Direct Time Integration Methods for Structural Acoustics and Far Field Scattering Computations

1995 ◽  
Vol 117 (4) ◽  
pp. 488-492 ◽  
Author(s):  
J. P. Wright
Keyword(s):  
Numerical Solutions ◽  
Time Integration ◽  
Three Dimensional ◽  
Spherical Geometry ◽  
Far Field ◽  
Structural Acoustics ◽  
Explicit Integration ◽  
Orthogonal Function ◽  
Preliminary Validation ◽  
Time Integration Methods

Procedures are described for doing: (a) transient three dimensional structural acoustics computations, using a combination of modal and finite element techniques in space and explicit integration in time; (b) far field scattering computations, using orthogonal function expansions and the method of characteristics. These methods are discussed and preliminary validation of the approach is accomplished by comparing numerical solutions with available analytic results for some problems in spherical geometry.

Correction of kinetic data in non-isothermal reactions with non-uniform temperatures: analytical treatments for spherical reactant masses

1981 ◽  
Vol 378 (1772) ◽  
pp. 27-60 ◽  
Keyword(s):  
Numerical Solutions ◽  
Three Dimensional ◽  
Spherical Geometry ◽  
Close Relation ◽  
Infinite Cylinder ◽  
Central Temperature ◽  
Temperature Position ◽  
Special Cases ◽  
Satisfactory Precision ◽  
Endothermic Reactions

Exothermic reactions normally proceed non-isothermally, and to treat them quantitatively allowance must be made for variations in internal temperature. Except for the special cases of the infinite slab and infinite cylinder, this has so far had to be done solely by numerical computation. The sphere and other finite, three-dimensional geometries have eluded analytical solution, even in the stationary state. The present paper presents two useful treatments for spherical geometry. They provide families of compact analytical expressions of very satisfactory precision for all the important aspects, errors rarely being greater than 2% right up to criticality. The aspects studied include temperature-position profiles, central temperature-excesses and surface temperature-gradients. (Both stable and unstable subcritical conditions as well as critical cases can be treated.) From these there follow values for rate-constant correction-factors f (and their reciprocals, the effectiveness factors η) and, in turn, the means of correcting errors in Arrhenius activation energies and in reaction orders - again in expressive, simple forms. In addition, explicit equations have been set out for the first time to cover the whole range of boundary conditions (arbitrary Biot number, β ) from Frank-Kamenetskii ( β → ∞) to Semenov ( β ═ 0) extremes. This is a major development, since mere tabulation of numerical solutions has hitherto been a formidable task. Calculating procedures are detailed in an Appendix. Endothermic reactions show self-cooling and self-repression instead of self-stimulation. These aspects are all expressively encapsulated in the change of sign of single coefficients in tidy equations. The two routes - quintic approximation (q. a.) and second-order reversion (r2) - can both be applied most conveniently in parametric form (this is necessary for r2 but not for q. a). The parameters l and x that appear naturally in the two treatments bear a close relation to corresponding integration constants encountered in one- and two-dimensional geometry. Each has striking physical significance, l being most directly related to the effectiveness factor and x to the central temperature excess.

Numerical Study of the Added Resistance of Ship Advancing in Waves Using Far-Field and Near-Field Methods

2015 ◽  
Author(s):  
D. C. Hong ◽  
J. G. Kim ◽  
K. H. Song ◽  
H. K. Lee
Keyword(s):  
Numerical Solutions ◽  
Near Field ◽  
Three Dimensional ◽  
Design Stage ◽  
Linear Potential ◽  
Far Field ◽  
Forward Speed ◽  
Added Resistance ◽  
Economic Point ◽  
Wave Range

When a ship advances in a seaway, it undergoes 6-degree-of-freedom motion. The ship motions and wave loads are very important from operability and survivability points of view. The resistance increase due to waves is also important from the economic point of view. Although the accurate prediction of these seakeeping characteristics should be done using the unsteady CFD computations, the analytical method based on the linear potential flow theory have been widely used to evaluate them at the early design stage since the latter does not require large computing resources. In the present paper, the added resistance of a ship advancing in waves was calculated using both Maruo’s far-field formula and the near-field method. The radiation-diffraction potential over the wetted surface of the ship has been obtained using the three-dimensional frequency-domain forward-speed free-surface Green function (Brard 1948) and the forward-speed Green integral equation (Hong 2000). Numerical solutions are obtained by making use of the 9-node second-order inner collocation boundary element method (Hong et al. 2014b). In the present paper, Maruo’s far-field formula was combined with the exact three-dimensional Kochin function so that the added resistance thereby obtained could show good comparison with experimental results over the entire wave range including the short wave range. The near-field added resistance is the time mean value of the 2nd-order forces acting on the advancing ship in waves. The time-mean hydrodynamic force, obtained by using direct integration of the hydrodynamic pressure due to the sum of the unsteady potential and steady potential approximated by the double-body potential over the wetted surface of the ship, was also presented. Comparison of the present far-field and simplified near-field numerical values and the experimental values reported by Journee (1992) of the added resistance for the Wigley ship models I and II has been made in order to find appropriate numerical values of the far-field added resistance over the entire frequency range of interest.

