arXiv subjects

Matthias Ruf

Publications and source records attributed to Matthias Ruf.

At least 19 recordsLinked to original sources

arXiv2024

POREMAPS: A finite difference based Porous Media Anisotropic Permeability Solver for Stokes flow

Porous materials are ubiquitous in various engineering and geological applications, where their permeability plays a critical role in viscous fluid flow and transport phenomena. Understanding and characterizing the microscale properties, the effective hydraulic parameters, and also the anisotropy of porous materials are essential for accurate modeling and predicting fluid flow behavior. The study pursues the Digital Rock Physics approach to retrieve intrinsic permeability and its evolution in anisotropic configurations of porous media, which are subjected to pore space alterations. Therefore, we discuss the development and implementation of a computational framework based on the finite difference method to solve the pseudo-unsteady Stokes equations for fluid flow on the pore scale. We present an efficient and highly parallelized implementation of this numerical method for large voxel-based data sets originating from different image-based experimental setups. A comprehensive variety of benchmarks has been conducted to assess and evaluate the performance of the proposed solver. The solver's compatibility with huge domain sizes generated by state-of-the-art imaging techniques is demonstrated. We investigate an open-cell foam undergoing deformation, observing that contrary to initial expectations, no anisotropy emerges. Further, we examine a microfluidic cell experiencing precipitation within its pore space, resulting in clear anisotropic development during the clogging process.

physics.flu-dyn↗

arXiv2024

Upper bounds for the homogenization problem in nonlinear elasticity: the incompressible case

We consider periodic homogenization of hyperelastic models incorporating incompressible behavior via the constraint $\det(\nabla u)=1$. We show that the 'usual' homogenized integral functional $\int W_{\rm hom}(\nabla u)\,dx$, where $W_{\rm hom}$ is the standard multicell-formula of non-convex homogenization restricted to volume preserving deformations, yields an upper bound for the $\Gamma$-limit as the scale of periodicity tends to zero.

math.AP↗

arXiv2023

On the Lavrentiev gap for convex, vectorial integral functionals

We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$ F: g+W_0^{1,1}(\Omega)^m\to\mathbb{R}\cup\{+\infty\},\qquad F(u)=\int_\Omega W(x,\mathrm{D} u)\,\mathrm{d}x, $$ where the boundary datum $g:\Omega\subset \mathbb{R}^d\to\mathbb{R}^m$ is sufficiently regular, $\xi\mapsto W(x,\xi)$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\leq d-1$, we assume $q$-growth from above with $q\leq \frac{(d-1)p}{d-1-p}$, while for $p>d-1$ we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the $x$-dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions.

math.AP↗

arXiv2023

New homogenization results for convex integral functionals and their Euler-Lagrange equations

We study stochastic homogenization for convex integral functionals $$u\mapsto \int_D W(ω,\tfrac{x}\varepsilon,\nabla u)\,\mathrm{d}x,\quad\mbox{where}\quad u:D\subset \mathbb{R}^d\to\mathbb{R}^m,$$ defined on Sobolev spaces. Assuming only stochastic integrability of the map $ω\mapsto W(ω,0,ξ)$, we prove homogenization results under two different sets of assumptions, namely $\bullet_1\quad$ $W$ satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate $W^*(\cdot,0,ξ)$ and a mild monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of $u$, $\bullet_2\quad$ $W$ is $p$-coercive in the sense $|ξ|^p\leq W(ω,x,ξ)$ for some $p>d-1$. Condition $\bullet_2$ directly improves upon earlier results, where $p$-coercivity with $p>d$ is assumed and $\bullet_1$ provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler-Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if $W(ω,x,ξ)$ is comparable to $W(ω,x,-ξ)$ in a suitable sense, we show that the homogenized integrand is differentiable.

math.AP↗

arXiv2023

Stochastic homogenization of functionals defined on finite partitions

We prove a stochastic homogenization result for integral functionals defined on finite partitions assuming the surface tension to be stationary and possibly ergodic. We also consider the convergence of boundary value problems when we impose a boundary value just on part of the boundary. As a consequence, we show that if the homogenized surface tension is isotropic, then one can obtain it by a multi-cell problem where Dirichlet boundary conditions are imposed only at the bottom and the top of the cube. We also show that this result fails in the general anisotropic case.

math.AP↗

arXiv2023

A spectral ansatz for the long-time homogenization of the wave equation

Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present work is a new ansatz for the long-time two-scale expansion inspired by spectral analysis. Based on this spectral ansatz, we extend and refine all previous results in the field, proving homogenization up to optimal timescales with optimal error estimates, and covering all the standard assumptions on heterogeneities (both periodic and stationary random settings).

math.AP↗

arXiv2022

Stochastic homogenization of degenerate integral functionals with linear growth

We study the limit behaviour of a sequence of non-convex, vectorial, random integral functionals, defined on $W^{1,1}$, whose integrands satisfy degenerate linear growth conditions. These involve suitable random, scale-dependent weight-functions. Under minimal assumptions on the integrand and on the weight-functions, we show that the sequence of functionals homogenizes to a non-degenerate functional defined on $BV$.

math.AP↗

arXiv2022

Detecting micro fractures: A comprehensive comparison of conventional and machine-learning based segmentation methods

