H.F. Maathuis
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1
Optimisation is at the heart of modern engineering. From reducing aircraft emissions to designing safer cars or tailoring drugs for specific diseases, the goal is to find the best solution among countless possibilities. Yet real-world systems are complex, and every design must meet strict constraints related to safety, performance, and physical laws. A design that performs well but violates just one constraint, such as structural failure during flight or non-compliance, is not desirable.
To assess a design’s performance, engineers rely on complex computer simulations that capture physical processes like drag or structural deformation of an aircraft. These simulations often behave like black boxes, as they are expensive to run and the relationship between inputs and outputs is typically non-linear and opaque. This makes exhaustive search of the design space impossible and necessitates data-efficient optimisation strategies.
Bayesian Optimisation (BO) has emerged as a state-of-the-art method for optimising expensive black-box functions, offering a principled way to make the most of limited data. It builds a probabilistic model of the system to guide evaluations efficiently, balancing exploration of uncertain regions with exploitation of promising designs. Although BO has been widely adopted across scientific and engineering domains, it continues to face significant challenges in scenarios that involve both high-dimensional input spaces and complex feasibility constraints. These settings form the primary focus of this thesis.
The first contribution of this work is to show why techniques that work in unconstrained settings, such as random subspace embeddings or simple model priors, often fail under constraints. To address this, the thesis introduces supervised subspace methods and revisits dimensionality-scaled priors that improve both robustness and feasibility discovery in constrained problems.
Second, it proposes scalable strategies to model thousands of constraints, which arise, for example, in structural or aerospace design. Rather than modelling each constraint separately, the thesis uses dimensionality reduction to reduce input and output dimensionality, making constrained optimisation tractable at scale.
Finally, it develops methods for multi-source optimisation, where both accurate and approximate models are available. A modelling framework captures their discrepancies and a novel acquisition strategy balances information gain, cost, and constraint satisfaction, accelerating convergence under tight budgets.
Together, these contributions extend the reach of BO to realistic, simulation-based engineering problems. The resulting tools are broadly applicable and help bridge the gap between theoretical advances in optimisation and the practical demands of high-stakes engineering design.
...
To assess a design’s performance, engineers rely on complex computer simulations that capture physical processes like drag or structural deformation of an aircraft. These simulations often behave like black boxes, as they are expensive to run and the relationship between inputs and outputs is typically non-linear and opaque. This makes exhaustive search of the design space impossible and necessitates data-efficient optimisation strategies.
Bayesian Optimisation (BO) has emerged as a state-of-the-art method for optimising expensive black-box functions, offering a principled way to make the most of limited data. It builds a probabilistic model of the system to guide evaluations efficiently, balancing exploration of uncertain regions with exploitation of promising designs. Although BO has been widely adopted across scientific and engineering domains, it continues to face significant challenges in scenarios that involve both high-dimensional input spaces and complex feasibility constraints. These settings form the primary focus of this thesis.
The first contribution of this work is to show why techniques that work in unconstrained settings, such as random subspace embeddings or simple model priors, often fail under constraints. To address this, the thesis introduces supervised subspace methods and revisits dimensionality-scaled priors that improve both robustness and feasibility discovery in constrained problems.
Second, it proposes scalable strategies to model thousands of constraints, which arise, for example, in structural or aerospace design. Rather than modelling each constraint separately, the thesis uses dimensionality reduction to reduce input and output dimensionality, making constrained optimisation tractable at scale.
Finally, it develops methods for multi-source optimisation, where both accurate and approximate models are available. A modelling framework captures their discrepancies and a novel acquisition strategy balances information gain, cost, and constraint satisfaction, accelerating convergence under tight budgets.
Together, these contributions extend the reach of BO to realistic, simulation-based engineering problems. The resulting tools are broadly applicable and help bridge the gap between theoretical advances in optimisation and the practical demands of high-stakes engineering design.
...
Optimisation is at the heart of modern engineering. From reducing aircraft emissions to designing safer cars or tailoring drugs for specific diseases, the goal is to find the best solution among countless possibilities. Yet real-world systems are complex, and every design must meet strict constraints related to safety, performance, and physical laws. A design that performs well but violates just one constraint, such as structural failure during flight or non-compliance, is not desirable.
