Lagrangian Mesh Research Articles

Dynamic response of spherical tanks subjected to the explosion of hydrogen-blended natural gas

Hydrogen is a renewable and clean energy and obtains increasing attention all around the world. Blending hydrogen into natural gas can promote hydrogen development and storage tank farms play an essential role in the storage of hydrogen-blended natural gas (HBNG). However, upon leakage from the tank, HBNG accumulates to create a vapor cloud and may explode, resulting in more severe consequences than natural gas. Furthermore, the explosion has the potential to trigger the domino effect in the spherical tank storage area, leading to multiple explosions. But little attention has been paid on this research issue. This article analyzes the dynamic response of the spherical tank exposed to HBNG explosions. The fluid–structure interaction theory is adopted to establish the whole finite element model. Meanwhile, the advanced Arbitrary Lagrangian-Eulerian (ALE) method can reduce mesh distortion and enhance computational accuracy, and the process of explosion is analyzed through the coupling of the Lagrangian mesh for the structure and the Eulerian mesh for the vapor cloud. Then, the algorithm is employed that couples shell elements with solid elements by utilizing a penalty function. Besides, the dynamic response of the spherical tank and pillar surfaces is analyzed in the trends of effective stress, displacement, velocity, and structural damage under various parameters, evaluating the energy absorbed by the spherical tank and pillar. Accordingly, the equipment damage caused by HBNG explosions considering possible damage caused by secondary explosions can be obtained. Then, the effects of the HBNG explosion chain on equipment are analyzed in tank areas. The dynamic response of structures in the explosion is closely related to the different hydrogen blended ratios, distances, and structural thickness, which significantly impact the effective stress of the spherical tank and pillar. Moreover, the damage caused by the HBNG explosion is more serious for pillars than the spherical tank, making the role of pillars crucial in supporting the spherical tank. The results obtained in this study can be used to support the design and safety operation of HBNG storage areas.

An arbitrary Lagrangian–Eulerian positivity-preserving finite volume scheme for radiation hydrodynamics equations in the equilibrium-diffusion limit

In this paper, an arbitrary Lagrangian–Eulerian (ALE) positivity-preserving finite volume scheme is constructed for radiation hydrodynamics equations (RHE) in the equilibrium-diffusion limit. Firstly, the integral form equations of RHE in the ALE framework are presented. Then, the operator-splitting method is applied to divide the equations into hyperbolic part and parabolic part. In addition, a second-order positivity-preserving finite volume scheme is constructed for the hyperbolic part based on MUSCL reconstruction. The vertex velocity is obtained by the predictor–corrector strategy. The Winslow method is applied to improve the quality of the Lagrangian mesh. Furthermore, a nonlinear positivity-preserving finite volume scheme suitable for distorted mesh is proposed for the parabolic part. Finally, some numerical examples are given to show the accuracy and reliability of the numerical scheme.

Rope–sheave contact transient analysis in hoisting operations with a bristle model and an arbitrary Lagrangian–Eulerian approach

AbstractThis paper describes the development of a computational model for the rope–sheave contact interaction in reeving systems when the ropes are modeled with an arbitrary Lagrangian–Eulerian approach. This discretization approach has been developed in previous publications as a general and systematic method for the modeling and simulation of reeving systems. However, the rope–sheave contact model was avoided assuming the no-slip contact condition. The contact model developed in this paper introduces specialized ALE-ANCF-cubic rope contact elements that are used to discretize the rope segment winded at the sheave. The contact is modeled using a set of virtual discrete bristles attached to material points in the mid-line of the rope in one end and in contact with the sheave in the other end. Therefore, a second Lagrangian mesh, apart of the ALE mesh used to discretize the rope, is used to define the fixed ends of the bristles. The kinematics and dynamics used to calculate the normal and tangential contact forces are described in detail. The contact model is 3D and can be used to analyze the contact with a sheave groove with arbitrary shape. The tangential contact force model can be used to describe stick and slip contact conditions and, to improve the simulation performance of the model, an LuGre regularization tangential contact force model is used. The rope-sheave contact model is used to analyze the behavior of a simple elevator system. The numerical results show that the static rope-sheave contact interaction agrees well with an analytical solution of the problem. Finally, the same elevator system is analyzed dynamically for a cabin ride of 8 meters with a steady velocity of 1 m/s. Results show that the normal and tangential contact forces during the steady velocity period are not so different from the static solution, but very different from the classical Creep Theory and Firbank’s Theory.

