A Fourier neural operator-based lightweight machine learning framework for topology optimization

In conventional level set methods, the slope of the level set function needs to be well controlled to maintain the numerical stability during the topology optimization process. One common solution is to regularize the level set function to be a signed distance function, which is usually achieved by periodically implementing the so called re-initialization scheme to force the level set function to gain the desired signed distance property. However, the re-initialization scheme will bring some unwanted drawbacks to the optimization process, such as zero level set drifting, time consuming etc. In addition, re-initialization is usually implemented outside the optimization loop, which will cause convergence issues. In this paper, a distance regularization functional is introduced to the structural topology optimization objective functional to ensure the signed distance property of the level set function near the structure boundaries. This functional can also keep the level set function to be constant-value at positions far away from the structural boundaries. The radial basis function (RBF) based parameterization technique together with the mathematical programming are utilized to improve the potential capability of handling multiple constraints for the topology optimization. The combination of these two techniques makes the level set based topology optimization be capable of handling complicated multi-constrained problems with higher numerical efficiency, leaving no compromise to multiple drawbacks. To demonstrate the validity of the proposed scheme, benchmark examples on minimum compliance structural optimization are employed. This type of problem is computed by the conventional level set method with the introduced distance regularization functional, the RBF based parametric level set and at last, the distance regularized RBF based parametric level set separately to demonstrate their differences.

This paper presents an application of topology optimization and response surface method to optimize the geometry of a bicycle crank arm and the experimental validation of it. This is purposely to reduce the crank arm mass and create a preliminary design of a lightweight structure necessary for the high-performance bicycle development. A three-dimensional bike crank arm model was made in the SpaceClaim software followed by a static finite element analysis using ANSYS Workbench 2019 R1. A multiple cycling load was applied simultaneously in seven crank angles of 30, 45, 60, 90, 120, 135, and 150° relative to the horizontal position to create the multiple loads to the crank. From there, topology optimization was then conducted to investigate the effect of mass constraint, stress constraint, angle of cycling, and crank materials on the topological pattern result. To minimize stress concentration at corners, a shape optimization using the response surface method was conducted and obtained the final geometry. From the result, it is shown that both optimization methods not only successfully reduce the crank arm mass and provide several optimum design options but also are able to reduce the maximum stress in the crank arm up to 20% after the optimization process. The experimental validation using a newly developed wireless measurement system shows a considerable agreement to the numerical results.

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.

