Computing stable configurations of confined smectic liquid crystals with a deep variational framework

I. INTRODUCTION

• Computing confined minima is difficult because geometries may be irregular or three-dimensional, the energy yields a fourth-order equation, and many layers produce highly oscillatory density fields.Neural variational methods can represent fields directly, but spectral bias may favor a nearly homogeneous, nonlayered state over high-frequency layering.Source: §I · p. 1
• The DVF represents coupled order parameters with a variational neural network on a regular reference domain, maps in physical confinement, and uses a temporary wavelength-promoting warmup penalty.The penalty is removed afterward, leaving the final configuration governed by the original mLdG energy.Source: §I · pp. 1–2
• The DVF recovers established SmA morphologies and predicts geometry-dependent structures, including chevron-like SmC layers in a tangent-anchored sphere.Its minima reproduce corner-localized defects in polygons, bipolar textures in spheres, and concentric layers under homeotropic anchoring. Irregular domains show geometry-dependent layer junctions or dislocations, while repeated optimizations find competing configurations; a tangent-anchored sphere supports chevron-like SmC layers.Source: §I · p. 2

II. MATHEMATICAL MODELS

• The mLdG model couples a symmetric, traceless tensor Q for orientational order with a scalar field u whose periodic modulation represents smectic density layers.The principal eigenvector of Q defines the director, while a local plane wave in u has layer normal k/|k| and spacing 2π/|k|.Source: §I · p. 1, §II · p. 2
• The total free energy combines nematic bulk and elastic energy, smectic bulk energy, and a coupling term linking orientational and positional order.The nematic term uses the standard Landau–de Gennes potential in 3D and its reduced counterpart in 2D.Source: §II · p. 2
• Phase-dependent coupling selects layer wavelength and orientation: SmA aligns the layer normal with the director, while SmC imposes a prescribed tilt angle θ0.Thus, the mLdG energy selects the layered modulation rather than imposing it externally.Source: §II · p. 3
• Boundary anchoring constrains Q while u is free and satisfies natural boundary conditions; increasing dimensionless confinement length λ at fixed q packs more layers into the normalized domain.The effective wave number is λq, the dimensionless spacing is 2π/(λq), and the increasingly oscillatory field requires higher numerical resolution.Source: §II · p. 3

III. DEEP VARIATIONAL FRAMEWORK

• The DVF represents coupled order parameters on a regular reference domain to address computational challenges in the mLdG model.These challenges include complex confinement geometries, higher-order derivatives, and strongly oscillatory smectic layers.Source: §III · p. 3
• Coordinate mappings incorporate physical confinement into the DVF’s representation on the regular reference domain.Source: §III · p. 3
• A Fourier-based architecture with a wavelength-guided warmup penalty facilitates recovery of oscillatory smectic layers.Source: §III · p. 3

A. Neural representation of the order parameters

• The network is evaluated on a regular reference domain to produce represented order parameters, with physical-domain fields defined through the coordinate mapping.Source: A. Neural representation of the order… · p. 3
• Free-energy minimization over network parameters replaces field minimization and yields a network representing a computed local energy minimum.The loss includes auxiliary terms specified separately in Sec. III C.Source: A. Neural representation of the order… · p. 3
• The Fourier-based architecture adapts the DVF representation from Ref. [19], using embedding, Fourier layers, and projection to produce positional and orientational fields.Following the Fourier Neural Operator design, its layers combine spectral convolution, pointwise transformation, and nonlinear activation; the orientational output uses a symmetry-preserving parameterization.Source: A. Neural representation of the order… · p. 3

