Finding Optimal Designs Research Articles

Tractable optimal experimental design using transport maps*

Abstract We present a flexible method for computing Bayesian optimal experimental designs (BOEDs) for inverse problems with intractable posteriors. The approach is applicable to a wide range of BOED problems and can accommodate various optimality criteria, prior distributions and noise models. The key to our approach is the construction of a transport-map-based surrogate to the joint probability law of the design, observational and inference random variables. This order-preserving transport map is constructed using tensor trains and can be used to efficiently sample from (and evaluate approximate densities of) conditional distributions that are required in the evaluation of many commonly-used optimality criteria. The algorithm is also extended to sequential data acquisition problems, where experiments can be performed in sequence to update the state of knowledge about the unknown parameters. The sequential BOED problem is made computationally feasible by preconditioning the approximation of the joint density at the current stage using transport maps constructed at previous stages. The flexibility of our approach in finding optimal designs is illustrated with some numerical examples inspired by disease modeling and the reconstruction of subsurface structures in aquifers.

A Double Exponential Particle Swarm Optimization with non-uniform variates as stochastic tuning and guaranteed convergence to a global optimum with sample applications to finding optimal exact designs in biostatistics

Nature-inspired metaheuristic algorithms, like Particle Swarm Optimization (PSO), are powerful general-purpose optimization tools but they invariably do not come with rigorous theoretical justifications and can fail to find a global optimum. By treating PSO as a random search optimization process and repairing the famous Global Search Convergence Theorem by imposing an additional condition in the proof, we create a novel theory-based algorithm called the Double Exponential Particle Swarm optimization algorithm (DExPSO) that converges to a global optimum. In particular, we show the common practice of using uniform variates as stochastic components in PSO and related algorithms does not satisfy the sufficient conditions in DExPSO and hence may provide a reason why PSO and other nature-inspired algorithms, like QPSO, LcRiPSO, and CSO can fail to converge. Additionally, in more complicated design problems, we show that DExpSO tends to converge to the support points of the optimal design more frequently and faster than PSO and its variants do. Moreover, there is a possibility to modify other PSO variants to DExPSO variants, and such hybridization offers promising improvement in the quality of the global search. Our applications include finding designs that minimize the integrated mean squared prediction error and locally D-optimal exact designs for a 68-compartmental model to assess radioactive particles retained in the human lung after exposure. Because PSO, and more generally, metaheuristics are used across disciplines, including ecology, pharmacokinetics and pharmacodynamics studies, agriculture, engineering, and computer science, there are potentially broad and deep implications of our results.

Robust design for mixture experiments: an efficient class of exchangeable designs for Scheffé polynomials

Modern industry, engineering, and science seek to explore new methods to determine the composition that optimally describes certain product features. The goal of mixture design is to determine the proportions that optimize some property of the response. Most practitioners adopt a classical design approach since they vaguely know the model form prior to running their experiments. This work is focused on finding optimal designs for Scheffé polynomials where the parametric description of the response function may be inadequate. Our purpose is to obtain an analytical solution to the continuous problem (when possible) and to otherwise provide a numerical alternative. Theoretical results are proven for binary blends, while other strategies based on imposing restrictions and numerical approaches are provided for ternary blends. A real example illustrates the favorability of the results for a three-salt mixture setup. The geometrical properties observed in the designs encouraged the investigation of a class of exchangeable designs. The results reveal that this class of restricted designs may be widely recommended for their simplicity, high performance, and computational efficiency.

Sensitivity analysis of natural convection in a porous cavity filled with nanofluid and equipped with horizontal fins using various optimization methods and MRT-LB

In the present study, natural convection heat transfer is investigated in a porous cavity filled with Cu/water nanofluid and equipped with horizontal fins. Optimization and sensitivity analysis of the fin’s geometry, porous medium and nanofluid properties to maximize heat transfer rate is the aim of this work. To achieve this purpose, a design space is created by input parameters which include length, number of fins, distance between fins, porosity, Darcy number and volumetric fraction of the nanoparticles. Several tools have been used to implement optimization methods including the Taguchi method (TM) for design points generation, sensitivity analysis of design variables by using signal-to-noise ratio (SNR) and analysis of variance (ANOVA), response surface method (RSM) for interpolation and regression by using nonparametric regression, and genetic algorithm (GA) for finding optimum design point. The double multi-relaxation time lattice Boltzmann method (MRT-LBM) is used to analyze and simulate the flow field and heat transfer in each design point. The results show that the optimal configuration leads to an average Nusselt number of 5.56. This optimal configuration is at the length of fins L/2, the number of fins 2, the distance between fins L/12, porosity 0.8, Darcy number 0.1, and the volumetric fraction of the nanoparticles 0.02. By using the SNR results, the Darcy number and the number of fins have the most and the least effect in maximizing the average Nusselt number, respectively. The ANOVA results and global sensitivity analysis (GSA) findings further validated this conclusion.

