Arbitrary Lagrangian Eulerian Formulation Research Articles

Numerical Modeling of Cone Penetration Test in Slightly Overconsolidated Clay with Arbitrary Lagrangian-Eulerian Formulation

In this paper the results of the cone penetration test (CPT) modeling with the arbitrary Lagrangian-Eulerian (ALE) formulation provided by Abaqus software package have been presented. The study compares the cone resistance and sleeve friction obtained in numerical analysis with values measured in soundings performed in the uniform layer of clayey soil in the Koszalin area. The clay layer was found to be slightly overconsolidated with OCR ranging from 3.5 to 4.5. The subsoil parameters used in the numerical model are based on laboratory tests data and the complementary CPT estimation. Evaluation of the most important factors influencing the numerical solution such as friction on the probe-soil interface and the undrained shear strength of the clay have been discussed. The possibilities of ALE method for soil parameters calibration are introduced and future challenges in large deformation problems modeling due to penetration issues are described.

Finite volume - space-time discontinuous Galerkin method for the numerical simulation of compressible turbulent flow in time dependent domains

The article is concerned with the numerical simulation of the compressible turbulent flow in time dependent domains. The mathematical model of flow is represented by the system of non-stationary Reynolds- Averaged Navier-Stokes (RANS) equations. The motion of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the RANS equations. This RANS system is equipped with two-equation k − ω turbulence model. These two systems of equations are solved separately. Discretization of the RANS system is carried out by the space-time discontinuous Galerkin method which is based on piecewise polynomial discontinuous approximation of the sought solution in space and in time. Discretization of the two-equation k − ω turbulence model is carried out by the implicit finite volume method, which is based on piecewise constant approximation of the sought solution. We present some numerical experiments to demonstrate the applicability of the method using own-developed code.

Numerical approximations of flow induced vibrations of vocal folds

The paper focus on mathematical modelling of incompressible fluid flow interacting with vibrations of an elastic vocal fold. The flow in moving domain is modelled by the incompressible Navier-Stokes equations written in the Arbitrary Lagrangian-Eulerian (ALE) form. The channel geometry is an approximation of the human glottal region. The flow model is coupled with a simplified structure model. The problem is mathematically described and the resulting fluid-structure interaction problem is discretized by a stabilized finite element method. A strong coupling algorithm is applied for solution of the coupled fluid-structure problem. The choice of boundary conditions is discussed, particularly the choice of different artificial inlet/outlet boundary conditions is described in details. The numerical results are shown.

Two-dimensional simulation by regularization of free surface viscoplastic flows with Drucker–Prager yield stress and application to granular collapse

This work is devoted to numerical modeling and simulation of granular flows relevant to geophysical flows such as avalanches and debris flows. We consider an incompressible viscoplastic fluid, described by a rheology with pressure-dependent yield stress, in a 2D setting with a free surface. We implement a regularization method to deal with the singularity of the rheological law, using a mixed finite element approximation of the momentum and incompressibility equations, and an arbitrary Lagrangian Eulerian (ALE) formulation for the displacement of the domain. The free surface is evolved by taking care of its deposition onto the bottom and of preventing it from folding over itself. Several tests are performed to assess the efficiency of our method. The first test is dedicated to verify its accuracy and cost on a one-dimensional simple shear plug flow. On this configuration we setup rules for the choice of the numerical parameters. The second test aims to compare the results of our numerical method to those predicted by an augmented Lagrangian formulation in the case of the collapse and spreading of a granular column over a horizontal rigid bed. Finally we show the reliability of our method by comparing numerical predictions to data from experiments of granular collapse of both trapezoidal and rectangular columns over horizontal rigid or erodible granular bed made of the same material. We compare the evolution of the free surface, the velocity profiles, and the static–flowing interface. The results show the ability of our method to deal numerically with the front behavior of granular collapses over an erodible bed.

