Thomas Lewis Hill

Indirect reduced-order models (ROMs) aim to dramatically reduce the cost of simulating nonlinear dynamic behaviour in large nonlinear finite element systems. Applied force based methods, most notably implicit condensation and expansion, are a classical approach. This work summarises and critiques the past 25 years of development whilst making the case for the continued relevance of the method in the context of the more recent and mathematically rigorous invariant manifold based reduction approach. To facilitate this, a novel applied force framework is developed: the Applied Force Reduction (AFR) method. This includes a more precise approach to evaluate the general accuracy of indirect ROMs through the distinction between verification (the correct implementation of the method) and validation (evaluating the accuracy of the ROM in replicating the full-order system). The former is the focus of this work, and a general verification algorithm is presented which is implemented in the new open source software package. The verification process involves studying the convergence of the underlying polynomial models to create a suitable dataset to inform the reduced-order model whilst minimising interactions with the FE model. This process is completely automated and allows for the efficient construction of ROMs capturing the dynamic behaviour of large systems. This efficiency is demonstrated through consideration of a curved double arch with over 1,000,000 degrees-of-freedom. A two degree-of-freedom ROM is constructed and verified in 60 min, on a desktop computer, and is able to accurately reproduce the internal resonance found in the full-order system.

Reduced-order modelling describes a powerful set of techniques for dramatically reducing the cost of dynamic simulations in complex engineering systems, with broad application for designing and understanding nonlinearity. For Reduced-Order Models (ROMs) to be useful in an industrial context, an analyst must have a high degree of confidence that the results reported by the ROM are reflective of the full-order system. Typically, some full-order results are required to demonstrate the accuracy of a particular ROM. For large and complex systems, this will significantly undercut the initial computational motivations for creating said ROM in the first place. This work presents a novel, full-order simulation-free validation method for applied force ROMs of conservative, geometrically nonlinear systems. This is achieved by testing prospective results to see if they could exist in a series of larger ROMs. If they can, then the smaller ROM has sufficiently approximated the relevant section of the full-order invariant manifold. This allows potential reduction bases to be rapidly tested and a valid, ideally small, basis chosen. This is especially valuable in an applied force context where any low-frequency mode could be required in the reduction basis. A two degree-of-freedom ROM, describing a 100,000 degree-of-freedom finite element model of a curved double arch exhibiting internal resonance, is constructed and fully validated in under 15 min on a desktop computer. Refining the mesh such that there are 1,000,000 degrees-of-freedom increases this time to 170 min. Equivalent full-order simulation validation would take days. Furthermore, no a priori assumption is made about the nature or even the presence of the internal resonance, which is not the case for other reduced-order approaches. This is an important step towards accessible and automatable reduced-order modelling software.

Indirect Reduced-Order Models (ROMs) aim to dramatically reduce the cost of simulating nonlinear dynamic behaviour in large nonlinear finite element systems. Applied force based methods, most notably implicit condensation and expansion, are a classical approach. This work summarises and critiques the past 25 years of development whilst making the case for the continued relevance of the method in the context of the more recent and mathematically rigorous invariant manifold based reduction approach. To facilitate this, a novel applied force framework is developed: the Applied Force Reduction (AFR) method. This includes a more precise approach to evaluate the general accuracy of indirect ROMs through the distinction between verification (the correct implementation of the method) and validation (evaluating the accuracy of the ROM in replicating the full-order system). The former is the focus of this work, and a general verification algorithm is presented which is implemented in the new open source AFR Tool software package. The verification process involves studying the convergence of the underlying polynomial models to select an optimal dataset that minimises interaction with the FE model. This process is completely automated and allows for the efficient construction of ROMs capturing the dynamic behaviour of large systems. This efficiency is demonstrated through consideration of a curved double arch with over 1,000,000 degrees-of-freedom. A two degree-of-freedom ROM is constructed and verified in 60 minutes, on a desktop computer, and is able to accurately reproduce the internal resonance found in the full-order system.

Reduced-order modelling describes a powerful set of techniques for dramatically reducing the cost of dynamic simulations in complex engineering systems, with broad application for designing and understanding nonlinearity. For Reduced-Order Models (ROMs) to be useful in an industrial context, an analyst must have a high degree of confidence that the results reported by the ROM are reflective of the full-order system. Typically, some full-order results are required to demonstrate the accuracy of a particular ROM. For large and complex systems, this will significantly undercut the initial computational motivations for creating said ROM in the first place. This work presents a novel, full-order simulation-free validation method for applied force ROMs. This is achieved by testing prospective results to see if they could exist in a series of larger ROMs. If they can, then the smaller ROM has sufficiently approximated the relevant section of the full-order invariant manifold. This allows potential reduction bases to be rapidly tested and a valid, optimal ROM chosen. This is especially valuable in an applied force context where any low-frequency mode could be required in the reduction basis. A two degree-of-freedom ROM, describing a 100,000 degree-of-freedom finite element model of a curved double arch exhibiting internal resonance, is constructed and fully validated in under 15 minutes on a desktop computer. Equivalent full-order simulation validation would take days. Furthermore, no a priori assumption is made about the nature or even the presence of the internal resonance, which is not the case for other reduced-order approaches. This is an important step towards accessible and automatable reduced-order modelling software.

