Information propagation characterizes how input correlations evolve across layers in deep neural networks. This framework has been well studied using mean-field theory, which assumes infinitely wide networks. However, these assumptions break down for practical, finite-size networks. In this work, we study information propagation in randomly initialized neural networks with finite width and reveal that the boundary between ordered and chaotic regimes exhibits a fractal structure.

The continuous casting tundish plays a critical role as a metallurgical reactor in the continuous casting process, with its flow characteristics serving as a key parameter in the production of high-quality steel. These characteristics are typically assessed through residence time distribution (RTD) curves. This study examines the flow behaviour in a single-strand continuous casting tundish through a combination of numerical simulations and experimental validation.

Reduced order models (ROMs) have been regarded as an efficient alternative to conventional high-fidelity methods for accelerating the design and optimization processes in engineering applications. Many industrial facilities feature repeating geometrical patterns or contain subregions governed by distinct physical phenomena, making them well-suited to Domain Decomposition (DD) techniques. The integration of a ROM and DD is promising to further reduce computational costs by constructing local ROMs and assembling them into global solutions.

The current study aims to develop a non-intrusive Reduced Order Model (ROM) to reconstruct the full temperature field for a large-scale industrial application based on both numerical and experimental datasets. The proposed approach is validated against a domestic refrigerator.

Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives.

In computational fluid dynamics, Large Eddy Simulation (LES) offers a compelling balance between accuracy and computational cost by resolving large-scale flow structures while modeling unresolved subgrid scales. However, its predictive capacity is critically dependent on the choice and calibration of subgrid-scale (SGS) models, which often involve problem-dependent parameters and exhibit intricate interactions with the numerical discretization.

The widespread adoption of embedded vision systems in industrial applications has highlighted the limitations of deep learning models, which are characterized by a high number of parameters. This is representing a significant concern within the scientific community due to the increased computational resources and memory required for training and inference of these models. Addressing this, we propose a flexible and effective methodology for neural network compression that integrates a pluggable dimensionality reduction layer with a Knowledge Distillation (KD) approach.

Parameter-dependent dynamical systems that exhibit bifurcations pose significant computational challenges, as traditional continuation methods require repeated, costly simulations across large ranges of parameter values to capture sudden qualitative changes in the solution. In this work, we propose a systematic approach to reconstruct the branches of the entire bifurcation diagram in a single numerical solver leveraging generalized Polynomial Chaos (PC) expansion.

The process of decision-making in football is characterized by a complex interplay between spatial positioning, opponent pressure, and player intent. This work introduces a Graph Neural Network (GNN) framework designed to predict Receiver Selection, the optimal passing target, by modeling on-field interactions as dynamic graphs. Each player is represented as a node with positional and contextual features, while potential passing lines form weighted edges characterized by distance, angle, and pressure metrics.

Semi-discrete optimal transport (SOT), which maps a continuous probability measure to a discrete one, is a fundamental problem with wide-ranging applications. Entropic regularization is often employed to solve the SOT problem, leading to a regularized (RSOT) formulation that can be solved efficiently via its convex dual. However, a significant computational challenge emerges when the continuous source measure is discretized via the finite element (FE) method to handle complex geometries or densities, such as those arising from solutions to Partial Differential Equations (PDEs).

Super-resolution (SR) techniques based on deep learning have recently emerged as a promising approach to enhance the spatial resolution of computational fluid dynamics simulations while containing computational cost. In this paper, we investigate several SR architectures to improve coarse-grid simulations of mesoscale atmospheric flows, with training data generated from simulations of the weakly compressible Euler equations.

In this paper, we develop and analyze a mixed finite element method for a nonlinear, higher-order model describing nonlinear wave phenomena and exhibiting important conservation properties. A central goal of our approach is to ensure that these properties are preserved at the discrete level while avoiding the challenges typically encountered when constructing finite element subspaces of $H^2$ as would be required in a standard continuous Galerkin framework. At the continuous level, we establish well-posedness and characterize the solution through energy laws and mass conservation.

We demonstrate exponential convergence of Reduced Order Model (ROM) approximations for mixed boundary value problems of the stationary, incompressible Navier-Stokes equations in plane, polygonal domains Ω. Admissible boundary conditions comprise mixed BCs, no-slip, slip and open boundary conditions, subject to corner-weighted analytic boundary data and volume forcing. The small data hypothesis is assumed to ensure existence of a unique weak solution in the sense of Leray-Hopf.

We present a novel approach to define the filter and relax steps in the evolve-filter-relax (EFR) framework for simulating turbulent flows. The EFR main advantages are its ease of implementation and computational efficiency. However, as it only contains two parameters (one for the filter step and one for the relax step) its flexibility is rather limited. In this work, we propose a data-driven approach in which the optimal filter is found based on DNS data in the frequency domain.

