Arbitrary Lagrangian Eulerian Formulation Research Articles

Flow topology and heat transfer analysis of slotted and axisymmetric synthetic impinging jets

This work is focused on exploring the flow and heat transfer capabilities for slotted and axisymmetric cavity configurations for synthetic impinging jets; the jet is enclosed between two isothermal parallel plates, impinging into the top hot plate. Numerical simulations using the Arbitrary Lagrangian-Eulerian formulation of the Navier-Stokes equations are performed for Reynolds numbers of 50, 300 and 500 and a jet formation criteria of 3. The pattern of the flow from both configurations differ, which result in different impinging behaviors. The slotted jet is wider than the axisymmetric one, showing an increased turbulent mixing and a lower velocity deficit. On the other hand, the narrower jet of the axisymmetric configuration results in less mixing and more penetration, with a higher velocity deficit. This leads to a higher heat transfer peak in the axisymmetric configuration with a faster decay when moving away from the jet centerline than observed for the slotted configuration.

A reconstructed discontinuous Galerkin method for compressible flows on moving curved grids

A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and invalid elements near moving boundaries.

An effective modeling approach based on the ALE and M-integral for simulating crack propagation under thermo-mechanical loadings

Traditional engineering materials suffer from the presence of internal cracks, originating either during production processes or usage, that tend to grow under the action of thermo-mechanical loading, thus compromising the overall integrity. A proper evaluation of fracture behavior is then essential to limit dramatic failures that may lead to unsafe conditions. The present work proposes an effective modeling strategy for simulating crack propagation mechanisms in linear elastic continuum media subjected to mechanical and thermal loadings. The model involves a classic FE framework enhanced by Moving Mesh (MM) techniques and Interaction Integral Method (M-integral). MM serves as a powerful tool for reproducing geometry evolution caused by crack advance. In particular, the MM uses the Arbitrary Lagrangian-Eulerian (ALE) formulation, which is effective to handle random cracks. In the proposed framework, the ALE moves the computational nodes around the crack front, starting from a referential coordinate system according to classic fracture criteria, which define crack advance conditions and propagation direction. Reliable movements depend upon proper values of Stress Intensity Factors (SIFs) at crack fronts. To this end, the method adopts the M-integral for extracting mixed-mode SIFs. Since the crack front moves during the propagation, the M-integral is expressed by using the ALE formulation, thus evaluating fracture variables on deformed elements without losing accuracy. The use of proper rezoning techniques ensures the consistency of the mesh motion, limiting mesh distortions and remeshing events. The reliability of the proposed model is assessed through comparisons with existing numerical strategies.

Inertial cell sorting of microparticle-laden flows: An innovative OpenFOAM-based arbitrary Lagrangian-Eulerian numerical approach.

The need for cell and particle sorting in human health care and biotechnology applications is undeniable. Inertial microfluidics has proven to be an effective cell and particle sorting technology in many of these applications. Still, only a limited understanding of the underlying physics of particle migration is currently available due to the complex inertial and impact forces arising from particle-particle and particle-wall interactions. Thus, even though it would likely enable significant advances in the field, very few studies have tried to simulate particle-laden flows in inertial microfluidic devices. To address this, this study proposes new codes (solved in OpenFOAM software) that capture all the salient inertial forces, including the four-way coupling between the conveying fluid and the suspended particles traveling a spiral microchannel. Additionally, these simulations are relatively (computationally) inexpensive since the arbitrary Lagrangian-Eulerian formulation allows the fluid elements to be much larger than the particles. In this study, simulations were conducted for two different spiral microchannel cross sections (e.g., rectangular and trapezoidal) for comparison against previously published experimental results. The results indicate good agreement with experiments in terms of (monodisperse) particle focusing positions, and the codes can readily be extended to simulate two different particle types. This new numerical approach is significant because it opens the door to rapid geometric and flow rate optimization in order to improve the efficiency and purity of cell and particle sorting in biotechnology applications.

