Mesh Motion Research Articles

Numerical Simulation of Drying Process within a Novel Rotary Drying Machine for Palm Oil Sludge

This study investigated how to model a centre-offset annular rotary drum using OpenFOAM and the meshing software GMSH. The diameter of the outer cylinder is 600 mm, diameter of the inner cylinder 200 mm, the centre-offset mm, and rotation rate rpm. When the centre-offset is zero, the quality of mesh is preserved as the drum rotates. Introduction of the offset causes the mesh to be deformed to the point of being unusable as the drum is rotated. The reason was found to be the fixed nodes adjacent to the walls of the outer and inner cylinder. These nodes only respect the motion of the wall they are adjacent to. To circumvent this, we separated the inner volume of the rotary drum to allow the implementation of an OpenFOAM dynamic mesh handling scheme called arbitrary mesh interface (AMI). Implementing AMI allows the quality of the mesh to be kept even when the inner cylinder is rotating under nonzero-offset conditions. This is because AMI permits the sliding of non-conforming meshes next to each other. This preserves the quality of the mesh and secures a reliable and reproducible dynamic mesh motion for the implementation of the drying process in the future.

A numerical failure analysis of nano-filled ultra-high-performance fiber-reinforced concrete structures via a moving mesh approach

This work proposes a novel FE-based model for simulating crack propagation mechanisms in Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) structures (typically exhibiting compressive strength over 150 MPa and tensile strength around 10 MPa) enhanced with nanofillers under general loading conditions. This model combines synergistically a Moving Mesh (MM) technique with the Interaction Integral (M-integral) method, and the R-curve concept. In particular, the MM technique, based on the Arbitrary Lagrangian-Eulerian formulation, is used for tracking geometry changes of the computational domain caused by crack advance, thus avoiding massive re-meshing operations. In particular, a mesh regularization technique based on proper rezoning equations is used to ensure that the mesh points move consistently. The M-integral method is used for extracting the crack front variables (i.e., the Stress Intensity Factors (SIFs)), required for the computation of the mixed-mode crack growth direction. Finally, the well-known R-curve concept allows the ductile behavior of nano-enhanced UHPFRC beams to be captured in a LEFM setting. In particular, the crack growth resistance, regarded as a function of the crack extension, is numerically derived via a calibration procedure starting from the available experimental data. In addition, to assess the reliability and effectiveness of the proposed approach, the results of numerical simulations are compared with experimental outcomes reported in the literature.

On the hydrodynamic response and slamming impact of a cylindrical FPSO in combined wave-current flows

Owing to the lower construction and transportation costs, cylindrical floating production storage offloading (CFPSO) has received extensive attention in the industry. Currently, attention has primarily been focused on the damping performance of heave plate while few studies have touched upon hydrodynamic characteristics under complex combined wave-current condition. In this paper, we numerically investigate the hydrodynamic response and slamming impact of a CFPSO under combined wave-current flows. The active wave generating-absorbing boundary condition (GABC) along with the buoyancy-modified k-omega SST turbulence model are utilized to generate high-precision waves and current based on the open source computational fluid dynamics (CFD) framework OpenFOAM. A self-developed six degree of freedom (6DOF) motion module is used for solving rigid body motion and updating mesh motion. The numerical results of motion response and impact pressure are compared with the experimental data to verify the accuracy of present simulations. The correlation between impact pressure and relative wave elevation and wave velocity is analyzed and three types of slamming events have been identified. The flow fields such as vorticity, pressure and streamlines in the vicinity of the CFPSO are also presented and discussed, which provides a reference for structural design for the CFPSO under complex sea conditions.

