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The purpose of this article is to compare the 3 important types of central composite designs (CCDs) consisting of central composite circumscribed design (CCCD), central composite inscribed design (CCID), and central composite face-centroid design (CCFD) in response surface methodology (RSM). The difference among these designs is the distance from the center design to the axial points. The comparison was performed across the full second-order response surface model and across a set of reduced models for 3, 4, and 5 design factors (k = 3, 4, and 5) including 1, 3, and 5 center runs (nc = 1, 3, and 5). This study used D-, A-, and G- optimality criteria to evaluate the performance of CCDs by presenting the design optimality criteria comparison ranking throughout the reduced-model subsets of 43, 224, and 839 models for 3, 4, and 5 design variables, respectively. The results showed that CCCD was superior to CCID and CCFD according to A- and D- optimality criteria, while CCCD and CCID performed better than CCFD based on G- optimality criterion over a set of reduced models for 3, 4, and 5 design factors. It was observed that D-, A-, and G- optimality efficiencies were robust to changes in the linear and cross-product terms and sensitive to deviation in the square terms. The study will provide recommendations to assist the experimenters in the choice of the best design among the candidate designs for practice applications when some model effects may be insignificant. [ABSTRACT FROM AUTHOR]

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expense to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms. [ABSTRACT FROM AUTHOR]

Summary: Modelling epidemics using classical population‐based models suffers from shortcomings that so‐called individual‐based models are able to overcome, as they are able to take into account heterogeneity features, such as super‐spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice. By combining the reinforcement learning philosophy with reduced models, we propose a numerical approach to determine optimal health policies for a stochastic individual‐based model taking into account heterogeneity in the population. More precisely, we introduce a deterministic reduced population‐based model involving a neural network, designed to faithfully mimic the local dynamics of the more complex individual‐based model. Then the optimal control is determined by sequentially training the network until an optimal strategy for the population‐based model succeeds in also containing the epidemic when simulated on the individual‐based model. After describing the practical implementation of the method, several numerical tests are proposed to demonstrate its ability to determine controls for models with contact heterogeneity. [ABSTRACT FROM AUTHOR]

The concept of equivalence of the realistic, initial reference model and the simplified, reduced model is proposed. In reduced models, the action of the soil on the structure is replaced by the action of a layer with prescribed properties, defined by a set of parameters. The main difficulty here is to find the parameter values required by the simplified theory. The subject of this work is to find the dependence of the parameters of the reduced model on the parameters of the full model, including realistic soil behavior, in order to ensure the equivalence of both models. We show the potential of the method by presenting two examples: Winkler and Pasternak's model of a plate on the ground. We assume that both models are equivalent if they give identical results (displacements) at a finite number of observation points. An artificial neural network (ANN) is built in order to approximate and record the dependence of the parameters of the reduced model (at the network output) from the parameters of the full model (given at the network input). The complex network acts as a formula that assigns the parameters of the reduced model to a realistic description of the soil structure that is used for finite element method (FEM) modeling. The formalism we propose is quite general and can be applied to many engineering problems. The presented procedure is entirely numerical; it allows to calculate the parameters of the reduced model without resorting to symbolic calculations or additional theoretical considerations.

The confinement of energetic particles (EPs) generated by fusion reactions and external heating methods is crucial for the performance of future fusion devices. However, EP transport can occur due to their interaction with electromagnetic perturbations, affecting heating efficiency and overall performance. Robust reduced models are needed to analyze stability and transport. This paper presents an automated IMAS-based workflow for analyzing the time-dependent stability of EP-driven modes, focusing on the linear properties of Toroidal Alfvén Eigenmodes (TAEs) in general tokamak geometry. The workflow utilizes efficient computational methods and reduced models to deliver fast and reproducible results. A demonstration of the workflow's effectiveness was performed, identifying key linear properties of TAEs in various simulated ITER scenarios. This approach represents a critical step toward developing tools for analyzing EP transport and optimizing the performance of future fusion reactors. [ABSTRACT FROM AUTHOR]