A holistic two-stage decision-making methodology for passive and active building design strategies under uncertainty

In the last decade, many studies focused on finding optimal design solutions considering trade-offs between different aspects of building design. Accordingly, multi-objective optimization (MOO) approaches have been increasingly applied in the building industry. However, certain aspects must be deepened to ensure a more effective decision-making process in the early planning phase. On the one hand, uncertainties should be considered before making decisions to ensure the robustness of the optimal solutions; on the other hand, decisions are made at different times in building planning, and the sequential order of making decisions should be modeled. This paper presents a holistic two-stage multi-objective stochastic optimization (MOSO)-II framework to minimize the environmental impact of global warming potential (GWP) and cost throughout the entire life cycle of a building (phases of A1-A3, B4, B6 and C3-C4), considering passive and active design strategies in two consecutive stages, under uncertainty. Herein, individual/use and political/market uncertainties are considered. As a proof of concept, the proposed framework is applied in a case study for a generic zone in a multi-family terraced residential building type with solid brick construction. The advantages of the proposed framework are validated by comparing it with alternative single-stage MOSO frameworks. Results show that the proposed two-stage MOSO-II framework can deliver a smaller range of solutions with a better performance in terms of lower GWP and cost. It indicates that the proposed framework can effectively assist planners in decision-making by reducing the effort in choosing the proper solution. Secondly, the results also emphasize the importance of passive design strategies in sustainable building planning. In addition, the energy mix structure and cost of energy sources should be carefully adjusted in the future to promote a more ecologically sustainable building design.

Optimal thickness distribution design for blending composite laminates using buckling factor prediction

This article introduces a two-stage optimization approach for finding optimal blended composite laminate designs where the optimal thickness distribution is predicted based on determined stacking sequences. At each iteration, the stacking sequences are first optimized and then utilized as inputs to predict the optimal regional thicknesses through CNN-based regional buckling factor prediction. By predicting the optimal thicknesses, the proposed process simplifies the highly constrained mixed-variable blending optimization problem, expands its option domain, and reduces the number of design variables, all of which improve the combinatorial optimization efficiency. An integration of the proposed approach with a design guideline-based Genetic Algorithm is presented. The approach is applied to solve an 18-panel benchmark blending problem, the acquired solutions are compared with those of previous studies in the literature. The obtained result highlights the significant enhancement in performance of the integrated method.

A versatile strategy for hybridizing small experimental and large simulation data: A case for ceramic tape-casting process

In manufacturing industry, finding optimal design parameters for targeted properties has traditionally been guided by trial and error. However, limited data availability to few hundreds sets of experimental data in typical materials processes, the machine-learning capabilities and other data-driven modeling (DDM) techniques are too far from it to be practical. In this study, we show how a versatile design strategy, tightly coupled with physics-based modeling (PBM) data, can be applied to small set of experimental data to improve the optimization of process parameters. Our strategy uses PBM to achieve augmented data that includes essential physics: in other words, the PBM data allows the inverse design model to ‘learn’ physics, indirectly. We demonstrated the accuracy of both forward-prediction and inverse-optimization have been dramatically improved with the help of PBM data, which are relatively cheap and abundant. Furthermore, we found that the inverse model with augmented data can accurately optimize process parameters, even for ones those were not considered in the simulation. Such versatile strategy can be helpful for processes/experiments for the cases where the number of collectable data is limited, which is most of the case in industries.

Nature-inspired Metaheuristics for finding Optimal Designs for the Continuation-Ratio Models

The continuation-ratio (CR) model is frequently used in dose response studies to model a three-category outcome as the dose levels vary. Design issues for a CR model defined on an unrestricted dose interval have been discussed for estimating model parameters or a selected function of the model parameters. This paper uses metaheuristics to address design issues for a CR model defined on any compact dose interval when there are one or more objectives in the study and some are more important than others. Specifically, we use an exemplary nature-inspired metaheuristic algorithm called particle swarm optimization (PSO) to find locally optimal designs for estimating a few interesting functions of the model parameters, such as the most effective dose ($MED$), the maximum tolerated dose ($MTD$) and for estimating all parameters in a CR model. We demonstrate that PSO can efficiently find locally multiple-objective optimal designs for a CR model on various dose intervals and a small simulation study shows it tends to outperform the popular deterministic cocktail algorithm (CA) and another competitive metaheuristic algorithm called differential evolutionary (DE). We also discuss hybrid algorithms and their flexible applications to design early Phase 2 trials or tackle biomedical problems, such as different strategies for handling the recent pandemic.

