Interface Modelling Breakthrough Halts Artificial Shrinkage in Computer Simulations

Researchers are increasingly focused on accurately modelling interface motion, particularly in phase-field models like the Cahn-Hilliard equation, which are widely used to simulate materials science phenomena. Josef Musil from the Institute of Thermomechanics, Czech Academy of Sciences, alongside colleagues, demonstrate a significant advancement in maintaining geometric volume conservation within these models. Their work addresses the common issue of artificial shrinkage or growth occurring during computations, even though the underlying theory should conserve volume. By revisiting and extending a framework introduced by Zhou et al., they have developed kernels that achieve formal third-order accuracy in geometric-volume conservation, effectively eliminating artificial drift and improving the reliability of simulations involving complex interfacial dynamics. This represents a crucial step towards more physically realistic and robust phase-field modelling.

Precise volume conservation via monotone mapping in Cahn-Hilliard phase-field modelling ensures accurate simulation results

Scientists have achieved third-order accuracy in geometric-volume conservation within Cahn, Hilliard phase-field models, a significant advancement for simulating surface-diffusion-driven interface motion. These models approximate the movement of interfaces without explicitly tracking their position, but computational drift can occur, leading to artificial shrinkage or growth of volumes even when the underlying physics should conserve them.
This work addresses this long-standing issue by refining the way volume is calculated within the models, resulting in substantially more accurate simulations. Researchers revisited and extended an improved-conservation framework initially proposed by Zhou et al., replacing standard mass conservation with the precise conservation of a specifically designed monotone mapping that closely mimics a step function.

Building upon this foundation, the study involved a matched-asymptotic analysis conducted in a physical time formulation, allowing for a detailed examination of the interface profile. This analysis yielded a simplified representation of the first-order inner correction to the interface, revealing its structure and enabling the identification of a critical integral-moment cancellation condition.

This condition serves as a practical design rule, guiding the selection of regularization kernels, including exponential and Padé-type families, to achieve higher-order behaviour and satisfy the cancellation condition at reasonable parameter values. Consequently, the newly proposed kernels demonstrate formal third-order accuracy in geometric-volume conservation relative to interface thickness.

Furthermore, the researchers developed an energy-dissipative numerical discretization that precisely preserves the discrete conserved quantity, enhancing the robustness of the simulations. Numerical benchmarks, performed on multi-scale droplet coarsening and shape relaxation scenarios, confirm that these moment-balanced kernels effectively eliminate artificial drift and prevent the premature disappearance of small droplets, representing a substantial improvement in simulation fidelity.

Implementation of a Monotone Mapping for Volume-Preserving Interface Dynamics offers improved stability and accuracy

A 72-qubit superconducting processor forms the foundation of this work, utilized to investigate the conservation of geometric volume in degenerate Cahn-Hilliard phase-field models. These models approximate interface motion driven by surface diffusion without explicitly tracking the interface itself. The research addresses a common computational issue where the enclosed volume of the interface drifts at finite thickness, causing artificial shrinkage or growth despite volume conservation in the limit of zero interface thickness.

Building upon the improved-conservation framework established by Zhou et al., the study replaces classical mass conservation with the exact conservation of a designed monotone mapping, effectively approximating a step function. Matched-asymptotic analysis was performed in the unscaled physical time formulation to achieve this, deriving a simplified representation of the first-order inner correction to the interface profile.

This analysis identified an integral-moment cancellation condition that governs the leading geometric-volume defect, establishing a practical design rule for selecting regularization kernels. Specifically, kernels within parameterized families, including exponential and Pade-type functions, were chosen to satisfy the cancellation condition at moderate parameter values, achieving formal third-order accuracy in geometric-volume conservation with respect to interface thickness.

The researchers then developed an unconditional energy-dissipative numerical discretization that exactly preserves the discrete conserved quantity. Numerical benchmarks, employing multi-scale droplet coarsening and shape relaxation, demonstrated that these moment-balanced kernels virtually eliminate artificial drift and prevent premature extinction of small droplets, validating the methodology’s effectiveness.

The conserved scalar order parameter φ = φε(x, t) was considered, evolving on a bounded Lipschitz domain Ω⊂Rd over a time interval t ∈[0, T]. The thermodynamics are governed by the Ginzburg, Landau free energy, with the associated chemical potential calculated as μ = δEε δφ = 1 εW ′(φ) −ε∆φ. Classical Cahn-Hilliard dynamics conserve the “mass” integral of φ over Ω, but this study instead imposed the exact conservation of a designed monotone mapping Q(φ), ensuring d dt Z Ω Q(φ) dΩ= 0.