An arbitrary Lagrangian–Eulerian method for fluid–structure interactions due to underwater explosions

Purpose This study aims to evaluate blast loads on and the response of submerged structures. Design/methodology/approach An arbitrary Lagrangian–Eulerian method is developed to model fluid–structure interaction (FSI) problems of close-in underwater explosions (UNDEX). The “fluid” part provides the loads for the structure considers air, water and high explosive materials. The spatial discretization for the fluid domain is performed with a second-order vertex-based finite volume scheme with a tangent of hyperbola interface capturing technique. The temporal discretization is based on explicit Runge–Kutta methods. The structure is described by a large-deformation Lagrangian formulation and discretized via finite elements. First, one-dimensional test cases are given to show that the numerical method is free of mesh movement effects. Thereafter, three-dimensional FSI problems of close-in UNDEX are studied. Finally, the computation of UNDEX near a ship compartment is performed. Findings The difference in the flow mechanisms between rigid targets and deforming targets is quantified and evaluated. Research limitations/implications Cavitation is modeled only approximately and may require further refinement/modeling. Practical implications The results demonstrate that the proposed numerical method is accurate, robust and versatile for practical use. Social implications Better design of naval infrastructure [such as bridges, ports, etc.]. Originality/value To the best of the authors’ knowledge, this study has been conducted for the first time.

Study on transient performance of tilting-pad thrust bearings in nuclear pump considering fluid-structure interaction

To study the lubrication performance of tilting-pad thrust bearing (TPTBs) during start-up in nuclear pump, a hydrodynamic lubrication model of TPTBs was established based on the computational fluid dynamics (CFD) method and the fluid-structure interaction (FSI) technique. Further, a mesh motion algorithm for the transient calculation of thrust bearings was developed based on the user defined function (UDF).The result demonstrated that minimum film thickness increases first and then decreases with the rotational speed under start-up condition. The influence of pad tilt on minimum film thickness is greater than that of collar movement at low speed, and the establishment of dynamic pressure mainly depends on pad tilt and minimum film thickness increases. As the increase of rotational speed, the influence of pad tilt was abated, where the influence of the moving of the collar dominated gradually, and minimum film thickness decreases. For TPTBs, the circumferential angle of the pad is always greater than the radial angle. When the rotational speed is constant, the change rate of radial angle is greater than that of circumferential angle with the increase of loading forces. This study can provide reference for improving bearing wear resistance.

Convergence of a scheme for an elastic flow with tangential mesh movement

Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of Deckelnick and Dziuk [Math. Comp. 78 (2009) 645–671] we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical results that support the theory.

A semi-implicit slip algorithm for mesh deformation in complex geometries, implemented in OpenFOAM

Many engineering applications of computational fluid dynamics (CFD) comprise extensive movement of objects that necessitate complex dynamic mesh treatments. In particular, the mesh motion process frequently requires a proper slipping of mesh points on highly curved surfaces. The currently available implementation of explicit slip boundary conditions in OpenFOAM fails to allow large deformations of the mesh without severely degrading the mesh quality and inverting some of the cells. Thus, a robust semi-implicit slip algorithm, based on the Laplacian smoothing methodology, is developed in the present work to tackle this issue. The algorithm is in fact performed in two steps, one explicit and one implicit. The OpenFOAM implementation of the algorithm includes different mesh motion solvers and boundary conditions, based on the displacement or velocity of points. The method is first verified using simple, yet relevant, test cases, and it is shown that the developed algorithm significantly outperforms some of the well-known proprietary CFD codes. Then, it is applied to a complex practical CFD case study. An engineering application that requires the features of the developed mesh motion algorithm is the transient operation of Kaplan turbines. These double-regulated machines simultaneously adjust the guide vane and runner blade angles while changing the operating condition. CFD simulations of such transient operations are highly complex, as they involve mesh deformation of the guide vane passage and simultaneous mesh deformation and rigid-body rotation of the runner blade passage. The mesh deformation requires points to slip on the curved hub and shroud surfaces while preserving the cell quality in tiny blade clearances. Therefore, the feasibility of the developed algorithm is evaluated for a load rejection sequence of a Kaplan turbine model. Program summaryProgram Title: Semi-implicit slip mesh motionCPC Library link to program files:https://doi.org/10.17632/wztc26vh7b.1Developer's repository link:https://github.com/salehisaeed/semiImplicitSlipLicensing provisions: GPLv3Programming language: C++Nature of problem: CFD simulations of numerous engineering fluid flows, such as transient operation of hydraulic turbines, involve an immensely complicated mesh motion process consisting of simultaneous mesh deformation and mesh slipping on highly curved surfaces. The available standard mesh motion methodology in OpenFOAM lacks some features to simulate this elaborate mesh motion. The introduced program addresses this problem by developing a new dynamic mesh algorithm.Solution method: The program implements a robust semi-implicit algorithm for slipping the mesh points on curved surfaces. The algorithm includes two steps, namely, an explicit step based on the general slip condition and an implicit step based on the Dirichlet condition. It employs the Laplacian smoothing equations to spread the mesh deformation into the domain. Additionally, a solid-body rotation may be added on top of the deformed mesh, which could be useful for modeling the runner region in transient operation of Kaplan turbines which contains simultaneous mesh deformation and solid-body rotation of the mesh.

