Abstract This paper explores the theoretical foundations and practical challenges of peridynamics as a nonlocal continuum mechanics method. We establish connections between classical continuum mechanics principles and peridynamics formulations, with a particular focus on understanding how the pairwise force function in peridynamics relates to stress concepts in local theories. The work addresses persistent numerical challenges in peridynamic simulations, including zero‐energy modes and instabilities in correspondence‐based formulations. We present bond‐associated modeling approaches that enhance stability while maintaining computational accuracy. The theoretical discussions are supplemented with discretization methods for spatial and temporal domains. Through illustrative examples of complex fracture phenomena, we demonstrate how these theoretical insights translate to practical applications, providing guidance on selecting appropriate peridynamic formulations for computational mechanics problems. The numerical comparison identifies the bond‐associated quadrature point formulation as the most accurate correspondence‐class approach in the elastic regime. Under progressive bond deletion, however, the shape tensor, on which this formulation relies, becomes ill‐conditioned, and the model loses stability. This necessitates the development of refined damage formulations within the correspondence framework.
The characteristics of microstructure morphology of micro-heterogeneous materials may vary over the macroscopic length scale and thus result in macroscopically distributed, uncertain material properties. Hence, multiscale approaches for the structural analysis of such materials, for example, in terms of the FE 2 $$ {\mathrm{FE}}^2 $$ -method, should not be based on a single representative volume element. In this contribution a method is proposed, which considers different, artificial statistically similar volume elements at each macroscopic integration point which mimic the microstructure variability of the real material as a random field. For this purpose, the microstructure variation of the real material is quantified first in terms of the distribution of a scalar measure containing deviations of statistical measures of higher order, and then this distribution is used to construct a set of artificial microstructures to be used as volume elements within the multiscale simulation. To avoid manual discretization of the large amount of statistically similar volume elements, the finite cell method is combined with concurrent computational homogenization following the FE 2 $$ {\mathrm{FE}}^2 $$ -method. The proposed method is demonstrated for two examples, a simpler tensile experiment for testing purposes and a simplified, idealized deep drawing process.
We present a workflow for the inverse design of architected materials with targeted effective mechanical properties. The approach leverages a low-dimensional descriptor space to represent the topology and morphology of complex mesostructures, enabling efficient navigation within the design space. Data for training is generated through numerical homogenization using a Fast Fourier Transform (FFT)-based solver, providing high-fidelity mappings from structural descriptors on the mesoscale to effective properties. A neural network (NN)-based surrogate model is trained to approximate this mapping. The inverse design task is then formulated as an optimization problem over the descriptor space, where gradient-based optimizers are applied to identify the descriptors, the inputs of the surrogate. We focus on the case of anisotropic linear elasticity and demonstrate the method using spinodoid architected materials, which offer tunable anisotropy and a low-dimensional descriptor space. The framework is validated for the inverse design targeting the anisotropic stiffness of a femoral bone sample. In addition, we propose a method to determine the anisotropy class of a given stiffness tensor. This enables a quantitative evaluation of how closely the bone's anisotropy class can be approximated by spinodoids. We analyze the influence of the optimization loss function on the inverse design outcome by comparing results across different losses. Ultimately, a logarithmic loss function is chosen, as it enables simultaneous optimization of the stiffness and compliance.
Metallic materials may show an ultra-fine lamellar morphology leading to desirable macroscopic mechanical properties. In this paper, an analytical method for modeling the size-dependent mechanical behavior of material systems with lamellar microstructure is proposed. The main contribution of this manuscript is the combination of the exact elastic localization relations for a periodic laminate having inhomogeneous phases with a gradient plasticity constitutive model. This allows a new formulation of the equations governing the evolution of the plastic deformation as a closed system of equations. The yield functions form a system of integro-differential equations that can be solved analytically. This new framework allows to model the size-dependent mechanical behavior of multiphase lamellar materials where the subdomains are elastoplastic single crystals. This is demonstrated using a two-phase laminate. In addition, bounds for the qualitative distributions of the plastic slip and the back stress are derived using a dimensionless quantity. The kinematic hardening due to the back stress depends on the lamella width and the slip system orientation. The back stress vanishes not only for increasing lamella widths but also in slip systems where the slip plane normal is parallel to the lamination direction.
