Lagrangian Eulerian Research Articles

Numerical Study of Air Cushion Effect in Notched Disk Water Entry Process Using Structured Arbitrary Lagrangian–Eulerian Method

This paper presents a novel numerical investigation into the air cushion effect and impact loads during the water entry of notched discs, utilizing the Structured Arbitrary Lagrangian–Eulerian (S-ALE) algorithm in LS-DYNA. Unlike prior studies that focused on smooth or unnotched geometries, the present study explores how varying notch parameters influence the fluid–solid coupling process during high-speed water entry. The reliability and accuracy of the computational method are validated through grid independence verification and comparisons with experimental data and empirical formulas. Systematic analysis of the effects of notch size, water entry velocity, and entry angle on the evolution of the free surface, impact loads, and structural responses uncovers several novel findings. Notably, increasing the notch diameter significantly enhances the formation and stability of the air cushion, leading to a considerable reduction in peak impact loads—a phenomenon not previously quantified. Additionally, higher water entry Froude numbers are shown to accelerate air cushion compression and formation, markedly affecting free surface morphology and force distribution. The results also reveal that varying the water entry angle alters the air cushion’s morphological characteristics, where larger angles induce a more pronounced but less stable air cushion, influencing the internal structural response differently across regions.

Impact of the Draft Plate on the Wall Erosion and Flow Field Stability of a Cyclone Separator

Cyclone separators are commonly employed in the mining, metallurgy and chemical industries due to their simple structure, easy maintenance and high recovery efficiency. However, with the wide application of cyclone separators, many problems have become exposed in their practical operation, restricting their development. Among these, wall erosion is becoming a significant problem. In this study, to resolve the problem of severe erosion on the walls, the Eulerian–Lagrangian framework was employed to investigate a cyclone separator with a draft plate at the inlet and to evaluate the effect of a draft plate with angles of 0°, 45° and 90° on the degree of erosion and the stabilization of flow fields. Moreover, after verifying the reliability of the numerical model via data from experiments, the characteristics of gas–solid flow were analyzed and the effects of the new structure on the degree of wear were investigated. The results demonstrated that unfavorable phenomena such as secondary flow and wall erosion generated during the operation could be mitigated by the draft plate. When the plate angle was 90°, the wall erosion was the lightest and the range of influence of the secondary flow was the smallest. When the plate angle was 45°, the comprehensive performance was the best, and there was a better balance between the energy loss and the degree of wall erosion. Therefore, the presence of the draft plate has a significant impact on the interaction of gas–solid phases in a cyclone separator.

A high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for one-dimensional compressible multi-material flows

In this paper, a high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian (ALE) discontinuous Galerkin scheme is proposed for one-dimensional compressible multi-material flows. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of fluid mixture. The mesh velocity in the ALE framework is obtained by using an adaptive mesh method that can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduce the numerical dissipation near material interfaces. Using this adaptive mesh, the resolution of solution near some special regions such as material interfaces can be improved effectively by our scheme. With the appropriate time step condition and using a bound-preserving and positivity-preserving limiter, our scheme can ensure the positivity of density and pressure and the boundness of volume-fraction, which further ensures the computational robustness and degree of confidence of simulations under large density or pressure ratios and so on. In general, our scheme can be applied to the simulations of compressible multi-material flows efficiently with the essentially non-oscillatory property and physical-constraints-preserving (bound-preserving and positivity-preserving) property, and its steps are more concise compared to some other methods such as the indirect ALE methods. Some examples are tested to demonstrate the accuracy, essentially non-oscillatory property and physical-constraints-preserving property of our scheme.

