1. Thermoelasticity with phase transitions for 4D printing: diffuse interface models and mathematical analysis
   
   [RGC project number (12302023)] Period: 01/01/2024 to 31/12/2026. Funding: $866,032 HKD
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   Abstract: Four dimensional (4D) printing refers to the newly developed field derived from 3D printing that fabricates objects using stimulus-responsive smart materials. These objects are capable of morphing into different configurations in response to various environmental stimuli, leading to structures with potential for self-assembly and multifunctionality. The shape-shifting behavior comes from the smart materials, such as shape memory alloys and shape memory polymers, which can be trained to memorize and cycle through multiple configurations. Although numerous working proofs of concepts have demonstrated promising capabilities in soft robotics, medical implants, textile, construction and automotive industries, 4D printing has yet to be part of the mainstream in manufacturing, as the transformations of the smart materials become less reliable over prolong usage. Such issues can be mitigated by developing 4D printing designs utilizing a mixture of different smart materials, and thus there is currently a pressing need to understand the interactions between different smart materials and how they influence the mechanical behavior of the overall structure.
   
   In this proposal we plan to develop new mathematical models for thermally activated shape memory polymers (SMP) that can exhibit phase transitions, which is one of the more commonly used stimulus-responsive materials in 4D printing. The novelty of our approach is to describe the phase transition from one type of SMP to a second type with a diffuse interface ansatz. The resulting thermoelastic models seem to be amenable to further theoretical investigations, which motivates us to perform mathematical analysis for settings associated to specific types of elastic deformations (such as bending, elongation and twisting) that constitute the standard shape-shifting behaviors encountered in 4D printing applications. We envision the proposed models and accompanying analysis can contribute towards resolving other challenges in this fast-growing interdisciplinary field.
   
   Research outputs:

   
   
   • X. Jin, K.F. Lam and C. Ye. Numerical analysis of a FE/SAV scheme for a Caginalp phase field model with mechanical effects in stereolithography


2. Phase field modeling, calibration and experimental design for Stereolithography in 3D printing
   
   [RGC project number (22300522)] Period: 01/01/2023 to 31/12/2025. Funding: $688,110 HKD
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   Abstract: Industry 4.0 is a concept that attempts to revolutionize the way we manufacture products in factories. Through recent progresses in Internet-of-Things, cloud computing and artificial intelligence, companies can now capitalize on real-time visibility of the status and progress occurring in production, and have begun gradually transitioning towards a new manufacturing format with minimal human input and increased automation. At the centre of this paradigm shift, 3D printing technologies are expected to play a significant role.
   
   Stereolithography, as one of the earliest 3D printing technologies, builds objects in a layer-by-layer fashion by using an ultraviolet laser to solidify liquid polymer resin. Its main application lies in crafting high resolution 3D models of anatomical regions for medical training and surgery planning. However, objects created from the same design can exhibit undesirable variations in mechanical properties due to small differences in the printing environment. These variations can be minimized by combining data collection with well-placed sensors and numerical prediction of mechanical properties from calibrated mathematical models, which then guide the appropriate adjustment of the construction apparatus.
   
   We plan to provide a mathematical framework of the above strategy for stereolithography. The main goals of this proposal are to develop and calibrate some novel mathematical models, and explore sensor configurations minimizing uncertainty in the collected data to facilitate model calibration. The key ingredient for the modeling is the phase field approach that produces mathematical descriptions amenable to further theoretical and numerical analysis. Then, we tackle the problems of calibration and sensor placement with the framework of optimal experimental design. The proposed modeling and accompanying investigations can also be applied to other 3D printing technologies, and thus we envision the ideas developed here can contribute towards resolving other challenges encountered in the advent of Industry 4.0.
   