We propose a reinforcement learning (RL) framework for the dynamic selection of the filter parameter in Evolve-Filter (EF) regularization strategies for incompressible turbulent flows. Instead of prescribing the filter radius heuristically, the RL agent learns to adaptively control the filtering intensity in time, balancing numerical stability and physical accuracy. The methodology is assessed on two benchmark problems with fundamentally different dynamics: flow past a cylinder and decaying homogeneous turbulence.

In recent years, large-scale numerical simulations played an essential role in estimating the effects of explosion events in urban environments, for the purpose of ensuring the security and safety of cities. Such simulations are computationally expensive and, often, the time taken for one single computation is large and does not permit parametric studies. The aim of this work is therefore to facilitate real-time and multi-query calculations by employing a non-intrusive Reduced Order Method (ROM). 

This work presents an overview of several nonlinear reduction strategies for data compression from various research fields, and a comparison of their performance when applied to problems characterized by diffusion and/or advection terms. We aim to create a common framework by unifying the notation referring to a common two-stage pipeline. At the same time, we underline their main differences and objectives by highlighting the diverse choices made for each stage.

This work presents a comprehensive framework for capturing bifurcating phenomena and detecting bifurcation curves in nonlinear multiparametric partial differential equations, where the system exhibits multiple coexisting solutions for given values of the parameters. Traditional continuation methods for one-dimensional parameterizations employ the previously computed solution as the initial guess for the next parameter value.

We present a comparative analysis of the chemo-mechanical mechanisms that drive spontaneous oscillations in two distinct active filamentous structures: photo-chemically deformable liquid crystal elastomer (LCE) rods and ATP-powered eukaryotic cilia. Using a unified framework of active planar rods, we develop simplified mathematical models for both systems. We reduce the governing partial differential equations to one-degree-of-freedom (1-DOF) nonlinear oscillators, each undergoing a supercritical Hopf bifurcation.

This special issue, dedicated to the inaugural edition of the ECCOMAS-IACM conference series Math 2 Product (M2P), contains eleven articles on frontier topics in Computational Science and Engineering, with a specific focus on next-generation methodologies for industrial problems and sustainability.

We propose an efficient hyper-reduced order model (HROM) designed for segregated finite-volume solvers in geometrically parametrized problems. The method follows a discretize-then-project strategy: the full-order operators are first assembled using finite volume or finite element discretizations and then projected onto low-dimensional spaces using a small set of spatial sampling points, selected through hyper-reduction techniques such as DEIM. This approach removes the dependence of the online computational cost on the full mesh size.

The complexity of the cardiovascular system needs to be accurately reproduced in order to promptly acknowledge health conditions; to this aim, advanced multifidelity and multiphysics numerical models are crucial. On one side, Full Order Models (FOMs) deliver accurate hemodynamic assessments, but their high computational demands hinder their real-time clinical application. In contrast, Reduced Order Models (ROMs) provide more efficient yet accurate solutions, essential for personalized healthcare and timely clinical decision-making.

Graph autoencoders have gained attention in nonlinear reduced-order modeling of parameterized partial differential equations defined on unstructured grids. Despite they provide a geometrically consistent way of treating complex domains, applying such architectures to parameterized dynamical systems for temporal prediction beyond the training data, i.e. the extrapolation regime, is still a challenging task due to the simultaneous need of temporal causality and generalizability in the parametric space.

In collaboration with industrial partners, this research leverages the concept of a “Digital Twin” to create a virtual replica of industrial machinery that monitors health, identifies anomalies, and predicts potential failures. Analysing time-series data from multiple sensors poses significant challenges, particularly as machines dynamically adjust their operating conditions to meet production demands, making traditional forecasting algorithms ineffective.

This work investigates model reduction techniques for nonlinear parameterized and time-dependent PDEs, specifically focusing on bifurcating phenomena in Computational Fluid Dynamics (CFD). We develop interpretable and non-intrusive Reduced Order Models (ROMs) capable of capturing dynamics associated with bifurcations by identifying a minimal set of coordinates. Our methodology combines the Sparse Identification of Nonlinear Dynamics (SINDy) method with a deep learning framework based on Autoencoder (AE) architectures.

In this paper, we introduce a data-driven filter to analyze the relationship between Implicit Large-Eddy Simulations (ILES) and Direct Numerical Simulations (DNS) in the context of the Spectral Difference method. The proposed filter is constructed from a linear combination of sharp-modal filters where the weights are given by a convolutional neural network trained to replicate ILES results from filtered DNS data. In order to preserve the compactness of the discretization, the filter is local in time and acts at the elementary cell level.

The Plateau–Rayleigh instability shows that a cylindrical fluid flow can be destabilized by surface tension. Similarly, capillary forces can make an elastic cylinder unstable when the elastocapillary length is comparable to the cylinder’s radius. While existing models predict a single isolated bulge as the result of an instability, experiments reveal a periodic sequence of bulges spaced out by thinned regions, a phenomenon known as beading instability. Most models assume that surface tension is independent of the deformation of the solid, neglecting variations due to surface stretch.

