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1. DG approach to large bending plate deformations with isometry constraint

   https://doi.org/10.1142/S0218202521500044  

   Bonito, Andrea; Nochetto, Ricardo H.; Ntogkas, Dimitrios (January 2021, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove [Formula: see text]-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments. 

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2. Modelling and simulation of the cholesteric Landau-de Gennes model

   https://doi.org/10.1098/rspa.2023.0813  

   Hicks, Andrew L; Walker, Shawn W (June 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   This paper discusses modeling and numerical issues in the simulation of the Landau--de Gennes (LdG) model of nematic liquid crystals (LCs) with cholesteric effects. We propose a fully-implicit, (weighted) $L^2$ gradient flow for computing energy minimizers of the LdG model, and note a time-step restriction for the flow to be energy decreasing. Furthermore, we give a mesh size restriction, for finite element discretizations, that is critical to avoid spurious numerical artifacts in discrete minimizers, particularly when simulating cholesteric LCs that exhibit ``twist.'' Furthermore, we perform a computational exploration of the model and present several numerical simulations in 3-D, on both slab geometries and spherical shells, with our finite element method. The simulations are consistent with experiments, illustrate the richness of the cholesteric model, and demonstrate the importance of the mesh size restriction. 

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3. Optimal Control of the Landau–de Gennes Model of Nematic Liquid Crystals

   https://doi.org/10.1137/22M1506158  

   Surowiec, Thomas M; Walker, Shawn W (August 2023, SIAM Journal on Control and Optimization)

   We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter $Q = Q(x)$. Equilibrium LC states correspond to $$Q$$ functions that (locally) minimize an LdG energy functional. Thus, we consider an $L^2$-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external ``force'' controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where $Q(x) = 0$) in desired locations, which is desirable in applications. 

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4. Analysis and application of a local discontinuous Galerkin method for the electromagnetic concentrator model

   https://doi.org/10.1002/num.23069  

   Huang, Yunqing; Li, Jichun; Liu, Xin (September 2023, Numerical Methods for Partial Differential Equations)

   Abstract In this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi‐discrete and full‐discrete LDG schemes. Numerical results are presented to justify the theoretical analysis and demonstrate the interesting wave concentration property by the electromagnetic concentrator. 

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5. Plate Theory for Metric-Constrained Actuation of Liquid Crystal Elastomer Sheets

   https://doi.org/10.1007/s10659-025-10127-7  

   Bouck, Lucas; Padilla-Garza, David; Plucinsky, Paul (March 2025, Journal of Elasticity)

   Abstract Liquid crystal elastomers (LCEs) marry the large deformation response of a cross-linked polymer network with the nematic order of liquid crystals pendent to the network. Of particular interest is the actuation of LCE sheets where the nematic order, modeled by a unit vector called the director, is specified heterogeneously in the plane of the sheet. Heating such a sheet leads to a large spontaneous deformation, coupled to the director design through a metric constraint that is now well-established by the literature. Here we go beyond the metric constraint and identify the full plate theory that underlies this phenomenon. Starting from a widely used bulk model for LCEs, we derive a plate theory for the pure bending deformations of patterned LCE sheets in the limit that the sheet thickness tends to zero using the framework of$$\Gamma $$ $\Gamma$-convergence. Specifically, after dividing the bulk energy by the cube of the thickness to set a bending scale, we show that all limiting midplane deformations with bounded energy at this scale satisfy the aforementioned metric constraint. We then identify the energy of our plate theory as an ansatz-free lower bound of the limit of the scaled bulk energy, and construct a recovery sequence that achieves this plate energy for all smooth enough midplane deformations. We conclude by applying our plate theory to a variety of examples. 

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