Benchmark Solutions Research Articles

A new model for fluid transients in piping systems taking into account the fluid–structure interaction

The present work extends a recently developed quasi-2D flow model for fluid transients in elastic pipes to accommodate fluid–structure interaction mechanisms. In this context, the primary goal of the proposed approach is to analyze energy transfer and dissipative effects in the fluid–pipe system. The fluid–structure interaction couples the flow dynamics with the axial movement of the tube, giving rise to friction, Poisson, and junction coupling mechanisms to take place. The mechanical model mathematical structure forms a quasi-linear hyperbolic system of partial differential equations for which approximated solutions are sought by employing the method of characteristics. The proposed model reproduces classical benchmark solutions and presents a good agreement with experimental data. In addition, the whole thermomechanical consistent framework in which the model is established allows an accurate description of the flow’s internal structure. This feature allows a better comprehension of the phenomenon when compared to the classic frictionless and quasi-steady-friction-based four-equation models. The present approach allows a better description of the friction coupling mechanism and unveils uneven shear stress distributions in the unsteady flow. As a result, the energy dissipation in the fluid is captured with accuracy. Due to the absence or limited ability to describe these dispersive and dissipative effects, the traditional approaches are proved to lose their accuracy as the fluid transient goes. The coupled and uncoupled models are also analyzed and are proven to respond differently due to localized pipe–fluid interface effects in addition to the bulk mechanisms of transfer of energy that occur differently in both approaches.

Meshless stochastic vibration for laminated quadrilateral plates considering thermal factor

This paper presents a node-based meshless model for investigating the stationary/non-stationary stochastic thermal response characteristics of laminated arbitrary quadrilateral plates excited by various random loads. The thermo-elastic theory is incorporated into Hamilton's principle for deriving the dynamic governing formulas of the arbitrary quadrilateral plate. The displacement variables of plate structure are approximated by two-dimensional Chebyshev meshless shape functions, which is combined with multipoint mapping theory to discretize the system equations. After that, the stochastic dynamic response solutions of quadrilateral plates considering different boundary cases are generated by introducing a pseudo excitation methodology (PEM). The applicability and predictive accuracy of the present formulation to thermal vibration behaviors of quadrilateral plates are elucidated by means of carrying out convergence clarification and sufficient comparative studies between the calculated results as well as reference solutions from available literature. Also, the effects of several main factors including laminated parameters, thermal effects, excitation loads, etc. on free vibration and stationary/non-stationary response properties of laminated quadrilateral plates are revealed, which may serve as benchmark solutions for evaluating the correctness of other analytical or numerical methods.

Achievable Rate Maximization for Underlay Spectrum Sharing MIMO System With Intelligent Reflecting Surface

In this letter, the achievable rate maximization problem is considered for intelligent reflecting surface (IRS) assisted multiple-input multiple-output (MIMO) systems in an underlay spectrum sharing scenario, subject to interference power constraints at the primary users. The formulated non-convex optimization problem is challenging to solve due to its non-convexity as well as coupling design variables in the constraints. Different from existing works that are mostly based on alternating optimization (AO), we propose a penalty dual decomposition based gradient projection (PDDGP) algorithm to solve this problem. We also provide a convergence proof and a complexity analysis for the proposed algorithm. We benchmark the proposed algorithm against two known solutions, namely a minimum mean-square error based AO algorithm and an inner approximation method with block coordinate descent. Specifically, the complexity of the proposed algorithm grows <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linearly</i> with respect to the number of reflecting elements at the IRS, while that of the two benchmark methods grows with the third power of the number of IRS elements. Moreover, numerical results show that the proposed PDDGP algorithm yields considerably higher achievable rate than the benchmark solutions.

Modeling subgrid-scale scalar dissipation rate in turbulent premixed flames using gene expression programming and deep artificial neural networks

In this present study, gene expression programing (GEP) has been used for training a model for the subgrid scale (SGS) scalar dissipation rate (SDR) for a large range of filter widths, using a database of statistically planar turbulent premixed flames, featuring different turbulence intensities and heat release parameters. GEP is based on the idea to iteratively improve a population of model candidates using the survival-of-the-fittest concept. The resulting model is a mathematical expression that can be easily implemented, shared with the community, and analyzed for physical consistency, as illustrated in this work. Efficient evaluation of the cost function and a smart choice of basis functions have been found to be essential for a successful optimization process. The GEP based model has been found to outperform an existing algebraic model from the literature. However, the optimization process was found to be quite intricate, and the SGS SDR closure turned out to be difficult. Some of these problems have been explained using the model-agnostic interpretation method, which requires the existence of a trained artificial neural network (ANN). ANNs are known for their ability to represent complex functional relationships and serve as an additional benchmark solution for the GEP based model.