The current work aims to incorporate physics-based loss in Physics Informed Neural Network (PINN) directly using the numerical residual obtained from the governing equation in any dicretized forward solver. PINN's major difficulties in coupling with external forward solvers arise from the inability to access the discretized form (Finite difference, finite volume, finite element, etc.) of the governing equation directly through the network and to include them in its computational graph.

This paper presents a projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP s) arising from cardiovascular (CV) applications. In real-life scenarios, accurately defining outflow boundary conditions in patient-specific models poses significant challenges due to complex vascular morphologies, physiological conditions, and high computational demands. These challenges make it difficult to compute realistic and reliable CV hemodynamics by incorporating clinical data such as 4D magnetic resonance imaging.

The design of compliant mechanisms is crucial in several technologies and relies on the availability of solutions for nonlinear structural problems. One of these solutions is given and experimentally validated in the present article for a compliant mechanism moving along a smooth curved profile. In particular, a deformable elastic rod is held by two clamps, one at each end. The first clamp is constrained to slide without friction along a curved profile, while the second clamp moves in a straight line transmitting its motion through the elastic rod to the first clamp.

The flow behavior in the continuous casting tundish plays a critical role in steel quality and is typically characterized via residence time distribution (RTD) curves. This study investigates the fluid flow behaviour in a single-strand tundish using numerical and experimental approaches. Full-order model (FOM) steady-state simulations were conducted under both isothermal and non-isothermal conditions to assess the influence of thermal buoyancy on the flow characteristics.

In this work, reduced order models are presented for fluid flows characterized by different Mach numbers, ranging from low-speed, highly viscous fluid flows to weakly compressible flows. To populate the initial database of high-fidelity solutions, a high-fidelity solver based on the discontinuous Galerkin method was used, given its capability to deal with both fluids at low Reynolds and convection-dominated problems.

The proper orthogonal decomposition has been applied on a full-scale horizontal-axis wind turbine to shed light on the wake characteristics behind the wind turbine. In reality, the blade tip experiences high deflections even at the rated conditions which definitely alter the wake flow field, and in the case of a wind farm, may complicate the inlet conditions of the downstream wind turbine.

We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses these issues by introducing a data augmentation strategy based on OT principles. The proposed framework generates interpolated solutions tracing geodesic paths in the space of probability distributions, enriching the training dataset for the ROM.

We propose and analyze an $H^2$-conforming virtual element method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \geq 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $\Omega$ and is rather elementary. We prove stability and error estimates in $H^2(\Omega)$, including the effect of quadrature, under minimal regularity of the data.

Numerical stabilization techniques are often employed in under-resolved simulations of convection-dominated flows to improve accuracy and mitigate spurious oscillations. Specifically, the evolve--filter--relax (EFR) algorithm is a framework which consists in evolving the solution, applying a filtering step to remove high-frequency noise, and relaxing through a convex combination of filtered and original solutions.

We present a novel approach to perform agglomeration of polygonal and polyhedral grids based on spatial indices. Agglomeration strategies are a key ingredient in polytopal methods for PDEs as they are used to generate (hierarchies of) computational grids from an initial grid. Spatial indices are specialized data structures that significantly accelerate queries involving spatial relationships in arbitrary space dimensions.

Forecasting atmospheric flows with traditional discretization methods, also called full order methods (e.g., finite element methods or finite volume methods), is computationally expensive. We propose to reduce the computational cost with a Reduced Order Model (ROM) that combines Extended Convolutional Autoencoders (E-CAE) and Reservoir Computing (RC).

It is well known that in the computational fluid dynamics simulations related to the cardiovascular system the enforcement of outflow boundary conditions is a crucial point. In fact, they highly affect the computed flow and a wrong setup could lead to unphysical results. In this chapter we discuss the main features of two different ways for the estimation of proper outlet boundary conditions in the context of hemodynamics simulations: on one side, a lumped parameter model of the downstream circulation and, on the other one, a technique based on optimal control.

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution.

This study introduces a first step for constructing a hybrid reduced-order models (ROMs) for segregated fluid-structure interaction in an Arbitrary Lagrangian-Eulerian (ALE) approach at a high Reynolds number using the Finite Volume Method (FVM). The ROM is driven by proper orthogonal decomposition (POD) with hybrid techniques that combines the classical Galerkin projection and two data-driven methods (radial basis networks , and neural networks/ long short term memory). Results demonstrate the ROM ability to accurately capture the physics of fluid-structure interaction phenomena.

We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement u are given in the quite general form u(x) ∈ F_x, where F_x is a closed set, and thus comprise confinement conditions, unilateral constraints, controlled displacement conditions, etc., both in the bulk and on the boundary of the body.

