and polarization conversion efficiencies of the nano-posts to be taken into account simultaneously in the optimization procedures. Therefore, it is used to find geometrical configurations of the nano-posts with suitable phase dispersions and maximized polarization conversion efficiencies. An achromatic metalens has a diameter of 40 μm. The average focal efficiency of this metalens is 53% in the spectrum of 531 nm to 780 nm by simulation, which is higher than the previously reported achromatic metalenses with average efficiencies of 20~36%. The result shows that the introduced method can effectively improve the focal efficiency of the broadband achromatic metalens.

Process-Specific Topology Optimization Method Based on Laser-Based Additive Manufacturing of AlSi10Mg Components: Material Characterization and Evaluation

In the laser powder bed fusion process (PBF-LB), components are built up incrementally by locally melting metal powder with a laser beam. This process leads to inhomogeneous material properties of the manufactured components. By integrating these specific material properties into a topology optimization algorithm, product developers can be supported in the early phases of the product development process, such as design finding. For this purpose, a topology optimization method was developed, which takes the inhomogeneous material properties of components fabricated in the PBF-LB process into account. The complex pore architecture in PBF-LB components was studied with micro-computed tomography (µCT). Thereby, three characteristic regions of different porosity were identified and analyzed. The effective stiffness in each of these regions was determined by means of resonant ultrasonic spectroscopy (RUS) as well as finite element analysis. Afterward, the effective stiffness is iteratively considered in the developed topology optimization method. The resulting design proposals of two optimization cases were analyzed and compared to design proposals derived from a standard topology optimization. To evaluate the developed topology optimization method, the derived design proposals were additionally manufactured in the PBF-LB process, and the characteristic pore architecture was analyzed by means of µCT.

Quantitative Load Dependency Analysis of Local Trabecular Bone Microstructure to Understand the Spatial Characteristics in the Synthetic Proximal Femur

Analysis of the dependency of the trabecular structure on loading conditions is essential for understanding and predicting bone structure formation. Although previous studies have investigated the relationship between loads and structural adaptations, there is a need for an in-depth analysis of this relationship based on the bone region and load specifics. In this study, the load dependency of the trabecular bone microstructure for twelve regions of interest (ROIs) in the synthetic proximal femur was quantitatively analyzed to understand the spatial characteristics under seven different loading conditions. To investigate the load dependency, a quantitative measure, called the load dependency score (LDS), was established based on the statistics of the strain energy density (SED) distribution. The results showed that for the global model and epiphysis ROIs, bone microstructures relied on the multiple-loading condition, whereas the structures in the metaphysis depended on single or double loads. These results demonstrate that a given ROI is predominantly dependent on a particular loading condition. The results confirm that the dependency analysis of the load effects for ROIs should be performed both qualitatively and quantitatively.

Two-stage neural network via sensitivity learning for 2D photonic crystal bandgap maximization

We propose a two-stage neural network method to maximize the bandgap of 2D photonic crystals. The proposed model consists of a fully connected deep feed-forward neural network (FNN) and U-Net, which are employed, respectively, to generate the shape function and learn the sensitivity. The shape is generated by the FNN during the entire optimization process, and obtaining the sensitivity can be split into two steps. In the first step of the optimization, the sensitivity is calculated by finite element analysis (FEA) and the result is used as a sample to train the U-Net. Second, the optimization procedure is adopted instead of FEA, where a trained U-Net is used to generate the corresponding sensitivity. The main advantage of such an approach is that the shape function and sensitivity can be obtained by neural networks without solving a partial differenital equation. Therefore, the computational cost can be reduced by the proposed method without using large training sets. The effectiveness of the proposed method is verified in the numerical experiments in terms of the optimized shape and time consumption.

An acoustic cloaking design based on topology optimization

In this work, we explain how to utilize the topology optimization method for the design of acoustic cloaks based on the principle of scattering cancellation. To take account of the challenging fabrication restriction, we impose boundary control inside the optimization objective function and enforce hyperbolic tangent projection to minimize the gray transition regions of the optimized design. In addition, a filter based on the Helmholtz differential equation is used to remove any tiny structures due to the effect of discretized grids. Then, we fabricate the designed cloaks and conduct the experiments in a couple of representative set-ups to validate the proposed design approach. The experiments are conducted inside both air and water. We found that the current cloaking design performs much better in air than in water and reveal the associated reason. Overall, this work paves the way for the acoustic cloaking design, fabrication, and experiments for future practical applications.

Sustainable 3D Printing of Recyclable Biocomposite Empowered by Flash Graphene

Sustainability of 3D printing can be reflected in three main aspects: deployment of renewable inks, recycling of printed products, and applications for energy- and materials- savings. In this work, we demonstrated sustainable vat-photopolymerization (VPP)-based 3D printing in a whole life-cycle process by developing a renewable ink made of soybean oil and natural polyphenols and recycling the ink for reprinting or converting printed biocomposite to flash graphene (FG) as reinforcing nanofillers in the biocomposite. We also realized its applications in fabricating lightweight, materials-saving 3D structures, acoustic metamaterials, and disposable microreactors for time-saving and efficiency-improving synthesis of metal-organic framework nanostructures. In addition to enhancing the tensile strength and Young's modulus of the biopolymers by 42% and 232% with only 0.6 wt % FG nanofillers, respectively, FG improved the printability of the ink in forming 3D tubular structures, which are usually very hard to be achieved in transparent resin. Success of this work will inspire further development for sustainability in 3D printing.