We present “on the fly” algorithmic criteria for the accuracy and stability (non-stiffness) of reduced models constructed with the quasi-steady state and partial equilibrium approximations. The criteria comprise those introduced in Goussis (Combust Theor Model 16:869–926, 2012) that addressed the case where each fast time scale is due to one reaction and a new one that addresses the case where a fast time scale is due to more than one reactions. The development of these criteria is based on the ability to approximate accurately the fast and slow subspaces of the tangent space. Their validity is assessed on the basis of the Michaelis–Menten reaction mechanism, for which extensive literature is available regarding the validity of the existing various reduced models. The criteria predict correctly the regions in both the parameter and phase spaces where each of these models is valid. The findings are supported by numerical computations at indicative points in the parameter space. Due to their algorithmic character, these criteria can be readily employed for the reduction of large and complex mathematical models. [ABSTRACT FROM AUTHOR]

Noise poses a challenge for learning dynamical-system models because already small variations can distort the dynamics described by trajectory data. This work builds on operator inference from scientific machine learning to infer low-dimensional models from high-dimensional state trajectories polluted with noise. The presented analysis shows that, under certain conditions, the inferred operators are unbiased estimators of the well-studied projection-based reduced operators from traditional model reduction. Furthermore, the connection between operator inference and projection-based model reduction enables bounding the mean-squared errors of predictions made with the learned models with respect to traditional reduced models. The analysis also motivates an active operator inference approach that judiciously samples high-dimensional trajectories with the aim of achieving a low mean-squared error by reducing the effect of noise. Numerical experiments with high-dimensional linear and nonlinear state dynamics demonstrate that predictions obtained with active operator inference have orders of magnitude lower mean-squared errors than operator inference with traditional, equidistantly sampled trajectory data. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive overview of the preliminary divertor design and plasma exhaust scenario for the reactor-class Spherical Tokamak for Energy Production project. Due to the smaller size of the machine, with a major radius less than half that of most DEMO concepts, the current design features a double-null divertor geometry, comprising tightly baffled extended outer legs and shorter inner legs approaching an X-divertor. Leveraging a significant database of SOLPS-ITER simulations, the exhaust operational space is mapped out, offering valuable insights into the plasma exhaust dynamics. An approach involving the validation of simple, yet robust models capable of accurately predicting key exhaust parameters is detailed, thereby streamlining the design process. The simple models are used to simulate the entire plasma scenario from the plasma current ramp-up, through the burning phase, to the plasma current ramp-down. Notably, the findings suggest that pronounced detachment, with peak heat loads below engineering limits and electron temperatures below 5 eV, is achievable with a divertor neutral pressure between 10 Pa and 15 Pa during the burning phase, and pressures below 5 Pa during the ramp-up to maximise the auxiliary current-drive efficiency. Throughout the scenario, an Ar concentration of ${\approx}3$ % in the scrape-off layer (SOL) is required, in combination with a core radiation fraction of 70% driven by intrinsic emission and extrinsic injection of Xe seeded fuelling pellets. However, significant uncertainties remain regarding key parameters such as the SOL heat flux width, Ar screening, and plasma kinetic effects.

Pressure vessels are one of the essential industrial tools for highpressure containments. Catastrophic failure of pressure vessels is detrimental to society. It is essential to design pressure vessels by selecting highstrength materials and analyzing them beyondworking loads to ensure safety. Liner less composite cylinders have gained importance in the pressure vessel industry owing to their high strength-to-weight ratios, corrosion resistance, etc. However accurate and efficient prediction of their mechanical properties was required. Finite element methods were employed for the structural analysis of reduced models. The three-dimensional shell structure of the Graphite/Epoxy composite system was analyzed using APDL. Appropriate boundary conditions were applied to 5x reduced models internally pressurized to 20 MPa. Suitable mesh size was selected through mesh independence and stress distributions were discussed for reduced models, especially for the inner two layers. Comparison with previous research confirmed the validity of models. 0.1o rotated strip of vessel gives accurate and conservative results. Tsai Wu, Tsai Hill, Maximum Shear Stress (Smax), and Von Mises were used to assess the failure of composite cylinders. Each of the failure criterion predicts the failure of the second layer for all the reduced models. [ABSTRACT FROM AUTHOR]

We introduce a new family of paraxial asymptotic models that approximate the Vlasov–Maxwell equations in non-relativistic cases. This formulation is $n$th order accurate in a parameter $\eta $, which denotes the ratio between the characteristic velocity of the beam and the speed of light. This family of models is interesting, first because it is simpler than the complete Vlasov–Maxwell equation and then because it allows us to choose the model complexity according to the expected accuracy.