Considering the effect of parameter changes in Finite-Element (FE) models requires repeated dynamic FE simulations, which is very common in fields such as optimisation. Reduced-order modelling is an effective method to alleviate computational costs in simulations. Indirect forced-based methods, such as the Implicit Condensation and Expansion (ICE), are widely used for constructing Reduced Order Models (ROMs), which can capture the dynamics of FE models if the accuracy of static fitting is ensured. In this paper, we show that variations in parameters within an FE model can be efficiently considered by ICE-based ROMs. A parameter analysis of a wing-inspired FE model is conducted to demonstrate the technique. The results show that, when varying parameters only related to nonconservative forces, the ROM can accurately capture the dynamics of the FE model within a wide parameter range whilst achieving high computational efficiency of around 2,000 times that of dynamic simulations in FE software. Regarding parameters related to conservative terms, the ROM needs to be reconstructed to adjust for the variations in these parameters, leading to additional costs. However, the ROMs are still around 10 times faster than dynamic FE simulations for the proposed FE model. Considering the results in the parameter analysis, we demonstrate the computational efficiency of ROMs in gradient estimation, and illustrate their potential in optimisation.

Understanding the behaviour and characteristics of the secondary radiation force (SRF) near boundaries is crucial for advancing ultrasonic manipulation. Investigating the development and interaction of the SRF near boundaries offers valuable insights into particle dynamics within confined environments. This study examines SRF generation through an induced boundary that perturbs the primary radiation force in a mid-focused standing wave field. The experiments are first conducted on an expanded polystyrene sphere in air, without a boundary. The pressure field is perturbed such that the central node oscillates at various frequencies using a stepped-sine method. This allows the natural frequency of a trapped object to be identified. In this case, the object’s movement is treated like a damped oscillator. Next, a boundary is introduced to observe its influence on the natural frequency of the particle and hence the restoring force coefficient. The inclusion of the boundary leads to a significant increase in natural frequency, and the presence of the SRF is quantified by comparing the natural frequency measured with and without the boundary. Notably, as the particle nears the boundary, it encounters an attractive SRF. To explore this, boundaries of different dimensions are tested, revealing that the attraction increases with the boundary size and the attraction distance slightly varies. These findings not only enhance theoretical models of ultrasonic manipulation but also enable more precise control in practical applications such as ultrasonic levitation, medical ultrasound, and microfluidics. The results provide a deeper understanding of SRF behaviour near boundaries, contributing to the advancement of both scientific knowledge and technological applications.

Snap-through is a nonlinear jump phenomenon that is increasingly being designed into engineering structures. The rich and varied dynamics of snap-through present a significant bottleneck in the design process as costly dynamic simulations are required to ensure predictable behaviour. The purpose of this research is to investigate the use of a recent projection-based, reduced-order modelling technique, namely, implicit condensation and expansion with inertial compensation, to reproduce the snap-through behaviours found in bistable finite element systems in analogous models with minimal degrees of freedom. In this work, the free response of a simple mass-spring oscillator is used to differentiate different types of snap-through. This provides a framework for discussing the dynamics of a family of buckled beams. The ability of the reduced-order models to faithfully reproduce these dynamics is investigated by testing the existence and stability of predicted periodic orbits. It is shown that the minimum projection basis required to capture snap-through accurately can be found through considering the minimum energy required for different types of snap-through. For bistable beams, this can be further simplified to just requiring an eigenvalue analysis at the most unstable equilibrium.

Numerous powerful methods exist for developing reduced-order models (ROMs) from finite element (FE) models. Ensuring the accuracy of these ROMs is essential; however, the validation using dynamic responses is expensive. In this work, we propose a method to ensure the accuracy of ROMs without extra dynamic FE simulations. It has been shown that the well-established implicit condensation and expansion (ICE) method can produce an accurate ROM when the FE model’s static behaviour are captured accurately. However, this is achieved via a fitting procedure, which may be sensitive to the selection of load cases and ROM’s order, especially in the multi-mode case. To alleviate this difficulty, we define an error metric that can evaluate the ROM’s fitting error efficiently within the displacement range, specified by a given energy level. Based on the fitting result, the proposed method provides a strategy to enrich the static dataset, i.e. additional load cases are found until the ROM’s accuracy reaches the required level. Extending this to the higher-order and multi-mode cases, some extra constraints are incorporated into the standard fitting procedure to make the proposed method more robust. A curved beam is utilised to validate the proposed method, and the results show that the method can robustly ensure the accuracy of the static fitting of ROMs.