Manipulating microfluidic freezing phenomena of electrolytes using electromagnetic stimuli

We investigate the transients of freezing of a flowing electrolyte between cold microfluidic domain walls. We explore the novel effect of morphing the solidification characteristics of the flow system using externally applied electric and magnetic fields. The spatiotemporal evolution of solid–liquid (S–L) interface is simulated using arbitrary Lagrangian–Eulerian approach. We show that the solidification rate is highly influenced by advection effects, and for increasingly high liquid flow rates, solidification stops way before the event of channel closure. Due to electro-magneto-hydrodynamic interactions, a substantial decrease in channel closing time is observed with imposed fields. However, a considerably large electric field may lead to a significant increase in Joule heating, which marginally increases the channel closing time. The magnetic field is observed to produce smoother, curved frozen layers. The electric field tends to enhance the solidification rates more uniformly along the channel length, resulting in much flatter S–L front, hence trapping lesser liquid at channel closure. The results reveal that freezing kinetics in microfluidic devices can be morphed by using electric and magnetic fields, governed by the electric field number (En) and Hartmann number (Ha). The findings may be crucial for phase-change based microfluidic-systems, micro-castings, flow freezing valves, and diodes.

Collar effect on the pier scour: Eulerian–Lagrangian approach

Collar effect on the pier scour: Eulerian–Lagrangian approach

Orientation of inertialess spheroidal particles in turbulent channel flow with spanwise rotation

The orientation dynamics of inertialess prolate and oblate spheroidal particles in a directly simulated spanwise-rotating turbulent channel flow has been investigated by means of an Eulerian–Lagrangian point-particle approach. The channel rotation and the particle shape were parameterized using a rotation number Ro and the aspect ratio λ, respectively. Eleven particle shapes 0.05 ≤ λ ≤ 20 and four rotation rates 0 ≤ Ro ≤ 10 have been examined. The spheroidal particles retained their almost isotropic orientation in the core region of the channel, despite the significant mean shear rate set up by the Coriolis force. Irrespective of channel rotation rate Ro, rod-like spheroids tend to align in the streamwise direction, while disk-like particles are oriented in the wall-normal direction. These trends were accentuated with increasing departure from sphericity λ = 1. The changeover from the isotropic orientation mode in the centre to the highly anisotropic near-wall orientation mode commenced further away from the suction-side wall with increasing Ro, whereas the particle orientations on the pressure side of the rotating channel remained essentially unaffected by Ro. We observed that the alignments of the fluid rotation vector with the Lagrangian stretching direction were similarly unaffected by the imposed system rotation, except that the de-alignment set in deeper into the core at high Ro. This contrasts with the well-known substantial impact of system rotation on the velocity and vorticity fields. Similarly, slender rods and flatter disks were aligned with the Lagrangian stretching and compression directions, respectively, for all Ro considered, except in the vicinity of the walls. The typical near-wall de-alignment extended considerably further away from the suction-side wall at high Ro. We conjecture that this phenomenon reflects a change in the relative importance of mean shear and small-scale turbulence caused by the Coriolis force. Preferential particle alignment with Lagrangian stretching and compression directions are known from isotropic and anisotropic turbulence in inertial reference systems. The present results demonstrate the validity of this principle also in a non-inertial system.

Treatment of inelastic material models within a dynamic ALE formulation for structures subjected to moving loads

AbstractThis article showcases the development of a dynamic Arbitrary Lagrangian Eulerian (ALE) formulation to account for inelastic material models within a finite element framework. Such a formulation is commonly utilized in research domains like fluid mechanics, fluid‐structure interaction, quasi static remeshing techniques, and quasi static load movement. The work at hand describes the application of the ALE formulation to efficiently analyse structures subjected to moving loads in the field of transient inelastic solid mechanics. In particular, structures such as pavements, gantry crane girders etc., which are subjected to moving loads, can be numerically simulated, and their transient response in the relevant region around the load can be obtained without relying on moving loads. The focus of this article is to facilitate the treatment of history variables stemming from inelastic material models. Of particular interest is the advection procedure required to transport the history variables through the mesh, as the material appears to flow through it. The mathematical framework necessary to treat this advection process is described in detail, considering a nonlinear viscoelastic material model on a neo‐Hookean base at finite deformations. Then, four methods for numerically achieving the advection are implemented within a transient finite element ALE formulation. These methods are compared against each other, and additionally with the conventional Lagrangian method for validation. The results demonstrate satisfactory agreement with conventional simulation methods, while offering a significant improvement in terms of computation speed. With the work at hand, the dynamic response of inelastic materials subjected to moving loads can be numerically simulated in a computationally efficient manner.