B. Coordinate mapping for complex domains

• A smooth one-to-one map x = G(ξ) relates the regular reference domain to the physical domain, and G−1 pulls the problem back to the reference domain.Source: B. Coordinate mapping for complex… · p. 3
• The network is evaluated on a uniform reference grid, with physical geometry incorporated through G; volume integrals transform with the corresponding Jacobian factor.Source: B. Coordinate mapping for complex… · p. 3
• The DVF schematic shows symmetry and tracelessness imposed on the orientational output and a loss combining energy, boundary-condition, Euler–Lagrange residual, and warmup terms.Source: Figure 1 · p. 4
• Order-parameter gradients and higher-order positional-field derivatives are computed on the reference grid, then converted to physical derivatives by the chain rule.This allows the free energy to be evaluated on the reference domain and differentiated with respect to network parameters.Source: B. Coordinate mapping for complex… · p. 4

C. Free-energy minimization and warmup penalty

• The Deep Ritz objective keeps normalized mLdG energy primary, with Q anchoring enforced by squared Dirichlet mismatch or, for 3D tangent anchoring, a degenerate planar surface potential.Source: C. Free-energy minimization and warmup… · p. 4
• A small-weight Euler–Lagrange residual regularizer suppresses defect-localized grid-scale artifacts without replacing energy as the selection criterion; unstable critical points may also have small residuals.Source: C. Free-energy minimization and warmup… · p. 4
• The warmup penalty counters network spectral bias toward smooth, nearly homogeneous positional fields, promoting high-frequency modulation at the wave number λq favored by mLdG energy.Source: C. Free-energy minimization and warmup… · p. 4
• Averaging the warmup target over sampled directions avoids choosing a preferred layer orientation, while its nonzero projection target disfavors u ≡ 0; the penalty decays to zero so the final state is governed by mLdG.Source: C. Free-energy minimization and warmup… · p. 4

IV. NUMERICAL RESULTS

• The frustrated energy landscape supports competing minima under identical conditions; reported configurations are the lowest computed-energy minima, but do not exhaust the landscape.Source: §IV · pp. 4–5
• In square-confined SmA, DRM and DVF without warmup remain nonlayered, whereas full DVF recovers a layered state and D-like texture, both retained after FD relaxation.Source: Figure 2 · p. 5
• For comparisons across 2D and 3D confinements, parameters include a = −5, b = 0, c = 5, q = 2π, B0 = 10^-3, B = 6400, C = 3500; most results use λ² = 30 or 50.Source: §IV · p. 5

A. Validation and SmA minima in square confinement

• At λ2 = 30, only the full DVF recovers a layered, lower-energy branch; the DRM and DVF without warmup remain nonlayered, showing the penalty changes the basin reached.The nonlayered DVF-without-warmup state has a small EL residual, while the DRM has both high energy and a large residual.Source: Table I · p. 5, A. Validation and SmA minima in square… · p. 5
• Finite-difference relaxation from the DVF solution preserves its layered D-like morphology and yields a stationary state with a positive-definite Hessian.The DVF and relaxed fields differ locally in Q and phase-sensitively in u, while retaining the overall morphology.Source: A. Validation and SmA minima in square… · p. 5
• Square minima evolve from WORS-like and nearly homogeneous at λ2 = 1, through layering onset at 4.38, to D-like layered states at 30 and 100.As λ increases, interior line defects become unfavorable and D-like minima preserve orientational and positional order through most of the bulk.Source: Figure 3 · p. 6, A. Validation and SmA minima in square… · p. 6

B. SmA in 2D irregular domains

• In the triangle, an interior −1/2 defect anchors a threefold layer junction, while nearly regular layers follow its edges.At both λ2 = 30 and 50, the polygonal minima reflect how tangent anchoring is accommodated at corners.Source: B. SmA in 2D irregular domains · p. 6, Figure 4 · p. 6
• In pentagons and hexagons, orientational mismatch concentrates at two splay corners, where layer terminations or insertions form boundary dislocations while central layers remain nearly ideal.These organizations persist at both λ2 values, with symmetry-related choices producing differently oriented minima.Source: B. SmA in 2D irregular domains · p. 6, Figure 4 · p. 6
• In the disk, tangent anchoring produces a bipolar orientational configuration with defects at opposite boundary poles and continuously bending smectic layers between them.The smooth boundary offers no corners to accommodate the rotation required by tangent anchoring.Source: B. SmA in 2D irregular domains · pp. 6–7
• In spindle-shaped domains, narrowing localizes strong director variation near the tips and leaves the elongated center with a more uniform director and straighter, more regularly spaced layers.The wide spindle instead has a bipolar-like defect pair and pronounced curvature and distortion of interior layers.Source: Figure 4 · p. 6, B. SmA in 2D irregular domains · p. 7