   Research outputs:

   
   
   • X. Jin, K.F. Lam and C. Ye. Numerical analysis of a FE/SAV scheme for a Caginalp phase field model with mechanical effects in stereolithography


3. Optimising the design of support structures in additive manufacturing with phase fields
   
   [RGC project number (12300321)] Period: 01/01/2022 to 31/12/2024. Funding: $391,015 HKD
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   Abstract: Additive manufacturing, also known as 3D printing, is a fast-growing technology that is making an appearance in various businesses and industries. It has the advantage of creating complex-shaped objects in a rapid, flexible and cheap manner, but despite its potential in a variety of areas such as biomedical implants, automotive and aerospace components, and even jewellery and food, it is held back by several factors that prevent real integration with commercial production line assemblies.
   
   One of these factors which we plan to explore in this project is a design issue related to support structures. As the name implies, support structures are used to prop up the 3D object during printing to maintain its intended shape and prevent deformations caused by external factors. However, support structures obviously add to the material costs, printing time, and there is a risk of inflicting damage to the object during their removal. Since support structures are absolutely necessary for general object designs, previous researchers have employed the framework of optimisation, particularly in the context of structural topology optimisation, with the aim of finding optimal designs that address some of the issues arising from support structures.
   
   The goal of this proposal is to develop an alternative optimisation approach for the design of support structures based on the concept of phase fields, which in the past has seen successful applications in tackling problems motivated from shape and topology optimisation. We believe our proposed phase field approach can offer both mathematical and numerical advantages over current methods; such as a rigorous analysis of the optimisation problem, derivation of optimality conditions, convergence of the numerical iterations, and an overall improvement in computational effort. Furthermore, we envision the ideas developed in this proposal can be extended beyond the issue of support structures, and begin to contribute towards resolving other challenges faced by the additive manufacturing industries.
   
   Research outputs:

   
   
   • H. Garcke, K.F. Lam, R. Nürnberg and A. Signori. Phase field topology optimisation for 4D printing. ESAIM - Control Optim. Calc. Var., 29 (2023) Article number: 24
   
   
   • H. Garcke, K.F. Lam, R. Nürnberg and A. Signori. Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies. Appl. Math. Optim., 87 (2023) Article number: 44


4. Modelling and analysis of diffuse interface models for two-phase micropolar fluid flows
   
   [RGC project number (14303420)] Period: 01/01/2021 to 31/12/2023. Funding: $555,754 HKD
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   Abstract: Micropolar fluids are among the simplest cases of fluids with microstructures, where each fluid particle has its own internal rotations. Examples include ferrofluids, blood flows, bubbly liquids and liquid crystals, all of which play significant and important roles in various industries and also in the human body. Therefore, it becomes essential for scientists to effectively capture the behaviour of these kinds of fluids with the appropriate mathematical models, and to better analyse and simulate their flow dynamics. The classical Navier-Stokes equations that are well-studied by the mathematical community is not a suitable description for micropolar fluids, as the internal rotations of each fluid particle are not accounted for.
   
   Through the works of A. Cemal Eringen and his coworkers, we now have mathematical models that extend the Navier-Stokes equations to capture these internal rotations. However, in many real world applications we often encounter mixtures of multiple kinds of fluids. In these situations we have to further extend previous models to account for the interactions between individual fluids that can influence the gross motion of the mixture. The main aims of this proposal are to develop new mathematical models for mixtures of micropolar fluids, study their mathematical properties, and derive stable and efficient fully discrete numerical schemes for simulations. The key ingredient is the diffuse interface methodology which offers a model that is amenable to further mathematical analysis and allows us to address the existence of weak solutions and the stability of discrete solutions. Also included in the modelling is the dynamics of the contact line that is created when two or more fluids meet at the same solid boundary. This describes wetting and dewetting processes crucial in modern-day applications such as inkjet printing and lubrication.
   
   Research outputs:

   
   


5. On Cahn-Hilliard models with singular potentials and source terms
   
   [RGC project number (14302319)] Period: 01/09/2019 to 31/08/2022. Funding: $502,444 HKD
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   Abstract: The Cahn-Hilliard (CH) equation is a fourth order nonlinear partial differential equation (PDE) originally used to model the separation of a mixture containing two phases of matter, such as water and oil, and is now an ubiquitous component of many mathematical models in applied sciences. It is formulated with a nonconvex potential function, usually taken as a smooth polynomial, that possesses two equal minima, say −1 and 1. Through the addition of appropriate source terms, the CH equation has seen recent applications in modelling the growth of tumours and in repairing damaged black-and-white images (also known as inpainting). However, an unfortunate feature with a polynomial potential is that the solution may not stay bounded in between −1 and 1, and undesirable effects, such as negative mass densities in tumour models or new shades of colour in black-and-white images, can occur.
   
