arXiv subjects

Marco F. P. ten Eikelder

Publications and source records attributed to Marco F. P. ten Eikelder.

11 recordsLinked to original sources

arXiv2026

A thermodynamically consistent framework for finite growth of multi-constituent mixtures with application to tumor growth

Biological tissues grow by continuously producing, transporting, and reorganizing multiple interacting constituents. These processes are intrinsically coupled to finite deformation and residual stress. Existing models typically capture either finite growth kinematics or multi-constituent transport, but rarely both within a thermodynamically consistent setting. In particular, existing approaches do not consistently link the volume created by finite growth to the mass produced for each individual constituent. In this work, we develop a general continuum framework that unifies finite growth kinematics and multiphase mixture theory for fully saturated multi-constituent mixtures containing an arbitrary number of dilute dissolved solutes. Formulated in a solid-skeleton-based description, the framework rests on constituent-wise balance laws and a free-energy dissipation principle, from which thermodynamically admissible constitutive closures are derived for all mass-exchange, transport, reaction, and growth processes. The central novelty of the framework is a coupling between growth-induced volume creation and constituent mass production, expressed through volume accumulation fractions that distribute the newly created volume among the constituents while preserving saturation. We cast the resulting model in a total Lagrangian mixed weak form and specialize the general theory to a four-constituent, two-solute model of avascular tumor growth that couples nutrient transport, waste production, phenotype transitions between proliferative, hypoxic, and necrotic cells, volume growth, elastic deformation, and growth-induced residual stress. The model is implemented within a finite element setting and its capabilities are demonstrated on representative benchmark problems.

physics.bio-ph↗

arXiv2026

An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows

This paper introduces a high order numerical framework for efficient and robust simulation of compressible flows. To address the inefficiencies of standard hybridized discontinuous Galerkin (HDG) methods in large scale settings, we develop a macro element HDG method that reduces global and local degrees of freedom by embedding continuous Galerkin structure within macro-elements. This formulation supports matrix free implementations and enables highly parallel local solves, leading to substantial performance gains and excellent scalability on modern architectures. To enhance robustness in under resolved or turbulent regimes, we extend the method using entropy variables and a flux differencing approach to construct entropy stable and kinetic energy preserving variants. These formulations satisfy a discrete entropy inequality and improve stability without compromising high order accuracy. We demonstrate the performance of the proposed method on benchmark problems including the inviscid isentropic vortex and the Taylor Green vortex in both inviscid and turbulent regimes. Numerical results confirm optimal accuracy, improved robustness, and up to an order of magnitude speedup over standard HDG methods. These developments mark a significant advancement in high order methods for direct numerical simulation (DNS) of compressible flows.

cs.CE↗

arXiv2026

Uni-Flow: a unified autoregressive-diffusion model for complex multiscale flows

Spatiotemporal flows govern diverse phenomena across physics, biology, and engineering, yet modelling their multiscale dynamics remains a central challenge. Despite major advances in physics-informed machine learning, existing approaches struggle to simultaneously maintain long-term temporal evolution and resolve fine-scale structure across chaotic, turbulent, and physiological regimes. Here, we introduce Uni-Flow, a unified autoregressive-diffusion framework that explicitly separates temporal evolution from spatial refinement for modelling complex dynamical systems. The autoregressive component learns low-resolution latent dynamics that preserve large-scale structure and ensure stable long-horizon rollouts, while the diffusion component reconstructs high-resolution physical fields, recovering fine-scale features in a small number of denoising steps. We validate Uni-Flow across canonical benchmarks, including two-dimensional Kolmogorov flow, three-dimensional turbulent channel inflow generation with a quantum-informed autoregressive prior, and patient-specific simulations of aortic coarctation derived from high-fidelity lattice Boltzmann hemodynamic solvers. In the cardiovascular setting, Uni-Flow enables task-level faster than real-time inference of pulsatile hemodynamics, reconstructing high-resolution pressure fields over physiologically relevant time horizons in seconds rather than hours. By transforming high-fidelity hemodynamic simulation from an offline, HPC-bound process into a deployable surrogate, Uni-Flow establishes a pathway to faster-than-real-time modelling of complex multiscale flows, with broad implications for scientific machine learning in flow physics.

