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Stefano Rebay - One of the best experts on this subject based on the ideXlab platform.
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modified extended bdf scheme for the discontinuous Galerkin Solution of unsteady compressible flows
International Journal for Numerical Methods in Fluids, 2014Co-Authors: Alessandra Nigro, Stefano Rebay, Antonio Ghidoni, Francesco BassiAbstract:SUMMARY In this paper, a high-order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the Solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second-order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG-MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade. Copyright © 2014 John Wiley & Sons, Ltd.
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spectral p multigrid discontinuous Galerkin Solution of the navier stokes equations
International Journal for Numerical Methods in Fluids, 2011Co-Authors: Francesco Bassi, N. Franchina, Antonio Ghidoni, Stefano RebayAbstract:Discontinuous Galerkin (DG) methods are very well suited for the construction of very high-order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high-order spectral element DG approximation of the Navier–Stokes equations coupled with a p-multigrid Solution strategy based on a semi-implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the Solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p-multigrid scheme with non-spectral elements and a spectral element DG approach with an implicit time integration scheme.
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optimal runge kutta smoothers for the p multigrid discontinuous Galerkin Solution of the 1d euler equations
Journal of Computational Physics, 2011Co-Authors: Francesco Bassi, Antonio Ghidoni, Stefano RebayAbstract:This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical Solution of the steady state Solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid Solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the Solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.
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Spectral p-multigrid discontinuous Galerkin Solution of the Navier-Stokes equations
International Journal for Numerical Methods in Fluids, 2011Co-Authors: Francesco Bassi, N. Franchina, Antonio Ghidoni, Stefano RebayAbstract:This work presents a family of original Runge-Kutta methods specifically designed to be effective relaxation schemes in the numerical Solution of the steady state Solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid Solution algorithm. The design criterion for the construction of the Runge-Kutta methods here developed is different form the one traditionally used to derive optimal Runge-Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the Solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented. © 2010 Elsevier Inc.
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Spectral p‐multigrid discontinuous Galerkin Solution of the Navier–Stokes equations
International Journal for Numerical Methods in Fluids, 2010Co-Authors: Francesco Bassi, N. Franchina, Antonio Ghidoni, Stefano RebayAbstract:Discontinuous Galerkin (DG) methods are very well suited for the construction of very high-order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high-order spectral element DG approximation of the Navier–Stokes equations coupled with a p-multigrid Solution strategy based on a semi-implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the Solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p-multigrid scheme with non-spectral elements and a spectral element DG approach with an implicit time integration scheme.
Francesco Bassi - One of the best experts on this subject based on the ideXlab platform.
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linearly implicit rosenbrock type runge kutta schemes applied to the discontinuous Galerkin Solution of compressible and incompressible unsteady flows
Computers & Fluids, 2015Co-Authors: Francesco Bassi, Antonio Ghidoni, Lorenzo Alessio Botti, Alessandro Colombo, Francesco Carlo MassaAbstract:In this work we investigate the use of linearly implicit Rosenbrock-type Runge-Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the Navier-Stokes equations. The final goal of this activity is the application of such schemes to the high-order accurate, both in space and time, simulation of turbulent flows. Besides being able to overcome the severe time step restriction of explicit schemes, Rosenbrock schemes have the attractive feature of requiring just one Jacobian matrix evaluation per time step, thus reducing the overall computational effort. Several high-order (up to sixth order) Rosenbrock schemes available in the literature have been implemented and evaluated on benchmark test cases of both compressible and incompressible flows. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an Appendix to the paper. An implementation of Rosenbrock schemes for systems of equations with a Solution dependent block diagonal matrix multiplying the time derivative is here proposed and described in detail. This can occur, for example, if sets of working variables different from the conservative ones are used in the compressible Navier-Stokes equations. In particular, we have found useful to employ primitive variables based on the logarithms of pressure and temperature in order to ensure the positivity of all thermodynamic variables at the discrete level. The best performing Rosenbrock scheme resulting from our analysis has then been applied to the Implicit Large Eddy Simulation of the transitional flow around the Selig-Donovan SD7003 airfoil.
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modified extended bdf scheme for the discontinuous Galerkin Solution of unsteady compressible flows
International Journal for Numerical Methods in Fluids, 2014Co-Authors: Alessandra Nigro, Stefano Rebay, Antonio Ghidoni, Francesco BassiAbstract:SUMMARY In this paper, a high-order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the Solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second-order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG-MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade. Copyright © 2014 John Wiley & Sons, Ltd.