To assess a design’s performance, engineers rely on complex computer simulations that capture physical processes like drag or structural deformation of an aircraft. These simulations often behave like black boxes, as they are expensive to run and the relationship between inputs and outputs is typically non-linear and opaque. This makes exhaustive search of the design space impossible and necessitates data-efficient optimisation strategies.
Bayesian Optimisation (BO) has emerged as a state-of-the-art method for optimising expensive black-box functions, offering a principled way to make the most of limited data. It builds a probabilistic model of the system to guide evaluations efficiently, balancing exploration of uncertain regions with exploitation of promising designs. Although BO has been widely adopted across scientific and engineering domains, it continues to face significant challenges in scenarios that involve both high-dimensional input spaces and complex feasibility constraints. These settings form the primary focus of this thesis.
The first contribution of this work is to show why techniques that work in unconstrained settings, such as random subspace embeddings or simple model priors, often fail under constraints. To address this, the thesis introduces supervised subspace methods and revisits dimensionality-scaled priors that improve both robustness and feasibility discovery in constrained problems.
Second, it proposes scalable strategies to model thousands of constraints, which arise, for example, in structural or aerospace design. Rather than modelling each constraint separately, the thesis uses dimensionality reduction to reduce input and output dimensionality, making constrained optimisation tractable at scale.
Finally, it develops methods for multi-source optimisation, where both accurate and approximate models are available. A modelling framework captures their discrepancies and a novel acquisition strategy balances information gain, cost, and constraint satisfaction, accelerating convergence under tight budgets.
Together, these contributions extend the reach of BO to realistic, simulation-based engineering problems. The resulting tools are broadly applicable and help bridge the gap between theoretical advances in optimisation and the practical demands of high-stakes engineering design.
To assess a design’s performance, engineers rely on complex computer simulations that capture physical processes like drag or structural deformation of an aircraft. These simulations often behave like black boxes, as they are expensive to run and the relationship between inputs and outputs is typically non-linear and opaque. This makes exhaustive search of the design space impossible and necessitates data-efficient optimisation strategies.
Bayesian Optimisation (BO) has emerged as a state-of-the-art method for optimising expensive black-box functions, offering a principled way to make the most of limited data. It builds a probabilistic model of the system to guide evaluations efficiently, balancing exploration of uncertain regions with exploitation of promising designs. Although BO has been widely adopted across scientific and engineering domains, it continues to face significant challenges in scenarios that involve both high-dimensional input spaces and complex feasibility constraints. These settings form the primary focus of this thesis.
The first contribution of this work is to show why techniques that work in unconstrained settings, such as random subspace embeddings or simple model priors, often fail under constraints. To address this, the thesis introduces supervised subspace methods and revisits dimensionality-scaled priors that improve both robustness and feasibility discovery in constrained problems.
Second, it proposes scalable strategies to model thousands of constraints, which arise, for example, in structural or aerospace design. Rather than modelling each constraint separately, the thesis uses dimensionality reduction to reduce input and output dimensionality, making constrained optimisation tractable at scale.
Finally, it develops methods for multi-source optimisation, where both accurate and approximate models are available. A modelling framework captures their discrepancies and a novel acquisition strategy balances information gain, cost, and constraint satisfaction, accelerating convergence under tight budgets.
Together, these contributions extend the reach of BO to realistic, simulation-based engineering problems. The resulting tools are broadly applicable and help bridge the gap between theoretical advances in optimisation and the practical demands of high-stakes engineering design.
Bayesian Optimisation (BO) is a sample-efficient method for optimising expensive black-box functions, making it particularly suitable for engineering problems where gradients are unavailable and evaluating the objective or constraints is computationally costly. However, such problems often involve high-dimensional inputs and a large number of constraints, posing significant challenges for standard BO frameworks. While prior research has addressed scalability with respect to high-dimensional inputs in constrained settings, efficiently handling large numbers of constraints, i.e. high-dimensional outputs, remains an open problem. This work introduces Autoencoder-Enhanced Joint Dimensionality Reduction for Constrained BO (AERO-BO), a framework that performs dimensionality reduction in both the input (design variable) and output (objective and constraint) spaces via autoencoders. These autoencoders are trained online, requiring no pre-training, and their respective latent representations are connected through Gaussian Processes, which serve as surrogate models during optimisation. By operating in a joint latent space, AERO-BO enables scalable and efficient optimisation in settings with hundreds of design variables and thousands of black-box constraints.
...