Computational Modeling of Cough Droplet Behavior in the Human Upper Respiratory Airway

Abstract This study investigated the interaction of cough droplets with airflow in a realistic human airway. The ultimate aim was to understand the behavior of cough droplets inside the airway and to assess the potential of droplets to be retained in the airway or transmitted to the lungs. A computational fluid dynamics (CFD) model, based on the Euler–Lagrangian framework, was employed to predict the two-phase, droplet-laden transient cough flow in a realistic three-dimensional (3D) human airway. The airway geometry was reconstructed from patient computed tomography (CT) scan dataset. The discrete phase model was used to track the motion of the droplets in the air flow. Two distinct cough profiles—a strong cough and a weak cough—acquired experimentally from human subjects, were used as input to simulate normal and disordered cough functions. The effects of cough strength and droplet size on droplet retention and aspiration in the airway were investigated. It was found that droplet retention was significantly higher for a weak cough compared to a strong cough. For a weak cough, the highest droplet retention percentage exceeded 60%, while for a strong cough, it was less than 20%. Larger sized droplets were more likely to be aspirated into the lungs, especially under weak cough conditions. In the case of weak cough, more than 5% of the 200 μm sized droplets were aspirated into the lungs, whereas for strong cough, aspiration was less than 2%.

Numerical evaluation of wear parameters using meta-models

AbstractWear parameters identified by linear friction tester (LFT) experiments overestimate the wear mass loss when applied on a laboratory abrasion & skid tester 100 (LAT100) simulation. On the other hand, the identification of wear parameters directly from the LAT100 experiment can be challenging since variables such as the contact area and the slip velocity are not measured experimentally and need to be assumed. To improve the identification process, numerical calibration is used as a more suitable approach. This approach relies on meta-models as a target function for minimization. The meta-models are generated using a sample of wear parameters applied to an LAT100 finite element modeling (FEM) to calculate the corresponding wear mass.. In this model, a transport velocity is defined for the rolling simulation, and an arbitrary lagrangian–eulerian (ALE) adaptive meshing approach is adopted for the wear modeling. For the wear model, a combination of archard’s wear model and schallamach’s abrasion law is used. This model contains a pair of wear parameters to be identified. The ALE adaptive meshing technique moves the nodes independently of the material. Since the mesh topology remains the same, failure of the simulation occurs if the wear volume loss exceeds that of the element. Meta-models are created to extend wear modeling beyond this failure. Once the meta-models are created, they are used as a target function for the minimization algorithm. The minimization algorithm aims to find the optimal wear parameters by minimizing the difference between experimentally observed and numerically produced wear mass loss. The minimization algorithm inputs a set of wear parameters into the meta-models which in turn yield a prediction of the wear mass loss. The process is carried out until an optimum parameter set is identified. Such an approach has a lower accuracy if the parameters are identified directly from the experiment using assumptions regarding the contact shear stress and the sliding velocity. Nonetheless, the main advantage of parameters identified using the meta-model approach is the usability of these parameters in an LAT100 model.

Computational fluid dynamics simulations of spray tests in a multicompartment construction with an Eulerian–Lagrangian approach

Computational fluid dynamics simulations of spray tests in a multicompartment construction with an Eulerian–Lagrangian approach

Evaluating the Seismic Resilience of Above-Ground Liquid Storage Tanks

Historical seismic events have repeatedly highlighted the susceptibility of above-ground liquid storage steel tanks, underscoring the critical need for their proper design to minimize potential damage due to seismic forces. A significant failure mechanism in these structures, which play essential roles in the extraction and distribution of various raw or refined materials—many of which are flammable or environmentally hazardous—is the dynamic buckling of the tank walls. This study introduces a numerical framework designed to assess the earthquake-induced hydrodynamic pressures exerted on the walls of cylindrical steel tanks. These pressures result from the inertial forces generated during seismic activity. The computational framework incorporates material and geometric nonlinearities and models the tanks using four-node shell elements with two-point integration, specifically Belytschko shell elements. The Arbitrary Lagrangian–Eulerian (ALE) method is employed to accommodate substantial structural and fluid deformations, enabling a full simulation of fluid–structure interaction through highly nonlinear algorithms. Experimental test data are utilized to validate the proposed modeling approach, particularly in replicating sloshing phenomena and identifying stress concentrations that may lead to wall buckling. The study further presents results from a parametric analysis that varies the height-to-radius and radius-to-thickness ratios of a typical anchored flat-bottomed tank, examining the seismic performance of this common storage system. These results provide insights into the relationship between tank properties and mechanical behavior under dynamic loading conditions.

