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F W Williams - One of the best experts on this subject based on the ideXlab platform.
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coupled bending torsional dynamic stiffness matrix of an axially loaded timoshenko beam element
International Journal of Solids and Structures, 1994Co-Authors: J R Banerjee, F W WilliamsAbstract:Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to Machine Accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.
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coupled bending torsional dynamic stiffness matrix for timoshenko beam elements
Computers & Structures, 1992Co-Authors: J R Banerjee, F W WilliamsAbstract:Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix elements of a uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. Application of the developed theory in the context of wings, blades and grillages is discussed with particular reference to an established algorithm. Programming the derived stiffness expressions on a VAX computer indicates about 87% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The correctness of the stiffness expressions is numerically checked up to Machine Accuracy against the corresponding stiffnesses from the inversion method. The stiffnesses are also checked up to nine figure Accuracy against those obtained from a comparable approximate method.
J R Banerjee - One of the best experts on this subject based on the ideXlab platform.
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coupled bending torsional dynamic stiffness matrix of an axially loaded timoshenko beam element
International Journal of Solids and Structures, 1994Co-Authors: J R Banerjee, F W WilliamsAbstract:Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to Machine Accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.
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coupled bending torsional dynamic stiffness matrix for timoshenko beam elements
Computers & Structures, 1992Co-Authors: J R Banerjee, F W WilliamsAbstract:Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix elements of a uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. Application of the developed theory in the context of wings, blades and grillages is discussed with particular reference to an established algorithm. Programming the derived stiffness expressions on a VAX computer indicates about 87% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The correctness of the stiffness expressions is numerically checked up to Machine Accuracy against the corresponding stiffnesses from the inversion method. The stiffnesses are also checked up to nine figure Accuracy against those obtained from a comparable approximate method.
Stephane Zaleski - One of the best experts on this subject based on the ideXlab platform.
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a mass momentum consistent volume of fluid method for incompressible flow on staggered grids
Computers & Fluids, 2021Co-Authors: T Arrufat, M Crialesiesposito, Daniel Fuster, Yue Ling, L C Malan, S Pal, Ruben Scardovelli, Gretar Tryggvason, Stephane ZaleskiAbstract:Abstract The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with capillary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d’Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to Machine Accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner “consistent” with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.
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a momentum conserving consistent volume of fluid method for incompressible flow on staggered grids
arXiv: Computational Physics, 2018Co-Authors: Daniel Fuster, T Arrufat, M Crialesiesposito, Yue Ling, S Pal, Ruben Scardovelli, Gretar Tryggvason, Leon Malan, Stephane ZaleskiAbstract:The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a consistent mass and momentum-conserving discretization of the Navier-Stokes equation. Incompressible flow with capillary forces is modelled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d'Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to Machine Accuracy provided incompressiblity is satisfied which leads to a method that is both momentum and mass-conserving. To improve the stability of these methods momentum fluxes are advected in a manner "consistent" with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a falling raindrop and for an atomizing flow in air-water conditions.
Jeffrey R Koseff - One of the best experts on this subject based on the ideXlab platform.
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a non staggered grid fractional step method for time dependent incompressible navier stokes equations in curvilinear coordinates
Journal of Computational Physics, 1994Co-Authors: Yan Zang, Robert L Street, Jeffrey R KoseffAbstract:A numerical method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations in curvilinear coordinates is presented. The non-staggered-grid method originally developed by C. M. Rhie and W. L. Chow (AIAAJ.21, 1525 (1983)) for steady state problems is extended to compute unsteady flows. In the computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the mid-point on their corresponding cell faces. The momentum equations are integrated semi-implicitly by the approximate factorization technique. The intermediate velocities are interpolated onto the faces of the control volume to form the source terms of the pressure Poisson equation, which is solved iteratively with a multigrid method. The compatibility condition of the pressure Poisson equation is satisfied in the same manner as in a staggered-grid method; mass conservation can be satisfied to Machine Accuracy. The pressure boundary condition is derived from the momentum equations. Solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable Accuracy and efficiency of the present method.
Eric Lamballais - One of the best experts on this subject based on the ideXlab platform.
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high order compact schemes for incompressible flows a simple and efficient method with quasi spectral Accuracy
Journal of Computational Physics, 2009Co-Authors: Sylvain Laizet, Eric LamballaisAbstract:In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier-Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and Accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the Machine Accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order Accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given Accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral Accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.