   A remedy is to employ singular potentials, such as a logarithmic-type function, to ensure that the solution to the model stays in between −1 and 1. For recent applications in tumour growth and inpainting, the new combination of singular potentials and source terms has not received much attention in the literature, and current analytical studies are mostly confined to establishing the existence of weak solutions. In this project, we plan to expand the scope of the analysis for these new Cahn-Hilliard models with singular potentials and source terms by investigating the issues of uniqueness and regularity of solutions, as well as addressing the existence of stationary solutions analytically and also numerically.
   
   Research outputs:

   
   
   • K.F. Lam and R. Wang. Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn-Hilliard systems with bounded mass source
   
   
   • A. Giorgini, K.F. Lam, E. Rocca and G. Schimperna. On the Existence of Strong Solutions to the Cahn-Hilliard-Darcy system with mass source. SIAM J. Math. Anal. (2021)
   
   
   • K.F. Lam. Global and exponential attractors for a Cahn–Hilliard equation with logarithmic potentials and mass source. J. Differential Equations, 312:237--275 (2022)
   
   
   • H. Garcke, K.F. Lam and A. Signori. Sparse Optimal Control of a Phase Field Tumor Model with Mechanical Effects. SIAM J. Control Optim., 59:1555--1580 (2021)
   
   
   • S. Frigeri, K.F. Lam and A. Signori. Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities. European Jnl. Appl. Math. (2021)
   
   
   • M. Ebenbeck and K.F. Lam. Weak and stationary solutions to a Cahn-Hilliard-Brinkman model with singular potentials and source terms. Adv. Nonlinear Anal., 10:24--65 (2020)


6. Mathematical studies of a phase field approach to shape optimization
   
   [RGC project number (14302218)] Period: 01/01/2019 to 31/12/2021. Funding: $456,452 HKD
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   Abstract: In shape optimization problems, the aim is to find a class of shapes that optimizes some prescribed goals under a partial differential equation (PDE) constraint. The classical formulation for such problems involves representing the unknown shape as a mathematical surface, and seeks to derive optimality conditions (equations/inequalities involving nonlinear PDEs). The optimal shapes can then be realized numerically by solving the optimality conditions. However, shape optimization problems are often severely ill-posed (as the existence of optimal shapes is not guaranteed in general) and can be computationally intensive (due to the requirement of a new approximation of the computational domain at every numerical iteration).
   
   We propose to use a phase field approach to overcome the above problems, since it has several important mathematical and numerical advantages over the classical approach; namely the phase field method can be justified (in the sense that existence of optimal shapes can be proved rigorously) and in addition both the PDE constraint and the optimality conditions are solved on the same fixed domain throughout the optimization procedure. Furthermore, by sending a small parameter to zero, the optimality conditions from the classical formulation are recovered. Hence, one may view the phase field approach as a consistent approximation for shape optimization problems. The goal of this proposal is to develop phase field approximations to study nonlinear inverse problems that can be casted as shape optimization problems.
   
   Research outputs:

   
   
   • H. Garcke, K.F. Lam, R. Nürnberg and A. Signori. Phase field topology optimisation for 4D printing. ESAIM - Control Optim. Calc. Var., 29 (2023) Article number: 24
   
   
   • H. Garcke, K.F. Lam, R. Nürnberg and A. Signori. Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies. Appl. Math. Optim., 87 (2023) Article number: 44
   
   
   • K.F. Lam and I. Yousept. Consistency of a phase field regularization for an inverse problem governed by a quasilinear Maxwell system. Inverse Problems, 36 (2020) 040511 (33pp)

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