A Dynamic Parallel Meshless Method for the Problems with Large-Scale Movable and Deformable Boundary

This paper puts forward a dynamic parallel meshless computing algorithm that efficiently solves flow fields with largescale motions of movable and deformable boundaries. The partition boundary is updated, as the moving boundary moves across the material interface. Meanwhile, the point clouds near the moving boundary are reconstructed. Our algorithm also solves the workload balance between nodes and information exchange in each region of the computational field, using the governing equations in the arbitrary Lagrangian-Eulerian (ALE) form. The AUFS scheme is extended to calculate the numerical convective flux. Take the interaction between a helium bubble and a shockwave as an example. Our algorithm is applied to compute the flow field with different numbers of discrete points (33,044 and 66,089) and partitions (2 and 4). The results show that our algorithm achieves an efficiency of over 80%, and captures the interaction between shockwaves and the bubble accurately. Hence, our parallel algorithm is suitable for solving problems with largescale motions of deformation boundaries. The research results shed new light on the calculation speed for similar problems.

A NURBS-based discontinuous Galerkin method for conservation laws with high-order moving meshes

The objective of the present work is to develop a new numerical framework for simulations involving deformable domains, in the specific context of high-order meshes consistent with Computer-Aided Design (CAD) representations. Thus, the proposed approach combines ideas from isogeometric analysis, able to handle exactly CAD-based geometries, and Discontinuous Galerkin (DG) methods with an Arbitrary Lagrangian-Eulerian (ALE) formulation, able to solve complex problems with moving grids. The resulting approach is a DG method based on rational Bézier elements, that can be easily constructed from Non-Uniform Rational B-Splines (NURBS), formulated in a general ALE setting. We focus here on applications in two-dimensional compressible flows, but the method could be applied to other models as well. Two verification exercises are conducted, to assess rigorously the properties of the method and the convergence rates for representations up to sixth order. Finally, three problems are analysed in depth, involving compressible Euler and Navier-Stokes equations, for an oscillating cylinder and a pitching airfoil. In particular, the convergence of flow characteristics is investigated, as well as the impact of using curved boundaries in the context of deformable domains.

High-order arbitrary Lagrangian–Eulerian discontinuous Galerkin methods for the incompressible Navier–Stokes equations

This paper develops robust discontinuous Galerkin methods for the incompressible Navier–Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian–Eulerian formulations are proposed in a unified framework for both coupled as well as projection or splitting-type Navier–Stokes solvers. The framework is flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. The Navier–Stokes equations with ALE transport term are solved on the deformed geometry storing one instance of the mesh that is updated from one time step to the next. Discretization in space is applied to the time discrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step. This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as is shown theoretically and demonstrated numerically by the example of the free-stream preservation test. We discuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projection-type methods and the ingredients needed to preserve high-order accuracy. We show numerically that the formulations proposed in this work maintain the formal order of accuracy of the Navier–Stokes solvers in both space and time. Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows. The implementation is based on fast matrix-free operator evaluation in order to devise computationally efficient algorithms.

Numerical analysis of high-speed water jet spot welding using the arbitrary Lagrangian-Eulerian (ALE) method

The high-speed water jet is used for different manufacturing processes such as cutting, shot peening, cleaning, machining, and spot welding. As water jet spot welding is a concise time welding process, numerical analysis can optimize it and examine the effect of input parameters on plastic deformation and stress distribution that would be difficult to measure experimentally. In this study, the entire water jet spot welding process of aluminum HS30 is modeled through ABAQUS/EXPLICIT based on an arbitrary Lagrangian-Eulerian (ALE) formulation which helps to alleviate many of the traditional Lagrangian-based implementation drawbacks. Three-dimensional (3D) models are used to evaluate the effect of water jet velocity, flyer stand-off distance (SD), and flyer thicknesses during the process. The weld’s quality in terms of failure or success is also investigated by using stress and strain measures. Numerical analysis of influence factors can be a promising avenue to avoid high-cost post-production inspection, which is intended for mass production.

Arbitrary Lagrangian-Eulerian formulations for cutting pattern generation of tensile membrane structures

ABSTRACT An application of Arbitrary Lagrangian-Eulerian (ALE) method is used to generate the cutting pattern shapes of membrane structures. The initial patterning panels are selected as the projection of three-dimensional (3D) membrane panels and are assumed as the reference configuration. While the Lagrangian displacements define the forced displacements from this configuration to the 3D configuration, the Eulerian terms describe the motion from this reference configuration to the current patterning panels. The cutting pattern panels can be determined by this process considering the effect of prescribed pre-stress and material properties. Several examples are performed to point out the effectiveness of the proposed method on simplifying the cutting pattern process and the reduction in the stress deviation from the intended pre-stress.