Studying porous rock materials with X-Ray Computed Tomography (XRCT) has been established as a standard procedure for the non-destructive visualization of flow and transport in opaque porous media. Despite the recent advances in the field of XRCT, some challenges still remain due to the inherent noise and imaging artefacts in the produced data. These issues become even more profound when the objective is the identification of fractures, and/or fracture networks. The challenge is the limited contrast between the regions of interest and the neighboring areas. This limited contrast can mostly be attributed to the minute aperture of the fractures. In order to overcome this challenge, it has been a common approach to apply digital image processing, such as filtering, to enhance the signal-to-noise ratio. Additionally, segmentation methods based on threshold-/morphology schemes can be employed to obtain enhanced information from the features of interest. However, this workflow needs a skillful operator to fine-tune its input parameters, and the required computation time significantly increases due to the complexity of the available methods, and the large volume of the data-set. In this study, based on a data-set produced by the successful visualization of a fracture network in Carrara marble with XRCT, we present the segmentation results from a number of segmentation methods. Three conventional and two machine-learning-based methods are evaluated. The segmentation results from all five methods are compared to each other in terms of segmentation quality and time efficiency. Due to memory limitations, and in order to accomplish a fair comparison, all the methods are employed in a 2D scheme. The output of the 2D U-net model, which is one of the adopted machine-learning-based segmentation methods, shows the best performance regarding the quality of segmentation and the required processing time.

cs.LG↗

arXiv2021

Visualizing particle networks in granular media by in situ X-ray computed tomography

In this contribution, cylindrical samples consisting of monodisperse soft (rubber) and stiff (glass) particles are pre-stressed under uniaxial compression. Acoustic P-waves at ultrasound frequencies are superimposed into prepared samples with different soft-stiff volume fractions. Earlier investigations showed the importance of particles networks, i.e. force chains, in controlling the effective mechanical properties of particulate systems. Measured P-wave modulus showed a significant decline while more soft particles are added due to a change in microstructure. However, for small contents of soft particles, it could be observed that the P-wave modulus is increasing. For the understanding of such kinds of effects, detailed insight into the microstructure of the system is required. To gain this information and link it to the effective properties, we made use of high-resolution micro X-ray Computed Tomography (micro-XRCT) imaging and combined it with the classical stiffness characterization. Both performed in situ meaning inside the laboratory-based XRCT scanner. With micro-XRCT imaging, the granular microstructure can be visualized in 3d and characterized subsequently. By post-processing of the data, the individual grains of the particulate systems could be uniquely identified. Finally, the contact network of the packings which connects the center of particles was established to demonstrate the network transition from stiff- to soft-dominated regimes. This has allowed for unprecedented observations and a renewed understanding of particulate systems. It has been demonstrated that micro-XRCT scans of particles packings can be analyzed and compared in 3d to gain extensive information on the scale of the single particles. Here, the in situ setup and workflow from the start of acquiring images in situ till the post-processing of the image data is explained and demonstrated by selected results.

cond-mat.mtrl-sci↗

arXiv2021

Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations

We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form \begin{equation*} c|ξA(ω,x)|^p\leq f(ω,x,ξ)\leq |ξA(ω,x)|^p+Λ(ω,x) \end{equation*} for some $p\in (1,+\infty)$ and with a stationary and ergodic diagonal matrix $A$ such that its norm and the norm of its inverse satisfy minimal integrability assumptions. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of $f$ with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.

math.AP↗

arXiv2021

Fluctuation estimates for the multi-cell formula in stochastic homogenization of partitions

In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.

math.AP↗

arXiv2021

Emergence of concentration effects in the variational analysis of the $N$-clock model

We investigate the relationship between the $N$-clock model (also known as planar Potts model or $\mathbb{Z}_N$-model) and the $XY$ model (at zero temperature) through a $Γ$-convergence analysis of a suitable rescaling of the energy as both the number of particles and $N$ diverge. We prove the existence of rates of divergence of $N$ for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the $N$-clock model in this regime features an energy which may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex-vortex interaction energy.

math-ph↗

arXiv2021

Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size

We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $δ$, the discretized functionals $Γ$-converge to the Mumford-Shah functional only if $δ\ll\varepsilon$, $\varepsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $Γ$-convergence result also for $δ\sim\varepsilon$, a regime at which the discretization with respect to a periodic lattice converges instead to an anisotropic version of the Mumford-Shah functional.

math.AP↗

arXiv2020

The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$

We study a nearest neighbors ferromagnetic spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of $N$ equi-spaced vectors, also known as $N$-clock model. We find a fast rate of divergence of $N$ with respect to the lattice spacing for which the $N$-clock model has the same discrete-to-continuum variational limit of the $XY$ model, in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of $N$ at which the coarse-grain limit does not detect topological defects, but it is instead a $BV$-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for $N$, whose analysis requires the aid of Cartesian currents.

math-ph↗

arXiv2020

Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle~$\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $Γ$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.

math-ph↗

arXiv2019

Discrete stochastic approximations of the Mumford-Shah functional

We propose a $Γ$-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension.

math.AP↗

arXiv2018

From statistical polymer physics to nonlinear elasticity

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the $Γ$-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the $Γ$-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).

math-ph↗

arXiv2018

Loss of strong ellipticity through homogenization in 2D linear elasticity: A phase diagram

Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Gutiérrez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.

math.AP↗