Numerical modeling of cutting force and chip formation during water Jet Assisted Machining of Ti17 based on SPH/FEM method

In this paper, we conduct a numerical simulation of a water jet-assisted machining process for a titanium alloy Ti17, with a focus on addressing the Fluid-Structure Interaction (FSI) problem arising during lubricated machining. The proposed coupling strategy employs the Smoothed Particle Hydrodynamics (SPH) free mesh method, coupled with an ordinary Lagrangian mesh. The SPH method is utilized to model the water jet, while the Lagrangian Finite Element Method (FEM) is employed to represent the workpiece and the tool. The hydrodynamic behavior of water is captured using the linear Huguenot form of the Mie-Greisen equation of state. Additionally, the Johnson-Cook constitutive law is applied to describe the thermoviscoplastic behavior of the titanium alloy. It is essential to note that thermal and mechanical effects were decoupled in this study. The investigation focuses on cutting force and chip morphology under two distinct cutting regimes: Dry Machining (DM) and Water Jet Assisted Machining, specifically with conventional lubrication (CL). The numerical simulations are compared with experimental results found in the literature, with an emphasis on cutting force and chip morphology. The obtained simulation results exhibit a favorable correlation with the experimental data. All numerical simulations were conducted using ABAQUS version 6.14.

Thermal-structural hybrid Lagrangian solver and numerical simulation-based correction of shape deformation of stainless-steel parts produced by laser powder bed fusion

An efficient thermal-structural numerical solver for Additive Manufacturing has been developed based on a modified Lagrangian approach to solve the energy conservation equations in differential form. The heat transfer is modeled using the finite difference method applied to a deforming Lagrangian mesh. The structural solver has been enhanced with the proposed effective quasi-elastic differential approach for modeling the elastoplastic behavior of materials. The algorithm is relatively simple to implement yet is highly effective. The solver can predict shape deformations of metal parts printed using the laser powder bed fusion technique. The second key capability of the solver is the auto-compensation of distortions of 3D-printed parts by proposing a corrected geometry of a surface to be printed, in order to ensure minimal deviation of the actual printed part from the desired one, even under non-optimal operating conditions or for complex shapes. All the simulation results have been verified in real-life experiments for 3D parts of sizes ranging from 10 to 15 mm up to 40 mm.

Bridging Eulerian and Lagrangian Poisson-Boltzmann solvers by ESES.

The Poisson-Boltzmann (PB) model is a widely used electrostatic model for biomolecular solvation analysis. Formulated as an elliptic interface problem, the PB model can be numerically solved on either Eulerian meshes using finite difference/finite element methods or Lagrangian meshes using boundary element methods. Molecular surface generators, which produce the discretized dielectric interfaces between solutes and solvents, are critical factors in determining the accuracy and efficiency of the PB solvers. In this work, we investigate the utility of the Eulerian Solvent Excluded Surface (ESES) software for rendering conjugated Eulerian and Lagrangian surface representations, which enables us to numerically validate and compare the quality of Eulerian PB solvers, such as the MIBPB solver, and the Lagrangian PB solvers, such as the TABI-PB solver. Furthermore, with the ESES software and its associated PB solvers, we are able to numerically validate an interesting and useful but often neglected source-target symmetric property associated with the linearized PB model.

Pulmonary vein flow split effects in patient-specific simulations of left atrial flow

Disruptions to left atrial (LA) blood flow, such as those caused by atrial fibrillation (AF), can lead to thrombosis in the left atrial appendage (LAA) and an increased risk of systemic embolism. LA hemodynamics are influenced by various factors, including LA anatomy and function, and pulmonary vein (PV) inflow conditions. In particular, the PV flow split can vary significantly among and within patients depending on multiple factors. In this study, we investigated how changes in PV flow split affect LA flow transport, focusing for the first time on blood stasis in the LAA, using a high-fidelity patient-specific computational fluid dynamics (CFD) model. We use an Immersed Boundary Method, simulating the flow in a fixed, uniform Cartesian mesh and imposing the movement of the LA walls with a moving Lagrangian mesh generated from 4D Computerized Tomography images. We analyzed LA anatomies from eight patients with varying atrial function, including three with AF and either a LAA thrombus or a history of Transient Ischemic Attacks (TIAs). Using four different flow splits (60/40% and 55/45% through right and left PVs, even flow rate, and same velocity through each PV), we found that flow patterns are sensitive to PV flow split variations, particularly in planes parallel to the mitral valve. Changes in PV flow split also had a significant impact on blood stasis and could contribute to increased risk for thrombosis inside the LAA, particularly in patients with AF and previous LAA thrombus or a history of TIAs. Our study highlights the importance of considering patient-specific PV flow split variations when assessing LA hemodynamics and identifying patients at increased risk for thrombosis and stroke. This knowledge is relevant to planning clinical procedures such as AF ablation or the implementation of LAA occluders.