Study on the oblique water entry impact performance of AUV under different launch conditions based on coupled FEM-ALE method

In this study, the hydrodynamic response of an autonomous underwater vehicle (AUV) subjected to oblique water entry impact is investigated by employing an explicit FEM technique with arbitrary Lagrangian–Eulerian formulation. The predictive ability of the employed numerical model is validated by correlating the cavity evolution of AUV and peak impact accelerations with the experimental outcomes of the previous research work. The influence of advection algorithms on the water entry phenomenon is studied. It is found that the first order donor cell advection scheme is more suitable for low speed impact problems as it is computationally more efficient. However, for high speed water entry simulations, the van Leer advection scheme gives more accurate results. Moreover, using different water entry angles, the effect of the angle of attack (AOA) and length of AUV and its head shape on the impact performances are investigated and discussed. The results show that the radial impact load is more likely to be influenced by AOA and the effect of AOA on the radial impact load is more significant for a small water entry angle, i.e., 30°, as compared to a large entry angle, i.e., 60°. It is also observed that AUV of a smaller length shows the ricochet behavior at a water entry angle of 30°. It is seen that the head shape has a substantial effect on the axial and radial impact loads and velocity attenuation of a vehicle under water. The conclusions drawn in this research work will be beneficial for the selection of appropriate initial launch conditions and for the designing of an AUV structure.

An energy stable one-field monolithic arbitrary Lagrangian–Eulerian formulation for fluid–structure interaction

In this article we present a one-field monolithic finite element method in the Arbitrary Lagrangian–Eulerian (ALE) formulation for Fluid–Structure Interaction (FSI) problems. The method only solves for one velocity field in the whole FSI domain, and it solves in a monolithic manner so that the fluid solid interface conditions are satisfied automatically. We prove that the proposed scheme is unconditionally stable, through energy analysis, by utilising a conservative formulation and an exact quadrature rule. We implement the algorithm using both F-scheme and d-scheme, and demonstrate that the former has the same formulation in two and three dimensions. Finally several numerical examples are presented to validate this methodology, including combination with remesh techniques to handle the case of very large solid displacement.

A high-order vertex-centered quasi-Lagrangian discontinuous Galerkin method for compressible Euler equations in two-dimensions

This article introduces a new vertex-centered quasi-Lagrangian discontinuous Galerkin method for two-dimensional compressible flows that is third-order accurate both in space and time. The computational domain is divided into structured quadrilateral cells with straight edges. Nodal control volumes with curved edges are constructed surrounding the grid vertices. The Euler equations in arbitrary Lagrangian-Eulerian (ALE) form are discretized on these nodal control volumes using a discontinuous Galerkin method. The time marching is implemented by the third-order strong-stability-preserving Runge-Kutta method. A compact HWENO reconstruction algorithm is used as limiter to eliminate spurious oscillations near discontinuities. The polynomial expression of fluid velocity defined in a nodal control volume is also obtained from the reconstruction procedure, and is used to calculate the moving velocity of corresponding grid vertex. In this way, the grid vertices are moved in a rigorously Lagrangian manner, although there are still mass fluxes between neighbouring nodal control volumes. Therefore the scheme is called as a quasi-Lagrangian one. The scheme is conservative for mass, momentum and total energy. Some numerical tests are carried out to demonstrate the accuracy and robustness of the scheme. It has a favorable qualitative behavior for discontinuous problems and optimal convergence rates for smooth problems.

Numerical Analysis of Shock Initiation on Highly Energetic Material LX-04

The paper presents investigation of shock initiation effects on highly energetic material LX-04, an HMX based one, by a blunt brass projectile using numerical modelling methods implemented in LS-Dyna software. Multi – Material Arbitrary Lagrangian Eulerian (MM-ALE) formulation has been used to provide the possibility of fulfilment of the elements with multiple materials accordingly to multiple phases of the composition detonated. The work is focused on LS-Dyna’s equation of the state keyword *EOS_IGNITON_AND_GROWTH_IN_HE. Proper comparison with experimental data is presented. The introduction into the subject of highly nonlinear transient dynamic finite element analysis, possibilities, and superiority of this kind of modelling are being discussed. Reasons of sustaining the need of data by numerical solutions rather than experimental handling in military applications are given. Some of typical technical issues, occurring during such a fast-performing phenomenon, and the need of experimental validation of the models are typed together with detonation-state determination’s observation capabilities.