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spectral p multigrid discontinuous Galerkin Solution of the navier stokes equations
International Journal for Numerical Methods in Fluids, 2011Co-Authors: Francesco Bassi, N. Franchina, Antonio Ghidoni, Stefano RebayAbstract:Discontinuous Galerkin (DG) methods are very well suited for the construction of very high-order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high-order spectral element DG approximation of the Navier–Stokes equations coupled with a p-multigrid Solution strategy based on a semi-implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the Solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p-multigrid scheme with non-spectral elements and a spectral element DG approach with an implicit time integration scheme.
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optimal runge kutta smoothers for the p multigrid discontinuous Galerkin Solution of the 1d euler equations
Journal of Computational Physics, 2011Co-Authors: Francesco Bassi, Antonio Ghidoni, Stefano RebayAbstract:This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical Solution of the steady state Solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid Solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the Solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.
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Spectral p-multigrid discontinuous Galerkin Solution of the Navier-Stokes equations
International Journal for Numerical Methods in Fluids, 2011Co-Authors: Francesco Bassi, N. Franchina, Antonio Ghidoni, Stefano RebayAbstract:This work presents a family of original Runge-Kutta methods specifically designed to be effective relaxation schemes in the numerical Solution of the steady state Solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid Solution algorithm. The design criterion for the construction of the Runge-Kutta methods here developed is different form the one traditionally used to derive optimal Runge-Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the Solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented. © 2010 Elsevier Inc.
Jaime Peraire - One of the best experts on this subject based on the ideXlab platform.
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Discontinuous Galerkin Solution of the Navier-Stokes Equations on
2020Co-Authors: Jaume Bonet, Jaime PeraireAbstract:We describe a method for computing time-dependent Solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a xed reference conguration and the time varying domain. By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference conguration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a xed reference conguration. The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-Kutta method. For general domain changes, the standard scheme fails to preserve exactly the free-stream Solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the exibility of the approach to handle high order approximations on complex geometries.
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An Embedded Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations
20th AIAA Computational Fluid Dynamics Conference, 2011Co-Authors: Jaime Peraire, Cuong Nguyen, Bernardo CockburnAbstract:In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the Solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin Solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the Solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, and offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method. {&}copy; 2010 by J. Periare, N.C. Nguyena and B. Cockburn.
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discontinuous Galerkin Solution of the navier stokes equations on deformable domains
Computer Methods in Applied Mechanics and Engineering, 2009Co-Authors: Perolof Persson, Jaume Bonet, Jaime PeraireAbstract:We describe a method for computing time-dependent Solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying domain, By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration, The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-Kutta method, For general domain changes, the standard scheme fails to preserve exactly the free-stream Solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high order approximations on complex geometries.
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Discontinuous Galerkin Solution of the Navier-Stokes equations on deformable domains
Computer Methods in Applied Mechanics and Engineering, 2009Co-Authors: Per O. Å. Persson, Jaume Bonet, Jaime PeraireAbstract:We describe a method for computing time-dependent Solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time-varying domain. By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration. The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-Kutta method. For general domain changes, the standard scheme fails to preserve exactly the free-stream Solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high-order approximations on complex geometries. © 2009 Elsevier B.V. All rights reserved.
Andre Teofilo Beck - One of the best experts on this subject based on the ideXlab platform.