In this article, the influence of large deformations on the effective permeability of a bicontinuous porous material is investigated. On the fine-scale, Neo–Hooke hyperelasticity is considered for the solid skeleton. In a third medium approach, we model the pore space as filled with a softer material of the same type. Fluid flow through the deformed pores is expressed in terms of a Stokes' flow model. The influence of the pore pressure on the deformation is neglected, resulting in a one-way coupling which allows for a sequential solution of the two physical problems. The framework of Variationally Consistent Homogenization is used to derive a two-scale formulation based on a Representative Volume Element (RVE) characterizing the microstructure. Finally, a two-step procedure to compute the deformation dependent permeability is established: Firstly, the deformation of the RVE for a given macroscale deformation gradient is computed. Secondly, sensitivities for the fluid flow through the deformed RVE are computed and used to determine the effective permeability tensor. A numerical study is conducted for sets of RVEs with the same material parameters and different porosity. For the case of uniaxial compression, a significant influence of the deformation on the effective permeability is observed: For a macroscale compression of 20 % $$ 20\% $$ , the effective permeability orthogonal to the compression direction is reduced by almost 60 % $$ 60\% $$ .
The objective of this contribution is to provide novel analytical estimates for the effective properties of micro-heterogeneous elastic solids accounting for interfacial effects, and to compare the results with computational simulations using the finite element method. The interphase transition zone between different constituents in the microstructure here is replaced by the recently proposed canonical interface model . The canonical interface model encompasses all previous interface models such as the general interface model, the cohesive interface model, and the elastic interface model. This manuscript presents a comprehensive study covering a broad range of interface parameters and stiffness ratios, demonstrating the utility of the proposed solutions in understanding and fine-tuning composites behavior.
We study signals that are sparse either on the vertices of a graph or in the graph spectral domain. Recent results on the algebraic properties of random integer matrices as well as on the boundedness of eigenvectors of random matrices imply two types of support size uncertainty principles for graph signals. Indeed, the algebraic properties imply uniqueness results if a sparse signal is sampled at any set of minimal size in the other domain. The boundedness properties of eigenvectors imply stable reconstruction by basis pursuit if a sparse signal is sampled at a slightly larger randomly selected set in the other domain.
Computational homogenization has become an attractive path to determine the macroscopic inelastic behavior of a structure from the knowledge of the behavior of its (usually simpler) constituents, using a Representative Volume Element (RVE) at the microscale. Since the numerical solution of both scales by means of the FE 2 method is prohibitively expensive, much research effort has been spent on the development of surrogate models, particularly by making use of data-driven machine learning methods. Due to the vast spectrum of proposed surrogate modeling approaches, the decision on the method of choice for potential applicants is difficult. While computational efficiency is usually used as the central performance measure, a careful distinction between on- and offline efforts is important. Moreover, the simulation context should also influence the selection. If, for instance, a thousand simulations are to be carried out for a single microstructure, the choice of approach may be different than if single, or a few, simulations are to be carried out for a thousand different microstructures. In the latter case, the flexibility and adaption effort of the method to new microstructures plays a key role. The present study aims to derive guidelines in this direction, by comparing different such approaches and carefully analyzing their performance in simulating inelastic macroscale behaviors, influenced by the morphology of 3D foam-like microstructures. Four formulations are considered: the Deshpande–Fleck phenomenological macroscale model, a general analytical surrogate model, a hybrid neural network surrogate model, and a hyper-reduced FE 2 approach. The methods are compared with respect to computational costs and predictive capabilities for two 3D benchmark scenarios.