This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition. Searching this approximation in a data-driven approach is formalized as attempting to solve a low-rank constrained optimization problem. This problem is non-convex, and state-of-the-art algorithms are all sub-optimal. This paper shows that there exists a closed-form solution, which is computed in polynomial time, and characterizes the ℓ 2 -norm of the optimal approximation error. The paper also proposes low-complexity algorithms building reduced models from this optimal solution, based on singular value decomposition or eigenvalue decomposition. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks. [ABSTRACT FROM AUTHOR]

Model reduction based on matrix interpolation provides an efficient way to compute the dynamic response of nearly periodic structures composed of substructures (cells) with varying properties. This may concern 2D or 3D substructures subjected to geometric modifications (mesh variations) or more classic dimension changes. An efficient interpolation strategy for a nearly periodic structure can be obtained by (i) interpolating reduced substructure matrices over a multi-dimensional parametric space and (ii) reducing the number of degrees of freedom at substructure boundaries by considering the interface modes of an equivalent purely periodic structure. In this paper, two basis enrichment techniques are proposed to improve the accuracy of the interpolation strategy. This consists in (i) considering high-order static modes, in addition to component modes, to express the reduced substructure matrices and (ii) adding static correction vectors into the basis of interface modes to account for the varying properties of the substructures. Numerical experiments are carried out which clearly highlight the relevance of the basis enrichment techniques for predicting the harmonic behavior of nearly periodic structures with 2D or 3D substructures. [ABSTRACT FROM AUTHOR]

• A reduced-order modelling technique when the orientation is not uniform is discussed. • The orientation depends on geometry and loading for instance the thin membranes. • This Fourier-based reduction technique is an extension of Ginzburg-Landau equation. • The obtained reduced macroscopic models can be discretized by finite elements. We discuss a reduced-order modeling technique based on Fourier series for membrane wrinkling when the orientation of the wrinkles is not uniform. Indeed, the orientation of the wrinkles depends on geometry and loading, for instance in the case of perforated membrane or with non uniform residual stresses. This Fourier-based reduction technique is an extension of the famous Ginzburg-Landau equation and it has been applied to the wrinkling of beams, plates, sandwich structures and film-substrate systems. The obtained reduced macroscopic models can be discretized by finite elements. In this paper, a finite element of type Discrete Kirchhoff Triangle (DKT18) is used in the numerical applications, the starting model being the Föppl von Karman (FvK) or Extended Föppl von Karman (EFvK) shell models. [ABSTRACT FROM AUTHOR]

Reduced model spaces, such as reduced bases and polynomial chaos, are linear spaces Vn of finite dimension n which are designed for the efficient approximation of certain families of parametrized PDEs in a Hilbert space V. The manifold M that gathers the solutions of the PDE for all admissible parameter values is globally approximated by the space Vn with some controlled accuracy εn, which is typically much smaller than when using standard approximation spaces of the same dimension such as finite elements. Reduced model spaces have also been proposed in [Y. Maday et al., Internat. J. Numer. Methods Ergrg., 102 (2015), pp. 933-965] as a vehicle to design a simple linear recovery algorithm of the state u ∈ M corresponding to a particular solution instance when the values of parameters are unknown but a set of data is given by m linear measurements of the state. The measurements are of the form ℓj(u), j = 1, ..., m, where the ℓj are linear functionals on V. The analysis of this approach in [P. Binev et al., SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 1-29] shows that the recovery error is bounded by μnεn, where μn = μ(Vn, W) is the inverse of an inf-sup constant that describe the angle between Vn and the space W spanned by the Riesz representers of (ℓ1, ..., ℓm). A reduced model space which is efficient for approximation might thus be ineffective for recovery if μn is large or infinite. In this paper, we discuss the existence and effective construction of an optimal reduced model space for this recovery method. We extend our search to affine spaces which are better adapted than linear spaces for various purposes. Our basic observation is that this problem is equivalent to the search of an optimal affine algorithm for the recovery of M in the worst case error sense. This allows us to perform our search by a convex optimization procedure. Numerical tests illustrate that the reduced model spaces constructed from our approach perform better than the classical reduced basis spaces. [ABSTRACT FROM AUTHOR]