Evaluation of Spatiotemporal Patterns of the Spinal Muscle Coordination Output during Walking in the Exoskeleton.

Recent advances in the performance and evaluation of walking in exoskeletons use various assessments based on kinematic/kinetic measurements. While such variables provide general characteristics of gait performance, only limited conclusions can be made about the neural control strategies. Moreover, some kinematic or kinetic parameters are a consequence of the control implemented on the exoskeleton. Therefore, standard indicators based on kinematic variables have limitations and need to be complemented by performance measures of muscle coordination and control strategy. Knowledge about what happens at the spinal cord output level might also be critical for rehabilitation since an abnormal spatiotemporal integration of activity in specific spinal segments may result in a risk for abnormalities in gait recovery. Here we present the PEPATO software, which is a benchmarking solution to assess changes in the spinal locomotor output during walking in the exoskeleton with respect to reference data on normal walking. In particular, functional and structural changes at the spinal cord level can be mapped into muscle synergies and spinal maps of motoneuron activity. A user-friendly software interface guides the user through several data processing steps leading to a set of performance indicators as output. We present an example of the usage of this software for evaluating walking in an unloading exoskeleton that allows a person to step in simulated reduced (the Moon’s) gravity. By analyzing the EMG activity from lower limb muscles, the algorithms detected several performance indicators demonstrating differential adaptation (shifts in the center of activity, prolonged activation) of specific muscle activation modules and spinal motor pools and increased coactivation of lumbar and sacral segments. The software is integrated at EUROBENCH facilities to benchmark the performance of walking in the exoskeleton from the point of view of changes in the spinal locomotor output.

Investigation on the performance of meshfree RBF based method for the solution of thin film flows over topographies through depth-averaged Momentum Integral Model

Motivated by the performance of meshfree Radial Basis Functions (RBF) based numerical methods in providing dependable and precise solutions to a multitude of situations, this investigation explores the possibility of employing RBF based schemes to determine the solution of thin film flows involving interface/free surface through depth-averaged momentum integral model. As a first step towards this goal, thin film flow (two and three dimensional) over substrates featuring topographies (trench, peak, step-up, step-down) is considered. The assessment is based on how well the steady-state solutions, encapsulated by RBF scheme, agree with the available solutions of the model equations based on lubrication approximation (LUB) and depth averaged forms (DAF) and solved by conventional grid-based methods and experimental observations. In order to gain confidence in our approach, RBF based meshfree scheme is applied to LUB and DAF models after confirming the qualitative and quantitative agreement of the predicted solutions for the lid-driven cavity problem with the benchmark solutions. The striking agreement of the steady state film thickness profiles obtained by the meshfree RBF scheme with the solutions predicted by grid based schemes and experimental observations demonstrate the expected efficiency of RBF based schemes in capturing the dynamics of the film thickness that is a characteristic for a film over the chosen topography. The study reveals the scope for extending the extent of free surface flows or flows involving interfaces that can be addressed using meshfree RBF based schemes. It is hoped that the study would throw more light on the advantages of employing RBF based meshfree schemes in fluid flow problems involving interfaces in order to keep up with the rapidly changing computational environment.

Graph decomposition for constructing blackstart restoration strategies in benchmark cases

This paper describes methods for constructing benchmark cases and solution strategies related to restoring an electric grid model from a blackstart condition. Existing public cases for grid restoration problems are limited in size and scope—this paper delineates the features required to create new synthetic, detailed datasets for this purpose. As part of validating these datasets, benchmark results are included for an electric grid restoration strategy based on a formulation of the restoration problem as a time sequence of ac power flow solutions. The algorithm produces a benchmark restoration sequence for an example 200-bus case, showing how critical loads can be restored within 45 min, the majority of the grid within 6 h, and all loads within 36 h. At the core of the solution strategy is a directed graph decomposition heuristic, as the algorithm builds a bus energization spanning forest. Because these datasets are built synthetically, they can have a high degree of realism in features such as geographic coordinates and modeling complexities, while avoiding concerns of data confidentiality, meaning that the data produced in this work can be made publicly available for the benefit of the research community.

On the Bövik-Benveniste methodology and related approaches for modelling thin layers.

