arXiv subjects

Malte A. Peter

Publications and source records attributed to Malte A. Peter.

15 recordsLinked to original sources

arXiv2026

Neural Network Enhanced Polyconvexification of Isotropic Energy Densities in Computational Mechanics

We present a neural network approach for fast evaluation of parameter-dependent polyconvex envelopes, which are crucial in computational mechanics. Our method uses a neural network architecture that inherently encodes polyconvexity in the main variable by combining a feature extraction layer that computes the minors function on the signed singular value characterisation of isotropic energy densities with a Partially Input Convex Neural Network (PICNN). The envelope inequality is weakly enforced by penalisation during training, as are the symmetries of the function. As a guiding example, we focus on a pseudo time incremental variational damage problem, which is parameter-dependent on previous time-step iterates, the deformation gradient and the internal variable. This problem is reformulated in terms of signed singular values and a splitting approach is applied to reduce the dimension of the parameter space, thereby making training more tractable. Numerical experiments show that the networks achieve favourable accuracy for engineering applications while providing high compression and significant speed-up over traditional polyconvexification schemes. Most importantly, the network adapts to varying physical or material parameters, enabling real-time polyconvexification in large-scale computational mechanics scenarios.

math.NA↗

arXiv2026

Effective permeabilities for flow through anisotropic microscopic geometries

This work develops a computational and theoretical framework for determining effective permeabilities in anisotropic microscopic geometries containing dense, fibre-like obstacles, motivated by the need to model flow in coiled aneurysm domains accurately. Building on homogenisation theory and fully resolved simulations in Representative Elementary Volumes (REVs), we validate the permeability model introduced in [C. Boutin, Study of permeability by periodic and self-consistent homogenisation. Eur. J. Mech. A Solids, 19(4):603-632, 2000] and propose a systematic methodology for capturing the directional variations induced by fibre orientation. The resulting permeability tensors are incorporated into macroscopic flow simulations based on the Darcy equation, enabling direct comparison of anisotropic and isotropic permeability models across several benchmark configurations. Our findings show that anisotropy has a significant impact on local flow direction and magnitude, generating directional permeability contrasts which cannot be reproduced by classical isotropic approximations. By integrating coil-induced microstructural effects into continuum-scale hemodynamic models, the proposed approach enables more realistic assessment of post-treatment aneurysm flow behaviour. Beyond this clinical application, the framework is broadly applicable to other biomedical and engineering systems involving fibrous or filamentous porous microstructures.

physics.flu-dyn↗

arXiv2026

Effective elastic wave transmission through a periodically voided interface

Effective interface conditions for a periodically voided thin layer separating two homogeneous bulk regions are derived for the elastic wave equation by taking the simultaneous limit of vanishing layer periodicity and layer thickness. The limit problems are obtained using the unfolding method for thin perforated domains. We consider three different scalings of the material parameters in the layer that characterise its stiffness, each leading to a distinct type of interface condition and requiring the solution of scaling-dependent cell problems. Depending on the scaling, the resulting effective model yields either a membrane equation or a Kirchhoff-Love plate equation. In the critical regime of reduced stiffness, the interface equation additionally depends on the microscopic variable. By selecting appropriate cell problems, this equation can be reformulated as an effective interface condition between the bulk domains.

math.AP↗

arXiv2025

Effective transmission through an interface with evolving microstructure

We study the asymptotic behaviour of a system of nonlinear reaction--diffusion--advection equations in a domain consisting of two bulk regions connected via microscopic channels distributed within a thin membrane. Both the width of the channels and the thickness of the membrane are of order $\varepsilon \ll 1$, and the geometry evolves in time in an a priori known way. We consider nonlinear flux boundary conditions at the lateral boundaries of the channels and critical scaling of the diffusion inside the layer. Extending the method of homogenisation in domains with evolving microstructure to thin layers, we employ two-scale convergence and unfolding techniques in thin layers to derive an effective model in the limit $\varepsilon \to 0$, in which the membrane is reduced to a lower-dimensional interface. We obtain jump conditions for the solution and the total fluxes, which involve the solutions of local, space--time-dependent cell problems in the reference channel.

math.AP↗

arXiv2025

Characterization of polyconvex isotropic functions

Polyconvexity is an important concept in the analysis of energies related to elasticity. A function $f \colon \R^{d\times d} \to \R$ is called polyconvex if it can be written as a convex function in the minors of the argument. We show that for isotropic functions it suffices to consider diagonal matrices. For $d=3$, this leads to a dimension reduction for the convex representative of $f$ from $\R^{19}$ to $\R^7$. Moreover, we present a new result for the polyconvexity of functions formulated in the principal invariant of the left or right stretch tensor.