C. SmA in 3D confinements

• Tangent anchoring localizes cube distortions near edges and corners, whereas the smooth sphere organizes frustration around antipodal defects and a curved smectic-layer stack.In the cube, the interior director is approximately aligned with a body diagonal; in the sphere, the strongest distortions occur near the two poles.Source: C. SmA in 3D confinements · p. 7
• Homeotropic anchoring in a sphere favors an approximately radial director and closed concentric, onion-like SmA layers rather than the tangent-anchored bipolar stack.The comparison shows that the same spherical confinement supports distinct layered organizations under different anchoring conditions.Source: C. SmA in 3D confinements · p. 7
• With vertical top and radial bottom anchoring in a cube, focal-conic-like layer curvature and a low-order biaxial reconstruction zone remain localized near the substrate.The upper domain retains an almost vertical director and nearly uniform horizontal layers; the reconstruction also persists after finite-difference relaxation.Source: C. SmA in 3D confinements · pp. 7–8

D. SmC in spherical confinement

• At θ0 = π/6, SmC retains a bipolar orientational texture but develops surface-localized chevron-like layer bending, unlike the curved, approximately parallel SmA stack.The calculations use the same preferred wave number q, so the restructuring is attributed to preferred director–layer tilt rather than imposed layer-spacing change.Source: Figure 6 · p. 8, D. SmC in spherical confinement · p. 8

V. DISCUSSION AND CONCLUSION

• The adapted DVF combines Fourier representation, coordinate mappings, and a wavelength-guidance warmup penalty; the square-domain ablation found warmup essential, and finite-difference relaxation confirmed minima persistence.The validation described here concerns the tested square-domain regime.Source: §V · p. 8
• Across computed states, geometry and anchoring organize the large-scale Q field while director–layer coupling links it to morphology, recovering WORS-like, D-like, bipolar, TFCD-like, and chevron-like structures.The discussion frames these results as evidence that geometry, anchoring, and director–layer coupling jointly control defect localization and layer morphology.Source: §V · p. 8
• The study covers static states in fixed domains over a finite parameter and boundary-condition range, with proposed extensions to other modulated-order models, shell geometries, and jointly optimized free boundaries.The discussion identifies cholesteric models and spherical shells as possible extensions, and proposes representing both order parameters and the free boundary with trainable functions.Source: §V · pp. 8–9

• Symmetry and tracelessness are enforced through tensor parameterizations, with two scalar fields for Q in 2D and five in 3D; the 3D potential uses temperature-dependent A and positive material parameters B and C.Source: Appendix A · p. 9
• In 2D, tr(Q^3)=0 and choosing A=−B^2/(2C) gives the quartic reduced Landau–de Gennes potential used in all 2D computations, distinct because the tensor itself is reduced.Source: Appendix A · p. 9
• The energies are nondimensionalized using characteristic length L and energy units KL^(d−2), with rescaled model parameters and b=0; the resulting dimensionless energies cover SmA and SmC.Source: Appendix A · p. 9
• A further positive energy normalization leaves minima unchanged for each fixed λ̄ and keeps reported values on a comparable scale; bars are dropped and E is used in loss functions and reported energies.Source: Appendix A · pp. 9–10

Coordinate mapping

• An invertible map x = G(ξ) sends the regular reference domain [0, 1]^d, d = 2, 3, to the physical domain, with boundary correspondence prescribed one-to-one.The construction follows the coordinate-transformation idea in Ref..Source: Appendix B · p. 10