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Chaos–Galerkin Solution of stochastic Timoshenko bending problems
Computers & Structures, 2020Co-Authors: Cláudio R. Ávila Da Silva, Andre Teofilo BeckAbstract:This paper presents an accurate and efficient Solution for the random transverse and angular displacement fields of uncertain Timoshenko beams. Approximate, numerical Solutions are obtained using the Galerkin method and chaos polynomials. The Chaos-Galerkin scheme is constructed by respecting the theoretical conditions for existence and uniqueness of the Solution. Numerical results show fast convergence to the exact Solution, at excellent accuracies. The developed Chaos-Galerkin scheme accurately approximates the complete cumulative distribution function of the displacement responses. The Chaos-Galerkin scheme developed herein is a theoretically sound and efficient method for the Solution of stochastic problems in engineering. (C) 2011 Elsevier Ltd. All rights reserved.Sao Paulo State Foundation for Research - FAPESP[2008/10366-4]National Council for Research and Development - CNPq[301679/2009-6
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Galerkin Solution of stochastic reaction diffusion problems
Journal of Heat Transfer-transactions of The Asme, 2013Co-Authors: C Avila R Da Silva, Andre Teofilo Beck, Admilson T Franco, Oscar Alfredo Garcia De SuarezAbstract:In this paper, the Galerkin method is used to obtain numerical Solutions to twodimensional steady-state reaction-diffusion problems. Uncertainties in reaction and diffusion coefficients are modeled using parameterized stochastic processes. A stochastic version of the Lax–Milgram lemma is used in order to guarantee existence and uniqueness of the theoretical Solutions. The space of approximate Solutions is constructed by tensor product between finite dimensional deterministic functional spaces and spaces generated by chaos polynomials, derived from the Askey–Wiener scheme. Performance of the developed Galerkin scheme is evaluated by comparing first and second order moments and probability histograms obtained from approximate Solutions with the corresponding estimates obtained via Monte Carlo simulation. Results for three example problems show very fast convergence of the approximate Galerkin Solutions. Results also show that complete probability densities (histograms) of the responses are correctly approximated by the developed Galerkin basis. [DOI: 10.1115/1.4023938]
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chaos Galerkin Solution of stochastic timoshenko bending problems
Computers & Structures, 2011Co-Authors: Claudio Avila R Da Silva, Andre Teofilo BeckAbstract:This paper presents an accurate and efficient Solution for the random transverse and angular displacement fields of uncertain Timoshenko beams. Approximate, numerical Solutions are obtained using the Galerkin method and chaos polynomials. The Chaos-Galerkin scheme is constructed by respecting the theoretical conditions for existence and uniqueness of the Solution. Numerical results show fast convergence to the exact Solution, at excellent accuracies. The developed Chaos-Galerkin scheme accurately approximates the complete cumulative distribution function of the displacement responses. The Chaos-Galerkin scheme developed herein is a theoretically sound and efficient method for the Solution of stochastic problems in engineering.
Mohammad I. Younis - One of the best experts on this subject based on the ideXlab platform.
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Multifrequency excitation of an inclined marine riser under internal resonances
Nonlinear Dynamics, 2020Co-Authors: Feras K. Alfosail, Mohammad I. YounisAbstract:We study the multifrequency excitation of an inclined marine riser under two-to-one and three-to-one internal resonances. The riser model accounts for the initial static deflection, self-weight, and mid-plane stretching nonlinearity. By tuning the initial applied tension and configuration angles of the riser, the ratio between its first and third natural frequencies approaches two. In another case, the ratio between its first and fifth natural frequencies approaches three. As recently revealed by experimental observations, a riser can experience multifrequency vortex-induced vibrations. Hence here, the excitation frequencies are tuned such that one frequency is near the first primary resonance, while the other frequency is near the second primary resonance. The multiple-timescale perturbation method is used to analyze the nonlinear motion of the riser considering the internal resonances. Frequency response results of the perturbation method are compared to a Galerkin Solution, which show good agreement. The perturbation results in the two-to-one internal resonance case demonstrate that increasing the forcing amplitude at the second primary resonance suppresses the energy exchange due to internal resonance and reduces the likelihood of Hopf bifurcations, while an opposite trend is observed in the three-to-one internal resonance case. Then, the dynamic Solutions of the modulation equations of the perturbation method are analyzed using the Floquet theory to examine the post-Hopf bifurcation response. The limit cycle responses in both internal resonance cases exhibit several period doubling bifurcations possibly leading to quasi-periodic and other complex motions, which can endanger the life of the riser.
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Two-to-one internal resonance of an inclined marine riser under harmonic excitations
Nonlinear Dynamics, 2019Co-Authors: Feras K. Alfosail, Mohammad I. YounisAbstract:In this paper, we study the two-to-one internal resonance of an inclined marine riser under harmonic excitations. The riser is modeled as an Euler–Bernoulli beam accounting for mid-plane stretching, self-weight, and an applied axial top tension. Due to the inclination, the self-weight load causes a static deflection of the riser, which can tune the frequency ratio between the third and first natural frequencies near two. The multiple-time-scale method is applied to study the nonlinear equation accounting for the system nonlinearity. The Solution is then compared to a Galerkin Solution showing good agreement. A further investigation is carried out by plotting the frequency response curves, the force response curves, and the steady-state response of the multiple-time-scale Solution, in addition to the dynamical Solution obtained by Galerkin, as they vary with the detuning parameters. The results reveal that the riser vibrations can undergo multiple Hopf bifurcations and experience quasi-periodic motion that can lead to chaotic behavior. These phenomena lead to complex vibrations of the riser, which can accelerate its fatigue failure.