Conservative Split-Explicit Time Integration Methods for the Compressible Nonhydrostatic Equations

2007 ◽  
Vol 135 (8) ◽  
pp. 2897-2913 ◽  
Author(s):  
J. B. Klemp ◽  
W. C. Skamarock ◽  
J. Dudhia
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Test Cases ◽  
Explicit Integration ◽  
Linear Dispersion ◽  
Acoustic Gravity Waves ◽  
Wide Range ◽  
Terrain Following ◽  
Integration Techniques ◽  
Time Integration Methods

Abstract Historically, time-split schemes for numerically integrating the nonhydrostatic compressible equations of motion have not formally conserved mass and other first-order flux quantities. In this paper, split-explicit integration techniques are developed that numerically conserve these properties by integrating prognostic equations for conserved quantities represented in flux form. These procedures are presented for both terrain-following height and hydrostatic pressure (mass) vertical coordinates, two potentially attractive frameworks for which the equation sets and integration techniques differ significantly. For each set of equations, the linear dispersion equation for acoustic/gravity waves is derived and analyzed to determine which terms must be solved in the small (acoustic) time steps and how these terms are represented in the time integration to achieve stability. Efficient techniques for including numerical filters for acoustic and external modes are also presented. Simulations for several idealized test cases in both the height and mass coordinates are presented to demonstrate that these integration techniques appear robust over a wide range of scales, from subcloud to synoptic.

Mathematical Model Investigation of Far-Field Transport of Ocean-Dumped Sewage Sludge Related to Remote Sensing

1982 ◽  
Vol 14 (3) ◽  
pp. 33-39
Author(s):  
C Y Kuo
Keyword(s):  
New York ◽  
Sewage Sludge ◽  
Laboratory Data ◽  
Three Dimensional ◽  
Approximate Model ◽  
Transport Processes ◽  
Far Field ◽  
New York Bight ◽  
Mean Current

An existing, three-dimensional, Eulerian-Lagrangian finite-difference model was modified and used to examine the far-field transport processes of dumped sewage sludge in the New York Bight. Both in situ and laboratory data were utilized in an attempt to approximate model inputs such as mean current speed, vertical and horizontal diffusion coefficients, particle size distributions, and specific gravities. Concentrations of the sludge near the sea surface predicted from the computer model were compared qualitatively with those remotely sensed.

Numerical solutions for strain-softening surrounding rock under three-dimensional principal stress condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sheng Yu-ming ◽  
Li Chao ◽  
Xia Ming-yao ◽  
Zou Jin-feng
Keyword(s):  
Finite Element ◽  
Principal Stress ◽  
Stress Condition ◽  
Strain Softening ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Strength Criterion ◽  
Surrounding Rock ◽  
Flow Rule ◽  
Finite Element Software

Abstract In this study, elastoplastic model for the surrounding rock of axisymmetric circular tunnel is investigated under three-dimensional (3D) principal stress states. Novel numerical solutions for strain-softening surrounding rock were first proposed based on the modified 3D Hoek–Brown criterion and the associated flow rule. Under a 3D axisymmetric coordinate system, the distributions for stresses and displacement can be effectively determined on the basis of the redeveloped stress increment approach. The modified 3D Hoek–Brown strength criterion is also embedded into finite element software to characterize the yielding state of surrounding rock based on the modified yield surface and stress renewal algorithm. The Euler implicit constitutive integral algorithm and the consistent tangent stiffness matrix are reconstructed in terms of the 3D Hoek–Brown strength criterion. Therefore, the numerical solutions and finite element method (FEM) models for the deep buried tunnel under 3D principal stress condition are presented, so that the stability analysis of surrounding rock can be conducted in a direct and convenient way. The reliability of the proposed solutions was verified by comparison of the principal stresses obtained by the developed numerical approach and FEM model. From a practical point of view, the proposed approach can also be applied for the determination of ground response curve of the tunnel, which shows a satisfying accuracy compared with the measuring data.

Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Imtiaz Ahmad ◽  
Aly R. Seadawy ◽  
Hijaz Ahmad ◽  
Phatiphat Thounthong ◽  
Fuzhang Wang
Keyword(s):  
Meshless Method ◽  
Material Science ◽  
Numerical Study ◽  
Research Work ◽  
Time Integration ◽  
Three Dimensional ◽  
Nuclear Material ◽  
Telegraph Equations ◽  
Model Equations ◽  
Derivatives Of

Abstract This research work is to study the numerical solution of three-dimensional second-order hyperbolic telegraph equations using an efficient local meshless method based on radial basis function (RBF). The model equations are used in nuclear material science and in the modeling of vibrations of structures. The explicit time integration technique is utilized to semi-discretize the model in the time direction whereas the space derivatives of the model are discretized by the proposed local meshless procedure based on multiquadric RBF. Numerical experiments are performed with the proposed numerical scheme for rectangular and non-rectangular computational domains. The proposed method solutions are converging quickly in comparison with the different existing numerical methods in the recent literature.

scholarly journals Numerical Simulation of the Coagulation Dynamics of Blood

2008 ◽  
Vol 9 (2) ◽  
pp. 83-104 ◽  
Author(s):  
T. Bodnár ◽  
A. Sequeira
Keyword(s):  
Blood Coagulation ◽  
Computational Study ◽  
Time Integration ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Clot Formation ◽  
Diffusion Reaction ◽  
Numerical Code ◽  
Injured Region ◽  
Reaction Equations

The process of platelet activation and blood coagulation is quite complex and not yet completely understood. Recently, a phenomenological meaningful model of blood coagulation and clot formation in flowing blood that extends existing models to integrate biochemical, physiological and rheological factors, has been developed. The aim of this paper is to present results from a computational study of a simplified version of this coupled fluid-biochemistry model. A generalized Newtonian model with shear-thinning viscosity has been adopted to describe the flow of blood. To simulate the biochemical changes and transport of various enzymes, proteins and platelets involved in the coagulation process, a set of coupled advection–diffusion–reaction equations is used. Three-dimensional numerical simulations are carried out for the whole model in a straight vessel with circular cross-section, using a finite volume semi-discretization in space, on structured grids, and a multistage scheme for time integration. Clot formation and growth are investigated in the vicinity of an injured region of the vessel wall. These are preliminary results aimed at showing the validation of the model and of the numerical code.

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