physics.flu-dyn↗

arXiv2026

Review of thermodynamic structures and structure-preserving discretisations of Cahn--Hilliard-type models

The Cahn-Hilliard equation and extensions, notably the Cahn-Hilliard-Darcy and Cahn-Hilliard-Navier-Stokes systems, provide widely used frameworks for coupling interfacial thermodynamics with flow. This review surveys the thermodynamic structures underlying these models, focusing on the formulation of free energy functionals, dissipation mechanisms, and variational principles. We compare structural properties, emphasizing how these models encode conservation laws and energy dissipation. A central theme is the translation of these thermodynamic structures into numerical practice by providing representative discretisation strategies that aim to preserve mass conservation, stability, and energy decay. Particular attention is paid to the trade-offs between accuracy, efficiency, and structure preservation in large-scale simulations.

math.NA↗

arXiv2025

Fast-Forward Lattice Boltzmann: Learning Kinetic Behaviour with Physics-Informed Neural Operators

The lattice Boltzmann equation (LBE), rooted in kinetic theory, provides a powerful framework for capturing complex flow behaviour by describing the evolution of single-particle distribution functions (PDFs). Despite its success, solving the LBE numerically remains computationally intensive due to strict time-step restrictions imposed by collision kernels. Here, we introduce a physics-informed neural operator framework for the LBE that enables prediction over large time horizons without step-by-step integration, effectively bypassing the need to explicitly solve the collision kernel. We incorporate intrinsic moment-matching constraints of the LBE, along with global equivariance of the full distribution field, enabling the model to capture the complex dynamics of the underlying kinetic system. Our framework is discretization-invariant, enabling models trained on coarse lattices to generalise to finer ones (kinetic super-resolution). In addition, it is agnostic to the specific form of the underlying collision model, which makes it naturally applicable across different kinetic datasets regardless of the governing dynamics. Our results demonstrate robustness across complex flow scenarios, including von Karman vortex shedding, ligament breakup, and bubble adhesion. This establishes a new data-driven pathway for modelling kinetic systems.

cs.LG↗

arXiv2025

Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model

Phase separation in binary mixtures, governed by the Cahn-Hilliard equation, plays a central role in interfacial dynamics across materials science and soft matter. While numerical solvers are accurate, they are often computationally expensive and lack flexibility across varying initial conditions and geometries. Neural operators provide a data-driven alternative by learning solution operators between function spaces, but current architectures often fail to capture multiscale behavior and neglect underlying physical symmetries. Here we show that an equivariant U-shaped neural operator (E-UNO) can learn the evolution of the phase-field variable from short histories of past dynamics, achieving accurate predictions across space and time. The model combines global spectral convolution with a multi-resolution U-shaped architecture and regulates translation equivariance to align with the underlying physics. E-UNO outperforms standard Fourier neural operator and U-shaped neural operator baselines, particularly on fine-scale and high-frequency structures. By encoding symmetry and scale hierarchy, the model generalizes better, requires less training data, and yields physically consistent dynamics. This establishes E-UNO as an efficient surrogate for complex phase-field systems.

cs.LG↗

arXiv2025

A simple, fully-discrete, unconditionally energy-stable method for the two-phase Navier-Stokes Cahn-Hilliard model with arbitrary density ratios

The two-phase Navier-Stokes Cahn-Hilliard (NSCH) mixture model is a key framework for simulating multiphase flows with non-matching densities. Developing fully discrete, energy-stable schemes for this model remains challenging, due to the possible presence of negative densities. While various methods have been proposed, ensuring provable energy stability under phase-field modifications, like positive extensions of the density, remains an open problem. We propose a simple, fully discrete, energy-stable method for the NSCH mixture model that ensures stability with respect to the energy functional, where the density in the kinetic energy is positively extended. The method is based on an alternative but equivalent formulation using mass-averaged velocity and volume-fraction-based order parameters, simplifying implementation while preserving theoretical consistency. Numerical results demonstrate that the proposed scheme is robust, accurate, and stable for large density ratios, addressing key challenges in the discretization of NSCH models.