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 Introduction
 With the rapid advancements in neurotechnological machinery and improved analytical insights from machine learning in neuroscience, the availability of big brain data has increased tremendously. Neurological health research is done using digitized brain data.[1] There must be adequate data governance to secure the privacy of subjects participating in brain research and treatments. If not properly regulated, the research methods could lead to significant breaches of the subject’s autonomy and privacy. This paper will address the necessity for neuroprotection laws, which effectively govern the use of big brain data to ensure respect for patient privacy and autonomy.
 Background
 Artificial intelligence and machine learning can be integrated with neuroscience big brain data to drive research studies. This integrative technology allows patterns of electrical activity in neurons to be studied in detail.[2]Specifically, it uses a robotic system which can reason, plan, and exhibit biologically intelligent behavior. Machine learning is a method of computer programming where the code can adapt its behavior based on big brain data.[3] The big brain data is the collection of large amounts of information for the purpose of deciphering patterns through computer analysis using machine learning.[4] The information that these technologies provide is extensive enough to allow a researcher to read a patient’s mind. AI and machine learning technologies work by finding the underlying structure of brain data, which is then described by patterns known as latent factors, eventually resulting in an understanding of the brain’s temporal dynamics.[5]
 Through these technologies, researchers are able to decipher how the human brain computes its performances and thoughts. However, due to the extensive and complex nature of the data processed through AI and machine learning, researchers may gain access to personal information a patient may not wish to reveal. From a bioethical lens, tensions arise in the realm of patient autonomy. Patients are not able to control the transmission of data from their brains that is analyzed by researchers. Governing brain data through laws may enhance the extent of patient privacy in the case where brain data is being used through AI technologies.[6] A responsible approach to governing brain data would require a sophisticated legal structure.
 Analysis
 Impact on Patient Autonomy and Privacy 
 In research pertaining to big brain data, the consent forms do not fully cover the vast amounts of information that is collected. According to research, personal data has become the most sought out commodity to provide content to corporations and the web-based service industry. Unfortunately, data leaks that release private information frequently occur.[7] The storage of an individual’s data on technologies accessible on the internet during research studies makes it vulnerable to leaks, jeopardizing an individual’s privacy. These data leaks may cause the patient to be identified easily, as the degree of information provided by AI technologies are personalized and may be decoded through brain fingerprinting methods.[8]
 There has been an extensive growth in the development and use of AI. It is efficient in providing information to radiologists who diagnose various diseases including brain cancer and psychiatric disease, and AI assists in the delivery of telemedicine.[9] However, the ethical pitfall of reduced patient autonomy must be addressed by analyzing current AI technologies and creating more options for patient preference in how the data may be used. For instance, facial recognition technology[10] commonly used in health care produces more information than listed in common consent forms, threatening to undermine informed consent. Facial recognition software collects extensive data and may disclose more information than a person would prefer to provide despite being a useful tool for diagnosing medical and genetic conditions.[11] In addition, people may not be aware that their images are being used to generate more clinical data for other purposes. It is difficult to guarantee the data is anonymized. Consent requirements must include informing people about the complexity of the potential uses of the data; software developers should maximize patient privacy.[12] Furthermore, there is a “human element” in the use of AI technologies as medical providers control the use and the extent to which data is captured or accessed through the AI technologies.[13] People must understand the scope of the technology and have clear communication with the physician or health care provider about how the medical information will be used. 
 Existing Laws for Brain Data Governance 
 A strict system of defined legal responsibilities of medical providers will ensure a higher degree of patient privacy and autonomy when AI technologies and data from machine learning are used. Governing specific algorithmic data is crucial in safeguarding a patient’s privacy and developing a gold standard treatment protocol following the procurement of the information.[14] Certain AI technologies provide more data than others, and legal boundaries should be established to ensure strong performance, quality control, and scope for patient privacy and autonomy. For instance, currently AI technologies are being used in the realm of intensive neurological care. However, there is a significant level of patient uncertainty about how much control patients have over the data’s uses.[15] Calibrated legal and ethical standards will allow important brain data to be securely governed and monitored.
 Once brain signals are recorded and processed from one individual, the data may be merged with other data in Brain Computer Interface Technology (BCI).[16] To ensure a right and ability to retrieve personal data or pull it from the collection, specific regulations for varying types of data are needed.[17] The importance of consent and patient privacy must be considered through giving patients a transparent view of how brain data is governed.[18] The legal system must address discriminatory issues and risks to patients whose data is used in studies. Laws like the General Data Protection Regulation (GDPR) and the California Consumer Privacy Protection Act (CCPA) can serve as effective models to protect aggregated data. These laws govern consumer information and ensure the compliance when personal data is collected.[19] California voters recently approved expansion of the CCPA to health data. The Washington Privacy Act, which would have provided rights to access, change, and withdraw personal data, failed to pass. Other states should improve privacy as well,[20] although a federal bill would be preferable. Scientists at the Heidelberg Academy of Sciences argue for data security to be governed in a manner that balances patient privacy and autonomy with the commercial interests of researchers.[21] The balance could be achieved through privacy protections like those in the Washington Privacy Act. Although the Health Insurance Portability and Accountability Act (HIPAA) provides an overall framework to deter the likelihood of dangers to patient protection and privacy, more thorough laws are warranted to combat pervasive data transfer and analysis that technology has brought to the health care industry.[22] Breaches of patient privacy under current HIPAA regulations include releasing patient information to a reporter without their consent and sending HIV data to a patient’s employer without consent.[23] HIPAA does not cover information being shared with outside contractors who do not have an agreement with technology companies to keep patient data confidential. HIPAA regulations also do not always address blatant breaches on patient data confidentiality.[24] Patients must be provided with methods to monitor the data being analyzed to be able to view the extent of private information being generated via AI technologies. In health research, the medical purposes of better diagnosis, earlier detection of diseases, or prevention are ethical justifications for the use of the data if it was collected with permission, the person understood and approved the uses of the data, and the data was deidentified.
 A standard governance framework is required in providing the fairest system of care to patients who allow their brain data to be examined. Informed consent in the neuroscience field could reaffirm the privacy and autonomy of patients by ensuring that they understand the type of information collected. Laws also could protect data after a patient’s death. Malpractice in the scope of brain data could give people a cause of action critical in safeguarding patient’s rights. Data breach lawsuits will become common but generally do not cover deidentified data that becomes part of big data collection. A more synchronized approach to the collection and consent process will encourage an understanding of how big data is used to diagnose and treat patients. Some altruistic people may even be more likely to consent if they know the largescale data collection is helpful to treat and diagnose people. Others should have the ability to opt out of sharing neurological data, especially when there is not certainty surrounding deidentification.[25]
 Conclusion
 Artificial intelligence and machine learning technologies have the potential to aid in the diagnosis and treatment of people globally by extracting and aggregating brain data specific to individuals. However, the secure use of the data is necessary to build trust between care providers and patients, as well as in balancing the bioethical principles of beneficence and patient autonomy. We must ensure the highest quality of care to patients, while protecting their privacy, informed consent, and clinical trust. More sophis

