Simulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards highorder discretization strategies in last decades. In this PhD thesis, a highorder adaptive hybidizable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible NavierStokes equations. Voigt notation for symmetric secondorder tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when loworder polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an elementbyelement superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilized to define a local error indicator to drive degree adaptivity procedures and accurately capture localized features of the flow. The resulting HDG solver is thus extended to the case of transient problems via highorder time integration schemes, namely the explicit singly diagonal implicit RungeKutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretization, a preconditioned iterative solver is proposed. This is critical in the context of highorder approximations in threedimensional domains leading to largescale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and threedimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.
Simulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards highorder discretization strategies in last decades. In this PhD thesis, a highorder adaptive hybidizable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible NavierStokes equations. Voigt notation for symmetric secondorder tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when loworder polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an elementbyelement superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilized to define a local error indicator to drive degree adaptivity procedures and accurately capture localized features of the flow. The resulting HDG solver is thus extended to the case of transient problems via highorder time integration schemes, namely the explicit singly diagonal implicit RungeKutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretization, a preconditioned iterative solver is proposed. This is critical in the context of highorder approximations in threedimensional domains leading to largescale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and threedimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.
Adaptive low and highorder hybridized methods for unsteady incompressible flow simulations
KARKOULIAS, ALEXANDROS
20200617
Abstract
Simulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards highorder discretization strategies in last decades. In this PhD thesis, a highorder adaptive hybidizable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible NavierStokes equations. Voigt notation for symmetric secondorder tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when loworder polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an elementbyelement superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilized to define a local error indicator to drive degree adaptivity procedures and accurately capture localized features of the flow. The resulting HDG solver is thus extended to the case of transient problems via highorder time integration schemes, namely the explicit singly diagonal implicit RungeKutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretization, a preconditioned iterative solver is proposed. This is critical in the context of highorder approximations in threedimensional domains leading to largescale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and threedimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.File  Dimensione  Formato  

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