Fast, accurate, and conservative geometry near moving fluid interfaces remains critical for multiphase simulation on unstructured meshes. At coarse resolution, maintaining low drift and geometric fidelity often requires substantial refinement or specialized reconstruction. We present a local quadric-patch curved-volume framework on tetrahedral meshes: crease-aware patches are fitted, canonicalized, classified, and evaluated by subtracting matched analytic sub-volumes to recover the missing curved patch volume for each piecewise-linear (PL) surface triangle, then split conservatively into vertex dual volumes. On analytic quadrics (e.g., spheres and ellipsoids) we achieve near-round-off accuracy at sub-millisecond per-element cost (single thread). In interface-only transients, cube-to-sphere relaxation reduces final enclosure error from 7.1% to about 0.04%, and Rayleigh l=2 oscillations reduce long-time volume drift by more than two orders at coarse resolution. The method provides a practical drop-in geometry module for curvature-corrected volume integration in multiphase solvers.
Explicit interface tracking algorithms are used to track the evolving geometry and topology of deforming material interfaces, most often using triangle meshes. In work spanning more than forty years, this powerful approach has been applied extensively across computational physics, materials science, engineering, computer graphics, computer vision, and more. As a result, many different techniques have been proposed to carry out topological changes (merging, splitting, etc.) of explicit interfaces. However, the interdisciplinary nature of this topic has also led to a relatively fragmented body of literature. The present work therefore provides a thorough survey of topology change mechanisms used in explicit interface tracking, describing and categorizing the existing methods and tracing their evolution across domains. We consider both the two-phase case and generalizations to three or more phases, and conclude with a discussion of potential directions for future investigation.
We propose and analyze a second-order accurate, predictor-corrector implicit-explicit Runge–Kutta (IMEXRK) framework for Allen–Cahn and Cahn–Hilliard equations with variable mobility. A first-order predictor yields an intermediate state, after which a two-stage, second-order IMEXRK corrector advances the solution while evaluating the mobility function explicitly and identically using the intermediate state in both stages. Thanks to the “frozen mobility” and stabilization terms, the original discrete energy dissipation law is established under an a priori maximum bound. However, justifying this bound is nontrivial, since the energy functional lacks H2 control and prevents direct Sobolev embedding. For constant mobility, we use coarse-refined estimates to derive global-in-time discrete H1 and H2 bounds under a mild time-step restriction τ≲ε6|lnε|−4, while a log-interpolation inequality gives the required maximum bound of O(ε−1|lnε|). The stabilization parameter is thus characterized as O(ε−2|lnε|2), without relying on global Lipschitz assumptions, a priori maximum bounds, or mesh-dependent time-step constraints. Numerical tests verify convergence and energy dissipation for variable-mobility gradient flows.
A fourth-order accurate, single-step method is presented for solving the homogeneous and isotropic Maxwell’s equations on overlapping grids in complex geometries. The scheme follows a Lax-Wendroff-type approach and is based on a Taylor expansion in time combined with five-point wide finite difference approximations in space. This formulation provides high-order accuracy in space and time. A von Neumann analysis on a uniform periodic grid shows stability for CFL numbers up to unity. In practice, the method exhibits robust stability for challenging overlapping grid configurations and supports comparable CFL limits. Additionally, on Cartesian grids, the method is shown to exactly preserve the discrete divergence-free constraint, so that an initially divergence-free field remains divergence-free throughout the computation. The solver is demonstrated on a variety of overlapping grid configurations with both periodic and perfect electrical conductor boundary conditions, verifying fourth-order convergence in all cases. These results provide a foundation for extensions to source-driven Maxwell problems and self-consistent plasma–fluid coupling, highlighting the potential of the approach for predictive computation in complex electromagnetic systems.