math.NA↗

arXiv2025

A Multiscale-Multiphysics Framework for Modeling Organ-scale Liver Regrowth

We present a framework for modeling liver regrowth on the organ scale that is based on three components: (1) a multiscale perfusion model that combines synthetic vascular tree generation with a multi-compartment homogenized flow model, including a homogenization procedure to obtain effective parameters; (2) a poroelastic finite growth model that acts on all compartments and the synthetic vascular tree structure; (3) an evolution equation for the local volumetric growth factor, driven by the homogenized flow rate into the microcirculation as a measure of local hyperperfusion and well-suited for calibration with available data. We apply our modeling framework to a prototypical benchmark and a full-scale patient-specific liver, for which we assume a common surgical cut. Our simulation results demonstrate that our model represents hyperperfusion as a consequence of partial resection and accounts for its reduction towards a homeostatic perfusion state, exhibiting overall regrowth dynamics that correspond well with clinical observations. In addition, our results show that our model also captures local hypoperfusion in the vicinity of orphan vessels, a key requirement for the prediction of ischemia or the preoperative identification of suitable cut patterns.

cs.CE↗

arXiv2024

The matrix-free macro-element hybridized Discontinuous Galerkin method for steady and unsteady compressible flows

The macro-element variant of the hybridized discontinuous Galerkin (HDG) method combines advantages of continuous and discontinuous finite element discretization. In this paper, we investigate the performance of the macro-element HDG method for the analysis of compressible flow problems at moderate Reynolds numbers. To efficiently handle the corresponding large systems of equations, we explore several strategies at the solver level. On the one hand, we devise a second-layer static condensation approach that reduces the size of the local system matrix in each macro-element and hence the factorization time of the local solver. On the other hand, we employ a multi-level preconditioner based on the FGMRES solver for the global system that integrates well within a matrix-free implementation. In addition, we integrate a standard diagonally implicit Runge-Kutta scheme for time integration. We test the matrix-free macro-element HDG method for compressible flow benchmarks, including Couette flow, flow past a sphere, and the Taylor-Green vortex. Our results show that unlike standard HDG, the macro-element HDG method can operate efficiently for moderate polynomial degrees, as the local computational load can be flexibly increased via mesh refinement within a macro-element. Our results also show that due to the balance of local and global operations, the reduction in degrees of freedom, and the reduction of the global problem size and the number of iterations for its solution, the macro-element HDG method can be a competitive option for the analysis of compressible flow problems.

cs.CE↗

arXiv2023

Connecting continuum poroelasticity with discrete synthetic vascular trees for modeling liver tissue

Computational simulations have the potential to assist in liver resection surgeries by facilitating surgical planning, optimizing resection strategies, and predicting postoperative outcomes. The modeling of liver tissue across multiple length scales constitutes a significant challenge, primarily due to the multiphysics coupling of mechanical response and perfusion within the complex multiscale vascularization of the organ. In this paper, we present a modeling framework that connects continuum poroelasticity and discrete vascular tree structures to model liver tissue across disparate levels of the perfusion hierarchy. The connection is achieved through a series of modeling decisions, which include source terms in the pressure equation to model inflow from the supplying tree, pressure boundary conditions to model outflow into the draining tree, and contact conditions to model surrounding tissue. We investigate the numerical behaviour of our framework and apply it to a patient-specific full-scale liver problem that demonstrates its potential to help assess surgical liver resection procedures

cs.CE↗

arXiv2020

Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model

We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche's method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche's method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection-diffusion equation, we develop a residual-based model that incorporates the non-vanishing fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.

math.NA↗