This paper reviews several leading approaches for asymptotic modelling of thin layers in elastostatics and wave propagation phenomena. The issues related to applications of the so-called 'equivalent' or 'effective' boundary conditions and their interpretations are highlighted. Comparative analysis of asymptotic models is performed for a two-dimensional elastostatic case using a novel complex variables-based modelling tool. Its implementation allows for straightforward derivations of higher order boundary conditions for problems with layers of arbitrary sufficiently smooth curvatures. Explicit expressions for the conditions up to the third order are provided. All models are tested using available benchmark solutions and the solutions for the limiting cases of the layer parameters. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Numerical simulation of three-fluid Rayleigh-Taylor instability using an enhanced Volume-Of-Fluid (VOF) model: New benchmark solutions

The main objective of the present study is to introduce two novel benchmark solutions namely: two-dimensional three-fluid Rayleigh-Taylor Instability problems, aiming to provide an up-to-date data set and a unique fundamental insight into morphology and hydrodynamic behavior of coupled Rayleigh-Taylor-Kelvin-Helmholtz instability phenomenon. To this end, the Volume-Of-Fluid (VOF) model is adopted to probe the complex configurations and kinetic processes of highly nonlinear multi-fluid flow problems with large topological changes and moving interfaces. However, to improve the performance and accuracy of the classical VOF model and preserve monotonicity for the density and viscosity, a novel high-order bounded advection scheme is first proposed in the context of the Total Variation Diminishing and Normalized Variable Diagram (TVD-NVD) constraints and then is utilized for the discretization of the convection terms in the Navier-Stokes and transport equations. To further increase the accuracy of the numerical simulations, the second-order PLIC-ELVIRA is implemented for the reconstruction of the physical discontinuity between phases and the determination of its curvature. Furthermore, to enhance the consistency and stability of the classical VOF model in handling incompressible multi-fluid flows, a novel semi-iterative pressure-velocity coupling algorithm is constructed by the combination of the standard PISO and SIMPLEC algorithms and is then applied to ensure mass conservation in each grid cell. To demonstrate the versatility and robustness of the proposed model in dealing with the multiphase flows involving large interface deformation and breaking phenomena, a series of canonical test cases such as dam-break over a dry bed with and without stationary obstacle, 2D three-fluid rising bubble, two-fluid and three-fluid Rayleigh-Taylor Instability are adopted. In the last stage, an improved VOF model is applied to solve two new three-fluid Rayleigh-Taylor Instability benchmark problems on a staggered grid system. The results of this study can provide a wide panorama on the improvements of standard VOF model and may be utilized as benchmark solutions for validation of various CFD tools or simply to understand more complex related multi-fluid flows.

3-D analytical solution of non-homogeneous transversely isotropic thick closed cylindrical shells

This paper presents an effective analytical method based on displacement potential functions (DPF) for solving 3D static problem of thick and multilayer transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. The governing equations are solved by the separation of variable method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. The analysis covers a straightforward solution process for handling problems on multilayered cylindrical shells of transversely isotropic material, adopting all boundary and continuity conditions. Extensive sets of general radial loads located on the inner and outer faces of the cylindrical shell may be stated and examined with in a systematic manner. Comparisons are performed to other existing analytical results for one and multilayered cylindrical shells, and show excellent agreement for different materials, thicknesses and aspect ratios of the shell. In addition, various more involved problems are studied and solved analytically for single and three-layered shells of transversely isotropic material with different sets of radial loading functions at the outer and inner shell surfaces. The results of the present study can be used as benchmark solutions for other studies.

Multiscale structural assessment of toroidal field coils via reduced-order models

In a tokamak fusion reactor the confinement of the plasma is guaranteed primarily by a series of Toroidal Field Coils (TFC). The TFCs generate a toroidal magnetic field, and must typically withstand large electromagnetic forces. A constant-tension theoretical approach is commonly applied to define an ideal TFC profile, equilibrating in-plane load distributions without shear and bending effects. This paper focuses on the effectiveness of reduced-order models to analyse the TFC mechanical response when also non-ideal shapes are considered. Different beam models are introduced by referring to different treatments of shear and bending effects. A bottom-up homogenisation technique is employed in order to account for the subscale arrangement of TFC components, allowing also to deduce quantitative indications on average localisation effects. The considered beam theories have been specialised to quantify the discrepancies in the macro-micro TFC mechanical response when arc-based approximations are considered instead of the ideal TFC profile. The successful comparison with numerical benchmark solutions, defined by detailed three-dimensional Finite Element models, confirm that beam-like approaches may be considered reliable in a preliminary design stage.