Simulation of 2D fluid–structure interaction in inviscid compressible flows using a cell-vertex central difference finite volume method

In the present study, the applicability and accuracy of a cell-vertex finite volume method developed are assessed in simulating 2D fluid–structure interaction in inviscid compressible flows where the nonlinear phenomena exist in both the unsteady transonic fluid flows and the large nonlinear deformation of solid structures. The unsteady Euler equations are considered as the governing equations of the fluid flow in the arbitrary Lagrangian–Eulerian form and the large nonlinear deformation of the solid structure is considered to be governed by the Cauchy equations in the total Lagrangian form. Both the domains are discretized by a second-order central-difference cell-vertex finite volume method in space on arbitrary quadrilateral grids and a second-order implicit method in time. For each time step, the resulting set of coupled implicit nonlinear equations in either domains is solved iteratively in the pseudo-time. The fluidic and structural parts are coupled in a partitioned manner for the simulation of fluid–structure interaction problems where at the interface the grids of the two domains are coincident. The formulation for the structural dynamics applied here can accurately simulate different types of boundary conditions. It can also properly model the panel thickness and therefore the stress distribution within the structure can be reasonably obtained. Some benchmark test cases are studied and the results obtained are compared with the available experimental/numerical results. Simplicity of the implementation, similarity of the discretization and accuracy of the solution demonstrate the capability and robustness of the solution methodology proposed in simulating fluid–structure interaction problems in compressible flows.

A consistent implementation for inelastic materials in an ALE formulation for steady state rolling contact

AbstractThe numerical analysis of rolling contact for rubber materials is a challenging task, especially due to the many nonlinearities inherent to the material, large deformations, friction, and energy dissipation, among others. Industrial applications can be found in ball bearings, rollers, and most commonly in tires of vehicles, applications where reliable numerical simulations lead to the improvement of durability, performance and safety. While a transient analysis stands as a practical and powerful tool for the simulation of rotating bodies, the large amount of computational resources required represents its biggest disadvantage. An alternative frequently used lays in a steady state simulation by means of an Arbitrary Lagrangian Eulerian (ALE) formulation, where the rotational velocity and axial loads are assumed to remain constant. Within this framework, the reference configuration is neither attached to the material particles nor fixed in space and special attention should be paid to the history variables of inelastic materials. In this work, a viscoelastic material model is implemented in an in‐house finite element code, based on a generalized Maxwell model. The implementation takes into consideration the contribution of all elements connected in circumferential direction and a consistent linearization is made for each of them, leading to an assembled stiffness matrix with more non‐zero values than a standard one. This approach is combined with smeared reinforcement embedded in base elements. The reinforcing layers are described by a hyperelastic material model, providing additional advantages for the modeling and simulation of reinforced rollers and tires. Numerical results for different examples show the capabilities of this implementation and the efficiency of the numerical algorithms is discussed. Important remarks and an outlook for further research concludes this presentation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interaction between a simplified soft palate and compressible viscous flow

Fluid–structure interaction in a simplified 2D model of the upper airways is simulated to study flow-induced oscillation of the soft palate in the pharynx. The goal of our research has been a better understanding of the mechanisms of the Obstructive Sleep Apnea Syndrome and snoring by taking into account compressible viscous flow. The inspiratory airflow is described by the 2D compressible Navier–Stokes equations, and the soft palate is modeled as a flexible plate by the linearized Euler–Bernoulli thin beam theory. Fluid–structure interaction is handled by the arbitrary Lagrangian–Eulerian formulation. The fluid flow is computed by utilizing 4th order accurate summation by parts difference operators and the 4th order accurate classical Runge–Kutta method which lead to very accurate simulation results. The motion of the cantilevered plate is solved numerically by employing the Newmark time integration method. The numerical schemes for the structure are verified by comparing the computed frequencies of plate oscillation with the associated second mode eigenfrequency in vacuum. Vortex dynamics is assessed for the coupled fluid–structure system when both airways are open and when one airway is closed. The effect of mass ratio, rigidity and damping coefficient of the plate on the oscillatory behavior is investigated. An acoustic analysis is carried out to characterize the acoustic wave propagation induced by the plate oscillation. It is observed that the acoustic wave corresponding to the quarter wave mode along the length of the duct is the dominant frequency. However, the frequency of the plate oscillation is recognizable in the acoustic pressure when reducing the amplitude of the quarter wave mode.