Heterogeneous materials are crucial to producing lightweight components, functional components, and structures composed of them. A crucial step in the design process is the rapid evaluation of their effective mechanical, thermal, or, in general, constitutive properties. The established procedure is to use forward models that accept microstructure geometry and local constitutive properties as inputs. The classical simulation‐based approach, which uses, for example, finite elements and FFT‐based solvers, can require substantial computational resources. At the same time, simulation‐based models struggle to provide gradients with respect to the microstructure and the constitutive parameters. Such gradients are, however, of paramount importance for microstructure design and for inverting the microstructure‐property mapping. Machine learning surrogates can excel in these situations. However, they can lead to unphysical predictions that violate essential bounds on the constitutive response, such as the upper (Voigt‐like) or the lower (Reuss‐like) bound in linear elasticity. Therefore, we propose a novel spectral normalization scheme that a priori enforces these bounds. The approach is fully agnostic with respect to the chosen microstructural features and the utilized surrogate model: It can be linked to neural networks, kernel methods, or combined schemes. All of these will automatically and strictly predict outputs that obey the upper and lower bounds by construction. The technique can be used for any constitutive tensor that is symmetric and where upper and lower bounds (in the Löwner sense) exist, that is, for permeability, thermal conductivity, linear elasticity, and many more. We demonstrate the use of spectral normalization in the Voigt–Reuss net using a simple neural network. Numerical examples on truly extensive datasets illustrate the improved accuracy, robustness, and independence of the type of input features in comparison to much‐used neural networks.
In this paper we establish for an intermediate Reynolds number domain the stability of N-front and N-back solutions for each N > 1 corresponding to traveling waves, in an experimentally validated model for the transition to turbulence in pipe flow proposed in [Barkley et al., Nature 526(7574):550-553, 2015]. We base our work on the existence analysis of a heteroclinic loop between a turbulent and a laminar equilibrium proved by Engel, Kuehn and de Rijk in [Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022], as well as some results from this work. The stability proof follows the verification of a set of abstract stability hypotheses stated by Sandstede in [SIAM Journal on Mathematical Analysis 29.1 (1998), pp. 183-207] for traveling waves motivated by the FitzHugh-Nagumo equations. In particular, this completes the first detailed analysis of Engel, Kuehn and de Rijk in [Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022] leading to a complete existence and stability statement that nicely fits within the abstract framework of waves generated by twisted heteroclinic loops.
The evolution of image halftoning, from its analog roots to contemporary digital methodologies, encapsulates a fascinating journey marked by technological advancements and creative innovations. Yet the theoretical understanding of halftoning is much more recent. In this article, we explore various approaches towards shedding light on the design of halftoning approaches and why they work. We discuss both halftoning in a continuous domain and on a pixel grid. We start by reviewing the mathematical foundation of the so-called electrostatic halftoning method, which departed from the heuristic of considering the back dots of the halftoned image as charged particles attracted by the grey values of the image in combination with mutual repulsion. Such an attraction-repulsion model can be mathematically represented via an energy functional in a reproducing kernel Hilbert space allowing for a rigorous analysis of the resulting optimization problem as well as a convergence analysis in a suitable topology. A second class of methods that we discuss in detail is the class of error diffusion schemes, arguably among the most popular halftoning techniques due to their ability to work directly on a pixel grid and their ease of application. The main idea of these schemes is to choose the locations of the black pixels via a recurrence relation designed to agree with the image in terms of the local averages. We discuss some recent mathematical understanding of these methods that is based on a connection to Sigma-Delta quantizers, a popular class of algorithms for analog-to-digital conversion.
We consider a local Cahn-Hilliard-type model for tumor growth as well as a nonlocal model where, compared to the local system, the Laplacian in the equation for the chemical potential is replaced by a nonlocal operator. The latter is defined as a convolution integral with suitable kernels parametrized by a small parameter. For sufficiently smooth bounded domains in three dimensions, we prove convergence of weak solutions of the nonlocal model towards strong solutions of the local model together with convergence rates with respect to the small parameter. The proof is done via a Gronwall-type argument and a convergence result with rates for the nonlocal integral operator towards the Laplacian due to Abels, Hurm arXiv:2307.02264.