We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM.

In this work, we present the modelling and numerical simulation of a molten glass fluid flow in a furnace melting basin. We first derive a model for a molten glass fluid flow and present numerical simulations based on the Finite Element Method (FEM). We further discuss and validate the results obtained from the simulations by comparing them with experimental results.

This chapter provides an extended overview about Reduced Order Models (ROMs), with a focus on their features in terms of efficiency and accuracy. In particular, the aim is to browse the more common ROM frameworks, considering both intrusive and data-driven approaches. We present the validation of such techniques against several test cases. The first one is an academic benchmark, the thermal block problem, where a Poisson equation is considered. Here a classic intrusive ROM framework based on a Galerkin projection scheme is employed.

In this manuscript, we combine non-intrusive reduced-order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM, able to accurately capture the flow dynamics in different physical settings. The flow prediction obtained using the mixed formulation is derived from a convex combination of the predictions of several previously trained reduced-order models (ROMs), with each model assigned a space-dependent weight.

The simulation of atmospheric flows by means of traditional discretization methods remains computationally intensive, hindering the achievement of high forecasting accuracy in short time frames. In this paper, we apply three reduced order models that have successfully reduced the computational time for different applications in computational fluid dynamics while preserving accuracy: Dynamic Mode Decomposition (DMD), Hankel Dynamic Mode Decomposition (HDMD), and Proper Orthogonal Decomposition with Interpolation (PODI).

Real-world applications of computational fluid dynamics often involve the evaluation of quantities of interest for several distinct geometries that define the computational domain or are embedded inside it. For example, design optimization studies require the realization of response surfaces from the parameters that determine the geometrical deformations to relevant outputs to be optimized.

Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models’ design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), which explores the synergies of active subspaces with supervised clustering techniques in order to carry out a more efficient dimension reduction in the parameter space.

This chapter focuses on the combination of reduced order models and data-driven techniques applied to the study of turbulent flows in order to improve the pressure and velocity accuracy of standard reduced order methods. We focus on reduced order models constructed by means of Proper Orthogonal Decomposition with Galerkin approach, enhanced with two different stabilization techniques: (i) the supremizer enrichment approach, (ii) the pressure Poisson equation approach.

Neuronal cultures have been a reference experimental model for several decades. However, 3D cell arrangement, spatial constraints on neurite outgrowth, and realistic synaptic connectivity are missing. The latter limits the study of structure and function in the context of compartmentalization and diminishes the significance of cultures in neuroscience. Approximating ex vivo the structured anatomical arrangement of synaptic connectivity is not trivial, despite being key for the emergence of rhythms, synaptic plasticity, and ultimately, brain pathophysiology.

We propose a regularization for Reduced Order Models (ROMs) of the quasi-geostrophic equations (QGE) to increase accuracy when the Proper Orthogonal Decomposition (POD) modes retained to construct the reduced basis are insufficient to describe the system dynamics. Our regularization is based on the so-called BV-alpha model, which modifies the nonlinear term in the QGE and adds a linear differential filter for the vorticity.

We present an a posteriori error analysis for the mixed virtual element method (mixed-VEM) applied to general second order elliptic equations. The resulting error estimator is of residual-type. Via the inclusion of a fully local postprocessing of the mixed-VEM solution, we show that the estimator provides a reliable and efficient control on the -norm error between the exact and the postprocessed flux. Numerical examples confirm the theoretical properties of the estimator, and show that it can be effectively used to drive an adaptive mesh refinement algorithm.

Underwater organisms exhibit sophisticated propulsion mechanisms, enabling them to navigate fluid environments with exceptional dexterity. Recently, substantial efforts have focused on integrating these movements into soft robots using smart shape-changing materials, particularly by using light for their propulsion and control. Nonetheless, challenges persist, including slow response times and the need of powerful light beams to actuate the robot. This last can result in unintended sample heating and potentially necessitate tracking specific actuation spots on the swimmer.

We present a velocity-based moving mesh virtual element method for the numerical solution of PDEs involving boundaries which are free to move. The virtual element method is used for computing both the mesh velocity and a conservative Arbitrary Lagrangian-Eulerian solution transfer on general polygonal meshes.

Parallel to the need for new technologies and renewable energy resources to address sustainability, the emerging field of Artificial Intelligence (AI) has experienced continuous high-speed growth in the application of its capabilities of modelling, managing, processing, and making sense of data in the entire areas related to the production and management of energy.

Multi-fidelity models are of great importance due to their capability of fusing information coming from different numerical simulations, surrogates, and sensors. We focus on the approximation of high-dimensional scalar functions with low intrinsic dimensionality. By introducing a low dimensional bias we can fight the curse of dimensionality affecting these quantities of interest, especially for many-query applications. We seek a gradient-based reduction of the parameter space through linear active subspaces or a nonlinear transformation of the input space.