Study of linear shaped charge (LSC) jet penetrating layered targets

This paper presents the results of an experimental and numerical study conducted to investigate the mechanism of penetration of linear shaped charges (LSCs) into double layered targets. A type of LSC causing less collateral damage in terms of jet residues was studied. The LSC consisted of a linear liner and case with the depth to diameter aspect ratio of 2.0. Anti-theft doors and spaced steel plates were used to represent double layered targets. The anti-theft door and the equivalent targets were penetrated with little jet residues behind the exit crater. The experiments were simulated using the commercially available software AutoDYN as a 2-D planer symmetry FEM model of the LSC penetrating a double layered target. An Eulerian mesh was used to model the LSCs whilst the double layered target was modelled using a Lagrangian mesh. A fully coupled algorithm was used for two different meshes. The data from the experiments (the shape of the breach and jet residues) were used to validate the numerical model. The validated numerical model was used to gain insights into the penetrating mechanisms of the LSC. The numerical results indicated that the penetration mechanisms of the two layers target were different. The first layer was penetrated by a continuous jet typical of a high-speed jet whereas the second layer was perforated by both a jet tip and slug characterized by a low-impact velocity projectile (slug).

Dynamic domain decomposition method based on weighted Voronoi diagrams

Dynamic load balancing during the high performance computing is an important part of parallel workflow. An efficient way to achieve the desired distribution of load during computation is to use the Voronoi diagram (VD) for the domain decomposition. The advantage of this approach is that it can be used in conjunction with various numerical methods and balancing strategies. Domain decomposition based on Voronoi diagrams can be used together with Eulerian meshes, Lagrangian meshes, and meshfree methods, such as molecular dynamic or smoothed-particle hydrodynamics (SPH). However, for the basic VD decomposition it may be difficult to obtain load balance when the adaptive mesh refinement (AMR) is used: because of the high heterogeneity of the load on the adapted mesh, the balancing algorithms may be unstable or have poor convergence.In the present study we propose an improvement of the ordinary VD domain decomposition method with using the weighted Voronoi diagrams. We demonstrate the possibilities of such domain decomposition to achieve a better balanced configuration on two- and three-dimensional problems using mesh configurations typical for computational fluid dynamics.

A particle finite element method based on Level–Set functions

Since the seminal work of Idelsohn, Oñate and Del-Pin (2004), the Particle Finite Element Method (PFEM) has relied on a Delaunay triangulation and the Alpha–Shape (AS) algorithm in the remeshing process. This approach guarantees a good quality of the Lagrangian mesh, but introduces a list of shortcomings that demand geometrical treatments tailored to each problem. In order to improve the remeshing process in PFEM, this work proposes the use of a Level–Set (LS) function instead of the Alpha–Shape algorithm. Since the Level–Set considers the boundary of the fluid and its interior, and not only a geometric criterion as does the Alpha–Shape, the proposed strategy (PFEM–LS) shows more robustness than the classical approach (PFEM–AS) owing to three main improvements. First, the LS function allows for a better control over the elements that are created during the fluid/fluid contact, which helps to reduce mass creation. Second, it helps to preserve the smoothness of the free surface and to reduce mass loss. Third, it allows the meshing of solitary particles that are detached from the free surface, which improves the representation of drops in PFEM. The methodology is presented and validated using free surface flow problems in 2D.

Численное моделирование пространственных задач гидродинамики с учетом упругих процессов

In the present work, the support operator method for spatial problems of elasticity theory is used to construct a finite-difference approximation of elastic forces on staggered Lagrangian meshes. For displacement vectors on irregular meshes, as applied to difference schemes, for problems of elasticity theory, the corresponding discrete operations have been developed. Taking into account the energy balance of the medium, the presented families of integrally consistent approximations of vector analysis operations are sufficient for numerical modeling of these processes. The resulting forces acting on the nodal domains of the medium are obtained explicitly in two-dimensional and three-dimensional geometry. Calculations are given for the propagation of sound waves in an aluminum three-dimensional orthogonal plate due to an end impact. On the example of numerical calculations, the invariance of the elastic force and energy characteristics of the medium during solid-state rotations is confirmed.