Numerical simulation for peristalsis of Quemada fluid: A dynamic mesh approach

Introduction: Flow dynamics due to the peristaltic pumping has been the topic of great interest for the researchers. But numerical and analytical analyses for the peristaltic motion are limited where flow domain is deformed real-time. Research on peristalsis has a limitation where theoretical aspects of walls motion are considered, neglecting the real time deformation of the walls.Objectives: This paper aims to propose a more reliable and accurate numerical methodology for peristaltic motions to address the above-mentioned challenge. Stream traces, velocities, and pressure drops along the tube is to be visualized more accurately.Methods: In present study a finite volume based dynamic mesh motion method is adopted to analyze the peristaltic motion of a non-Newtonian Quemada fluid in an axisymmetric channel. The walls and interior domain of the channel is dynamically deformed for a sinusoidal wave traveling on boundary.Results: Simulation of unsteady flow behavior for time t=0s to 2s and amplitude ratio Φ=0.2,0.4,and0.6. predicts fluid trapping phenomenon. Rotation of fluid particles is more prominent for higher amplitude ratios. Pressure gradient increases with increasing amplitude ratios.Conclusion: A novel dynamic mesh method is proposed for peristaltic pumping. It provides more accurate and more physical results for stream traces; pressure drops and velocities along the tube. A limited case of the study validates the theoretical and analytical results already presented in literature; hence the method is reliable.

Impact of cell-sized confinement on particle transport within a motor-driven cytoskeletal network.

Intracellular transport of small particles and macromolecules is affected by the degree of crowding, confinement, and motor-driven activity within the cell. In vitro studies have examined how transport dynamics depend on each of these factors. But studies have primarily focused on these factors in isolation. Previously, we showed how crowding and confinement synergistically couple to slow the diffusion of DNA molecules. Here, we explore the combined effects of crowding by a composite network of actin and microtubule filaments, activity driven by molecular motors, and confinement by a lipid membrane to cell-sized volumes. We quantify transport of tracer particles within this tunable in vitro environment using differential dynamic microscopy (DDM) and single particle tracking. We simultaneously measure the dynamics and spatial organization of the composite network to link transport dynamics to network properties. To what degree tracers exhibit subdiffusive motion due to caging within the network mesh or superdiffusive motion due to motor activity depends on network composition, motor concentration, and confinement size.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for two‐dimensional compressible flows on adaptive quadrilateral meshes

AbstractIn this article, a discontinuous Galerkin (DG) scheme on the adaptive quadrilateral meshes is proposed to simulate two‐dimensional compressible flows in the direct arbitrary Lagrangian–Eulerian (ALE) framework. In our scheme, the Euler equations are discretized in the reference element with the help of a bilinear map. A kind of Taylor expansion basis functions in the reference element is used to construct the interpolation polynomials of variables. We describe the property that the material derivatives of the basis functions used in the DG discretization are equal to zero, with which the scheme is simplified. Furthermore, the mesh velocity in our ALE framework is obtained by implementing the approach of mesh movement based on the variational principle from [Adaptive mesh methods for one‐ and two‐dimensional hyperbolic conservation laws. SIAM J Numer Anal. 2003;41:487–515]. This approach of mesh movement automatically concentrates the mesh nodes near the regions with large gradient values of the variables and can greatly improve the resolution of the solution near these regions. In addition, a WENO (weighted essentially non‐oscillatory) reconstruction helps our scheme remove the numerical oscillations. Some numerical examples are presented to demonstrate the accuracy, high resolution, and robustness of our scheme.