This study proposes a novel material point method (MPM) for thin Reissner–Mindlin (RM) shells. The weak form of the RM shell is derived on 2D curved surfaces using Hamilton’s principle, and then the formulations in the 2D surface space are transformed to 3D Cartesian space in which the MPM background grid is defined. The proposed MPM shell method describes shell dynamics using a unified equation that is compatible with either the updated or total Lagrangian framework. The discretization of the weak form is performed with a general interpolation material point (GIMP) approach. Newmark-β time integration is adopted, which allows a much larger time step than the widely used update-stress-last (USL) scheme. For numerical stability, stress points are employed, and the gradients of shell field values are calculated on the curvilinear surface using the kernel function method. At stress points, stress tensors are updated incrementally, considering large deformations and rotations. Furthermore, an anti-hourglass term is proposed to further improve the numerical stability of computing pseudo-normals of the shell. Self- or multi-body contact of the shell can be treated easily, inheriting MPM’s natural handling of collisions and topological changes. A set of benchmark cases demonstrates the robustness and accuracy of the proposed method. Notably, the MPM shell method can produce highly accurate predictions with an MPM grid cell size much larger than the shell thickness, which eliminates the need for a very fine grid to resolve stress in the thickness direction, as in traditional MPM. To the authors’ knowledge, this is the first MPM work that uses thin shell theory and avoids relying on a meshed discretization of the shell.
In geological CO₂ sequestration, effective pressure management is crucial to prevent caprock failure and reduce the risk of leakage. Controlled brine extraction offers a practical mechanism to regulate pressure while increasing storage capacity. However, optimizing coupled injection and extraction strategies typically relies on repeated evaluations of high-fidelity reservoir simulators, leading to prohibitive computational costs.We propose a sequential multifidelity optimization (SMFO) framework that combines a physics-based reduced-order model (CGNet) with a high-fidelity simulator to accelerate well-control optimization. The CGNet model is obtained through aggressive coarsening of the fine-scale reservoir model and retains the dominant flow connectivities of the original model. Implemented within a differentiable simulator (MRST), the CGNet model is calibrated using adjoint-based misfit minimization to reproduce the dynamic responses of the high-fidelity simulator. Within the SMFO framework, the low-fidelity model is used to efficiently explore the control space during early optimization stages, while the high-fidelity simulator is employed in later iterations to ensure solution accuracy.Application to the Johansen formation benchmark demonstrates that the reduced CGNet model achieves a sevenfold reduction in simulation time while preserving key dynamic features, including saturation evolution and well responses. Scaled mismatches remain below 10⁻², and plume similarity metrics indicate strong agreement with high-fidelity results (SSIM > 0.93, RMSE < 0.06). When embedded in the SMFO framework, the proposed approach attains the same optimal net present value as direct high-fidelity optimization while reducing total CPU time by more than 50%. These results indicate that differentiable physics-based surrogates, when combined with gradient-based optimization, provide an effective and scalable approach for computational efficiency reservoir management in large-scale CO₂ storage applications.
Numerical simulations of high-speed compressible flow problems—such as transitional or turbulent compressible flows and other complex scenarios—require high-order methods with high resolution. These problems pose significant challenges for conventional low-order numerical methods. High-order methods, known for their robustness and accuracy in resolving a broad range of temporal and spatial scales, are therefore essential. Specifically, a high-order upwind multi-layer compact (MLC) scheme with spectral-like resolution was developed by Bai (2019) for such simulations. This scheme features a multi-layer formulation that simultaneously evolves both the solution and its spatial derivatives. While the MLC scheme performs well for smooth flows, its linear formulation is not well suited for flows with discontinuities, such as shock waves, which frequently arise in supersonic compressible flows. In contrast, the nonlinear Weighted Essentially Non-Oscillatory (WENO) scheme effectively captures shock waves while maintaining high-order accuracy in smooth regions. Building on the WENO framework, the Hermite WENO (HWENO) scheme was developed to achieve comparable or higher accuracy with narrower stencils. HWENO reconstruction uses both function values and their first derivatives to maintain compact stencils, making it highly compatible with the MLC scheme for handling shock-containing flows. This study aims to enhance the MLC scheme with shock-capturing capability through the HWENO reconstruction process, thereby developing a new shock-capturing MLC-HWENO scheme. Unlike the traditional finite difference HWENO scheme proposed by Liu and Qiu (2015), which applies nonlinear reconstruction to both the governing and derivative equations, the proposed MLC-HWENO scheme applies it only to the governing equation. The fluxes in the derivative equations are directly approximated using the MLC scheme’s linear procedure. Compared with the original finite difference HWENO scheme, the new MLC-HWENO formulation offers a simpler structure, improved computational efficiency, reduced numerical dissipation, and higher accuracy. Numerical experiments are conducted to demonstrate the accuracy, efficiency, and high-resolution capabilities of the proposed scheme.