Optimum design of pin-jointed aluminum structures to AA-ASD using three Meta-heuristic algorithms

The main motivation for the current study is to investigate the weight advantage of the aluminum material on pin-jointed (truss) structures. For this purpose, optimization algorithms that find the most suitable structural designs with aluminum and steel materials have been developed. The criterion for finding optimum designs is to minimize the weight of the structure and met the design constraints. The constraint functions are derived according to the “American Institute of Steel Structures – Allowable Stress Design” (AISC-ASD) and “Aluminum Association - Allowable Stress Design” (AA-ASD) design codes. To increase the reliability of the obtained results, three robust methods named the Artificial Bee Colony (ABC), Biogeography Based Optimization (BBO), and Brainstorming Optimization (BSO) algorithms are utilized for the proposed optimization problem. Four real-scale pin-jointed structures are used to research the weight advantage of aluminum structures. Three of these structures are real-scale truss structures that were previously optimized as made of steel elements by other metaheuristics. The last pin-jointed structure is the real case project which is modeled and optimized for the first time in this study. According to the obtained results, significant optimal design differences are determined between the construction of the tackled pin-jointed structures with aluminum or steel structural elements. Considering other advantages such as transportation and maintenance, using aluminum material in pin-jointed structures is recommended.

Toward more numerically stable computations of optimal experimental designs

The optimal experimental design theory is based on working with the so-called information matrix which measures the amount of information provided by a design. As a consequence, the crucial expressions that are evaluated in algorithms for finding optimal designs generally involve the information matrix. However, we show that these expressions can be based on a matrix of a simpler form, which we denote as half-information matrix. In particular, we show that by using suitable matrix decompositions, many algorithms can be formulated just in the terms of with no need to compute the information matrix. Crucially, working with the half-information matrix is more stable: its condition number is the square root of that of the information matrix. As a consequence, stabilized versions of algorithms (i.e. those based on working with ) can be applied to much more ill-conditioned models. We show on examples for both exact and approximate designs that the stabilized algorithms work on models that are unfeasible for the standard versions. We also show that the use of suitable rank-one updates of the utilized matrix decompositions can lead to a significant speed-up of the exchange-type algorithms.

Design, properties, and applications of non-pneumatic tires: A review

The interest in airless or non-pneumatic tire (NPT) structures emerges due to potential performance and environmental advantages, such as reduced rolling resistance and improved recyclability, when compared with traditional rubber-based pneumatic tires. NPTs can already be found in commercial applications such as forklifts, electric scooters, and wheelchairs. Even so, up to the knowledge of authors, there is no known established large-scale production of NPTs for most applications. This work presents a systematic literature review regarding the mechanical design and development of NPTs. A summary of structural characteristics and mechanical behaviour analysis is provided. Relevant standards, analysis strategies, materials, production techniques, and the end-of-life of tires are addressed. In the end, the authors debate the current challenges and prospects of these emergent products. Furthermore, this research aims at being used as a tool, potentiating the development of innovative and efficient, airless tire solutions. The review presented clarifies that many academic efforts are focused on finding optimal functional designs, which are frequently complex and physically demanding. These geometrically complex design proposals are, in most cases, composed of intricate cellular structures and spokes arrangements. Based on the current lack of knowledge, authors identify key research areas to be further developed such as the production, performance, and reliability of NPTs. It is debatable whether there will ever be a shortcoming of the rubber tire, or a time when fully recyclable tires are a societal obligation. Nevertheless, the number of reasons to promote the development of a reliable replacement to the traditional tire is far from scarce.

On the design of paired comparison experiments with application

In practice, paired comparison experiments involving pairs of either full or partial profiles are frequently used. When all attributes have a general common number of levels, the problem of finding optimal designs is considered in the presence of a second-order interaction model. In this setting, the D-optimal designs for the second-order interaction model have both types of pairs in which either all attributes have different levels or approximately half of the attributes are different. The proposed optimal designs can be used as a benchmark to compare any design for estimating main effects and two- and three-attribute interactions. A practical situation that incorporates the corresponding second-order interactions is covered.