Bayesian Optimisation (BO) is a sample-efficient method for optimising expensive black-box functions, making it particularly suitable for engineering problems where gradients are unavailable and evaluating the objective or constraints is computationally costly. However, such problems often involve high-dimensional inputs and a large number of constraints, posing significant challenges for standard BO frameworks. While prior research has addressed scalability with respect to high-dimensional inputs in constrained settings, efficiently handling large numbers of constraints, i.e. high-dimensional outputs, remains an open problem. This work introduces Autoencoder-Enhanced Joint Dimensionality Reduction for Constrained BO (AERO-BO), a framework that performs dimensionality reduction in both the input (design variable) and output (objective and constraint) spaces via autoencoders. These autoencoders are trained online, requiring no pre-training, and their respective latent representations are connected through Gaussian Processes, which serve as surrogate models during optimisation. By operating in a joint latent space, AERO-BO enables scalable and efficient optimisation in settings with hundreds of design variables and thousands of black-box constraints.
Design optimization offers the potential to develop lightweight aircraft structures with reduced environmental impact. Due to the high number of design variables and constraints, these challenges are typically addressed using gradient-based optimization methods to maintain efficiency, however overlooking the global design space. Moreover, gradients are frequently unavailable. Bayesian optimization presents a promising gradient-free alternative, enabling sample-efficient global optimization through probabilistic surrogate models. Although Bayesian optimization has shown its effectiveness for problems with a small number of design variables, it struggles to scale to high-dimensional problems, particularly when incorporating large-scale constraints. This challenge is especially pronounced in aeroelastic tailoring, where directional stiffness properties are integrated into the structural design to manage aeroelastic deformations and enhance both aerodynamic and structural performance. Ensuring the safe operation of the system requires simultaneously addressing constraints from various analysis disciplines, making global design space exploration even more complex. This study seeks to address this issue by employing high-dimensional Bayesian optimization combined with dimensionality reduction to tackle the optimization challenges in aeroelastic tailoring. The proposed approach is validated through experiments on a well-known benchmark case, as well as its application to the aeroelastic tailoring problem, demonstrating the feasibility of Bayesian optimization for high-dimensional problems with large-scale constraints.
...
Design optimization offers the potential to develop lightweight aircraft structures with reduced environmental impact. Due to the high number of design variables and constraints, these challenges are typically addressed using gradient-based optimization methods to maintain efficiency, however overlooking the global design space. Moreover, gradients are frequently unavailable. Bayesian optimization presents a promising gradient-free alternative, enabling sample-efficient global optimization through probabilistic surrogate models. Although Bayesian optimization has shown its effectiveness for problems with a small number of design variables, it struggles to scale to high-dimensional problems, particularly when incorporating large-scale constraints. This challenge is especially pronounced in aeroelastic tailoring, where directional stiffness properties are integrated into the structural design to manage aeroelastic deformations and enhance both aerodynamic and structural performance. Ensuring the safe operation of the system requires simultaneously addressing constraints from various analysis disciplines, making global design space exploration even more complex. This study seeks to address this issue by employing high-dimensional Bayesian optimization combined with dimensionality reduction to tackle the optimization challenges in aeroelastic tailoring. The proposed approach is validated through experiments on a well-known benchmark case, as well as its application to the aeroelastic tailoring problem, demonstrating the feasibility of Bayesian optimization for high-dimensional problems with large-scale constraints.
Design optimisation potentially leads to lightweight aircraft structures with lower environmental impact. Due to the high number of design variables and constraints, these problems are ordinarily solved using gradient-based optimisation methods, leading to a local solution in the design space while the global space is neglected. Bayesian Optimisation is a promising path towards sample-efficient, global optimisation based on probabilistic surrogate models. While for problems with a low number of design variables, Bayesian Optimisation methods have demonstrated their strength, the scalability to high-dimensional problems while incorporating large-scale constraints is still lacking. Especially in aeroelastic tailoring where directional stiffness properties are embodied into the structural design of aircraft, to control aeroelastic deformations and to increase the aerodynamic and structural performance, the safe operation of the system needs to be ensured by involving constraints resulting from different analysis disciplines. Hence, a global design space search becomes even more challenging. The present study attempts to tackle the problem by using high-dimensional Bayesian Optimisation in combination with a dimensionality reduction approach to solve the optimisation problem occurring in aeroelastic tailoring, presenting a novel approach for high-dimensional problems with large-scale constraints. Experiments on well-known benchmark cases with black-box constraints show that the proposed approach can incorporate large-scale constraints.