Support vector machines (SVMs) are an important tool for performing classification on scattered data, where one usually has to deal with many data points in high-dimensional spaces. We propose solving SVMs in primal form using feature maps based on trigonometric functions or wavelets. In small dimensional settings the fast Fourier transform (FFT) and related methods are a powerful tool in order to deal with the considered basis functions. For growing dimensions the classical FFT-based methods become inefficient due to the curse of dimensionality. Therefore, we restrict ourselves to multivariate basis functions, each of which only depends on a small number of dimensions. This is motivated by the well-known sparsity of effects and recent results regarding the reconstruction of functions from scattered data in terms of truncated analysis of variance (ANOVA) decompositions, which makes the resulting model even interpretable in terms of importance of the features as well as their couplings. The usage of small superposition dimensions has the consequence that the computational effort no longer grows exponentially but only polynomially with respect to the dimension. In order to enforce sparsity regarding the basis coefficients, we use the frequently applied ℓ 2 $$ {\ell}_2 $$ -norm and, in addition, ℓ 1 $$ {\ell}_1 $$ -norm regularization. The found classifying function, which is the linear combination of basis functions, and its variance can then be analyzed in terms of the classical ANOVA decomposition of functions. Based on numerical examples we show that we are able to recover the signum of a function that perfectly fits our model assumptions. Furthermore, we perform classification on different artificial and real-world data sets. We obtain better results with ℓ 1 $$ {\ell}_1 $$ -norm regularization, both in terms of accuracy and clarity of interpretability.
AbstractAll digital objects that result from the modeling and simulation field are valid sets of research data. In general, research data are the result of intense intellectual activity that is worth communicating. This communication is an essential research practice that, whether with the aim of understanding, critiquing or further developing results, smoothly leads to collaboration, which not only involves discussions, and sharing institutional resources, but also the sharing of data and information at several stages of the research process. Data sharing is intended to improve and facilitate collaboration but quickly introduces challenges like reproducibility, reusability, interoperability, and standardization. These challenges are deeply rooted in an apparent reproducibility standard, about which there is a debate worth considering before emphasizing how the modeling and simulation workflow commonly occurs. Although that workflow is almost natural for practitioners, the sharing practices still require special attention because the principles (known as FAIR principles) that guide research practices towards data sharing also guide the requirements for machine actionable results. The FAIR principles, however, do not address the actual implementation of the data sharing process. This implementation requires careful consideration of characteristics of the sharing platforms for benefiting the most of the data sharing activity. This article serves as an invitation to integrate data sharing practices into the established routines of researchers and elaborates on the perspectives, and guidelines surrounding data sharing implementation.
AbstractComputational fluid dynamics (CFD) carry the potential to provide detailed insights into intraventricular hemodynamics and complement in vivo flow measurement techniques. A variety of CFD approaches emerged in recent years, mostly building solely on medical image data as patient‐specific input. While the utilized medical imaging method and chosen CFD approach both influence the computed hemodynamics, thereto related differences are rarely investigated. The present study addresses this issue with an inter‐(imaging)‐modality and inter‐model comparison of intracardiac flow computations. Magnetic resonance imaging (MRI) and transthoracic echocardiography (TTE) data of a volunteer were acquired and used to reconstruct the anatomical structures. For each modality, the reconstructed shapes were applied in two previously introduced CFD approaches to compute whole‐cycle ventricular flow patterns. While both methods involved benefits and challenges, similar valve velocities were computed, being in accordance with in vivo 4D flow MRI and pulsed‐wave Doppler velocity measurements (systolic peak velocity: 1.24–1.26 m/s (MRI), 0.9–1.25 m/s (TTE); diastolic peak velocity: 0.54 m/s (MRI), 0.59–0.75 m/s (TTE)). A detailed flow analysis with vortex formation, kinetic energy, and mid‐ventricular velocities indicated the computed inter‐modality differences to be larger than inter‐method ones. Quantitatively, this could be observed in the direct flow rate ( inter‐modality: 13, inter‐method, 3). These results help to gain trust in CFD approaches to compute intraventricular flow and emphasize the importance of standardized input data. Future studies, however, should consider a broader data base.
