A physics-based model order reduction framework for nonlinear contact problems
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In many engineering applications, Finite Element (FE) models are vital for simulating real-world phenomena. However, resolving complex geometries or capturing nonlinear behaviour often makes these FE models computationally expensive. This challenge is particularly critical in scenarios that require repeated simulations, such as design optimisation, parametric studies, or uncertainty quantification. In cases such as digital twin applications, near-real-time simulations are needed, further increasing the computational burden. Model Order Reduction (MOR) techniques aim to address these challenges by reducing the dimensionality of the original Full Order Model (FOM), thereby producing a Reduced Order Model (ROM) that is significantly smaller and computationally more efficient, while retaining the core physics of the FOM. Most conventional MOR techniques are data-driven. The most widely used among them, Proper Orthogonal Decomposition (POD), depends on the simulation data of the very FOM it seeks to reduce. While effective for interpolation within the well-sampled parameter space, POD is generally unreliable for extrapolation to unseen parameter ranges. As an alternative, the primary objective of this thesis is to develop a physics-based MOR framework that does not rely on any training data. The proposed approach utilises Modal Derivatives (MDs), which are the first-order derivatives of vibration modes, and has been demonstrated to capture nonlinear behaviour within a reduced-order basis. In this thesis, this framework is adapted to address quasi-static geometrically nonlinear solid mechanics and unilateral finite deformation contact problems. Initially, in Chapter 2, the concept of MDs is explained in detail and applied to quasi-static geometrically nonlinear problems. To ensure an orthogonal projection and, in turn, numerical stability of the reduction, these MDs are orthogonalised using the modified Gram-Schmidt process. An online greedy selection is then employed to identify the most significant orthogonalised MDs, achieving an efficient reduction. This approach is validated through various test cases, demonstrating its validity and effectiveness in different scenarios. Chapter 3 extends this framework to handle unilateral finite deformation contact problems where contact constraints are enforced using the Lagrange multipliers. In addition to the orthogonalisation and greedy basis selection strategies introduced earlier in Chapter 2, an exclusion domain is introduced, based on the location of maximum shear stress from Hertzian Contact theory. Within this exclusion domain, the displacement degrees of freedom, along with the Lagrange multipliers enforcing the contact constraints, are maintained at full FE resolution. In contrast, the remaining displacement degrees of freedom are reduced using the earlier framework. This hybrid, MD-based strategy performs well across various unilateral benchmark contact problems, highlighting the effectiveness of the chosen approach. Although the techniques established in Chapters 2 and 3 achieve reduced dimensionality, the resulting improvements in simulation time remain modest. The principal challenge is that, despite reducing the primary variable (displacement), computing the tangent stiffness matrix still requires integration over the entire FE mesh. This integration step remains a significant bottleneck in reducing nonlinear FE problems. To overcome this and accelerate simulations, the developed MOR techniques are combined with a hyper-reduction strategy. Specifically, Energy Conserving Sampling and Weighing (ECSW) is employed, which selects a subset of elements for integration, allowing for the efficient computation of reduced quantities. This combined approach is tested in the next chapter through a Bayesian inference case study, which requires multiple simulations and serves as an ideal validation for the MOR framework enhanced with hyper-reduction. In this context, two quasi-static, geometrically nonlinear problems from Chapter 2 are utilised, providing uncertainty quantification with significantly faster runtimes. In the penultimate chapter of the thesis, all the developed tools are integrated to achieve hyper-reduction of unilateral finite deformation frictional contact problems. Coulomb friction is incorporated into the contact formulation established in Chapter 3. The ECSW hyper-reduction technique is adapted accordingly to include the exclusion zone defined for reducing contact problems. Two examples are considered: first, one of the benchmark problems used in Chapter 3 is revisited to validate this improved framework. Then, a more complex 3D case involving the movement of a catheter inside a soft-tissue-like arterial tube is examined. The results show that the proposed physics-based model order reduction, combined with ECSW, can effectively capture complex, nonlinear, and contact phenomena with considerably lower computational cost, making it a promising tool for large-scale simulations in computational mechanics. Overall, the results presented in this thesis demonstrate the potential of a fully physics-based model order reduction framework, combined with ECSW hyper-reduction, to efficiently simulate non-linear contact FE problems. By eliminating reliance on training data, the developed methods based on MDs achieve substantial computational savings while preserving critical mechanical responses. The successful application to both benchmark and complex use cases underscores the robustness and versatility of the approach, making it a promising tool for faster simulation responses.
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