A desirable feature in multidisciplinary design optimization (MDO) is to achieve an acceptable preliminary or initial design in an expeditiously fashion. One way to pursue this goal is to describe the structural boundary using a small number of design variables. In this paper we propose an approach to structural optimization that enhances the performance of multidisciplinary design tools because it greatly decreases the number of design variables assigned to structural deflnition. This approach is based on the Level Set Methods and uses the coe‐cients of the Fourier series expansion of the level set function as design variables. The global character of these coe‐cients and the fact that the level set function is real-valued provide the means to reduce the design space dimension. Within the level set framework, the structure is conflned to the region in which the level set function is negative. In this way, the zero level set of this function implicitly describes the structural boundary, which can undergo topological changes during the optimization process as the function evolves. The proposed methodology also averts successive mesh generation. We show that an interface can be described to any desirable accuracy using flnitely many Fourier modes of the signed distance to the interface. The total error is inversely proportional to the number of Fourier modes. We present two validation examples in topology optimization, and use these cases to study the efiects of the methodology parameters on the results. Finally, we propose two examples of application to the enhancement of aircraft roll maneuvers. In the flrst, the goal is to determine the layout of reinforcements in the front spar of a wing-aileron that maximizes the aileron reversal speed, subject to a constraint in the amount of reinforcing material. We also propose the design of a morphing airfoil-aileron conflguration capable of changing its mean camber to maximize the aileron efiectiveness through the activation of a system of actuators, given a restriction on power consumption.

Automatic differentiation (AD) was introduced into topology optimization (TO) more than two decades ago to compute accurate gradients through complex computational workflows. Nevertheless, its adoption within the TO community has remained limited, largely due to the strong reliance on adjoint-based sensitivity analysis—which typically offers superior memory efficiency and runtime performance—and the practical difficulties of integrating large-scale simulations into specialized AD frameworks. The recent rise of machine learning (ML) has opened new opportunities for TO through the advanced AD capabilities of modern ML frameworks such as and . A growing body of work at the intersection of ML and TO now focuses on tightly coupling ML components with classical TO workflows. Neural TO is a prominent example, in which an untrained neural network parameterizes the material density field and optimization proceeds over the network parameters. To enable such ML–TO hybrid workflows, a deeper understanding of how AD systems operate in these frameworks is essential. This article explains the practical principles of AD in modern ML frameworks and their relation to classical adjoint-based sensitivity analysis. We present implementation strategies for wrapping essential operations—such as finite element solvers—into AD-compatible components without reimplementing them from scratch. These ideas are illustrated through two compact code examples: a classical TO pipeline with selectively AD-wrapped components and a neural TO workflow.