A projection-based time-segmented reduced order model for fluid-structure interactions

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (two/three dimension) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

Seafloor Landing Dynamics of a Novel Underwater Robot Using ALE Algorithm

Seafloor landing research is crucial for underwater robots tasked with long-term fixed-point observation missions, particularly those requiring prolonged operations on the seabed. Developing safe and stable landing technology is essential for the success of these missions. This paper introduces a novel underwater robot by establishing its dynamic model and using the arbitrary Lagrangian–Eulerian (ALE) method to conduct a comprehensive study of its deep-sea landing process. Prior to detailed parameter analysis, the ALE algorithm’s effectiveness and accuracy were validated through experimental data and simulation results from previous studies. This study examines the penetration depth and force conditions during the landing process by varying factors such as seabed soil properties, landing velocity, and the robot’s mass. Findings indicate that seabed soil properties significantly influence penetration depth and pressure, with variations in soil density and strength affecting the robot’s landing behavior. In contrast, the robot’s mass has a relatively minor effect, suggesting that the choice of structural materials has limited impact on penetration depth and pressure during landing. Additionally, landing velocity was found to significantly affect penetration depth and pressure; higher velocities result in greater penetration depths and pressures. This highlights the importance of controlling landing velocity to minimize impact forces and protect the robot. The results of this study provide theoretical support and data for developing deep-sea landing technology for novel underwater robots and inform the selection of structural materials, ensuring the successful execution of long-term fixed-point observation missions in deep-sea environments.

Blast resistance of CFRP composite strengthened masonry arch bridge under close-range explosion

Carbon fiber reinforced polymers (CFRP) are recognized for their exceptional strength-to-weight ratio. They offer a viable and effective solution for strengthening and retrofitting masonry bridges, helping to extend their service life, improve structural performance, and meet modern safety and load requirements. Wrapping of CFRP around masonry elements can enhance their confinement and ductility. This flexibility plays a crucial role in preventing sudden brittle failure, allowing for controlled deformation, which is essential for blast resistance. Additionally, CFRP materials possess the ability to flex and absorb energy, which proves beneficial in containing and redistributing forces generated during an explosion, consequently reducing the risk of catastrophic failure. This study employed the coupled Eulerian–Lagrangian (CEL) technique available in the finite element software Abaqus/Explicit to simulate the blast loads. Various detonation scenarios were considered, taking into account factors such as location and their impacts on bridge structures. A detailed micro-model was developed using finite element software and accurate geometric data acquired from FARO laser scanning of the case study. The properties of masonry units and backfill were characterized using the Johnson-Holmquist II damage model and Mohr–Coulomb criteria. The Jones-Wilkins-Lee equation of state (EOS) was applied to replicate the behavior of trinitrotoluene (TNT). In accordance with the JH-II model, the researchers formulated a VUMAT code. The study examined the distinct damage mechanisms and overall structural responses of bridges. By evaluating the blast resistance of individual bridge models, the most critical scenarios were pinpointed. Carbon Fiber Reinforced Polymer (CFRP) was then utilized as a method to fortify bridges against blast loads. A comparison was made between the damage propagation before and after the reinforcement.