Introducing an extension of a recently derived dimension-reduced model for an infinitely deep inviscid and irrotational layer, a two-layer system is examined in the present paper. A second thin viscous layer is added on top of the original one-layer system. The set-up is a combination of a long-wave approximation (upper layer) and a deep-water approximation (lower layer). Linear stability analysis shows the emergency of Rayleigh–Taylor and Kelvin–Helmholtz instabilities. Finally, numerical solutions of the model reveal spatial and temporal pattern formation in the weakly nonlinear regime of both instabilities. This article is part of the theme issue 'Stokes at 200 (Part 1)'. [ABSTRACT FROM AUTHOR]

In this work, we present a new conceptual model to describe fluid flow in a porous media system in presence of a large fault. Geological faults are often modeled simply as interfaces in the rock matrix, but they are complex structures where the high strain core is surrounded by the so called damage zones, characterized by the presence of smaller fractures which enhance the permeability of the medium. To obtain reliable simulation outcomes, the damage zone as well as the fault, have to be accurately described. The new model proposed in this work considers both these regions as lower dimensional and embedded in the rock matrix. The model is presented, analyzed, and tested in several configurations to prove its robustness and ability to capture many important features, such as high contrast and heterogeneity of permeability. [ABSTRACT FROM AUTHOR]

Edge codes such as SOLPS coupled to neutral codes such as EIRENE have become so comprehensive and sophisticated that they now constitute, in effect, ‘code-experiments’ that, as for actual experiments, can benefit from interpretation using simple models and conceptual frameworks, i.e. reduced models. The first task is the identification of options for the reduced model control parameters that are best suited for control of the action of the divertor, i.e. for control of target power loading and sputter–erosion, primarily. A strong correlation between the electron temperature at the divertor target, T _e,t , and the neutral deuterium D _2 density at the target, n _D2,t , flux-tube resolved, has recently been reported for a number of code studies including SOLPS-4.3 modeling of a set of ∼50 ITER baseline cases: Q _DT = 10, q _95 = 3, P _SOL = 100 MW, metallic walls, and Ne seeding (Pitts et al 2019). Part A of the present study reports new results for largely the same ITER cases, confirming the strong correlation reported earlier between local values of T _e,t , and (i) n _D2,t , and (ii) normalized volumetric losses of power and pressure in the divertor. Strong correlations have now also been found, and are reported here for the first time, between T _e,t and all of the divertor target quantities of practical interest. A physical explanation for this surprising result has not as yet been fully identified; nevertheless it has encouraging implications for reduced modeling of the ITER divertor. For such ITER conditions, (i) the global Ne injection rate, Inj _Ne (Ne s ^−1 ), and (ii) the electron temperature at the location on the target where the peak power deposition occurs, T _e,t @q _⊥,pk (eV), are found to be promising reduced model control parameters. In this part B, a reduced model for the ITER divertor is developed and described in detail, based on reversed-direction two point modeling, Rev2PM. The input to the reduced model is a value of the variable pair $\left({T}_{\text{e},\mathrm{t}}@{q}_{\perp ,\mathrm{p}\mathrm{k}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ for a chosen case and the output are values of the various target as well as divertor-entrance quantities of practical interest, e.g. q _⊥,pk , the electron density at the X-point, n _e,Xpt , etc. The reduced model was quantitatively characterized using one half of the code cases; it was then used to successfully predict (replicate) the code values of e.g. n _e,Xpt for the other half.

