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Walter Boscheri - One of the best experts on this subject based on the ideXlab platform.
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High Order Direct Arbitrary-Lagrangian–Eulerian (ALE) Finite Volume Schemes for Hyperbolic Systems on Unstructured Meshes
Archives of Computational Methods in Engineering, 2017Co-Authors: Walter BoscheriAbstract:In this work we develop a new class of high order accurate Arbitrary-Lagrangian–Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral Meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space–time finite element predictor , which performs a one-step time discretization. Specifically, we adopt a discontinuous Galerkin predictor which can handle stiff source terms that might produce jumps in the local space–time solution. Since we are dealing with moving Meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space–time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space–time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space–time nodes. The computational Mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire Mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous Mesh configuration at any time level, the Mesh motion is evaluated by assigning each node of the computational Mesh with a unique Velocity vector at each timestep. The nodal solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the Mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local Mesh Velocity can be chosen independently from the local fluid Velocity. Once the vertex Velocity and thus the new node location has been determined, the old element configuration at time $$t^n$$ t n is connected with the new one at time $$t^{n+1}$$ t n + 1 with straight edges to represent the local Mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space–time divergence operator and is then integrated on a moving space–time control volume. We adopt a linear parametrization of the space–time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical magneto-hydrodynamics equations and the non-conservative seven-equation Baer–Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Finally we briefly present some variants of the algorithm that aim at improving the overall computational efficiency.
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high order direct arbitrary lagrangian eulerian ale finite volume schemes for hyperbolic systems on unstructured Meshes
Archives of Computational Methods in Engineering, 2017Co-Authors: Walter BoscheriAbstract:In this work we develop a new class of high order accurate Arbitrary-Lagrangian–Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral Meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space–time finite element predictor, which performs a one-step time discretization. Specifically, we adopt a discontinuous Galerkin predictor which can handle stiff source terms that might produce jumps in the local space–time solution. Since we are dealing with moving Meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space–time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space–time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space–time nodes. The computational Mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire Mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous Mesh configuration at any time level, the Mesh motion is evaluated by assigning each node of the computational Mesh with a unique Velocity vector at each timestep. The nodal solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the Mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local Mesh Velocity can be chosen independently from the local fluid Velocity. Once the vertex Velocity and thus the new node location has been determined, the old element configuration at time \(t^n\) is connected with the new one at time \(t^{n+1}\) with straight edges to represent the local Mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space–time divergence operator and is then integrated on a moving space–time control volume. We adopt a linear parametrization of the space–time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical magneto-hydrodynamics equations and the non-conservative seven-equation Baer–Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Finally we briefly present some variants of the algorithm that aim at improving the overall computational efficiency.
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An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics
International Journal for Numerical Methods in Fluids, 2016Co-Authors: Walter BoscheriAbstract:Summary In this paper we present a new family of direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured Meshes in multiple space dimensions. The scheme is designed to be high order accurate both in space and time and the Mesh motion, which provides the new Mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space-time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy [26, 25] that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. According to [19, 20] we rely on an element-local space-time Galerkin finite element predictor on moving curved Meshes to obtain a high order accurate one-step time discretization, while the Mesh Velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the Mesh quality. Next, the old Mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space-time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt the quadrature-free integration proposed in [21], in which the space-time boundaries of the control volumes are split into simplex sub-elements that yield constant space-time normal vectors and Jacobian matrices. In this way the integrals over the simplex sub-elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics (HD) equations) as well as to the magnetohydrodynamics (MHD) equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm [19, 20] that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the three-dimensional case. This article is protected by copyright. All rights reserved.
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High-Order Arbitrary Lagrangian Eulerian One-Step WENO Finite Volume Schemes on Unstructured Meshes for Conservative and Nonconservative Hyperbolic Balance Laws
2014Co-Authors: Walter Boscheri, Michael Dumbser, O. ZanottiAbstract:In this work we present a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) onestep finite volume schemes for the solution of nonlinear systems of conservative and nonconservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral Meshes, respectively. High order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. The entire Mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. We underline that our scheme is supposed to be an ALE algorithm, where the local Mesh Velocity can be chosen independently from the local fluid Velocity. Once the vertex Velocity and thus the new node location has been determined, the old element configuration at time t n is connected with the new one at time t n+1 with straight edges to represent the local Mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. Specifically, the nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving spacetime control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes.
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High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows
arXiv: Numerical Analysis, 2013Co-Authors: Michael Dumbser, Walter BoscheriAbstract:In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a nonlinear WENO reconstruction operator on unstructured triangular Meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the Mesh Velocity can be chosen independently of the fluid Velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.