Numerical treatment of frictional contact in ALE formulation for disc brake assemblies

The Arbitrary-Lagrangian-Eulerian (ALE) formulation is a popular approach for finite element (FE) models governed by large rigid body movement. It facilitates the decomposition of guiding motions and relative deformation inside the model by considering the main motion of the system in the form of a material flow inside the FE mesh. This way, similar to fluid dynamics, an observer focused on spatial positions rather than individual material particles is introduced. Among others, this approach serves as an established tool in the simulation of tire rolling contact and its dynamics. However, while allowing for specific mesh refinement and simplifying issues such as finite displacements due to rigid body motions, it complicates contact computations. The history of contacting particles is not inherently tracked, demanding a special treatment of their relative displacement for evaluating their frictional contact state. The handling of stationary problems in this context is well-studied for the examples of elastic-rigid rolling contact, including inelastic material properties and thermo-mechanical coupling. This work attends to the remaining area of frictional contact in transient ALE problems and extends the methods towards elastic-elastic contact coupling in a three dimensional framework. To this end, classic Zero-Thickness contact elements, whose application in these surroundings is enabled by the relative kinematic ALE framework in the first place, are extended towards ALE compatibility. The proposed approach involves a slip-based contact formulation including inelastic slip variables and is applied in the field of disc brake assemblies. In here, several stationary and transient studies of emerging contact patterns and limit cycles are accomplished, outlining the capabilities of the proposed implementation.

A scalable framework for the partitioned solution of fluid–structure interaction problems

In this work, we present a scalable and efficient parallel solver for the partitioned solution of fluid–structure interaction problems through multi-code coupling. Two instances of an in-house parallel software, TermoFluids, are used to solve the fluid and the structural sub-problems, coupled together on the interface via the preCICE coupling library. For fluid flow, the Arbitrary Lagrangian–Eulerian form of the Navier–Stokes equations is solved on an unstructured conforming grid using a second-order finite-volume discretization. A parallel dynamic mesh method for unstructured meshes is used to track the moving boundary. For the structural problem, the nonlinear elastodynamics equations are solved on an unstructured grid using a second-order finite-volume method. A semi-implicit FSI coupling method is used which segregates the fluid pressure term and couples it strongly to the structure, while the remaining fluid terms and the geometrical nonlinearities are only loosely coupled. A robust and advanced multi-vector quasi-Newton method is used for the coupling iterations between the solvers. Both the fluid and the structural solver use distributed-memory parallelism. The intra-solver communication required for data update in the solution process is carried out using non-blocking point-to-point communicators. The inter-code communication is fully parallel and point-to-point, avoiding any central communication unit. Inside each single-physics solver, the load is balanced by dividing the computational domain into fairly equal blocks for each process. Additionally, a load balancing model is used at the inter-code level to minimize the overall idle time of the processes. Two practical test cases in the context of hemodynamics are studied, demonstrating the accuracy and computational efficiency of the coupled solver. Strong scalability test results show a parallel efficiency of 83% on 10,080 CPU cores.

EdgeCFD: a parallel residual-based variational multiscale code for multiphysics

High fidelity multiphysics simulations are ubiquitous in science and engineering but still face many challenges to run efficiently in today's supercomputers. This work reports advanced technologies present in EdgeCFD, a hybrid parallel variational multiscale multiphysics finite element software capable of running industrial problems involving turbulent incompressible, compressible, free-surface flows, advection-diffusion of multiple scalar fields, fluid-object interaction using an Arbitrary Lagrangian-Eulerian formulation, and, more recently, used to supporting uncertainty quantification. We also show the strategies to support the solution of large scale problems, including in-situ visualisation and how EdgeCFD handles computational steering. Industrial applications in the oil and gas industry are shown to demonstrate EdgeCFD capabilities. We present the analysis of vortex-induced vibration in cylindrical risers with appendages to reduce drag and lift, and the simulation of successive discharges of turbidity currents, a critical process for understanding the formation of oil reservoirs.