An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems

The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high-order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary Lagrangian–Eulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit Runge–Kutta scheme is used for the temporal discretization. Quantities of interest that take the form of space–time integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. These quantities of interest and their gradients are used in the context of gradient-based PDE-constrained optimization.The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible Navier–Stokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradients computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration.

A multi-dimensional finite volume cell-centered direct ALE solver for hydrodynamics

In this paper we describe a second order multi-dimensional scheme, belonging to the class of direct Arbitrary Lagrangian–Eulerian (ALE) methods, for the solution of non-linear hyperbolic systems of conservation law. The scheme is constructed upon a cell-centered explicit Lagrangian solver completed with an edge-based upwinded formulation of the numerical fluxes, computed from the MUSCL-Hancock method, to obtain a full ALE formulation. Numerical fluxes depend on nodal grid velocities which are either set or computed to avoid most of the mesh problems typically encountered in purely Lagrangian simulations. In order to assess the robustness of the scheme, most results proposed in this paper have been obtained by computing the grid velocities as a fraction of the Lagrangian nodal velocities, the ratio being set before running the test case. The last part of the paper describes preliminary results about the triple point test case run in the ALE framework by computing the grid velocities with the fully adaptive Large Eddy Limitation (L.E.L.) method proposed in [1]. Such a method automatically computes the grid velocities at each node defining the mesh from the local characteristics of the flow. We eventually discuss the advantages and the drawback of the coupling.

Finite Element Modelling and Residual Stress Prediction in End Milling of Ti6Al4Valloy

Titanium and its alloys are materials that exhibit unique combination of mechanical and physical properties that enable their usage in various fields. In spite of having a lot of advantages, their usage is limited because they are difficult to machine due to their inherent properties of high specific heat capacity, reactivity with tool and low thermal conductivity thereby causing excessive tool wear. To facilitate the process of machining, it becomes necessary to find out and relieve the residual stress caused during machining. Since experiments cannot be performed for each instance, creation of an FE model is desirable. In this paper a finite element analysis (FEA) of the machining of Ti6Al4V for different cutting speeds is presented. A 3D finite element model is developed with the Titanium alloy (Ti6Al4V) as the workpiece and a four flute carbide tip end mill cutter as the tool to predict the residual stress developed within the titanium alloy after machining. The finite element model utilises the Johnson-Cook model to depict the plasticity and the damage criteria and implements the Arbitrary Lagrangian Eulerian (ALE) formulation to increase the accuracy of the model. The FE model has been developed and the findings are presented. The results indicate that residual stresses are maximum at the surface and decrease linearly along the depth and increase as the cutting speed and depth of cut are increased.

Forced pitch motion of wind turbines

The possibility of a wind turbine entering vortex ring state during pitching oscillations is explored in this paper. The aerodynamic performance of the rotor was computed using the Helicopter Multi-Block flow solver. This code solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. A 10-MW wind turbine was put to perform yawing and pitching oscillations suggesting the partial vortex ring state during pitching motion. The results also show the strong effect of the frequency and amplitude of oscillations on the wind turbine performance.

A Coupled Floating Offshore Wind Turbine Analysis with High-fidelity Methods

This paper presents results of numerical computations for floating off-shore wind turbines using, as an example, a machine of 10-MW rated power. The hydrodynamic loads on the support platform are computed using the Smoothed Particle Hydrodynamics method, which is mesh-free and represents the water and floating structures as a set of particles. The aerodynamic loads on the rotor are computed using the Helicopter Multi-Block flow solver. The method solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. The loosely coupled algorithm used in this work is described in detail and the obtained results are presented.

Computational comparison of aortic root stresses in presence of stentless and stented aortic valve bio-prostheses

We provide a computational comparison of the performance of stentless and stented aortic prostheses, in terms of aortic root displacements and internal stresses. To this aim, we consider three real patients; for each of them, we draw the two prostheses configurations, which are characterized by different mechanical properties and we also consider the native configuration. For each of these scenarios, we solve the fluid–structure interaction problem arising between blood and aortic root, through Finite Elements. In particular, the Arbitrary Lagrangian–Eulerian formulation is used for the numerical solution of the fluid-dynamic equations and a hyperelastic material model is adopted to predict the mechanical response of the aortic wall and the two prostheses. The computational results are analyzed in terms of aortic flow, internal wall stresses and aortic wall/prosthesis displacements; a quantitative comparison of the mechanical behavior of the three scenarios is reported. The numerical results highlight a good agreement between stentless and native displacements and internal wall stresses, whereas higher/non-physiological stresses are found for the stented case.