In this paper, a high-order discontinuous Galerkin (DG) method is designed for volume-fraction models of compressible multi-material flows. To achieve high-resolution discontinuity capturing, a subcell limiter is incorporated, which suppresses spurious oscillations while introducing minimal numerical dissipation through a high-order weighted essentially non-oscillatory (WENO) evolution on the subcells. By combining a quasi-conservative formulation with appropriate numerical fluxes and subcell WENO reconstructions, the proposed scheme preserves the Abgrall condition and maintains equilibrium states with constant volume fractions. In addition, a bound-preserving limiter is applied, and the admissibility of the updated cell averages is rigorously established using the geometric quasilinearization approach. A comprehensive set of two- and three-dimensional benchmark tests demonstrates the accuracy and robustness of the method.
Physics-informed neural networks (PINNs) face significant challenges in solving singularly perturbed problems, which are characterized by sharp gradients within thin boundary layers. Such problems are ubiquitous in fluid mechanics, heat transfer, and solid mechanics. In this paper, we propose the composite-expansion physics-informed neural network (CE-PINN), a novel framework inspired by the method of composite expansions. CE-PINN is built upon a two-stage workflow for universality and efficiency. The offline stage performs a one-time training of multiple sub-networks to generate solutions for asymptotic terms independent of the perturbation parameter, ε. Subsequently, the online stage leverages these pre-trained sub-networks to efficiently construct the specific composite solution for any arbitrary small ε. CE-PINN advances existing methods with three key advantages: (1) it transforms the singularly perturbed problem into simpler sub-problems for improved trainability; (2) it offers a systematic path to higher-order asymptotic solutions without complex matching; and (3) it eliminates the need for time-consuming retraining for a range of small ε values. We validate CE-PINN on a suite of benchmark problems, including linear and nonlinear cases in one and two dimensions. Results demonstrate that our method achieves superior accuracy and computational efficiency over standard PINN and other PINN-based approaches, establishing a systematic “train once, solve for any small ε” workflow.
The Galerkin difference method is a type of finite element methods where Lagrange shape functions are piecewise but span over multiple cells. With this, it allows to choose time-step sizes in explicit temporal methods independently of the polynomial order of the shape functions, which is a strong benefit compared to the standard finite element method, where the time-step sizes need to be reduced with increasing polynomial degree to maintain stability. This property has been proven to be valid for advection and wave equations. In this work, we extend the methodology for the linear advection equation on immersed domains, i.e., consider cut Galerkin difference methods. We analytically and numerically show that Galerkin difference methods are applicable to immersed domains and time-step sizes are comparable to the uncut case, i.e., are independent of the polynomial order.
This study presents a coupled thermo-mechanical SPH framework for simulating high-speed multimaterial impacts involving large deformations. The framework consistently integrates reaction heat release, heat conduction, and temperature-dependent constitutive laws and equations of state into the governing equations, enabling accurate modeling of plastic flow, thermal response, and damage evolution. For metallic materials, the Johnson-Cook constitutive model and failure criterion are employed, while a friction- and viscosity-dominated post-failure response is introduced to represent mechanical behavior after melting and fragmentation. To address tensile instability in conventional SPH methods, a novel physically based pressure-limiting technique coupled with temperature evolution is developed. Furthermore, a dynamic particle activation strategy is proposed to efficiently resolve only the shock-affected regions, greatly enhancing computational efficiency. Numerical results show that the proposed framework can robustly capture interface evolution, temperature distribution, and failure patterns in high-speed multimaterial impacts. The findings highlight the crucial role of thermal effects in impact-induced damage processes. Overall, the developed model achieves high accuracy and stability under extreme strain-rate conditions, offering a reliable computational tool for investigating coupled thermal-mechanical behavior and damage mechanisms in high-speed impact events.