Lateral migration of a deformable fluid particle in a square channel flow of viscoelastic fluid

Cross-stream migration of a deformable fluid particle is investigated computationally in a pressure-driven channel flow of a viscoelastic fluid via interface-resolved simulations. Flow equations are solved fully coupled with the Giesekus model equations using an Eulerian–Lagrangian method and extensive simulations are performed for a wide range of flow parameters to reveal the effects of particle deformability, fluid elasticity, shear thinning and fluid inertia on the particle migration dynamics. Migration rate of a deformable particle is found to be much higher than that of a solid particle under similar flow conditions mainly due to the free-slip condition on its surface. It is observed that the direction of particle migration can be altered by varying shear thinning of the ambient fluid. With a strong shear thinning, the particle migrates towards the wall while it migrates towards the channel centre in a purely elastic fluid without shear thinning. An onset of elastic flow instability is observed beyond a critical Weissenberg number, which in turn causes a path instability even for a nearly spherical particle. An inertial path instability is also observed once particle deformation exceeds a critical value. Shear thinning is found to be suppressing the path instability in a viscoelastic fluid with a high polymer concentration whereas it reverses its role and promotes path instability in a dilute polymer solution. It is found that migration of a deformable particle towards the wall induces a secondary flow with a velocity that is approximately an order of magnitude higher than the one induced by a solid particle under similar flow conditions.

Active control of transonic airfoil flutter using synthetic jets through deep reinforcement learning

This paper presents a novel framework for the active control of transonic airfoil flutter using synthetic jets through deep reinforcement learning (DRL). The research, conducted in a wide range of Mach numbers and flutter velocities, involves an elastically mounted airfoil with two degrees of freedom of pitching and plunging oscillations, subjected to transonic flow conditions at varying Mach numbers. Synthetic jets with zero-mass flux are strategically placed on the airfoil's upper and lower surfaces. This fluid–structure interaction (FSI) problem is treated as the learning environment and is addressed by using the arbitrary Lagrangian–Eulerian lattice Boltzmann flux solver (ALE-LBFS) coupled with a structural solver on dynamic meshes. DRL strategies with proximal policy optimization agents are introduced and trained, based on the velocities probed around the airfoil and the dynamic responses of the structure. The results demonstrate that the pitching and plunging motions of the airfoil in the limited cycle oscillation (LCO) can be effectively alleviated across an extended range of Mach numbers and critical flutter velocities beyond the initial training conditions for control onset. Furthermore, the aerodynamic performance of the airfoil is also enhanced, with an increase in lift coefficient and a reduction in drag coefficient. Even in previously unseen environments with higher flutter velocities, the present strategy is achievable satisfactory control results, including an extended flutter boundary and a reduction in the transonic dip phenomenon. This work underscores the potential of DRL in addressing complex flow control challenges and highlights its potential to expedite the application of DRL in transonic flutter control for aeronautical applications.

Study on the jetting characteristics of an underwater explosion bubble collapsing near a floating body

This study explores the dynamic behavior and jet characteristics of underwater explosion (UNDEX) bubble oscillating near a rigid floating body using the arbitrary Lagrangian–Eulerian (ALE) method. Experiments on UNDEX bubble oscillating in a free field or oscillating near a rigid floating body in an explosion tank are used to validate the effectiveness of the ALE method in simulating the behaviors of high pressure bubble oscillating near a boundary in water. The numerical results are in good agreement with the experimental data. On this basis, the distribution of the field pressure and velocity of the oscillating bubble are further analyzed in detail. The evolution characteristics of the bubble jets are discussed for various values of the stand-off distance and explosion attack angle. The results reveal that a bubble produces two jet patterns for close stand-off distances (from γD=0.800 to γD=1.336) and attack angles of 0°, 45°, 75°, and 90°. The first bubble jet results in an annular splitting of the bubble, while the second jet is pointed toward the floating body. The aim of this study is to provide a reference for further understanding the jet dynamics of UNDEX bubble collapsing near a structure and the effective attack on ship sides.