A Consistent Implementation of Inelastic Material Models into Steady State Rolling

ABSTRACT Rolling contact is an important aspect in tire design, and reliable numerical simulations are required in order to improve the tire layout, performance, and safety. This includes the consideration of as many significant characteristics of the materials as possible. An example is found in the nonlinear and inelastic properties of the rubber compounds. For numerical simulations of tires, steady state rolling is an efficient alternative to standard transient analyses, and this work makes use of an Arbitrary Lagrangian Eulerian (ALE) formulation for the computation of the inertia contribution. Since the reference configuration is neither attached to the material nor fixed in space, handling history variables of inelastic materials becomes a complex task. A standard viscoelastic material approach is implemented. In the inelastic steady state rolling case, one location in the cross-section depends on all material locations on its circumferential ring. A consistent linearization is formulated taking into account the contribution of all finite elements connected in the hoop direction. As an outcome of this approach, the number of nonzero values in the general stiffness matrix increases, producing a more populated matrix that has to be solved. This implementation is done in the commercial finite element code ANSYS. Numerical results confirm the reliability and capabilities of the linearization for the steady state viscoelastic material formulation. A discussion on the results obtained, important remarks, and an outlook on further research conclude this work.

Local–global strategy for the prediction of residual stresses in FSW processes

This work describes the local–global strategy proposed for the computation of residual stresses in friction stir welding (FSW) processes. A coupling strategy between the analysis of the process zone nearby the pin tool (local level analysis) and the simulation carried out for the entire structure to be welded (global level analysis) is implemented to accurately predict the temperature histories and, thereby, the residual stresses in FSW. As a first step, the local problem solves the material stirring as well as the heat generation induced by the pin and shoulder rotation at the heat affected zone. The Arbitrary Lagrangian Eulerian (ALE) formulation is adopted to deal with the rotation of complex pin shapes. A thermo-rigid-viscoplastic constitutive law is employed to characterize the viscous flow of the material, driven by the high-strain rates induced by the FSW process. A mixed temperature–velocity–pressure finite element technology is used to deal with the isochoric nature of the strains. The output of this local analysis is the heat generated either by plastic dissipation or by friction, and it is used as the power input for the welding analysis at structural (global) level. The global problem is tackled within the Lagrangian framework together with a thermo-elasto-viscoplastic constitutive model. In addition, in this case, the mixed temperature–displacement–pressure format is introduced to deal with the deviatoric nature of the plastic strains. The outcomes of this analysis are the distortions and the residual stresses after welding. The material used in this work is stainless steel 304 L; however, the methodology presented is applicable to a wide range of materials. The proposed numerical strategy is validated by the experimental evidence.

A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes

We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results.

Optimization Strategy for a Shrouded Turbine Blade Using Variable-Complexity Modeling Methodology

This paper presents the development of a collaborative optimization framework in combination with a variable-complexity modeling technique for the multidisciplinary coupling analysis and design of a shrouded turbine blade. The multidisciplinary optimization design of the shrouded turbine blade involves a high-fidelity detailed computational model and medium-fidelity models, which can become prohibitively expensive. In this investigation, a variable-complexity modeling methodology is introduced, where low-fidelity models and a scaling function are used to approximate the medium- and high-fidelity models through the optimizers in an inner-loop optimization to reduce computational expense. The optimization framework developed includes the collaborative optimization process, parametric modeling of the shrouded turbine blade, fluid–structure interaction solver using arbitrary Lagrangian–Eulerian formulation, an adaptive hexahedral structure mesh generator by establishing virtual blocks and parametric fixed points, and a variable-complexity modeling method combining the multiplicative and additive corrections to manage three levels of fidelity models. On the shrouded turbine-blade design problem, it achieves a feasible optimizer only calling nine high-fidelity analyses. Response surface model variation and cross-validation tests are performed to verify the predictive power of the response surface model in the multidisciplinary design optimization process of the shrouded turbine blade.