Michael Dumbser - One of the best experts on this subject based on the ideXlab platform.
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High-Order Arbitrary Lagrangian Eulerian One-Step WENO Finite Volume Schemes on Unstructured Meshes for Conservative and Nonconservative Hyperbolic Balance Laws
2014Co-Authors: Walter Boscheri, Michael Dumbser, O. ZanottiAbstract:In this work we present a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) onestep finite volume schemes for the solution of nonlinear systems of conservative and nonconservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral Meshes, respectively. High order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. The entire Mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. We underline that our scheme is supposed to be an ALE algorithm, where the local Mesh Velocity can be chosen independently from the local fluid Velocity. Once the vertex Velocity and thus the new node location has been determined, the old element configuration at time t n is connected with the new one at time t n+1 with straight edges to represent the local Mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. Specifically, the nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving spacetime control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes.
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High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows
arXiv: Numerical Analysis, 2013Co-Authors: Michael Dumbser, Walter BoscheriAbstract:In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a nonlinear WENO reconstruction operator on unstructured triangular Meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the Mesh Velocity can be chosen independently of the fluid Velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.
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arbitrary lagrangian eulerian one step weno finite volume schemes on unstructured triangular Meshes
Communications in Computational Physics, 2013Co-Authors: Walter Boscheri, Michael DumbserAbstract:In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular Meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular Meshes with straight edges. This is possible in the ALE framework, which allows a local Mesh Velocity that is different from the local fluid Velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.
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High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: Applications to compressible multi-phase flows
Computers & Fluids, 2013Co-Authors: Michael Dumbser, Walter BoscheriAbstract:Abstract In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a non-linear WENO reconstruction operator on unstructured triangular Meshes. High order accuracy in time is obtained via a local Lagrangian space–time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space–time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian–Eulerian (ALE) method, where the Mesh Velocity can be chosen independently of the fluid Velocity. The new scheme is applied to the full seven-equation Baer–Nunziato model of compressible multi-phase flows with relaxation source terms in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer–Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.
Simon Guerdoux - One of the best experts on this subject based on the ideXlab platform.
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A 3D numerical simulation of different phases of friction stir welding
Modelling and Simulation in Materials Science and Engineering, 2009Co-Authors: Simon Guerdoux, L. FourmentAbstract:An adaptive arbitrary Lagrangian–Eulerian formulation is developed to compute the material flow and the temperature evolution during the three phases of the friction stir welding (FSW) process. It follows a splitting approach: after the calculations of the Velocity/pressure and temperature fields, the Mesh Velocity is derived from the domain boundary evolution and from an adaptive refinement criterion provided by error estimation, and finally state variables are remapped. In this way, the unilateral contact conditions between the plate and the tool are accurately taken into account, so allowing one to model various instabilities that may occur during the process, such as the role played by the plunge depth of the tool on the formations of flashes, the possible appearance of non-steady voids or tunnel holes and the influence of the threads on the material flow, the temperature field and the welding efforts. This formulation is implemented in the 3D Forge3 FE software with automatic reMeshing. The non-steady phases of FSW can so be simulated, as well as the steady welding phase. The study of different process conditions shows that the main phenomena taking place during FSW can be simulated with the right sensitivities.
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3D numerical simulation of the three stages of Friction Stir Welding based on friction parameters calibration
International Journal of Material Forming, 2008Co-Authors: L. Fourment, Simon GuerdouxAbstract:International audienceAn Arbitrary Lagrangian Eulerian (ALE) formulation was developed to simulate the different stages of the Friction Stir Welding (FSW) process with the FORGE3® F.E. software. A splitting method was utilized: a) the material Velocity/pressure and temperature fields are calculated, b) the Mesh Velocity is derived from the domain boundary evolution and an adaptive refinement criterion provided by error estimation, c) P1 and P0 variables are remapped. The proposed ALE formulation is applied to FSW simulation. Steady state welding, but also transient phases are simulated, showing good robustness and accuracy of the developed formulation. Friction parameters are identified for an Eulerian steady state simulation by comparison with experimental results. Simulations of the transient plunge and welding phases help to better understand the deposition process that occurs at the trailing edge of the probe, and in particular possible void formation. Flexibility and robustness of the model allows investigating the influence of threads and tooling designs
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3D numerical simulation of the three stages of Friction Stir Welding based on friction parameters calibration
International Journal of Material Forming, 2008Co-Authors: Lionel Fourment, Simon GuerdouxAbstract:An Arbitrary Lagrangian Eulerian (ALE) formulation was developed to simulate the different stages of the Friction Stir Welding (FSW) process with the FORGE3® F.E. software. A splitting method was utilized: a) the material Velocity/pressure and temperature fields are calculated, b) the Mesh Velocity is derived from the domain boundary evolution and an adaptive refinement criterion provided by error estimation, c) P1 and P0 variables are remapped. The proposed ALE formulation is applied to FSW simulation. Steady state welding, but also transient phases are simulated, showing good robustness and accuracy of the developed formulation. Friction parameters are identified for an Eulerian steady state simulation by comparison with experimental results. Simulations of the transient plunge and welding phases help to better understand the deposition process that occurs at the trailing edge of the probe, and in particular possible void formation. Flexibility and robustness of the model allows investigating the influence of threads and tooling designs.