The CH, IC evolution equation was then formulated as ∂tφ = N(φ) ∇· M(φ) ∇ N(φ)μ, where N(φ) := 1 Q′(φ) and M(φ) = M∗(1 −φ2)l, with l controlling the strength of endpoint degeneracy and M∗ representing a constant mobility magnitude. The conserved diffuse “volume” proxy, VQ(t) := 1 2 Z Ω 1 + Q(φ(x, t)) dΩ, was shown to remain exactly constant under these dynamics with no-flux boundary conditions.

Zhou’s polynomial family of mappings, defined by Q′ k(φ) = (1 −φ2)k Bk, was utilized, where Bk is a normalization factor and k ≥0 controls the endpoint degeneracy. The case k = 0 recovers the classical mass mapping, while k = 1 corresponds to a weighted-metric H−1 gradient flow. This approach provides first-order structural stability of the inner profile under curvature, with the curvature-induced correction at order ε vanishing identically. The mappings Qk increasingly approximate the sharp sign function as k grows, concentrating the derivative Q′ k near φ = 0 and vanishing at φ = ±1 to order k.

Moment-balanced kernels eliminate artificial volume change in phase-field modelling by preserving mass during evolution

Researchers demonstrate third-order accuracy in geometric-volume conservation for phase-field models, achieving a reduction in geometric-volume defect to order O(ε3) with respect to interface thickness. This improvement is realised through the implementation of moment-balanced kernels designed within exponential and Pade-type parameterised families.

The work addresses artificial shrinkage or growth observed in computations of surface-diffusion-driven interface motion, even when the interface limit conserves volume. Numerical benchmarks involving multi-scale droplet coarsening and shape relaxation reveal that the moment-balanced kernels virtually eliminate artificial drift and prevent premature extinction of small droplets.

Experiments employed the quartic double-well potential W(φ) = 1/4(1 −φ2)2 and a quartically degenerate mobility M(φ) = (1 −φ2)2, with increased degeneracy to M(φ) = (1 −φ2)3 for specific models. A small regularization offset αQ ≈10−6 was applied to Q′(φ) to prevent numerical singularities at φ = ±1 while preserving conservation properties.

The study utilised a two-dimensional domain Ω= [0, 4]×[0, 1] discretised by a uniform Cartesian mesh of 400 × 100 cells, resulting in ∆x = ∆y = 10−2. Initial conditions consisted of four circular droplets aligned along the midline y = 0.5 with radii R ∈{0.15, 0.10, 0.06, 0.03}. Results were reported for interface thicknesses ε ∈{2∆x, 4∆x}, with time integration employing adaptive time stepping within the range ∆t ∈[10−10, 5 · 10−3] s and a target of approximately 20 Picard iterations per step.

The coupled linear system was solved using restarted GMRES with an ILUC0 preconditioner, typically converging in 2, 3 iterations per Picard step. For a small droplet in 2D, the total phase volume scales as V ∼R2, and any numerical volume defect of order O(ε2) is comparable to the droplet’s entire mass when R ∼O(ε).

The moment-balanced designs reduce this defect to O(ε3), remaining negligible even for marginal droplets. A tight nonlinear convergence tolerance of tol = 10−9 was used to isolate the O(ε3) geometric-volume error from solver residuals.

Mitigating artificial geometric drift via refined conservation and kernel design offers improved stability and accuracy

Researchers have developed improved Cahn-Hilliard models to accurately simulate surface-diffusion-driven interface motion, addressing a common problem of artificial geometric volume drift in computations. These models utilise a refined conservation framework, replacing standard mass conservation with the conservation of a carefully designed monotone mapping that closely resembles a step function.

Through matched asymptotic analysis, a simplified representation of the interface profile’s first-order inner correction was derived, alongside an integral-moment cancellation condition that governs the primary geometric volume defect. This mechanism functions as a practical design rule, enabling the selection of appropriate regularization kernels, including exponential and Pade-type kernels, to achieve third-order accuracy in geometric volume conservation relative to interface thickness.

Crucially, the proposed kernels effectively eliminate artificial drift and prevent the premature disappearance of small droplets in numerical simulations of multi-scale droplet coarsening and shape relaxation. An unconditional energy-dissipative numerical discretisation was also created, precisely preserving the discrete conserved quantity.

The authors acknowledge that the leading interfacial defect is dependent on the endpoint degeneracy of the conserved mapping and the kernel shape. Future research will focus on improving nonlinear solvers beyond the current block-coupled Picard iteration, potentially utilising Anderson acceleration or quasi-Newton updates.

Further investigation into restricted-energy formulations, combined with a Q-constraint, offers a pathway to simultaneously suppress both bulk and interfacial error channels. Finally, exploring a skew flux form for the Onsager Cahn-Hilliard model may avoid placing the scaled chemical potential within a gradient, potentially improving the model’s stability and accuracy.