High-order ALE gas-kinetic scheme with WENO reconstruction

In this paper, a high-order multi-dimensional gas-kinetic scheme is presented for both inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation. Compared with the traditional ALE method, the flow variables are updated in the finite volume framework, and the rezoning and remapping steps are not required. The two-stage fourth-order method is used for the temporal discretization, and the second-order gas-kinetic solver is applied for the flux evaluation. In the two-stage method, the spatial reconstruction is performed at the initial and intermediate stages, and the computational meshes are determined by the mesh velocity. In the moving mesh procedure, the mesh may distort severely and the mesh quality is reduced. To achieve the accuracy and improve the robustness, the newly developed WENO method [40] on quadrilateral meshes is adopted at each stage. The Gaussian quadrature is used for flux calculation. For each Gaussian point, the WENO reconstruction is performed in local moving coordinate, where the variation of mesh velocity along cell interface is taken into account. Numerical examples are presented to validate the performance of current scheme, where the mesh adaptation method and the cell centered Lagrangian method are used to provide mesh velocities.

An efficient multibody dynamic model of arresting cable systems based on ALE formulation

Arresting cable systems (ACS) are widely used in aircraft carriers to decelerate an aircraft with high landing velocity in a limited runway length. Considering the complexity of the arresting process, it is extremely challenging to accurately and efficiently predict system dynamic behaviors, such as the arresting distance of aircrafts. In this paper, a comprehensive multibody dynamic model of ACS is proposed, where Arbitrary Lagrangian Eulerian (ALE) formulation is adopted to efficiently simulate the tailhook/pulley-cable moving contact in ACS. In our model, a moving fine mesh of the ALE cable, which tracks the real-time position of the tailhook, is used to accurately capture the contact behaviors between them, while evenly-distributed coarse meshes are used in non-contact areas, thereby greatly reducing DOFs and contact pairs. Additionally, the pulley joint is used to simulate the cable-sheave moving contact, where contact details are completely neglected and the ALE cable node is fixed to the dimensionless sheave point. Finally, the accuracy and efficiency of the proposed method are verified through three numerical tests with correlation to the conventional Lagrangian formulation.

Arbitrary Lagrangian–Eulerian formulation of lattice Boltzmann model for compressible flows on unstructured moving meshes

We propose the application of the arbitrary Lagrangian–Eulerian (ALE) technique to a compressible lattice Boltzmann model for the simulation of moving boundary problems on unstructured meshes. To that end, the kinetic equations are mapped from a moving physical domain into a fixed computational domain. The resulting equations in the computational domain are then numerically solved using the second-order accurate finite element reconstruction on an unstructured mesh. It is shown that the problem regarding the geometric conservation law (GCL), which needs a special treatment in the ALE Navier–Stokes solvers, does not appear here and the model satisfies the GCL exactly. The model is validated with a set of simulations including uniform flow preservation and compressible flow past an airfoil in plunging and pitching motion at different Mach numbers. It is demonstrated that the results are in good agreement with the experimental and other available numerical results in the literature. Finally, in order to show the capability of the proposed solver in simulating high-speed flows, transonic flow over pitching airfoil is investigated. It is shown that the proposed model is able to capture the complex characteristics of this flow, which involves multiple weak shock waves interacting with the boundary and shear layers.

Numerical study of an interlayer effect on explosively welded joints

Explosive welding is a useful method for joining similar or dissimilar metal alloys that are not easily joined by any other means of welding. Insertion of an interlayer between two-parent plates is beneficial to control the welding parameters and improve the welding quality. In order to investigate the importance of interlayer during the explosive welding process, this paper presents the numerical simulation of explosive welding of Ti6Al4V-SS304 in the presence of different interlayer materials, i.e. SS304, Ti6Al4V, CP-Ti, Cu, Al. To understand the whole welding process with efficient computational time, coupled smoothed particle hydrodynamics -Euler -Arbitrary Lagrangian-Eulerian formulation was opted. Results were analyzed on the basis of different material interlayers. Welding parameters, i.e. pressure, temperature, plastic strain and interface morphology revealed that interlayer improved the welding quality. Furthermore, as an interlayer, the materials with high strength showed high welding strength.