Physicists routinely need probabilistic models for a number of tasks such as parameter inference or the generation of new realizations of a field. Establishing such models for highly non-Gaussian fields is a challenge, especially when the number of samples is limited. In this paper, we introduce scattering spectra models for stationary fields and we show that they provide accurate and robust statistical descriptions of a wide range of fields encountered in physics. These models are based on covariances of scattering coefficients, i.e. wavelet decomposition of a field coupled with a pointwise modulus. After introducing useful dimension reductions taking advantage of the regularity of a field under rotation and scaling, we validate these models on various multiscale physical fields and demonstrate that they reproduce standard statistics, including spatial moments up to fourth order. The scattering spectra provide us with a low-dimensional structured representation that captures key properties encountered in a wide range of physical fields. These generic models can be used for data exploration, classification, parameter inference, symmetry detection, and component separation., (© The Author(s) 2024. Published by Oxford University Press on behalf of National Academy of Sciences.)

The goal of the paper is to automatize the construction and parameterization of kinetic reaction mechanisms that can describe a set of experimentally measured concentration versus time curves. Using the framework and theorems of formal reaction kinetics, first, we build a set of possible mechanisms with a given number of measured and unmeasured (real or fictitious) species and reaction steps that fulfill some chemically reasonable requirements. Then we fit all the corresponding mass‐action kinetic models and offer the best one to the chemist to help explain the underlying chemical phenomenon or to use it for predictions. We demonstrate the use of the method via two simple examples: on an artificial, simulated set of data and on a small real‐life data set. The method can also be used to do a kind of lumping to generate a model that can reproduce the simulation results of a detailed mechanism with less species and thereby can largely accelerate spatially inhomogeneous simulations. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating parameter sensitivities for stochastic models of multiscale reaction networks. These sensitivity values are important for model analysis, and the methods that currently exist for sensitivity estimation mostly rely on simulations of the stochastic dynamics. This is problematic because these simulations become computationally infeasible for multiscale networks due to reactions firing at several different timescales. However it is often possible to exploit the multiscale property to derive a "model reduction" and approximate the dynamics as a Piecewise deterministic Markov process, which is a hybrid process consisting of both discrete and continuous components. The aim of this paper is to show that such PDMP approximations can be used to accurately and efficiently estimate the parameter sensitivity for the original multiscale stochastic model. We prove the convergence of the original sensitivity to the corresponding PDMP sensitivity, in the limit where the PDMP approximation becomes exact. Moreover, we establish a representation of the PDMP parameter sensitivity that separates the contributions of discrete and continuous components in the dynamics and allows one to efficiently estimate both contributions. [ABSTRACT FROM AUTHOR]

Biological ion channels and synthetic nanopores are responsible for passive transport of ions through a membrane between two compartments. Modeling these ionic currents is especially amenable to reduced models because the device functions of these pores, the relation of input parameters (e.g., applied voltage, bath concentrations) and output parameters (e.g., current, rectification, selectivity), are well defined. Reduced models focus on the physics that produces the device functions (i.e., the physics of how inputs become outputs) rather than the atomic/molecular-scale physics inside the pore. Here, we propose four rules of thumb for constructing good reduced models of ion channels and nanopores. They are about (1) the importance of the axial concentration profiles, (2) the importance of the pore charges, (3) choosing the right explicit degrees of freedom, and (4) creating the proper response functions. We provide examples for how each rule of thumb helps in creating a reduced model of device behavior.