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain-decomposition (DD) methods and reduced-order modelling (ROM). In particular, we consider an optimisation-based domain-decomposition algorithm for the parameter-dependent stationary incompressible Navier-Stokes equations.

Dynamic mode decomposition (DMD) has recently become a popular tool for the nonintrusive analysis of dynamical systems. Exploiting proper orthogonal decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial bases evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context.

Tubular structures made of elastic helical fibers are widely found in nature and in technology. The complex and highly nonlinear mechanical properties of such assemblies have been understood either through minimal models or through complex simulations describing each individual fiber and their interactions. Here, inspired by Chebyshev’s geometric model of nets, we propose and experimentally validate a modeling framework that treats tubular braided meshes as continuum surfaces corresponding to the virtual envelope defined by the fibers.

Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones.

To shed light on the effect of the icing phenomenon on the vertical-axis wind turbine (VAWT) wake characteristics, we present a high-fidelity computational fluid dynamics simulation of the flow field of H-Darrieus turbine under the icing conditions. To address continuous geometry alteration due to the icing and predefined motion of the VAWT, a pseudo-steady approach proposed by Baizhuma et al. [“Numerical method to predict ice accretion shapes and performance penalties for rotating vertical axis wind turbines under icing conditions,” J. Wind Eng. Ind. Aerodyn.

In this study, we present a parallel immersed boundary strategy that uses Nitsche's method (noted NIB) to weakly impose on a given fluid the boundary conditions associated with a solid of arbitrary shape and motion. Specific details of the software implementation, as done in the software Lethe, are discussed. We verify the NIB method and compare it with other methods in the literature on the well-established test cases of Taylor-Couette flow and von Karman vortex street.

Motivated by pine cones and plant seeds that open or close in response to changes in humidity thanks to the shrinking or swelling of hygroscopic bilayers, and given the recent interest in artificial systems that mimic the hygroscopic motility of seeds, we consider bilayers consisting of two adhering elongated thin layers made of hygroscopic gels, modeled as active, transversally isotropic elastic materials. The direction of transverse isotropy is set as one of the fibers that are present in both biological and synthetic materials.

Partial differential equations can be used to model many problems in several fields of application including, e.g., fluid mechanics, heat and mass transfer, and electromagnetism. Accurate discretization methods (e.g., finite element or finite volume methods, the so-called full order models) are widely used to numerically solve these problems.

In this work, we propose a model order reduction framework to deal with inverse problems in a non-intrusive setting. Inverse problems, especially in a partial differential equation context, require a huge computational load due to the iterative optimization process. To accelerate such a procedure, we apply a numerical pipeline that involves artificial neural networks to parametrize the boundary conditions of the problem in hand, compress the dimensionality of the (full-order) snapshots, and approximate the parametric solution manifold.

The investigation of fluid-solid systems is very important in a lot of industrial processes. From a computational point of view, the simulation of such systems is very expensive, especially when a huge number of parametric configurations needs to be studied. In this context, we develop a non-intrusive data-driven reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method for Computational Fluid Dynamics (CFD) - Discrete Element Method (DEM) simulations.

We present a Large Eddy Simulation (LES) approach based on a nonlinear differential low-pass filter for the simulation of two-dimensional barotropic flows with under-refined meshes. For the implementation of such model, we choose a segregated three-step algorithm combined with a computationally efficient Finite Volume method. We assess the performance of our approach with the classical double-gyre wind forcing benchmark.

In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions.

Convolutional Neural Network (CNN) is one of the most important architectures in deep learning. The fundamental building block of a CNN is a trainable filter, represented as a discrete grid, used to perform convolution on discrete input data. In this work, we propose a continuous version of a trainable convolutional filter able to work also with unstructured data. This new framework allows exploring CNNs beyond discrete domains, enlarging the usage of this important learning technique for many more complex problems.

The focus of this work is on the application of classical Model Order Reduction techniques, such as Active Subspaces and Proper Orthogonal Decomposition, to Deep Neural Networks. We propose a generic methodology to reduce the number of layers in a pre-trained network by combining the aforementioned techniques for dimensionality reduction with input-output mappings, such as Polynomial Chaos Expansion and Feedforward Neural Networks.

Tumble dryers offer a fast and convenient way of drying textiles independent of weather conditions and therefore are frequently used in ordinary households. However, artificial drying of textiles consumes considerable amounts of energy, approximately 8.2 percent of the residential electricity consumption is for drying of textiles in northern European countries (Cranston et al., 2019). Several authors have investigated the aspects of the clothes drying cycle with experimental and numerical methods to understand and improve the process.

This paper provides an overview of the new features of the finite element library deal.II, version 9.5.

This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier–Stokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain’s configuration on the position of the bifurcation points.