Combining Finite Element and Machine Learning Methods to Predict Structures of Architectured Interlocking Ceramics

Attaining optimum structural ceramic designs calls for an extensive search in a vast design space. Herein, the thermomechanical properties of interlocked ceramics are evaluated and an approach to assist their design under thermal shock loading is proposed. A combination of finite‐element (FE) and machine learning (ML) methods is used to simulate behaviors of systems and then to sweep the vast domain of input combinations to determine the best‐performing designs, respectively. First, FE modeling is done using a limited number of interlocking architectures with different design parameters via Comsol Multiphysics. The simulation data is used for training ML algorithms. Of the examined ML algorithms, Gaussian process regression (GPR), extreme gradient boosting (XGB), and neural networks (NN) more accurately predict the thermomechanical responses of the interlocking ceramics. After validation, the combination of FE and ML approaches is applied to thermal shielding and heat sink applications to find the optimal interlocked ceramics in terms of minimal out‐of‐plane deformation and maximal heat absorption, respectively. The results show the success of the approach in finding optimum designs in a space of more than 2 million cases. The striking success of the ML approach implies its promising potential for predicting physical properties of ceramics.

Optimal test-control block designs with unequal block sizes and correlated errors

The purpose of the present article is to introduce theoretical and algorithmic approaches to address the problem of finding optimal test-control incomplete block designs with unequal block sizes where intra-block observations are correlated. Theoretical approach is used to find Etc -optimal designs analytically. In addition, due to the computational complexity of theoretical methods, in this article a two-phase optimization algorithm is proposed to construct ϕ -optimal or nearly ϕ -optimal designs. The effectiveness of the proposed algorithm is validated by comparing our results with optimal designs presented in several prior studies. Our algorithm has the advantages of being independent from the sizes of blocks, structure of correlation, and the optimality criteria. Moreover, it takes only a few minutes to obtain the optimal designs.

A method for global potential assessment of roof integrated two-stage solar concentrators (TSSCs) at district scale

This paper proposes a core method for integrating two-stage solar concentrators (TSSCs) as roof-integrated energy supply systems at the district scale. However, the performance of these systems not only depends on good configurations but also on optimal building typologies. The design process of buildings-integrated TSSCs on a district scale reflects a complex multi-criteria problem, where several conflicting concerns from multiple stakeholders need to be addressed. Thus, the proposed method aims to support the multi-criteria decision-making process in the early-stage design of sustainable districts utilizing TSSC technology. The method contributes to the design of the TSSCs relative to district designs. Our method enables design generation according to a set of decision variables related to the district (roof type and slope, orientation, building height, width, and length), and TSSC (type, geometric ratio, separation distance between mirrors, modules number, and solar cell size). The method uses a parametric modeling approach combined with a multi-objective optimization algorithm (NSGA-II) that enables design optimization of the district and TSSCs in two consecutive steps. The two-step design optimization allows finding optimal district designs for maximizing cumulative direct normal irradiance (DNI) and minimizing the total energy demand, and TSSC designs for maximizing average monthly load match index (av.LMI) (a ratio of energy yield to demand) and minimizing the average covered roof area occupied by modules. We validated the proposed method in an illustrative case study of ‘Buckower Felder’, a district in the city of Berlin (Germany). The illustrative application shows that our method enables the performance-driven, generative design of both; district and building-integrated TSSCs and searches for the most appropriate solutions i.e., urban designs for maximum DNI and minimum demand, and TSSC designs for maximum av.LMI and minimum covered roof area. Results indicate that there is a trade-off between objectives, where district design with high DNI (>4500 kW/m2/h/month) reflects high energy demand (8.6e + 05 kWh/month). Similarly, TSSC designs ensuring high av.LMI (>1.0) require a large roof area (>47 %). Given these trade-offs, the method can meaningfully support the decision-making process. Thus, the method allows for large-scale integration of TSSCs with buildings at an urban scale and supports solar energy planning and energy transition policies for sustainable districts.

Numerical Methods for Finding [formula omitted]-optimal Designs Analytically

The traditional way in statistics to find optimal designs for regression models is an analytical approach. Technical conditions that may be restrictive in practice are sometimes imposed to obtain the analytical results. Even then, the mathematical technique is invariably not amendable to find an optimal design under a different criterion or for the same criterion with a slightly changed model, suggesting that developing flexible and effective algorithms to search for the optimum is very useful. In particular, numerical results from an algorithm can be helpful to find analytical descriptions of optimal designs. As an example, particle swarm optimization has been shown to be quite effective for finding optimal designs for hard design problems and this paper demonstrates how its output can be used to find new analytic A-optimal approximate designs for the Gamma and inverse Gaussian models, each with the inverse link function. The methodology is quite general and may be applied to find analytical A-optimal designs for other models, like the Poisson model with the log link function, or other types of optimal designs.

