Robert Klöfkorn

Andreas Dedner, Birane Kane, Robert Klöfkorn et al.

In this paper we present a framework for solving two phase flow problems in porous media. The discretization is based on a Discontinuous Galerkin method and includes local grid adaptivity and local choice of polynomial degree. The method is implemented using the new Python frontend Dune-FemPy to the open source framework Dune. The code used for the simulations is made available as Jupyter notebook and can be used through a Docker container. We present a number of time stepping approaches ranging from a classical IMPES method to fully coupled implicit scheme. The implementation of the discretization is very flexible allowing for test different formulations of the two phase flow model and adaptation strategies.

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

We develop structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility. The proposed SWIPD-L and SIPGD-L methods incorporate parametrized mobility fluxes with edge-wise mobility treatments for enhanced coercivity-stability control. We prove coercivity for the generalized trilinear form and demonstrate optimal convergence rates while preserving mass conservation, energy dissipation, and the discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L methods confirm similar stability. Validation on $hp$-adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss.

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

The Cahn-Hilliard-Navier-Stokes (CHNS) system utilizes a diffusive phase-field for interface tracking of multi-phase fluid flows. Recently structure preserving methods for CHNS have moved into focus to construct numerical schemes that, for example, are mass conservative or obey initial bounds of the phase-field variable. In this work decoupled implicit-explicit formulations based on the Discontinuous Galerkin (DG) methodology are considered and compared to existing schemes from the literature. For the fluid flow a standard continuous Galerkin approach is applied. An adaptive conforming grid is utilized to further draw computational focus on the interface regions, while coarser meshes are utilized around pure phases. All presented methods are compared against each other in terms of bound preservation, mass conservation, and energy dissipation for different examples found in the literature, including a classical rising droplet problem.

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

We propose a structure-preserving discontinuous Galerkin scheme for the Cahn--Hilliard equations with degenerate mobility based on the Symmetric Weighted Interior Penalty formulation. By evaluating the mobility at cell averages rather than as a piecewise polynomial, the proposed scheme preserves strict degeneracy and yields a coercivity constant that is independent of the mobility, removing the need for regularisation. Moreover, we establish existence of discrete solutions even with degeneracy via a Leray--Schauder fixed-point argument, and show that the scheme satisfies a provable discrete maximum principle at arbitrary polynomial order $p$ when combined with the Zhang--Shu scaling limiter for $p > 0$ and from the scheme alone for $p = 0$. Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for $p \geq 1$ and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order $p+1$ in $L^2$ and validate structure-preserving properties with numerical results.

Zisheng Yao, Yuhe Zhang, Zhe Hu et al.

The unprecedented X-ray flux density provided by modern X-ray sources offers new spatiotemporal possibilities for X-ray imaging of fast dynamic processes. Approaches to exploit such possibilities often result in either i) a limited number of projections or spatial information due to limited scanning speed, as in time-resolved tomography, or ii) a limited number of time points, as in stroboscopic imaging, making the reconstruction problem ill-posed and unlikely to be solved by classical reconstruction approaches. 4D reconstruction from such data requires sample priors, which can be included via deep learning (DL). State-of-the-art 4D reconstruction methods for X-ray imaging combine the power of AI and the physics of X-ray propagation to tackle the challenge of sparse views. However, most approaches do not constrain the physics of the studied process, i.e., a full physical model. Here we present 4D physics-informed optimized neural implicit X-ray imaging (4D-PIONIX), a novel physics-informed 4D X-ray image reconstruction method combining the full physical model and a state-of-the-art DL-based reconstruction method for 4D X-ray imaging from sparse views. We demonstrate and evaluate the potential of our approach by retrieving 4D information from ultra-sparse spatiotemporal acquisitions of simulated binary droplet collisions, a relevant fluid dynamic process. We envision that this work will open new spatiotemporal possibilities for various 4D X-ray imaging modalities, such as time-resolved X-ray tomography and more novel sparse acquisition approaches like X-ray multi-projection imaging, which will pave the way for investigations of various rapid 4D dynamics, such as fluid dynamics and composite testing.