According to the requirements of the aerospace industry for high strength, high stiffness, and lightweight structural parts, topology optimization has been proved to be an effective product design method. As one of the most conceptual and prospective structural optimization design methods, topology optimization intends to seek the optimal layout of materials in an allowed design region under a given load and boundary conditions. Thus, the object of study in the article is the method of topological optimization of aircraft structures. The goal of this article is to analyze the existing approaches, algorithms, as well as application of the method of topological optimization in the aerospace field in applied problems. The tasks are to describe the existing various approaches methods, features, and research directions of topological optimization as well as to study the possibility of application in the manufacturing process of composite structures. The following results were obtained. The optimization methods are briefly explained and compared, and the advantages and limitations of each approach are discussed. The various ways of simultaneous optimization of fiber orientation and structural topology were described and analyzed. The features of different methods of continuous fiber orientation optimization method were reviewed. The discrete fiber orientation optimization methods were represented. The possibility of multi-scale concurrent topological optimization was described. The combination of topology optimization and additive manufacturing was considered. Finally, the topology optimization of FRC structures which have been resolved in literature are reviewed and the potential research fields requiring more investigation are pointed out. Conclusions. In the article, a comprehensive review of the topology optimization design of FRC structures was presented. The promising way is to combine topology optimization with additive manufacturing techniques. However, these proposed methods may not suitable for other more complex problems, such as bucking stability and natural frequency. Hence, the topology optimization design of complex FRC components under complicated conditions is the main challenge in the future. This can be a new trend in the topology design of FRC structures.

Chemical heuristics have been fundamental to the advancement of chemistry and materials science. These heuristics are typically established by scientists using knowledge and creativity to extract patterns from limited datasets. Machine learning offers opportunities to perfect this approach using computers and larger datasets. Here, we discuss the relationships between traditional heuristics and machine learning approaches. We show how traditional rules can be challenged by large-scale statistical assessment and how traditional concepts commonly used as features are feeding the machine learning techniques. We stress the waste involved in relearning chemical rules and the challenges in terms of data size requirements for purely data-driven approaches. Our view is that heuristic and machine learning approaches are at their best when they work together.

Topology optimization (TO) is currently a focal point for researchers in the field of structural optimization, with most studies concentrating on single-loading conditions. However, real engineering structures often have to work under various loading conditions. Approaches addressing multiple-loading conditions often necessitate subjective input in order to determine the importance of each loading condition, aiming for a compromise between them. This paper proposes a so-called bisection constraint method (BCM), offering a unique, user-preference-independent solution for TO problems amidst multiple-loading conditions. It is well-known that minimizing the system’s compliance is commonly used in TO as the objective. Generally, compliance is not as sufficient as stress to be used as a response to evaluate the performance of structures. However, formulations focusing on minimizing stress levels usually pose significant difficulties and instabilities. On the other hand, the compliance approach is generally simpler and more capable of providing relatively sturdy designs. Hence, the formulation of min–max compliance is used as the target problem formulation of the proposed method. This method attempts to minimize compliance under only one loading condition while compliances under the remaining loading conditions are constrained. During the optimization process, the optimization problem is automatically reformulated with a new objective function and a new set of constraint functions. The role of compliance under different loading conditions, i.e., whether it is to be treated as an objective or constraint function, might be changed throughout the optimization process until convergence. Several examples based on the solid isotropic material with penalization (SIMP) approach were conducted to illustrate the validity of the proposed method. Furthermore, the general effectiveness of the compliance approach in terms of stress levels is also discussed. The calculation results demonstrated that while the compliance approach is effective in several cases, it proves ineffective in certain scenarios.

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending.

A global optimization method for topology optimization using a genetic algorithm is proposed in this paper. The genetic algorithm used in this paper is assisted by the Kriging surrogate model to reduce computational cost required for function evaluation. To validate the global topology optimization method in flow problems, this research works on two single-objective optimization problems, where the objective functions are to minimize pressure loss and to maximize heat transfer of flow channels, and the multi-objective optimization problem, which combines these two problems. The shape of flow channels is represented by the level set function, and the pressure loss and the temperature of the channels are evaluated by the Building-Cube Method (BCM), which is a Cartesian-mesh CFD approach. The proposed method resulted in an agreement with previous study in the single-objective problems in its topology, and achieved global exploration of non-dominated solutions in the multi-objective problem.

Topological shape optimization of geometrically nonlinear structures using level set method

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.