Optimizing Buildings’ Life Cycle Performance While Allowing Diversity in the Early Design Stage

The main considerations in the early stage of architectural design are usually related to form and function. At the same time, with the growing concern regarding energy saving and carbon emission reduction, the parameters for the construction and physical quality of buildings are receiving more attention at the conceptual and schematic design stages. Diverse design options can emerge with the large number of variables to be considered in these stages. Moreover, the combined efforts to reduce buildings’ life cycle environmental impacts and cost, as well as the non-linear and often tradeoff relationship between the two objectives, make finding optimal design solutions for buildings’ life cycle performance complicated. Previous studies have established workflows to optimize buildings’ life cycle energy consumption, GWP, and/or cost; however, architectural design diversity has not been sufficiently discussed at the same time. In this study, a parametric optimization design process is established, aiming at minimizing the building’s operational energy consumption, life cycle environmental impacts, and life cycle cost. The setting of variables, as well as the workflows of the optimization process, is discussed from the perspective of both life cycle performance and architectural design diversity. A small-scale exhibition hall in China’s cold climate zone is selected as a case study. To approach the best design process applicable to this case, the optimal solution sets from different workflows under different variable settings are compared. The results show that by setting geometric and material variables in different steps in the entire optimization process, the resulting solutions can be a balance of architectural design and performance. In this case study, optimizing all of the design variables in one-step turned out to provide the best balance between design diversity and life cycle performance in the early design stage.

Application of computational modeling to study the parameters of dusty air flows in subway

To date, the subway is the most promising type of urban public transport, which is used daily by thousands of passengers in the world. In modern subways, the problem is the content of suspended dust in the air, which exceeds the maximum permissible concentration. Especially dangerous is the dust fraction PM2.5 or less, capable of penetrating through the biological barriers of the human body. This article discusses the results of a number of studies aimed at determining the concentration, fractional and chemical composition of such dust in subway structures. One of the ways to combat dustiness of the air in subways is its filtration. The authors propose the use of filters installed in the jetty ventilation breakdowns, the air flow through which is provided by the piston action of trains. Determination of optimal design and operational parameters of such filters is carried out using computer modeling by the finite volume method. The article provides an overview of the existing computational mathematical models of the movement of dusty air as a multiphase flow, and considers the possibility of their application to the conditions of underground subway structures.

Generative Design by Reinforcement Learning: Enhancing the Diversity of Topology Optimization Designs

Generative design refers to computational design methods that can automatically conduct design exploration under constraints defined by designers. Among many approaches, topology optimization-based generative designs aim to explore diverse topology designs, which cannot be represented by conventional parametric design approaches. Recently, data-driven topology optimization research has started to exploit artificial intelligence, such as deep learning or machine learning, to improve the capability of design exploration. This study proposes a reinforcement learning (RL) based generative design process, with reward functions maximizing the diversity of topology designs. We formulate generative design as a sequential problem of finding optimal design parameter combinations in accordance with a given reference design. Proximal Policy Optimization is used as the learning framework, which is demonstrated in the case study of an automotive wheel design problem. To reduce the heavy computational burden of the wheel topology optimization process required by our RL formulation, we approximate the optimization process with neural networks. With efficient data preprocessing/augmentation and neural architecture, the neural networks achieve a generalized performance and symmetricity-reserving characteristics. We show that RL-based generative design produces a large number of diverse designs within a short inference time by exploiting GPU in a fully automated manner. It is different from the previous approach using CPU which takes much more processing time and involving human intervention.

A methodological approach for the optimal design of the toroidal field coils of a tokamak device using artificial intelligence

As prototypes of future commercial tokamaks, DEMOs nuclear fusion power plants are expected to be able to produce cost-effective electrical power. In this view, an optimized design becomes crucial in the whole engineering workflow. Up to now, the design of one of the most critical components, the cross-section of each of the toroidal field coils inner leg winding pack, was performed using a sequential trial-and-error procedure. In this work, a novel comprehensive approach is proposed to include all the main design aspects into a unified tool taking advantage of artificial neural networks for faster computation in finding optimal design configurations. This procedure overcomes several difficulties including dealing with both real-valued and discrete design variables, the significant CPU-time of magneto-structural analysis and also guarantees the optimality for the winding pack configuration. The proposed methodology was demonstrated for the 2019 ENEA DEMO configuration which includes 16 toroidal field coils, made-up of 6 Nb3Sn double layers and a Wind and React manufacturing technique.