Soft actuators typically require time-varying or spatially modulated control to be operationally effective. The scope of the present paper is to show, theoretically and experimentally, that a natural way to overcome this limitation is to exploit mechanical instabilities. We report experiments on active filaments of polyelectrolyte (PE) gels subject to a steady and uniform electric field. A large enough intensity of the field initiates the motion of the active filaments, leading to periodic oscillations.

In this work, a non-intrusive data-driven ROM based on a POD–ANN approach is developed for fast and reliable numerical simulation of blood flow patterns occurring in a patient-specific coronary system when an isolated stenosis of the LMCA occurs. A CABG performed with the LITA on the LAD is analyzed. The introduction of a patient-specific configuration is an attractive element of this work because it makes possible to establish personalized clinical treatment. In addition, a FFD technique is used, which gives the opportunity to deform directly the mesh and not only the geometry.

This work recasts time-dependent optimal control problems governed by partial differential equations in a Dynamic Mode Decomposition with control framework. Indeed, since the numerical solution of such problems requires a lot of computational effort, we rely on this specific data-driven technique, using both solution and desired state measurements to extract the underlying system dynamics.

In this paper, we present recent efforts to develop reduced order modeling (ROM) capabilities for spectral element methods (SEM). Namely, we detail the implementation of ROM for both continuous Galerkin and discontinuous Galerkin methods in the spectral/hp element library Nektar++. The ROM approaches adopted are intrusive methods based on the proper orthogonal decomposition (POD). They admit an offline-online decomposition, such that fast evaluations for parameter studies and many-queries are possible.

Nowadays, the shipbuilding industry is facing a radical change toward solutions with a smaller environmental impact. This can be achieved with low emissions engines, optimized shape designs with lower wave resistance and noise generation, and by reducing the metal raw materials used during the manufacturing. This work focuses on the last aspect by presenting a complete structural optimization pipeline for modern passenger ship hulls which exploits advanced model order reduction techniques to reduce the dimensionality of both input parameters and outputs of interest.

As a major breakthrough in artificial intelligence and deep learning, Convolutional Neural Networks have achieved an impressive success in solving many problems in several fields including computer vision and image processing. Real-time performance, robustness of algorithms and fast training processes remain open problems in these contexts. In addition object recognition and detection are challenging tasks for resource-constrained embedded systems, commonly used in the industrial sector.

Active materials have emerged as valuable candidates for shape morphing applications, where a body reconfiguration is achieved upon triggering its active response. Given a desired shape change, a natural question is to compare different morphing mechanisms to select the most effective one with respect to an optimality criterion. We introduce an optimal control problem to determine the active strains suitable to attain a target equilibrium shape while minimizing the complexity of the activation.

Left ventricular assist devices (LVADs) are used to provide haemodynamic support to patients with critical cardiac failure. Severe complications can occur because of the modifications of the blood flow in the aortic region. In this work, the effect of a continuous flow LVAD device on the aortic flow is investigated by means of a non-intrusive reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method based on radial basis functions (RBF).

This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermo-mechanical problems in a finite element setting. We apply Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim to compare POD-G and POD-ANN in terms of relevant features including errors and computational efficiency.

The control of shape in active structures is a key problem for the realization of smart sensors and actuators, which often draw inspiration from natural systems. In this context, slender structures, such as thin plates, have been studied as a relevant example of shape morphing systems where curvature is generated by in-plane incompatibilities. In particular, in hydrogel plates these incompatibilities can be programmed at fabrication time, such that a target configuration is attained at equilibrium upon swelling or shrinking.

In this work, an innovative model is proposed as a design tool to predict both the inner and outer radii in rolled structures based on polydimethylsiloxane bilayers. The model represents an improvement of Timoshenko’s formula taking into account the friction arising from contacts between layers arising from rolling by more than one turn, hence broadening its application field towards materials based on elastomeric bilayers capable of large deformations.

Objective. Intrafascicular peripheral nerve implants are key components in the development of bidirectional neuroprostheses such as touch-enabled bionic limbs for amputees. However, the durability of such interfaces is hindered by the immune response following the implantation.

In this work we briefly present a thermomechanical model that could serve as starting point for industrial applications. We address the non-linearity due to temperature dependence of material properties and heterogeneity due to presence of different materials. Finally a numerical example related to the simplified geometry of blast furnace hearth walls is shown with the aim of assessing the feasibility of the modelling framework.

We consider an optimal flow control problem in a patient-specific coronary artery bypass graft with the aim of matching the blood flow velocity with given measurements as the Reynolds number varies in a physiological range. Blood flow is modelled with the steady incompressible Navier-Stokes equations. The geometry consists in a stenosed left anterior descending artery where a single bypass is performed with the right internal thoracic artery.

This work deals with optimal control problems as a strategy to drive bifurcating solution of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to change the solution profile and the stability of state solution branches.