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

Facilitating Large-Amplitude Motions of Wave Energy Converters in OpenFOAM by a modified Mesh Morphing Approach

High-fidelity simulations using computational fluid dynamics (CFD) for wave-body interaction are becoming increasingly common and important for wave energy converter (WEC) design. The open source finite volume toolbox OpenFOAM is one of the most frequently used platforms for wave energy. There are currently two ways to account for moving bodies in OpenFOAM: (i) mesh morphing, where the mesh deforms around the body; and (ii) an overset mesh method where a separate body mesh moves on top of a background mesh. Mesh morphing is computationally efficient but may introduce highly deformed cells for combinations of large translational and rotational motions. The overset method allows for arbitrarily large body motions and retains the quality of the mesh. However, it comes with a substantial increase in computational cost and possible loss of energy conservation due to the interpolation. In this paper we present a straightforward extension of the spherical linear interpolation (SLERP) based mesh morphing algorithm that increase the stability range of the method. The mesh deformation is allowed to be interpolated independently for different modes of motion, which facilitates tailored mesh motion simulations. The paper details the implementation of the method and evaluates its performance with computational examples of a cylinder with a moonpool. The examples show that the modified mesh morphing approach handles large motions well and provides a cost effective alternative to overset mesh for survival conditions.

Aeroelastic Analysis Using Deforming Cartesian Grids

Ongoing work in air-vehicle design illustrates the potential of advanced concepts to provide significant improvements in efficiency; but with their incorporation of lightweight flexible structures, such configurations may require active control systems to ensure reliability and safety. However, many contemporary analysis methods are inefficient for aeroelastic analysis and design of such configurations. This paper describes the development of a new approach that automates the geometry setup, mesh generation, and assembly of fluid–structural coupling interfaces to enable efficient aeroelastic and aeroservoelastic analysis of advanced concepts. The core elements for this approach are a cut-cell Cartesian grid-based computational fluid dynamics solver, a nonlinear beam element structural model, a conservative fluid–structural interface treatment, and the formulation and implementation of a new deforming grid capability within the cut-cell Cartesian grid solver. Herein, emphasis is on this latter component with detailed description given of the mesh motion strategy, evaluation of fluxes and structural loads at the surface, and computation of geometrical properties such as cell volume, directed face areas, centroids, and motion-induced fluxes for deforming Cartesian grids required to advance the flow states. Aeroelastic simulations exercising the capability show favorable agreement with data and predictions in the literature for subsonic and supersonic applications.

A high order positivity-preserving polynomial projection remapping method

In this paper, we develop a high-order positivity-preserving polynomial projection remapping method based on the L2 projection for the discontinuous Galerkin (DG) scheme. Combined with the Lagrangian type DG scheme and the rezoning strategies, we present an indirect arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method. By clipping precisely the intersections between the old distorted mesh and the new rezoned mesh, our remapping method is high-order accurate and has no limitation for the mesh movements, so it is suitable for the large deformable problems. A positivity-preserving limiter is also added for the physical variables in computational fluid dynamics without losing the original high-order accuracy and conservation. A multi-resolution weighted essentially non-oscillatory (WENO) limiter is adopted to overcome numerical oscillations and it can keep the original high-order accuracy in the smooth region. This WENO limiter combines several different degrees of polynomials which are the local L2 projections of the original polynomial with nonlinear weights calculated by their smoothness, therefore, it is highly parallel efficient. The properties of positivity-preserving, non-oscillation and high-order accuracy of the remapping method will be shown by a variety of numerical experiments on one, two and three dimensional unstructured meshes. The performance of the ALE-DG scheme with rezoning and remapping is also tested for the Euler system in one and two dimensions.