Transformed derivatives

• The Jacobian and its inverse convert reference-grid derivatives to physical first derivatives, while the volume element scales by |det J|.The Jacobian is assumed nonsingular at quadrature points.Source: Appendix B · p. 10
• Second-order derivatives include both the transformed reference Hessian and a term accounting for local curvature of the coordinate map.Together with the first-order formulas, these provide physical derivatives from reference-grid quantities once the forward map is known.Source: Appendix B · p. 10

Implementation

• The framework uses these formulas to evaluate Q gradient terms and higher-order derivatives of u for regular and irregular confinement geometries.Square and cube geometries are evaluated directly on uniform grids; mapped geometries use a uniform reference grid, differing only by an affine reference-coordinate change.Source: Appendix B · p. 10

1. Boundary conditions

• Anchoring is imposed on Γbc, the anchored part of ∂Ω, by prescribing Qbc in most calculations.The supplied passage introduces, but does not include, the specific prescription.Source: §1 · p. 10
• The prescribed targets include 2D tangent anchoring and 3D homeotropic anchoring on the sphere, with local tangents applied edgewise on polygons and the sphere’s outward radial normal.The outward radial normal specifies the spherical target.Source: §1 · p. 10
• The TFCD-inducing cube applies anchoring only on its top and bottom faces, using vertical and in-plane radial directions; its four lateral faces remain unanchored.The anchored boundary portion Γbc therefore consists of the top and bottom faces.Source: §1 · pp. 10–11
• Three-dimensional tangent anchoring uses a degenerate planar surface loss, placing a uniaxial director in the tangent plane without selecting an in-plane direction.It is used for the tangent-anchored cube and the SmA and SmC spheres.Source: §1 · p. 11
• No boundary term is imposed on u; because the mLdG energy depends on D2u, u instead satisfies a natural boundary condition.The traces of v and ∂νv are unrestricted.Source: §1 · p. 11
• Neither u nor ∂νu is prescribed at ∂Ω, so Q is explicitly anchored while u’s boundary behavior is determined variationally.The traces of v and ∂νv are unrestricted.Source: §1 · p. 11

2. Euler–Lagrange residual penalty

• The EL penalty is the squared Euclidean norm of residuals, whose common zero set is the EL system.It measures stationarity with respect to the scalar fields produced by the network.Source: §2 · p. 11
• For mapped geometries, derivatives and integration are evaluated using Appendix B.The parameterization uses m = 2 fields in 2D and m = 5 in 3D.Source: §2 · p. 11
• SmA residuals include orientational and positional EL equations, while the SmC formulation introduces an auxiliary scalar field before defining its residuals.Equation (C3) converts the compact stationary operators into the componentwise residuals used in code and Table I.Source: §2 · p. 11

Appendix D: Implementation details

• DVF computations use four Fourier layers with 64 hidden channels, GELU activation, and 16 retained Fourier modes in each direction.Grids use 129 points per direction in 2D and 65 in 3D, with fields and derivatives evaluated in double precision [48].Source: Appendix D · pp. 11–12
• AdamW uses a 10^-3 learning rate and 10^-2 weight decay for 5000 iterations in 2D and 10000 in 3D, with gradient norm clipped at 1.Each 2D setting uses 100 independent initializations and each 3D setting uses 50; figures show the lowest-energy minimum found.Source: Appendix D · p. 12
• Loss weights are fixed at wbc = 100 and wEL = 0.1, while the warmup penalty uses settings (50, 0.01) in 2D and (80, 0.02) in 3D.Its weight is time-dependent, and the directional average in Lp is estimated using 32 independently sampled directions per iteration.Source: Appendix D · p. 12
• Three-dimensional computations additionally use temporary orientational guidance during the first 500 iterations.The passage describes this as analogous to guidance used for phases with narrow at-…Source: Appendix D · p. 12