Dynamic SDN-Based Radio Access Network Slicing With Deep Reinforcement Learning for URLLC and eMBB Services

Radio access network (RAN) slicing is a key technology that enables 5G network to support heterogeneous requirements of generic services, namely ultra-reliable low-latency communication (URLLC) and enhanced mobile broadband (eMBB). In this paper, we propose a two time-scales RAN slicing mechanism to optimize the performance of URLLC and eMBB services. In a large time-scale, an SDN controller allocates radio resources to gNodeBs according to the requirements of the eMBB and URLLC services. In a short time-scale, each gNodeB allocates its available resources to its end-users and requests, if needed, additional resources from adjacent gNodeBs. We formulate this problem as a non-linear binary program and prove its NP-hardness. Next, for each time-scale, we model the problem as a Markov decision process (MDP), where the large-time scale is modeled as a single agent MDP whereas the shorter time-scale is modeled as a multi-agent MDP. We leverage the exponential-weight algorithm for exploration and exploitation (EXP3) to solve the single-agent MDP of the large time-scale MDP and the multi-agent deep Q-learning (DQL) algorithm to solve the multi-agent MDP of the short time-scale resource allocation. Extensive simulations show that our approach is efficient under different network parameters configuration and it outperforms recent benchmark solutions.

Analytical and numerical solutions to describe water table fluctuations due to canal seepage and time-varying recharge

Abstract Hybrid finite analytic solution (HFAS), Galerkin's method based finite element solution (FES) and fully implicit finite difference solution (FIFDS) of one dimensional nonlinear Boussinesq equation and Analytical solution of Boussinesq equation linearized by Baumann's transformation (analytical solution I, AS I) as well as linearized by Werner's transformation (analytical solution II, AS II) were employed to obtain water table rise in a horizontal unconfined aquifer lying between two canals located at finite distance having different elevations and subjected to various patterns of recharge, i.e. zero recharge, constant recharge, as well as time varying recharge. Considering HFAS as benchmark solution, water table in mid region as obtained from FES followed by FIFDS was observed quite close to that obtained from HFAS and as per L2 and Tchebycheff norms computation, it was ranked at first and second place, respectively. Both AS I and AS II predicted higher water table at t = 5 days but at t = 10 days, AS I predicted lower and AS II predicted higher water table at all distances due to linearization effect. So, analytical solutions of linearized Boussinesq equation were rated lower than numerical solutions of nonlinear Boussinesq equation.

Postbuckling and postbuckled vibration behaviour of imperfect trapezoidal sandwich plates with FG-CNTRC face sheets under nonuniform loadings

The present work investigates the postbuckling, and postbuckled vibration behaviour of initially imperfect trapezoidal sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets and FG porous metal foam core under the influence of non-uniform edge compression. The plate's kinematic assumptions are based on a refined higher order theory and the strain-displacement relations include von Karman assumptions for geometrical nonlinearity. The weak form of governing equations derived using Hamilton's principle is transformed into a discretized form of algebraic equations using the element free Galerkin (EFG) method in conjunction with moving kriging (MK) interpolation functions. The pre-buckling stresses are determined using static analysis to evaluate accurate critical buckling loads. Modified Riks technique is used to trace nonlinear equilibrium paths. Parametric studies include the effect of CNT distribution in face sheets, porosity distribution in the core layer and edge loading conditions on the nonlinear stability and vibration behaviour of sandwich plates. New results on trapezoidal sandwich plates with initial imperfections, hitherto not found in the literature, are presented for the first time, which can be used as benchmark solutions for further research.

Active control of nonlinear coupled transient vibrations of multifunctional sandwich plates with agglomerated FG-CNTs core/magneto-electro-elastic facesheets

This article investigates the controlled nonlinear transient response of sandwich plates made of magneto-electro-elastic (MEE) facesheets and agglomerated carbon nanotube (ACNT) core. The control is achieved by active constrained layer damping (ACLD) treatment which consists of piezoelectric (PE) and viscoelastic (VE) layers. The agglomeration is mathematically modelled using the Eshelby–Mori–Tanaka approach. The governing equations are derived under the framework of finite element (FE) methods using Hamilton’s principle and condensation technique. The VE patch of the ACLD treatment is modelled through the Golla–Hughes–McTavish method. This evaluation has considered two states of agglomerations, such as complete and partial agglomeration. The proposed FE model facilitates the integration of the three-field (magnetic, electric and elastic) coupling with the agglomeration and electromagnetic circuit effects, which makes this work interesting. It acts as a ready tool for the realistic design of a smart structure and system. In addition, the computational efforts and efficiency of the proposed FE model prove to be better than the commercial numerical software. Extensive numerical examples are solved to assess the influence of CNT distributions, electromagnetic circuits, coupling, mechanical boundary conditions, thickness configuration and ACLD patch position. This work serves as a novel benchmark solution for further research on the active control of agglomerated CNT structures.

Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first- and second-order stepping, as well as flows with one, two or three phases.

Solving a new variant of the capacitated maximal covering location problem with fuzzy coverage area using metaheuristic approaches

The Maximal Covering Location Problem (MCLP) is concerned with the optimal placement of a fixed number of facilities to cover the maximum number of customers. This article considers a new variant of MCLP where both the coverage radii of facilities and the distance between customer and facility are fuzzy. Moreover, the finite capacity of each facility is considered. We call this problem the capacitated MCLP with fuzzy coverage area (FCMCLP), and it is formulated as a 0–1 linear programming problem. In this article, two classical metaheuristics: particle swarm optimization, differential evolution, and two new-generation metaheuristics: artificial bee colony algorithm, firefly algorithm, are proposed for solving FCMCLP. Each of the customized metaheuristics utilizes a greedy deterministic heuristic to generate their initial populations. They also incorporate a local neighborhood search to improve their convergence rates. New instances of FCMCLP are generated from the traditional MCLP instances available in the literature, and IBM’s CPLEX solver is used to generate benchmark solutions. An experimental comparative study among the four customized metaheuristics is described in this article. The performances of the proposed metaheuristics are also compared with the benchmark solutions obtained from CPLEX.

Galerkin neural network approximation of singularly-perturbed elliptic systems

We consider the neural network approximation of systems of partial differential equations exhibiting multiscale features such as the Reissner–Mindlin plate model which poses significant challenges due to the presence of boundary layers and numerical phenomena such as locking. This work builds on the basic Galerkin Neural Network approach established in Ainsworth and Dong (2021) for symmetric, positive-definite problems. The key contributions of this work are (1) the analysis and comparison of several new least squares-type variational formulations for the Reissner–Mindlin plate, and (2) their numerical approximation using the Galerkin Neural Network approach. Numerical examples are presented which demonstrate the ability of the approach to resolve multiscale phenomena such for the Reissner–Mindlin plate model for which we develop a new family of benchmark solutions which exhibit boundary layers.

Development of the Dynamic Stiffness Method for the Out-of-Plane Natural Vibration of an Orthotropic Plate

In this present paper, the dynamic stiffness method (DSM) was formulated to investigate the out-of-plane natural vibration of a thin orthotropic plate using the classical plate theory (CPT). Hamilton’s principle was implemented to derive the governing differential equation of motion for free vibration of the orthotropic plate for Levy-type boundary conditions. The Wittrick–Williams (W–W) algorithm was used as a solution technique to compute the natural frequencies of a thin orthotropic plate for different boundary conditions, aspect ratios, thickness ratios, and modulus ratios. The obtained results are compared with the results by the finite element method using commercial software (ANSYS and those available) in the published literature. The presented results by the dynamic stiffness method can be used as a benchmark solution to compare the natural frequencies of orthotropic plates.

Novel MILP and CP models for distributed hybrid flowshop scheduling problem with sequence-dependent setup times

As regards distributed hybrid flow shop scheduling with sequence-dependent setup times (DHFSP-SDST), three novel mixed-integer linear programming (MILP) models and a constraint programming (CP) model are formulated for the same-factory and different-factory environments. The three novel MILP models are based on two different modeling ideas. The existing MILP model and the three proposed MILP models are compared in detail from several aspects, such as binary decision variables, continuous decision variables, constraints, solution performance and solution time. By solving the benchmarks in existing studies, the effectiveness and superiority of the proposed MILP and CP models are proved. Experimental results show that the MILP model of sequence-based modeling idea performs best, the MILP model of adjacent sequence-based modeling idea takes the second place and the existing MILP model of position-based modeling idea performs worst. The CP model is more efficient and effective than MILP models. In addition, compared with the existing meta-heuristic algorithms (e.g., DABC and IABC), the proposed MILP models prove the optimal solutions of 37 instances and improve 17 current best solutions. The CP model solves all the 45 instances to optimality and improves 19 current best solutions for benchmarks in the existing studies