Toward design optimization of a Pelton turbine runner

The objective of the present paper is to propose a strategy to optimize the performance of a Pelton runner based on a parametric model of the bucket geometry, massive particle based numerical simulations and advanced optimization strategies to reduce the dimension of the design problem. The parametric model of the Pelton bucket is based on four bicubic Bezier patches and the number of free parameters is reduced to 21. The numerical simulations are performed using the finite volume particle method, which benefits from a conservative, consistent, arbitrary Lagrangian Eulerian formulation. The resulting design problem is of High-dimension with Expensive Black-box (HEB) performance function. In order to tackle the HEB problem, a preliminary exploration is performed using 2'000 sampled runners geometry provided by a Halton sequence. A cubic multivariate adaptive regression spline surrogate model is built according to the simulated performance of these runners. Moreover, an original clustering approach is proposed to decompose the design problem into four sub-problems of smaller dimensions that can be addressed with more conventional optimization techniques.

Arbitrary Lagrangian Eulerian Formulation for Sloshing Tank Analysis in Nuclear Engineering

Fluid-structure interaction plays an important role in nuclear engineering design, where several numerical and experimental tests need to be performed on new tank design before getting into the production process. The design can be performed for fluid storage tanks that require knowledge of sloshing frequencies and hydrodynamic pressure distribution on the structure. These can be very useful for engineers and designers to define appropriate material properties and shell thickness of the structure to be resistant under seismic loading. Data presented in current tank seismic design codes such as Eurocode are based on simplified assumptions for the geometry and material tank properties. Fuel tanks may undergo different types of loading, including seismic loading, where the behavior of storage tanks includes material nonlinearities, which are caused by material yielding. The Arbitrary Lagrangian Eulerian formulation based on finite element analysis presented in the paper takes into account material properties of the structure as well as the complex geometry of the tank. The formulation uses a moving mesh with a mesh velocity defined through the structure motion. In this paper, we use different approaches to solve a fluid-structure coupling problem. The first one uses the full Navier-Stokes equation for the fluid with projection method, and the second approach uses potential flow theory. The problem consists of a sloshing deformable tank submitted to acceleration loading.

A coupled meshfree-mesh-based solution scheme on hybrid grid for flow-induced vibrations

In this paper, a coupled meshfree-mesh-based fluid solver is employed for flow-induced vibration problems. The fluid domain comprises of a hybrid grid which is formed by generating a body conformal meshfree nodal cloud around the solid object and a static Cartesian grid which surrounds the meshfree cloud in the far field. The meshfree nodal cloud provides flexibility in dealing with solid motion by moving and morphing along with the solid boundary without necessitating re-meshing. The Cartesian grid, on the other hand, provides improved performance by allowing the use of a computationally efficient mesh-based method. The flow equations, in arbitrary Lagrangian–Eulerian formulation, are solved by a local radial basis function in finite difference mode on moving meshfree nodes. Conventional finite differencing is used over the static Cartesian grid for flow equations in Eulerian formulation. The equations for solid motion are solved using a classic Runge–Kutta method. Closed coupling is introduced between fluid and structural solvers by using a sub-iterative prediction–correction algorithm. In order to reduce computational overhead due to sub-iterations, only near-field flow (in the meshfree zone) is solved during the inner iterations. The full fluid domain is solved during outer (time step) iterations only when the convergence at the solid–fluid interface has already been reached. In the meshfree zone, adaptive sizing of the influence domain is introduced to maintain suitable number of neighbouring particles. The use of a hybrid grid has been found to be useful in improving the computational performance by faster computing over the Cartesian grid as well as by reducing the number of computations in the fluid domain during fluid–solid coupling. The solution scheme was tested for problems relating to flow-induced cylindrical and airfoil vibration with one and two degrees of freedom. The results are found to be in good agreement with previous work and experimental results.