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system. [ABSTRACT FROM AUTHOR]

Highly concentrated moving nonlinearities are extremely difficult to solve numerically. The Selective Laser Melting Additive Manufacturing process is a problem of this kind. A material global-local scheme is proposed, which consists in describing the neighbourhood of the heat source by a moving local domain while the material phase fractions are represented in a global domain. The equations of the non-linear thermal problem are defined on the local domain only, assuming that the local domain is large enough to capture the most important variations of the temperature field. Additionally, a Hyper-Reduced-Order Model (HROM) is proposed for the local domain problem. The performance is studied by solving a SLM problem taken from the literature. [ABSTRACT FROM AUTHOR]

This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and global well-posedness results for the model equations. Moreover, we present reduced models in various parameter regimes for the approximate description of the motion of typical wave profiles and discuss the mathematically rigorous justification of the validity of these models. [ABSTRACT FROM AUTHOR]

The present paper introduces an iterative methodology to progressively reduce building simulation model complexity with the aim of identifying potential trade-offs between computational requirements (i.e., model complexity) and energy estimation accuracy. Different levels of model complexity are analysed, from commercial building energy simulation tools to low order calibrated thermal networks models. Experimental data from a residential building in Germany were collected and used to validate two detailed white-box models and a simplified white-box model. The validation process was performed in terms of internal temperature profiles and building thermal energy demand predictions. Synthetic profiles were generated from the validated models and used for calibrating high order models. A reduction (trimming) procedure was applied to reduce the model complexity using an energy performance criterion prior to model trimming. The proposed methodology has the advantage of keeping the physical structure of the original RC model, thus enabling the use of the trimmed lumped parameter building model for other applications. The analysis showed that it is possible to reduce the model complexity by half, while keeping the accuracy above 90% for the targeted building.

A method is presented for deriving reduced models for fluid flows over highly curved substrates with wider applicability and accuracy than existing models in the literature. This is done by reducing the Navier-Stokes equations to a novel system of boundary layer like equations in a general geometric setting. This is accomplished using a new, relaxed set of scalings that assert only that streamwise variations are "slow". These equations are then solved using the method of weighted residuals, which is demonstrated to be applicable regardless of the geometry selected. A large number of results in the literature can be derived as special cases of our general formulation. A few of the more interesting cases are demonstrated. Finally, the formulation is applied to two thick annular flow systems as well as a conical system in both linear and nonlinear regimes, which traditionally has been considered inaccessible to such reduced models. Comparisons are made with direct numerical simulations of the Stokes equations. The results indicate that reduced models can now be used to model systems involving thick liquid layers. [ABSTRACT FROM AUTHOR]

Recent particulate flow research using a discrete element simulation-dynamical systems approach is described. The simulation code used is very efficient and the mathematical model is an integro-partial differential equation. Examples are presented to show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

We formulate a new free-boundary type mathematical model describing the interaction between a shell and a mesh-like structure consisting of thin rods. Composite structures of this type arise in many applications. One example is the interaction between vascular walls treated with vascular devices called stents. The new model embodies two-way coupling between a two-dimensional (2D) Naghdi type shell model and a 1D network model of curved rods, describing no-slip and balance of contact forces and couples (moments) at the contact interface. The work presented here provides a unified framework within which 3D deformation of various composite shell-mesh structures can be studied. In particular, this work provides the first dimension reduction-based fully coupled model of mesh-reinforced shells. Using rigorous mathematical analysis based on variational formulation and energy methods, the existence of a unique weak solution to the coupled shell-mesh model is obtained. The existence result shows that weaker solution spaces than the classical shell spaces can be used to obtain existence, making this model particularly attractive to finite element method based computational solvers, where Lagrangian elements can be used to simulate the solution. An example of such a solver was developed within Freefem++ and applied to study mechanical properties of four commercially available coronary stents as they interact with the vascular wall. The simple implementation, low computational costs, and low memory requirements make this newly proposed model particularly suitable for fast algorithm design and for the coupling with fluid flow in fluid-composite structure interactions problems. [ABSTRACT FROM AUTHOR]