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.

Additive Manufacturing of Biomaterials-Design Principles and Their Implementation

Additive manufacturing (AM, also known as 3D printing) is an advanced manufacturing technique that has enabled progress in the design and fabrication of customised or patient-specific (meta-)biomaterials and biomedical devices (e.g., implants, prosthetics, and orthotics) with complex internal microstructures and tuneable properties. In the past few decades, several design guidelines have been proposed for creating porous lattice structures, particularly for biomedical applications. Meanwhile, the capabilities of AM to fabricate a wide range of biomaterials, including metals and their alloys, polymers, and ceramics, have been exploited, offering unprecedented benefits to medical professionals and patients alike. In this review article, we provide an overview of the design principles that have been developed and used for the AM of biomaterials as well as those dealing with three major categories of biomaterials, i.e., metals (and their alloys), polymers, and ceramics. The design strategies can be categorised as: library-based design, topology optimisation, bio-inspired design, and meta-biomaterials. Recent developments related to the biomedical applications and fabrication methods of AM aimed at enhancing the quality of final 3D-printed biomaterials and improving their physical, mechanical, and biological characteristics are also highlighted. Finally, examples of 3D-printed biomaterials with tuned properties and functionalities are presented.

Infill Microstructures for Additive Manufacturing

Additive Manufacturing (AM) is a well-known and rapidly advancing method, especially in the manufacturing of high-strength and lightweight microstructures. Utilizing AM, it is possible to fabricate any structure as complicated as it is. For an efficient and cost-effective printing, a critical parameter is the infill, which can be characterized from an easy 2D shape to high complexity. At the same time, Topology Optimization (TO) is an appropriate method to create high-strength and mass optimized microstructure lattices. In the current study, TO starts from a solid cubic volume of 15 × 15 mm, and different boundary conditions of two new cellular microstructures designed with 0.4 and 0.1 relative density are applied, respectively. The adopted TO method was Solid Isotropic Material with Penalization (SIMP), which predicts an optimal material distribution within a given design domain. TO methods do not check other characteristics of the structure, such as anisotropy. To evaluate and characterize the optimized microstructure, a general purpose homogenization method is utilized to calculate the Zener ratio and the elastic modulus. Using Fused Filament Fabrication (FFF), which is a material extrusion 3D printing method, lattice structure samples are fabricated and then tested in compression and tensile strength tests. The comparative results from the homogenization study showed that both microstructures have anisotropic behavior and an accepted response in the stress test similar to the homogenized material. The experimental results show that the mechanical behavior of the lattice structure changes significantly when the cell mapping angle differs.

Efficient Representation and Optimization for TPMS-Based Porous Structures

In this approach, we present an efficient topology and geometry optimization of triply periodic minimal surfaces (TPMS) based porous shell structures, which can be represented, analyzed, optimized and stored directly using functions. The proposed framework is directly executed on functions instead of remeshing (tetrahedral/hexahedral), and this framework substantially improves the controllability and efficiency. Specifically, a valid TPMS-based porous shell structure is first constructed by function expressions. The porous shell permits continuous and smooth changes of geometry (shell thickness) and topology (porous period). The porous structures also inherit several of the advantageous properties of TPMS, such as smoothness, full connectivity (no closed hollows), and high controllability. Then, the problem of filling an object's interior region with porous shell can be formulated into a constraint optimization problem with two control parameter functions. Finally, an efficient topology and geometry optimization scheme is presented to obtain optimized scale-varying porous shell structures. In contrast to traditional heuristic methods for TPMS, our work directly optimize both the topology and geometry of TPMS-based structures. Various experiments have shown that our proposed porous structures have obvious advantages in terms of efficiency and effectiveness.

Integrated Development of a Topology-Optimized Compliant Mechanism for Precise Positioning

A scheme for modelling and controlling a two-dimensional positioning system with a topology-optimized compliant mechanism is presented. The system is designed to ensure a relatively large workspace and exhibit robustness against system nonlinearities. A detailed design procedure based on topology optimization is presented, and a nonlinear description of the designed mechanism is developed as a starting point for further precise position control. The theoretical model is shown to be suitable for a considerably larger working range without losing consistency. A backstepping controller is employed to manipulate the nonlinearities in the model resulting from the geometrical and material nonlinearity of the mechanical structure. The hysteresis of the piezoelectric actuator is also taken into consideration. An experimental verification of the controller demonstrates that the proposed design approach improves the performance of compliant mechanism and satisfies the needs for precision positioning.