...
Design optimisation potentially leads to lightweight aircraft structures with lower environmental impact. Due to the high number of design variables and constraints, these problems are ordinarily solved using gradient-based optimisation methods, leading to a local solution in the design space while the global space is neglected. Bayesian Optimisation is a promising path towards sample-efficient, global optimisation based on probabilistic surrogate models. While for problems with a low number of design variables, Bayesian Optimisation methods have demonstrated their strength, the scalability to high-dimensional problems while incorporating large-scale constraints is still lacking. Especially in aeroelastic tailoring where directional stiffness properties are embodied into the structural design of aircraft, to control aeroelastic deformations and to increase the aerodynamic and structural performance, the safe operation of the system needs to be ensured by involving constraints resulting from different analysis disciplines. Hence, a global design space search becomes even more challenging. The present study attempts to tackle the problem by using high-dimensional Bayesian Optimisation in combination with a dimensionality reduction approach to solve the optimisation problem occurring in aeroelastic tailoring, presenting a novel approach for high-dimensional problems with large-scale constraints. Experiments on well-known benchmark cases with black-box constraints show that the proposed approach can incorporate large-scale constraints.
Exploring multi-fidelity aeroelastic tailoring
Prospect and model assessment
The design and optimisation of aircraft wings are critical tasks in aerospace engineering, requiring a balance between structural integrity, aerostructural performance, and manufacturability. This multifaceted challenge involves the interplay of various disciplines, each with distinct parameters and constraints. Traditional design approaches often fall short, necessitating advanced methodologies like Multidisciplinary Design Optimisation (MDO). MDO integrates aerodynamic, structural, and manufacturability analyses to explore a vast design space and identify optimal solutions that meet performance, safety, and cost criteria. Advancements in manufacturing technologies and material sciences have led to the increased use of composite materials, which offer an excellent weight-to-strength ratio. Aeroelastic Tailoring, which incorporates directional stiffness into structural design, further enhances performance. This study employs lamination parameters to efficiently represent composite layups within a gradient-based optimisation process, aiming to minimise weight while ensuring feasibility across multiple constraints. The work highlights the challenge of optimising aircraft designs using multiple models of varying fidelity. Traditional sequential optimisation approaches, which progressively integrate disciplines, may miss potential superior designs due to limited initial information. Instead, concurrent optimisation schemes are explored, utilising both low-fidelity (beam-based) and high-fidelity (shell-based) models. This approach promises structural feasibility, reduces computational costs, and incorporates high-fidelity information early in the design process. The envisioned methodology bridges different design stages, enabling better overall aircraft performance. By aligning and comparing a beam-based and shell-based model, the study explores their use in multi-fidelity optimisation. The results demonstrate the feasibility and benefits of this approach, offering a robust framework for future aircraft design projects.
...
The design and optimisation of aircraft wings are critical tasks in aerospace engineering, requiring a balance between structural integrity, aerostructural performance, and manufacturability. This multifaceted challenge involves the interplay of various disciplines, each with distinct parameters and constraints. Traditional design approaches often fall short, necessitating advanced methodologies like Multidisciplinary Design Optimisation (MDO). MDO integrates aerodynamic, structural, and manufacturability analyses to explore a vast design space and identify optimal solutions that meet performance, safety, and cost criteria. Advancements in manufacturing technologies and material sciences have led to the increased use of composite materials, which offer an excellent weight-to-strength ratio. Aeroelastic Tailoring, which incorporates directional stiffness into structural design, further enhances performance. This study employs lamination parameters to efficiently represent composite layups within a gradient-based optimisation process, aiming to minimise weight while ensuring feasibility across multiple constraints. The work highlights the challenge of optimising aircraft designs using multiple models of varying fidelity. Traditional sequential optimisation approaches, which progressively integrate disciplines, may miss potential superior designs due to limited initial information. Instead, concurrent optimisation schemes are explored, utilising both low-fidelity (beam-based) and high-fidelity (shell-based) models. This approach promises structural feasibility, reduces computational costs, and incorporates high-fidelity information early in the design process. The envisioned methodology bridges different design stages, enabling better overall aircraft performance. By aligning and comparing a beam-based and shell-based model, the study explores their use in multi-fidelity optimisation. The results demonstrate the feasibility and benefits of this approach, offering a robust framework for future aircraft design projects.