Comparison of Material Point Method and Finite Element Method for Post-Failure Large Deformation Geotechnical Analysis

Finite Element Method (FEM) has been the state-of-the-art method in geotechnical analysis since it first formulated in the 40s. It capable to handle Multiphysics simulation, soil-structure and soil-water interaction, and time history analysis. Though powerful, the standard Lagrangian FEM suffers mesh distortion when handling large strain deformation problem. This mesh entanglement problem makes post-failure analysis is considerably challenging to model if not impossible to do using FEM. The Material Point Method (MPM) then later introduced to solve these large strain deformation problems. Adapted from the Particle in Cell (PIC) method, MPM is a hybrid method that combines Eularian and Lagrangian approach by utilizing moving material points which are moving over spatially fixed computational mesh. This approach enables MPM to calculate not only fluid mechanics such in PIC but also solid mechanics and its intermediatory states. To demonstrate the capability of MPM and its consistency with FEM in geotechnical analysis, this article presents a comparison of FEM and MPM analysis on a hypothetical slope using Mohr-Coulomb constitutive model. The simulation shows that both FEM and MPM analyses are consistent to each other especially in small strain scheme. However, in large strain deformation, MPM is still able to get convergent result while FEM is not. The MPM simulation is also able to animate post failure behavior clearly, calculate post-failure strains and stresses distribution, and present final geometry of the model.

An improved immersed boundary method by coupling of the multi-direct forcing and Fourier pseudo-spectral methods

The present paper aims to report an improve on the development of the IMERSPEC methodology, which is a methodology that couples the Fourier pseudo-spectral and immersed boundary methods, which can now solve flows over complex geometries. In the present work, the Lagrangian mesh that represents the immersed body inside the flow does not have to coincide with the Eulerian mesh. Furthermore, a high accuracy for the resultant methodology is achieved. These two important improvements arise from the development of the multi-direct forcing method and used to calculate the force field of the immersed boundary method. In order to verify the IMERSPEC methodology, a manufactured solution technique was used. The results of a refined mesh were employed, and different distribution and interpolation functions were used and analysed. Finally, flows over a backward-facing step geometry were simulated in two dimensions. This type of flow presents several physical features, for example, detachment and reattachment of the boundary layer and vortex generation, which are difficult to capture using numerical solutions. A comparison with the experimental data is presented and shows that the IMERSPEC methodology is promising for computational fluid dynamics applications and its analysis.

An accurate and stable alternating directional moving particle semi-implicit method for incompressible flow simulation

As a Lagrangian mesh free method, Moving Particle Semi-implicit (MPS) method can easily handle complex incompressible flow with free surface. However, some deficiencies of MPS method such as inaccurate results, unphysical pressure oscillation and particle thrust near free surface still required to be further resolved. In this paper, an improved MPS method based on Crank-Nicolson time scheme, named Alternating Direction Moving Particle Semi-implicit (ADMPS) method, is proposed. In addition, a time integral source term (TIS) is developed to suppress unphysical pressure oscillation and a free-surface revise (FR) technique is proposed to avoid particle thrust near free surface. Enhancement of accuracy and stability by these improvements is proved by hydrostatic and dam breaking benchmarks, and the numerical results of the ADMPS-FR-TIS model show good agreement with the hydrostatic analytical solution and the dam breaking experiment data.

On the Lagrangian-Eulerian coupling in the immersed finite element/difference method

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method's performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretization combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta function). Earlier work with this methodology suggested that such coarse structural meshes can yield improved accuracy for shear-dominated cases and, further, found that accuracy improves when the structural mesh spacing is increased. However, these results were limited to simple test cases that did not include substantial pressure loading on the structure. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations in a broader range of tests. Our results indicate that kernels satisfying a commonly imposed even–odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy this condition. We also find that narrower kernels are more robust, in the sense that they yield results that are less sensitive to relative changes in the Eulerian and Lagrangian mesh spacings, and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.