Recent and on-going advances in mathematical methods and analysis techniques, coupled with the experimental and computational capacity to capture detailed flow structure at increasingly large Reynolds numbers, afford an unprecedented opportunity to develop realistic models of high Reynolds number turbulent wall-flow dynamics. A distinctive attribute of this new generation of models is their grounding in the Navier–Stokes equations. By adhering to this challenging constraint, high-fidelity models ultimately can be developed that not only predict flow properties at high Reynolds numbers, but that possess a mathematical structure that faithfully captures the underlying flow physics. These first-principles models are needed, for example, to reliably manipulate flow behaviours at extreme Reynolds numbers. This theme issue of Philosophical Transactions of the Royal Society A provides a selection of contributions from the community of researchers who are working towards the development of such models. Broadly speaking, the research topics represented herein report on dynamical structure, mechanisms and transport; scale interactions and self-similarity; model reductions that restrict nonlinear interactions; and modern asymptotic theories. In this prospectus, the challenges associated with modelling turbulent wall-flows at large Reynolds numbers are briefly outlined, and the connections between the contributing papers are highlighted. This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’. [ABSTRACT FROM AUTHOR]

This paper presents analyses of reduced order models (ROM) for wave propagation in media containing material defects. Parametric ROMs based on POD (Proper Orthogonal Decomposition) are used here to take into account parameters that characterize local defects such as center position, extension and magnitude (localized loss of stiffness). Local and adaptive (based on the Grassmann interpolation) POD strategies are used. Sequential analyses regarding parametric dependence with one, two and three parameters are performed along with accuracy and speed up as a function of model order. Furthermore, the accuracy of ROMs is also investigated for uncertainty quantification analyses. Our results show that the choice of ROM technique is not straightforward and the best solution, which is case dependent, is not always the most complex one. • Parametric Reduced Order Models (p-ROM) are analyzed for wave propagation. • Parametric ROM based on POD are considered. • A set of three parameters that characterize local damage are taken into account. • Accuracy and speed-up performance are investigated for three types of p-ROM. • The choice for a specific p-ROM is case dependent. [ABSTRACT FROM AUTHOR]

For design of vertical silos walls involving the storage of bulk solids to be safe and reliable, it is important knowing the largest possible number of variables such as: flow properties, silo geometry and pattern of flow desired. In order to validate the theories of flow prediction and design of conical hoppers, the flow properties of two bulk solids were determined, the theories of Jenike's flowability and Enstad and Walker for hopper design were analyzed and the results were compared with those experimentally obtained in a reduced model of a semicircular-section silo. Results show that Enstad theory for the hopper design is adequate to occur mass flow inside the silo, and for the sizing of the discharge outlet, the Walker's theory was closer to the appropriate than Jenike's theory, which was higher around 100% than the experimental hopper outlet.

A review of nonlinear water wave problems in complex domains is presented. These include non-smooth highly variable bottom topographies and boundaries as for example forked channel regions. The physical and mathematical relevance of each case is discussed. The Schwarz–Christoffel mapping is the common modeling tool for these different problems. The mapping produces conformal coordinates which allow the problems' dimension to be reduced and the weakly nonlinear partial differential system of equations to be simplified. Usually conformal mapping is adopted while preserving the Laplace operator. The main focus in the present review is to highlight some water wave problems where the underlying differential operators are not invariant under the conformal change of coordinates. The Schwarz–Christoffel mapping is used to produce a boundary fitted, curvilinear conformal coordinate system. The models reported include solitary waves on branching channels, and its reduction to nonlinear waves on graphs, as well as three dimensional surface-waves propagating over highly variable ridges. Of particular interest is nonlinear water waves on graphs, which has important applications and a scarce mathematical literature. Conformal mapping plays a crucial role regarding a systematic incorporation of forking angles into the equations, as not done before, and deducing a generalized compatibility condition at the graph nodes. • Conformal mapping is used irrespective of operator invariance. • Conformal coordinates as boundary fitted coordinates in intricate domains. • Application to nonlinear water waves on graphs. • Incorporation of forking angles and generalized compatibility condition at nodes. [ABSTRACT FROM AUTHOR]