math.AP↗

arXiv2024

Rigorous derivation of an effective model for coupled Stokes advection, reaction and diffusion with freely evolving microstructure

We consider the homogenisation of a coupled Stokes flow and advection-reaction-diffusion problem in a perforated domain with an evolving microstructure of size $\varepsilon$. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer having a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection-reaction-diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for $\varepsilon \to 0$, based on $\varepsilon$-uniform a priori estimates. As a result of the limit passage, the processes on the macroscale are described by an advection-reaction-diffusion problem coupled to Darcy's equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells, which depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro-macro model. For pure reactive-diffusive transport coupled with microscopic domain evolution but without advective transport, homogenisation results have recently been presented. We extend these models by advective transport which is driven by the Stokes equation in the a priori unknown evolving pore domain.

math.AP↗

arXiv2024

A Darcy law with memory by homogenisation for evolving microstructure

We consider the homogenisation of the instationary Stokes equations in a porous medium with an a-priori given evolving microstructure. In order to pass to the homogenisation limit, we transform the Stokes equations to a domain with a fixed periodic microstructure. The homogenisation result is a Darcy-type equation with memory term and has the form of an integro-differential equation. The evolving microstructure leads to a time and space dependent permeability coefficient and the local change of the porosity causes an additional source term for the pressure.

math.AP↗

arXiv2023

Computational polyconvexification of isotropic functions

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope of a function on a $d$-dimensional manifold, which is easily realized by standard algorithms. The significant speedup associated with the dimensional reduction from $d^2$ to $d$ is demonstrated in a series of numerical experiments.

math.NA↗

arXiv2023

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation by rank-one convexification prevents non-existence of minimizers and mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

cs.CE↗

arXiv2022

Homogenisation of local colloid evolution induced by reaction and diffusion

We consider the homogenisation of a coupled reaction-diffusion process in a porous medium with evolving microstructure. A concentration-dependent reaction rate at the interface of the pores with the solid matrix induces a concentration-dependent evolution of the domain. Hence, the evolution is fully coupled with the reaction-diffusion process. In order to pass to the homogenisation limit, we employ the two-scale-transformation method. Thus, we homogenise a highly non-linear problem in a periodic and in time cylindrical domain instead. The homogenisation result is a reaction-diffusion equation, which is coupled with an internal variable, representing the local evolution of the pore structure.

math.AP↗

arXiv2022

RayleighBloch waves above the cut-off

Extensions of Rayleigh-Bloch waves above the cut-off frequency are studied via the discrete spectrum of a transfer operator for a generalised channel containing a single cylinder. Their wavenumbers are shown to become complex-valued and an additional pair of wavenumbers to appear. For small to intermediate radius values, the extended Rayleigh-Bloch waves are shown connect the Neumann and Dirichlet trapped modes, then embed in the continuous spectrum. Rayleigh-Bloch waves vanish as frequency increases but reappear at high frequencies for small and large cylinders. The existence and properties of the Rayleigh-Bloch waves are connected with finite-array resonances.

physics.flu-dyn↗

arXiv2021

Homogenisation of the Stokes equations for evolving microstructure

We consider the homogenisation of the Stokes equations in a porous medium which is evolving in time. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables to model a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit with the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method and construct a family of $\varepsilon$-scaled operators $\operatorname{div}_\varepsilon^{-1}$, which are right-inverse to the corresponding divergences. The homogenisation result is a new version of Darcy's law. It features a time- and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic compressibility condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume.

math.AP↗

arXiv2019

Broadband control of water wave energy amplification in chirped arrays

Water waves in natural environments are typically broadband, nonlinear and dynamic phenomena. Taking concepts developed for slow light in optics, we address the challenge of designing arrays to control the spatial distribution of wave energy, and amplify target frequencies at specified locations. Experiments on incident waves interacting with a chirped array of eight vertical cylinders demonstrate significant amplifications as predicted numerically, and provide motivation for application to energy harvesting.

physics.flu-dyn↗

arXiv2018

Graded resonator arrays for spatial frequency separation and amplification of water waves

A structure capable of substantially amplifying water waves over a broad range of frequencies at selected locations is proposed. The structure consists of a small number of C-shaped cylinders arranged in a line array, with the cylinder properties graded along the array. Using linear potential-flow theory, it is shown that the energy carried by a plane incident wave is amplified within specified cylinders, for wavelengths comparable to the array length, and for a range of incident directions. Transfer matrix analysis is used to attribute the large amplifications to excitation of Rayleigh--Bloch waves and gradual slowing down of their group velocity along the array.

physics.flu-dyn↗

arXiv2017

A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.

math.AP↗