Abstract This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand? effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a fluid-structure interaction problem, we aim at generalizing previous works towards a more reliable description of the physics involved. In particular, we provide several insights on how the introduction of an elastic structure influences the bifurcating behavior.

We extend the applicability of the popular interior penalty discontinuous Galerkin method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarilyshaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations.

Adherent cells seeded on substrates spread and evolve their morphology while simultaneously displaying motility. Phenomena such as contact guidance, viz. the alignment of cells on patterned substrates, are strongly linked to the coupling of morphological evolution with motility. Here, we employ a recently developed statistical thermodynamics framework for modelling the non-thermal fluctuating response of cells to probe this coupling. This thermodynamic framework is first extended via a Langevin style model to predict temporal responses of cells to unpatterned and patterned substrates.

In this work we present an integrated computational pipeline involving several model order reduction techniques for industrial and applied mathematics, as emerging technology for product and/or process design procedures. Its data-driven nature and its modularity allow an easy integration into existing pipelines. We describe a complete optimization framework with automated geometrical parameterization, reduction of the dimension of the parameter space, and non-intrusive model order reduction such as dynamic mode decomposition and proper orthogonal decomposition with interpolation.

We investigate various data-driven methods to enhance projection-based model reduction techniques with the aim of capturing bifurcating solutions. To show the effectiveness of the data-driven enhancements, we focus on the incompressible Navier-Stokes equations and different types of bifurcations. To recover solutions past a Hopf bifurcation, we propose an approach that combines proper orthogonal decomposition with Hankel dynamic mode decomposition. To approximate solutions close to a pitchfork bifurcation, we combine localized reduced models with artificial neural networks.

This paper focuses on reduced-order models (ROMs) built for the efficient treatment of PDEs having solutions that bifurcate as the values of multiple input parameters change. First, we consider a method called local ROM that uses k-means algorithm to cluster snapshots and construct local POD bases, one for each cluster. We investigate one key ingredient of this approach: the local basis selection criterion.

Reduced order modeling (ROM) techniques are numerical methods that approximate the solution of parametric partial differential equation (PDE) by properly combining the high-fidelity solutions of the problem obtained for several configurations, i.e. for several properly chosen values of the physical/geometrical parameters characterizing the problem. In this contribution, we propose an efficient non-intrusive data-driven framework involving ROM techniques in computational fluid dynamics (CFD) for hemodynamics applications.

ATHENA is an open source Python package for reduction in parameter space. It implements several advanced numerical analysis techniques such as Active Subspaces (AS), Kernel-based Active Subspaces (KAS), and Nonlinear Level-set Learning (NLL) method. It is intended as a tool for regression, sensitivity analysis, and in general to enhance existing numerical simulations’ pipelines tackling the curse of dimensionality.

In this work, the Immersed Boundary Method (IBM) with feedback forcing introduced by Goldstein et al. (1993) and often referred in the literature as the Virtual Boundary Method (VBM), is addressed. The VBM has been extensively applied both within a Spectral and a Finite Difference (FD) framework. Here, we propose to combine the VBM with a computationally efficient Finite Volume (FV) method. We will show that for similar computational configurations, FV and FD methods provide significantly different results.

This paper provides an overview of the new features of the finite element library deal.II, version 9.3.

We present a stabilized POD-Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space.

In the present work, we investigate a cut finite element method for the parameterized system of second-order equations stemming from the splitting approach of a fourth order nonlinear geometrical PDE, namely the Cahn-Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry -- and mesh -- characteristic, through which, one can avoid e.g.

We develop a Reduced Order Model (ROM) for a Large Eddy Simulation (LES) approach that combines a three-step algorithm called Evolve-Filter-Relax (EFR) with a computationally efficient finite volume method. The main novelty of our ROM lies in the use within the EFR algorithm of a nonlinear, deconvolution-based indicator function that identifies the regions of the domain where the flow needs regularization.

We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for an implementation of the Leray model that combines a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients.

We propose a new algorithm for adaptive finite element methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by the application of a smoother.

We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another.

Growing plant shoots exhibit spontaneous oscillations that Darwin observed, and termed ‘circumnutations’. Recently, they have received renewed attention for the design and optimal actuation of bioinspired robotic devices. We discuss a possible interpretation of these spontaneous oscillations as a Hopf-type bifurcation in a growing morphoelastic rod. Using a three-dimensional model and numerical simulations, we analyse the salient features of this flutter-like phenomenon (e.g.

We present a three-dimensional morphoelastic rod model capable to describe the morphogenesis of growing plant shoots driven by differential growth. We discuss the evolution laws for endogenous oscillators, straightening mechanisms, and reorientations to directional cues, such as gravitropic reactions governed by the avalanche dynamics of statoliths. We use this model to investigate the role of elastic deflections due to gravity loading in circumnutating plant shoots.