Application of an overset grid method for the performance analysis of flapping airfoils

With the growing importance of renewable energy, the interest in energy-harvesting technologies based on flow-induced vibrations of airfoils has been rekindled in the past few years. Compared to conventional turbines, these devices are centrifugal stress-free and hence they are structurally more robust. They are also environmentally friendly due to the relatively low speeds, thus reducing the impact on flying animals. In order to numerically investigate these types of devices, mesh motion techniques must be used so that the computational grid can adapt to the time-varying shape of the domain and boundaries, thus preserving its robustness and quality. During the last years, many different dynamic mesh approaches have been developed to better predict the behaviour of a flow interacting with solid moving bodies. The aim of the present work is to apply and validate the overset grid method offered by the OpenFOAM software. In this type of mesh, one or more grid blocks are allowed to overlap with other sets of cells to solve the fluid domain with moving bodies, thus showing many potential advantages compared to the other existing dynamic mesh approaches. In the first part of this work, a numerical investigation of the flow over a stationary SD 7003 airfoil at low Reynolds number has been performed using an unsteady RANS approach. The computed results have then been compared to experimental and high-fidelity computational data available from the literature in order to validate the presented model in terms of numerical domain configuration, mesh refinement, and turbulence modelling. Subsequently, unsteady RANS simulations have been performed on the SD 7003 and NACA 0012 airfoils for two different amplitudes of flapping motion, i.e., 30° and 45°. Finally, the results obtained with the OpenFOAM overset grid solver have been compared with the numerical and experimental benchmarks presented by Kurtulus, showing an excellent agreement with both computed and measured data.

A unified approach for the solution of fluid‐solid interaction problems with hyperelastic deformation in internal flows

AbstractIn the single domain method for solving fluid–solid interaction (FSI) problems, a unified formulation is used for the entire computational domain. In such monolithic FSI solvers, all of the governing equations are solved simultaneously. In the present study, the single domain method is further extended to an interface‐tracking FSI solver which accounts for mesh movement via an Arbitrary Lagrangian–Eulerian (ALE) description of the governing equations. The focus is on internal flow problems with large deformation. Pressure and velocity are selected as the dependent variables for both solid and fluid parts of the computational domain. A distinguishing feature of the proposed method is that the governing equations at the interface are discretized in a conservative manner. Interfacial boundary conditions are enforced via a pressure–velocity splitting method to convert the kinematic and dynamic conditions at the interface into pressure–velocity relations. A PISO‐like procedure is used to solve the discretized equations. In order to evaluate the proposed solver, strongly‐coupled FSI benchmark test cases are employed. The performance of the proposed method and computational results are also compared with those obtained by a conventional partitioned solver. The results show that the proposed solver provides more accurate results on a coarser mesh compared to the benchmark solutions. The proposed method is also capable of solving strongly coupled problems for which the partitioned solver fails to converge.

Study on the Nonlinear Dynamic Behavior of Rattling Vibration in Gear Systems

To reveal the nonlinear dynamic behavior of gear rattling vibration caused by gear backlash, a 2-DOF oscillator model with spring and damping elements was established. Based on the theory of discontinuous dynamical systems, the phase plane of gear motion was divided into three parts: the domain of tooth surface meshing motion, the domain of free motion and the domain of tooth back meshing motion. Introducing the global mapping and local mapping dynamics method, the process of gear teeth from impact to meshing and then impact and meshing was accurately described. The influence of different restitution coefficients on gear impact-meshing motion was studied by numerical simulation. The results showed that the grazing bifurcation caused by gear backlash will lead to complex mapping structures of the system and even chaos. The restitution coefficient directly affects the impact-meshing behavior. The introduction of meshing stiffness and restitution coefficient can reasonably characterize the elastic deformation and energy loss during gear meshing, which provides a theoretical model for the application of the theory of discontinuous dynamical systems to a more complex multi-degree of freedom flexible contact gear transmission system.