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3D numerical simulation of the three stages of Friction Stir Welding based on friction parameters calibration
International Journal of Material Forming, 2008Co-Authors: Lionel Fourment, Simon GuerdouxAbstract:An Arbitrary Lagrangian Eulerian (ALE) formulation was developed to simulate the different stages of the Friction Stir Welding (FSW) process with the FORGE3® F.E. software. A splitting method was utilized: a) the material Velocity/pressure and temperature fields are calculated, b) the Mesh Velocity is derived from the domain boundary evolution and an adaptive refinement criterion provided by error estimation, c) P1 and P0 variables are remapped. The proposed ALE formulation is applied to FSW simulation. Steady state welding, but also transient phases are simulated, showing good robustness and accuracy of the developed formulation. Friction parameters are identified for an Eulerian steady state simulation by comparison with experimental results. Simulations of the transient plunge and welding phases help to better understand the deposition process that occurs at the trailing edge of the probe, and in particular possible void formation. Flexibility and robustness of the model allows investigating the influence of threads and tooling designs.
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Numerical simulation of the friction stir welding process
2007Co-Authors: Simon GuerdouxAbstract:This work presents the development of a numerical tool. An Arbitrary Lagrangian Eulerian (ALE) formulation is implemented in the 3D FORGE3® F.E. software to simulate the different stages of the Friction Stir Welding (FSW) process. A splitting method is utilized:a) the material Velocity/pressure and temperature fields are calculated, b) the Mesh Velocity is derived from the domain boundary evolution and an adaptive refinement criterion provided by error estimation, c) nodal and P0 variables are remapped. Different Velocity computations and remap techniques are investigated, providing significant advantages with respect to more standard approaches. Improvement is also brought to the contact algorithm through a tool smoothing procedure. These proposed enhancements have been tested and applied on industrial cases.Steady state welding, but also transient welding phases are simulated, exhibiting good robustness and accuracy of the developed ALE formulation. On the first hand, friction parameters are identified using Eulerian steady welding state simulations by comparison with experimental results. On the second hand, one major interest of the ALE model being the possibility to simulate void formation at the tool/workpiece interface, the transient plunge and welding phases are modeled. Their simulations can thus help to better understand the mechanisms of the deposition process that occurs at the trailing edge of the probe in order to obtain sound and defect-free welds. Finally, the flexibility and robustness of the model allows the investigation of new tooling designs influence in the deposition process.
Dominique Pelletier - One of the best experts on this subject based on the ideXlab platform.
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High‐order time integrators for front‐tracking finite‐element analysis of viscous free‐surface flows
International Journal for Numerical Methods in Fluids, 2015Co-Authors: L. Charlot, Dominique Pelletier, Stephane Etienne, Alexander Hay, André GaronAbstract:Summary This paper proposes implicit Runge–Kutta (IRK) time integrators to improve the accuracy of a front-tracking finite-element method for viscous free-surface flow predictions. In the front-tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian–Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second-order accurate time integrators at best. In this paper, we present a finite-element ALE formulation in which a consistent evaluation of the Mesh Velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high-order fixed Mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L-stable IRK time integrators for the incompressible Navier–Stokes equations to yield high-order time-accurate free-surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth-order accuracy in time. The proposed free-surface front-tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright © 2015 John Wiley & Sons, Ltd.
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Verication of a Free Surface Adaptive Finite Element Solution Algorithm
49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2011Co-Authors: Lise Charlot, Stephane Etienne, Dominique PelletierAbstract:This paper presents a verification of an adaptive finite element algorithm for the computation of free surface flows. We use an arbitrary Lagrangian Eulerian description for the computation of the free surface. Hence, the Mesh follows the deformations of the free surface. A pseudo-solid approach is used to compute the Mesh deformations. Special attention must be payed to the variational formulation of the Navier-Stokes equations and to the computation of the Mesh Velocity to satisfy the Geometric Conservation Law (GCL). The GCL ensures that the fixed Mesh order of accuracy of the time integrator is preserved on moving Meshes. Thus a 3 rd order accurate implicit Runge Kutta scheme has been used. The implementation of the free surface boundary condition is then verified with the method of the manufactured solution. The solution is computed with an adaptive finite element code. An analytical solution of the free surface and Navier-Stokes equations is developed. It is used as a reference to verify the spatial and temporal convergence rates of the method. Other test cases are also presented.
