Finite-element methods for fourth-order problems and smectic A liquid crystals
Bibliographic record
Abstract
In recent years, energy-minimization finite-element methods have been proposed for the computational modelling of equilibrium states of several types of liquid crystals (LCs) [4, 34, 110]. This thesis is particularly interested in the models of smectic A liquid crystals, based on the free-energy functionals proposed by Pevnyi, Selinger, and Sluckin [112], and by Xia et al. [138]. The Euler-Lagrange equations for these models include fourth-order terms acting on the smectic order parameter (or density variation of the LC) and second-order terms acting on the Q-tensor or director field. Thus, we first focus extensively on finite-element methods for fourth-order problems. These methods include (i) C¹-continuous elements with a nonsymmetric Nitsche-type penalty method to weakly impose the essential boundary conditions, (ii) a nonsymmetric version of the C⁰ interior penalty method, where the nonsymmetric forms are used to guarantee optimal convergence rates in terms of h ≤ 1 and q ≈ 40, where h and q are the refinement level and the smectic wavenumber that prescribes a preferred wavelength for the solution of 2π=q respectively, and (iii) mixed finite-element methods based on introducing the gradient of the solution as an explicit variable and constraining its value using a Lagrange multiplier, that are symmetric and allow us to strongly impose the essential boundary conditions. Preliminary experiments show that the mixed formulations may be advantageous over the other methods, in the sense that we can construct efficient preconditioners for these discretizations. Therefore, we consider a four-field formulation for models of smectic A liquid crystals, approximating the smectic order parameter, its gradient, the Lagrange multiplier, and the Q-tensor. Then, we focus on the construction of solvers for the nonlinear systems that result from the discretization of these models. We consider a Newton-Krylov- Multigrid approach, using Newton's method to linearize the systems, and developing monolithic geometric multigrid preconditioners for the resulting saddle-point systems with vertex-based patch relaxation schemes.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".