A new transient model for motion and variation of gas phase in pipelines applying collocated multi-grid method with the Eulerian-Lagrangian approach

This paper investigates the gas phase, including bubbles' and gas pockets' migration and evolution in gas-liquid flow in the pipelines, where the safe operation is severely influenced by the gas accumulations. To investigate the formation, as well as location and amount of these gas accumulations, a new transient model applied with the Population Balance Model (PBM), gas pockets capturing and tracking model are proposed, which are discretized and calculated in the collocated multi-grid with Eulerian-Lagrangian coupling scheme. The flow field of the pipeline is modeled with the Eulerian scheme, while the bubbles' migration, gas pockets' capturing and tracking can be modeled in the Lagrangian mesh. To couple the description of bubble-dimension and that of pipe-dimension, the Rayleith-Plesset equation and the Smooth Particle Hydrodynamics (SPH) are applied, thus, the changeable characteristics of bubbles along the pipes can be described. Based on that, the bubbles' transition to the gas pocket under specific conditions can be predicted. And the subsequent capturing and tracking process will work with the help of the collocated multi-grid, which provides the accurate and parallel calculation. The relative parameters are extracted from the published experimental data, and the simulation results are in good agreement with the measured results. Meanwhile, the case studies show the model has the prediction ability of the gas phase's migration and evolution, as well as the formation and location of gas accumulations in the pipeline.

Numerical Simulation of Two-Phase Water-Oil Flow in a Horizontal Pipe Using the Smoothed Particle Hydrodynamics Method

In this work, the numerical code DualPhysics, based on the Lagrangian particle and mesh free method Smoothed Particle Hydrodynamics, has been employed to solve the slightly compressible isothermal two-phase water-oil flow. The continuity and momentum equations are solved, and we used the modified Tait equation of state to determine the pressure. To validate the numerical code, we solved the modified Couette flow of two fluids. As a practical case, we solved the isothermal and two-dimensional two-phase water-oil flow. The mixing of the fluids occurs after passing through a 45 degree Y junction placed at the entrance of the horizontal pipeline. Results showed the potential for using the numerical code, although some modifications and alterations are still necessary to solve practical problems.

Partial optimal transport for a constant-volume Lagrangian mesh with free boundaries

This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a set of convex cells called a Laguerre diagram, parameterized by the position of N points in 3D and N additional parameters that control the volumes of the cells. These parameters are found as the (unique) solution of a convex optimization problem – semi-discrete Monge-Ampère equation – stemming from optimal transport theory. In this article, this setting is extended to objects with free boundaries and arbitrary topology, evolving in a domain of arbitrary shape, by solving a partial optimal transport problem. The resulting Lagrangian scheme makes it possible to accurately control the volume of the object, while precisely computing the intersections with the domain boundary, the interactions, the collisions, and the changes of topology.

An immersed boundary-lattice Boltzmann framework for fully resolved simulations of non-spherical particle settling in unbounded domain

Particle settling at moderate to high Reynolds number takes a considerable distance to reach a periodical or statistically steady regime. Hence, hardware memory limitations in fully resolved simulations constrain the maximum domain size for this flow class. Due to the locality in most of its algorithms, the Lattice Boltzmann Method (LBM) is increasingly popular for CFD studies. In the present paper, a domain transferring scheme is implemented in LBM, enabling simulations of particle motion in a virtually infinite domain, and it is combined with a high-quality Lagrangian mesh algorithm to be solved with the Immersed Boundary Method (IBM). A thorough mesh generation procedure for ellipsoidal particles is disclosed, as well as an extension of the internal mass compensation strategy of Suzuki and Inamuro (2011). Comparison with analytical results shows that the numerical model appropriately describes the particle rotation and can predict a terminal velocity close to Stokes solution. The numerical results of a buoyant sphere moving diagonally presented remarkable concordance with experimental data. Also, an excellent agreement with a numerical study of oblate spheroids settling in a vast domain was found. The domain transferring scheme reduced the memory demand in one order of magnitude.

A Full Coupled Numerical Method for Dynamic Response of Metro Tunnel Subjected to Surface Explosion

Terrorist blast attacks have increased in recent years. Hand bombs are one of the means for terrorist operations because of their dangerous progressive damages. In this paper, a full coupled numerical method is adopted to study the dynamic response of a metro tunnel in the sandy loam. The numerical model is developed using LS-DYNA and will be able to present a realistic behavior for the physics of this phenomenon. In the current study, the ALE method has been used. The air, explosive charge, and soil are considered as ALE’s parts; while, the structure of the tunnel has Lagrangian mesh. Two paths have been studied in the longitudinal and the circular directions for assessing tunnel lining safety. In the free-field state, the accuracy of the model is verified by comparing the peak pressure and acceleration in the soil with the empirical predictions available in the literature. The safety assessment has been done according to blast vibration criteria. The tunnel would not be safe, as per the PPV standard, under the condition of w=500kg and R=4m. Tunnel crowns are the most vulnerable areas while the peak particle velocity is 19cm/s with maximum permanent vertical deformation.