We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We couple the perturbed inverse iteration approach with mesh refinement strategy based on residual estimators. We demonstrate our approach on model problems in two and three dimensions.

This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation.

PyGeM is an open source Python package which allows to easily parametrize and deform 3D object described by CAD files or 3D meshes. It implements several morphing techniques such as free form deformation, radial basis function interpolation, and inverse distance weighting. Due to its versatility in dealing with different file formats it is particularly suited for researchers and practitioners both in academia and in industry interested in computational engineering simulations and optimization studies.

In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property.

In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique.

The goal of this paper is to test solids4Foam, the fluid-structure interaction (FSI) toolbox developed for foam-extend (a branch of OpenFOAM), and assess its flexibility in handling more complex flows. For this purpose, we consider the interaction of an incompressible fluid described by a Leray model with a hyperelastic structure modeled as a Saint Venant-Kirchhoff material. We focus on a strongly coupled, partitioned fluid-structure interaction (FSI) solver in a finite volume environment, combined with an arbitrary Lagrangian-Eulerian approach to deal with the motion of the fluid domain.

A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier--Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes.

Many active materials used in shape-morphing respond to an external stimulus by stretching or contracting along a director field. The programming of such actuators remains complex because of the single degree of freedom (the orientation) in local actuation. Here, texturing this field in zigzag patterns is shown to provide an extended family of biaxial active stretches out of an otherwise single uniaxial active deformation, opening a larger parameter space. By further modulating the zigzag patterns at the larger scale of the structure, its deployed shape can be controlled.

In this work we present an advanced computational pipeline for the approximation and prediction of the lift coefficient of a parametrized airfoil profile. The non-intrusive reduced order method is based on dynamic mode decomposition (DMD) and it is coupled with dynamic active subspaces (DyAS) to enhance the future state prediction of the target function and reduce the parameter space dimensionality.

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution.

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters.

We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences.

In this work, we study the mechanics of metamaterial sheets inspired by the pellicle of Euglenids. They are composed of interlocking elastic rods which can freely slide along their edges. We characterize the kinematics and the mechanics of these structures using the special Cosserat theory of rods and by assuming axisymmetric deformations of the tubular assembly. Through an asymptotic expansion, we investigate both structures that comprise a discrete number of rods and the limit case of a sheet composed by infinitely many rods.

We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained.

The J-integral is formulated in a direct manner for a gel consisting of a cross-linked polymer network and a mobile solvent. The form of the J-integral is given for a formulation that exploits the Helmholtz energy density of the gel and expressions are provided for it in both the unswollen reference configuration of the polymer network and in the current swollen configuration of the gel when small strains are superimposed on the swollen state.

Understanding the mechanics of brain embryogenesis can provide insights on pathologies related to brain development, such as lissencephaly, a genetic disease which causes a reduction of the number of cerebral sulci. Recent experiments on brain organoids have confirmed that gyrification, i.e. the formation of the folded structures of the brain, is triggered by the inhomogeneous growth of the peripheral region. However, the rheology of these cellular aggregates and the mechanics of lissencephaly are still matter of debate.

In this work, we present an approach for the efficient treatment of parametrized geometries in the context of POD-Galerkin reduced order methods based on Finite Volume full order approximations. On the contrary to what is normally done in the framework of finite element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the Discrete Empirical Interpolation Method (D-EIM) to handle together non-affinity of the parametrization and non-linearities.

The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels.

Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease. In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters.

We present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization.

We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization.

In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them.

Polymer gel plates may be programmed to morph into three-dimensional configurations upon swelling. An effective strategy to control such shape transformations consists in patterning the in-plane cross-linking density of the polymer network to realize non-homogeneous swelling. In general, one needs to solve an inverse problem to determine the shear modulus field that produces a given target shape. Here, we propose a computational framework for the solution of such an inverse problem, which we validate against two benchmark problems, i.e. making cones and saddles from gel disks.

We prove a basic error contraction result of an adaptive discontinuous Galerkin method for an elliptic interface problem. The interface conditions considered model mass transfer of solutes through semi-permeable membranes and other filtering processes. The adaptive algorithm is based on a residual-type a posteriori error estimator, with a bulk refinement criterion.

Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of ‘active matter’ in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano- and micromachines have been proposed.

In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively.

A posteriori error estimates in the $L_{\infty}(\mathcal{H})$ - and $L_2(\mathcal{V})$-norms are derived for fullydiscrete space-time methods discretising semilinear parabolic problems; here $\mathcal{V} \hookrightarrow \mathcal{H} \hookrightarrow \mathcal{V}^*$ denotes a Gelfand triple for an evolution partial differential equation problem. In particular, an implicit-explicit variable order ( $h p$-version) discontinuous Galerkin timestepping scheme is employed, in conjunction with conforming finite element discretisation in space.