A coupled machine-learning-individual-based model for migration dynamics simulation: A case study of migratory fish in fish passage facilities

The extensive development of hydropower projects has notably changed the ecohydrological conditions of fish habitats, affecting fish behavior, including habitat usage and migration, to varying extents. Understanding fish migration dynamics is essential for quantitatively assessing the impact of ecological restoration measures on migratory fish. However, no model has yet demonstrated sufficient accuracy to be considered valuable in ecological restoration engineering. To address this issue, in this article, a coupled machine-learning-individual-based model (ML-IBM) consisting of random forest (RF) and Eulerian–Lagrangian–agent method (ELAM) is constructed for predicting fish migration, aiming to find effective fish passage solutions before implementation. In this study, the passage data of ya-fish (Schizothorax prenanti) in vertical slot fishways (VSFs) is compiled to train ML-IBM to simulate fish migration in fish passage facilities. In movement prediction, the accuracy of swimming behavior classification reaches 83.4 %, and the R² for swimming speed regression exceeds 0.77. Compared with other state-of-the-art migration dynamic models, the proposed ML-IBM achieves the lowest root mean square error (RMSE) of 7.35 and a mean absolute error (MAE) of 6.26 in migration simulation results. Further, RF is used to quantitatively calculate the importance of input features. The contributions of each feature are analyzed and discussed from a hydrodynamic perspective, with the importance ranked as follows: flow velocity (FV) > turbulent kinetic energy (TKE) > total hydraulic strain (THS). This approach enhances the interpretability of the model and provides further insights into the mechanism of fish migration. The results presented in this study have significant implications for informing decision-making in the management of living resources and guiding engineering design processes.

Exploring arteriolar atherosclerosis: laminar blood flow across stenosis with fluid-structure interaction and gravitational effects

Abstract In response to the unanswered relevant questions surrounding atherosclerosis, it becomes imperative to investigate arterioles using sophisticated mathematical modelling techniques to shed light on critical stress and strain patterns influenced by gravity. The primary objective of this study is to scrutinize flow characteristics and probe stress and strain distributions experienced by the intima layer of arterioles, encompassing coronary, renal, cerebral, mesenteric, and pulmonary arteries, under gravitational forces. This investigation employs a fluid-structure interaction methodology utilizing arbitrary Eulerian–Lagrangian formulation. The study delves into blood flow characteristics within coronary, renal, cerebral, mesenteric, and pulmonary arterioles using the fluid-structure interaction technique, employing an arbitrary Eulerian–Lagrangian formulation. It thoroughly examines various biomechanical parameters such as the Cauchy–Green stress tensor, Principal strain, Piola–Kirchoff stress tensor, deformation tensor, and volume strain along the intima layer under the gravitational influence, elucidating vulnerable regions prone to endothelial dysfunction. Higher values of δV are found at the left shoulder and in the intima’s post stenosis area due to the pressure gradient along the flow channel, whereas other intima regions show a null volume strain. A thorough understanding of stress distribution is essential to create focused therapies to lessen vascular health problems. The stress in the post-stenosis region seems to affect the endothelial layer to a significant extent.

An immersed multi-material arbitrary Lagrangian–Eulerian finite element method for fluid–structure-interaction problems

Fluid–structure-interaction (FSI) phenomena are widely concerned in engineering practice and challenge current numerical methods. In this article, the finite element method is strongly coupled with the multi-material arbitrary Lagrangian–Eulerian (MMALE) method to develop a monolithic FSI method named the immersed multi-material arbitrary Lagrangian–Eulerian finite element method (IALEFEM). By immersing the finite elements in the MMALE computational grid, the fluid–solid interface is directly tracked by the element boundary with accurate normal directions. The fluid–structure-interaction is implicitly implemented by assembling the nodal variables and updating the Lagrangian momentum equation on the MMALE grid. Combining the advantages of both MMALE and FEM with the immersed boundary method, the IALEFEM is effective for solving complicated FSI problems with multi-material fluid flow. A slip fluid–structure-interaction method is also proposed to enhance the computational accuracy in simulating FSI problems with significantly different velocity fields. The accuracy and effectiveness of the IALEFEM are verified and validated by several benchmark numerical examples including the shock-cylinder obstacle interaction, flexible panel deformation induced by shock wave, dam break problem with large structural deformation, water entry of a wedge, fragmentation of a cylinder shell induced by blast, response of elastic plate subjected to spherical near-field explosion and structural damage of open-frame building under blast loading.