The occurrence of in-stent restenosis following percutaneous coronary intervention highlights the need for the creation of computational tools that can extract pathophysiological insights and optimize interventional procedures on a patient-specific basis. In light of this, a modeling framework encompassing the chemo-mechano-biological interactions in the arterial wall and the effects of hemodynamic perturbations is introduced in this work.
Forward-backward parabolic equations have been studied since the 1980s, but a mathematically rigorous picture is still far from being established. As quite a number of new papers have appeared recently, we review in this work the current state of the art. We focus our analysis on the status quo regarding the three most common types of regularizations, namely semidiscretization, the viscous approximation, and regularization with higher order spatial derivatives.
SummaryThe SIMulation supported LIVer Assessment for donor organs (SimLivA) project aims to develop a mathematical model to accurately simulate the influence of mechanical alterations in marginal liver grafts (specifically steatotic ones) and cold ischemia on early ischemia‐reperfusion injury (IRI) during liver transplantation. Our project tackles significant research challenges, including the co‐development of computational methodologies, experimental studies, clinical processes, and technical workflows. We aim to refine a continuum‐biomechanical model for enhanced IRI prediction, collect pivotal experimental and clinical data, and assess the clinical applicability of our model. Our efforts involve augmenting and tailoring a coupled continuum‐biomechanical, multiphase, and multi‐scale partial differential equation‐ordinary differential equation (PDE‐ODE) model of the liver lobule, allowing us to numerically simulate IRI depending on the degree of steatosis and the duration of ischemia. The envisaged model will intertwine the structure, perfusion, and function of the liver, serving as a crucial aid in clinical decision‐making processes. We view this as the initial step towards an in‐silico clinical decision support tool aimed at enhancing the outcomes of liver transplantation. In this paper, we provide an overview of the SimLivA project and our preliminary findings, which include: a cellular model that delineates critical processes in the context of IRI during transplantation; and the integration of this model into a multi‐scale PDE‐ODE model using a homogenized, multi‐scale, multi‐component approach within the Theory of Porous Media (TPM) framework. The model has successfully simulated the interconnected relationship between structure, perfusion, and function—all of which are integral to IRI. Initial results show simulations at the cellular scale that describe critical processes related to IRI during transplantation. After integrating this model into a multiscale PDE‐ODE model, first simulations were performed on the spatial distribution of key functions during warm and cold ischaemia. In addition, we were able to study the effect of tissue perfusion and temperature, two critical parameters in the context of liver transplantation and IRI.
Predicting the long-term success of endovascular interventions in the clinical management of cerebral aneurysms requires detailed insight into the patient-specific physiological conditions. In this work, we not only propose numerical representations of endovascular medical devices such as coils, flow diverters or Woven EndoBridge but also outline numerical models for the prediction of blood flow patterns in the aneurysm cavity right after a surgical intervention. Detailed knowledge about the post-surgical state then lays the basis to assess the chances of a stable occlusion of the aneurysm required for a long-term treatment success. To this end, we propose mathematical and mechanical models of endovascular medical devices made out of thin metal wires. These can then be used for fully resolved flow simulations of the post-surgical blood flow, which in this work will be performed by means of a Lattice Boltzmann method applied to the incompressible Navier-Stokes equations and patient-specific geometries. To probe the suitability of homogenized models, we also investigate poro-elastic models to represent such medical devices. In particular, we examine the validity of this modeling approach for flow diverter placement across the opening of the aneurysm cavity. For both approaches, physiologically meaningful boundary conditions are provided from reduced-order models of the vascular system. The present study demonstrates our capabilities to predict the post-surgical state and lays a solid foundation to tackle the prediction of thrombus formation and, thus, the aneurysm occlusion in a next step.
This review provides an introduction to - and overview of - the current state of the art in neural-network based regularization methods for inverse problems in imaging. It aims to introduce readers with a solid knowledge in applied mathematics and a basic understanding of neural networks to different concepts of applying neural networks for regularizing inverse problems in imaging. Distinguishing features of this review are, among others, an easily accessible introduction to learned generators and learned priors, in particular diffusion models, for inverse problems, and a section focusing explicitly on existing results in function space analysis of neural-network-based approaches in this context.