A high order positivity-preserving conservative WENO remapping method based on a moving mesh solver

In this paper, a high order accurate positivity-preserving conservative remapping algorithm is developed. Quadrilateral meshes in two dimensions are used as examples. This remapping method is based on the numerical solution of the trivial equation ∂u∂t=0 on a moving mesh, which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=T. A high order finite volume scheme on the moving mesh is used to solve this problem. Specifically, we adopt the multi-resolution weighted essentially non-oscillatory (WENO) method for the spatial discretization and a strong stability preserving (SSP) Runge-Kutta method for the temporal discretization. The remapping algorithm is high order accurate under very mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which can always be satisfied with a suitable choice of the final pseudo-time T. Furthermore, we design our remapping algorithm to have positivity-preserving property by using the linear scaling positivity-preserving limiter so that the algorithm could ensure the positivity-preserving property of relevant physical variables and maintain conservation and original order of accuracy. A series of numerical experiments are given to demonstrate the properties of our remapping algorithm such as high order accuracy, essentially non-oscillatory performance, positivity-preserving and high computational efficiency.

Formation of orogenic gold deposits by progressive movement of a fault-fracture mesh through the upper crustal brittle-ductile transition zone

Orogenic gold deposits are comprised of complex quartz vein arrays that form as a result of fluid flow along transcrustal fault zones in active orogenic belts. Mineral precipitation in these deposits occurs under variable pressure conditions, but a mechanism explaining how the pressure regimes evolve through time has not previously been proposed. Here we show that extensional quartz veins at the Garrcon deposit in the Abitibi greenstone belt of Canada preserve petrographic characteristics suggesting that the three recognized paragenetic stages formed within different pressure regimes. The first stage involved the growth of interlocking quartz grains competing for space in fractures held open by hydrothermal fluids at supralithostatic pressures. Subsequent fluid flow at fluctuating pressure conditions caused recrystallization of the vein quartz and the precipitation of sulfide minerals through wall-rock sulfidation, with some of the sulfide minerals containing microscopic gold. These pressure fluctuations between supralithostatic to near-hydrostatic conditions resulted in the post-entrapment modification of the fluid inclusion inventory of the quartz. Late fluid flow occurred at near-hydrostatic conditions and resulted in the formation of fluid inclusions that have not been affected by post-entrapment modification as pressure conditions never returned to supralithostatic conditions. This late fluid flow is interpreted to have formed the texturally late, coarse native gold that occurs along quartz grain boundaries and in open spaces. The systematic evolution of the pressure regimes in orogenic gold deposits such as Garrcon can be explained by relative movement of fault-fracture meshes across the base of the upper crustal brittle-ductile transition zone. We conclude that early vein quartz in orogenic deposits is precipitated at near-lithostatic conditions whereas the paragenetically late gold is introduced at distinctly lower pressure.

Micro-patterned PVDF membranes and magnetically induced membrane vibration system for efficient membrane bioreactor operation

Polyvinylidene difluoride (PVDF) based membranes have been used extensively for wastewater treatment due to their chemical and mechanical stability. However, membrane fouling remains a critical factor limiting the widespread use of membrane processes. To mitigate membrane fouling, patterned PVDF membranes were prepared via spray-modified non-solvent induced phase separation (s-NIPS) which resulted in membranes with large pattern heights of 750 ± 100 μm. Patterned membrane modules were then subjected to magnetically induced membrane vibration (MMV) applied perpendicular to the direction of the pattern to reduce the foulant deposition and delay the irreversible fouling in a submerged membrane bioreactor (MBR). Mere patterning resulted in a 70–300% decrease in membrane fouling rate and a doubled critical flux as compared to the flat counterpart. The introduction of membrane vibration further improved the membrane anti-fouling potential by prevention of foulant deposition through improved hydrodynamic conditions near the membrane surface and efficient removal of foulant deposition. Introducing an axial vibration of 5 Hz frequency dropped the fouling rate by an additional 40–60% and the critical flux for irreversible fouling was almost tripled from 15 L m−2 h−1 for the flat, non-vibrating membrane to 40 L m−2 h−1 for the s-NIPS patterned, vibrating membrane. A computational fluid dynamics (CFD) simulation with mesh motion at the patterned membrane interface confirmed the formation of vortices due to vibration, which are convected away from the membrane surface. The CFD results provided a sound basis for understanding the effect of membrane topography combined with vibration. These results suggest reduced fouling without the need for intense air scouring or frequent cleaning which can result in energy- and cost-effective filtration during MBR operation.