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Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow
Journal of Computational Physics, 2009Co-Authors: S. ítienne, André Garon, Dominique PelletierAbstract:This paper takes a fresh look at the geometric conservation law (GCL) from the perspective of the finite element method (FEM) for incompressible flows. The GCL arises naturally in the context of Arbitrary Lagrangian Eulerian (ALE) formulations for solving problems on deforming domains. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed Mesh accuracy when applied to deforming Meshes. A review of the literature shows that while many authors use an ALE FEM, few of them discuss the GCL issues. We show how a fixed Mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming Meshes and preserve its fixed Mesh high order temporal accuracy. An appropriate construction of the divergence of the Mesh Velocity guarantees GCL compliance while a separate construction of the Mesh Velocity itself allows the time-integrator to deliver its fixed Mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. It also explains why high order temporal accuracy is achieved with a conservative formulation of the incompressible Navier-Stokes equations, while only first order time accuracy is observed with the non-conservative formulation and all time-integrators investigated here. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming Mesh. In all cases studied, the proposed reconstructions of the Mesh Velocity and its divergence for the conservative formulation lead to optimal time accuracy on deforming grids.
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Geometric Conservation Law and Finite Element Methods for 3D unsteady Simulations of Incompressible Flow
46th AIAA Aerospace Sciences Meeting and Exhibit, 2008Co-Authors: Dominique PelletierAbstract:´This paper takes a fresh look at the Geometric Conservation Law (GCL) from the perspective of Arbitrary Lagrangian Eulerian (ALE) finite element methods for solving 3-D incompressible viscous flows problems on deforming domain. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed Mesh accuracy when applied to deforming Meshes. We show how a fixed Mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming Meshes and preserve its fixed Mesh high order temporal accuracy. An appropriate construction of the divergence of the Mesh Velocity guarantees GCL compliance while a separate construction of the Mesh Velocity itself allows the time-integrator to deliver its fixed Mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming Mesh. In all cases studied, the proposed reconstructions of the Mesh Velocity and its divergence lead to optimal time accuracy on deforming grids.
Seonho Cho - One of the best experts on this subject based on the ideXlab platform.
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adjoint shape design sensitivity analysis of fluid solid interactions using concurrent Mesh Velocity in ale formulation
Finite Elements in Analysis and Design, 2014Co-Authors: Hong-lae Jang, Seonho ChoAbstract:Abstract A coupled variational equation for fluid–solid interaction (FSI) problems is derived using a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, a traction continuity condition at interfaces, and a pseudo-equilibrium equation for Mesh Velocity. The moving boundary in arbitrary Lagrangian–Eulerian (ALE) formulation is included in the variational equations by the Mesh Velocity obtained from a displacement-loaded pseudo-structural problem at a concurrent configuration, which eventually facilitates to derive shape design sensitivity. A continuum-based adjoint shape sensitivity is derived under ALE formulation, which turns out to be very accurate and efficient due to the utilization of converged tangent and the linearity of both adjoint and sensitivity equations. Through numerical examples, the obtained sensitivity is verified in terms of accuracy and efficiency compared with finite difference sensitivity and further applied to the shape optimization problem of finding a stiff structure while satisfying a volume constraint.
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Adjoint shape design sensitivity analysis of fluid–solid interactions using concurrent Mesh Velocity in ALE formulation
Finite Elements in Analysis and Design, 2014Co-Authors: Hong-lae Jang, Seonho ChoAbstract:Abstract A coupled variational equation for fluid–solid interaction (FSI) problems is derived using a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, a traction continuity condition at interfaces, and a pseudo-equilibrium equation for Mesh Velocity. The moving boundary in arbitrary Lagrangian–Eulerian (ALE) formulation is included in the variational equations by the Mesh Velocity obtained from a displacement-loaded pseudo-structural problem at a concurrent configuration, which eventually facilitates to derive shape design sensitivity. A continuum-based adjoint shape sensitivity is derived under ALE formulation, which turns out to be very accurate and efficient due to the utilization of converged tangent and the linearity of both adjoint and sensitivity equations. Through numerical examples, the obtained sensitivity is verified in terms of accuracy and efficiency compared with finite difference sensitivity and further applied to the shape optimization problem of finding a stiff structure while satisfying a volume constraint.