Idealized models of reduced complexity are important tools to understand key processes underlying a complex system. In climate science in particular, they are important for helping the community improve our ability to predict the effect of climate change on the earth system. Climate models are large computer codes based on the discretization of the fluid dynamics equations on grids of horizontal resolution in the order of 100 km, whereas unresolved processes are handled by subgrid models. For instance, simple models are routinely used to help understand the interactions between small-scale processes due to atmospheric moist convection and large-scale circulation patterns. Here, a zonally symmetric model for the monsoon circulation is presented and solved numerically. The model is based on the Galerkin projection of the primitive equations of atmospheric synoptic dynamics onto the first modes of vertical structure to represent free tropospheric circulation and is coupled to a bulk atmospheric boundary layer (ABL) model. The model carries bulk equations for water vapor in both the free troposphere and the ABL, while the processes of convection and precipitation are represented through a stochastic model for clouds. The model equations are coupled through advective nonlinearities, and the resulting system is not conservative and not necessarily hyperbolic. This makes the design of a numerical method for the solution of this system particularly difficult. Here, we develop a numerical scheme based on the operator time-splitting strategy, which decomposes the system into three pieces: a conservative part and two purely advective parts, each of which is solved iteratively using an appropriate method. The conservative system is solved via a central scheme, which does not require hyperbolicity since it avoids the Riemann problem by design. One of the advective parts is a hyperbolic diagonal matrix, which is easily handled by classical methods for hyperbolic equations, while the other advective part is a nilpotent matrix, which is solved via the method of lines. Validation tests using a synthetic exact solution are presented, and formal second-order convergence under grid refinement is demonstrated. Moreover, the model is tested under realistic monsoon conditions, and the ability of the model to simulate key features of the monsoon circulation is illustrated in two distinct parameter regimes. [ABSTRACT FROM AUTHOR]

The definition of Statistical Energy Analysis (SEA) models for large complex structures is highly conditioned by the classification of the structure elements into a set of coupled subsystems and the subsequent determination of the loss factors representing both the internal damping and the coupling between subsystems. The accurate definition of the complete system can lead to excessively large models as the size and complexity increases. This fact can also rise practical issues for the experimental determination of the loss factors. This work presents a formulation of reduced SEA models for incomplete systems defined by a set of effective loss factors. This reduced SEA model provides a feasible number of subsystems for the application of the Power Injection Method (PIM). For structures of high complexity, their components accessibility can be restricted, for instance internal equipments or panels. For these cases the use of PIM to carry out an experimental SEA analysis is not possible. New methods are presented for this case in combination with the reduced SEA models. These methods allow defining some of the model loss factors that could not be obtained through PIM. The methods are validated with a numerical analysis case and they are also applied to an actual spacecraft structure with accessibility restrictions: a solar wing in folded configuration. [ABSTRACT FROM AUTHOR]

In countries with tropical climates, such as Brazil, the high summer temperatures associated with high relative humidities are a stress factor in animal production. Excessive heat within a poultry facility causes a reduction in feed intake and production, and increased bird mortality. With the knowledge that about 75% of the radiant heat load within a facility comes from the roof, it is necessary to study alternatives that can minimize this radiation. Thus, the objective of the present work is to analyze the thermal environment inside physical broiler housing models constructed on a reduced scale (1:10), where the thermal comfort was evaluated by the black globe humidity index (BGHI) and radiant heat load (RHL). Five models built with different roofing materials were evaluated. Based on the results, it can be concluded that roofs constructed with the channel clay roofing tile (TB30), natural fiber tiles with their external side painted white (TFVP15) and tiles made of recycled long-life packaging (TLV15), provide better thermal environments within the models. [ABSTRACT FROM AUTHOR]

Many examples of multi-stable shell structures have been recently proposed with the underlying hypothesis of the shell being completely free on its boundary. We describe a class of shallow shells which are bistable after one of their sides is completely clamped. This result, which has relevant technological implications, is achieved by a suitable design of the initial, stress-free, shape. [ABSTRACT FROM AUTHOR]

Kinetic characteristics of catalytic reactions in open gradientless systems are studied based on full and reduced systems of ordinary differential equations under steady-state conditions for some of the reactants. By the example of two-step reactions constituting the basis of most of the multistep catalytic processes equilibrium and nonequilibrium properties are compared and the error of application of the steady-state approximation to describing the kinetics of these reactions within the framework of various reduced models is examined. [ABSTRACT FROM AUTHOR]