Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

scholarly journals Dynamic faulting studied by a finite difference method

1982 ◽  
Vol 72 (2) ◽  
pp. 345-369
Author(s):  
Jean Virieux ◽  
Raul Madariaga
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Maximum Stress ◽  
Numerical Solutions ◽  
Shear Crack ◽  
Numerical Results ◽  
Three Dimensions ◽  
Shear Cracks ◽  
Stress Criterion

Abstract We have developed a finite difference method that is especially adapted to the study of dynamic shear cracks. We studied a number of simple earthquake source models in two and three dimensions with special emphasis on the modeling of the stress field. We compared our numerical results for semi-infinite and self-similar shear cracks with the few exact solutions that are available in the literature. We then studied spontaneous rupture propagation with the help of a maximum stress criterion. From dimensional arguments and a few simple examples, we showed that the maximum stress criterion depended on the physical dimensions of the fault. For a given maximum stress intensity, the finer the numerical mesh, the higher the maximum stress that had to be adopted. A study of in-plane cracks showed that at high rupture velocities, the numerical results did not resolve the stress concentration due to the rupture front from the stress peak associated with the shear wave propagating in front of the crack. We suggest that this is the reason why transonic rupture velocities are found in the numerical solutions of in-plane faulting when the rupture resistance is rather low. Finally, we studied the spontaneous propagation of an initially circular rupture. Two distinct modes of nucleation of the rupture were studied. In the first, a plane circular shear crack was formed instantaneously in a uniformly prestressed medium. After a while, once stress concentrations had developed around the crack edge, the rupture started to grow. In the second type of nucleation, a preexisting circular crack became unstable at time t = 0 and started to grow. The latter model appeared to us as a more realistic simulation of earthquake triggering. In this case, the initial stress was nonuniform and was the static field of the preexisting fault.

An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

2015 ◽  
Vol 7 (3) ◽  
pp. 295-322 ◽  
Author(s):  
Valeria Boccardo ◽  
Eduardo Godoy ◽  
Mario Durán
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Half Space ◽  
Plane Surface ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Quadratic Functional ◽  
Infinite Plane ◽  
Equilibrium Equations ◽  
Finite Element Software

AbstractThis paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.

Reflection of Longitudinal Micro-Rotational Wave at Viscoelastically Supported Boundary of Micropolar Half-Space

2016 ◽  
Vol 34 (3) ◽  
pp. 243-255 ◽  
Author(s):  
P. Zhang ◽  
P.-J. Wei ◽  
Y.-Q. Li
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Energy Flux ◽  
Incident Wave ◽  
Amplitude Ratio ◽  
Numerical Results ◽  
Elastic Parameters ◽  
Rotational Wave ◽  
Serial Connection ◽  
Delay Effects

AbstractThe reflection of longitudinal micro-rotational wave at the viscoelastically supported boundary of micropolar half-space is studied in this paper. The viscoelastic boundary is described by spring-dashpot model with parallel or serial connection. Both the spring and the dashpot contribute to the displacements and micro-rotation and the boundary conditions include the force stress and couple stress components. From the boundary conditions, the amplitude ratios and phase shifts of reflection waves with respect to the incident wave are obtained. Further, the energy flux ratios of the reflection waves to the incident wave are estimated. In order to validate the numerical results, the energy flux conservation with consideration of the energy dissipation of dashpot is used. Based on the numerical results, the influences of elastic parameters and viscous parameters are studied, respectively. It is found that the elastic parameters and the viscous parameters have evident influences on the amplitude ratio, the phase shift and the energy partition. The causes resulting in these deviations are related with the instantaneous elasticity of spring and the time-delay effects of dashpot.

Development of Three-Dimensional PML Boundary Conditions for Aeroacoustics Applications

2002 ◽  
Vol 1 (3) ◽  
pp. 307-327
Author(s):  
J-F. Dietiker ◽  
K.A. Hoffmann ◽  
M. Papadakis ◽  
R. Agarwal
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Perfectly Matched Layer ◽  
Optimum Combination ◽  
Curvilinear Coordinates ◽  
Main Program ◽  
Governing Equations ◽  
Compact Finite Difference ◽  
Finite Difference Formulation

Perfectly Matched Layer (PML) boundary conditions are derived in generalized curvilinear coordinates for three-dimensional aeroacoustic applications. The resulting governing equations are solved numerically by a four-stage Runge-Kutta scheme, with 4th/6th order compact finite difference formulation. The PML equations are programmed in a subroutine, which is easily incorporated to the main program LINEULER (Linearized Euler's equation solver). Two and three-dimensional benchmarks problems are solved to investigate the efficiency and accuracy of the PML boundary conditions. Investigations on the PML parameters have been conducted to determine the optimum combination of parameters used in the computations.

Thermoelasticity of an Anisotropic Half Space

1974 ◽  
Vol 41 (4) ◽  
pp. 935-940 ◽  
Author(s):  
J. Padovan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Half Space ◽  
Boundary Value ◽  
Integral Transforms ◽  
Numerical Results ◽  
Heat Sinks ◽  
Material Anisotropy ◽  
Body Forces

The thermoelasticity of an anisotropic Hookean half space is studied herein. A solution based on successive integral transforms is developed. The solution can handle arbitrary thermal and mechanical boundary conditions together with distributed body forces and heat sinks or sources. To illustrate the substantial effects of material anisotropy, a specific boundary-value problem is solved. Numerical results based on the solution are presented. These illustrate the significant effects of both thermal and mechanical material anisotropy.

scholarly journals Computation of 3D Flow Phenomena in Axial Flow Compressor Blade Rows

1991 ◽  
Author(s):  
M. Janssen ◽  
H.-J. Dohmen ◽  
K. G. Grahl
Keyword(s):  
Flow Field ◽  
Boundary Conditions ◽  
Axial Flow ◽  
Compressor Blade ◽  
Numerical Results ◽  
Curvilinear Coordinates ◽  
Simple Method ◽  
Flow Development ◽  
Blade Row ◽  
Flow Effects

The main subject of the present publication is the comparison of results achieved with a 3D-partially parabolic calculation procedure and experimental data for the three dimensional flow in stationary and rotating blade rows of axial flow compressors. To set up the numerical solution procedure, the Navier-Stokes Equations are written in finite difference form by applying the control-volume approach. The turbulent flow effects are taken into account by using the standard k—ε model for the calculation of the turbulent viscosity. For precisely introducing the boundary conditions for arbitrary geometries, the differential equations are transformed to a body-fitted coordinate system by a very simple method. To construct the physical mesh, the nonorthogonal curvilinear coordinates are taken as solutions of a suitable elliptic boundary value problem. The abilities of the developed computer program are shown by comparing experimental and numerical results for three applications. The first, most simple case deals with the flow development in an isolated, stationary blade row of cylindrical blades and uniform boundary conditions upstream of the blade row. A more complex flow is regarded by calculating the flow field through highly turned, custom tailored airfoils working in a multistage environment. The flow conditions upstream of the flow domain under consideration show a well developed end wall boundary layer at the hub, leading to a strongly skewed inflow due to the superimposed tangential velocity component of the rotor upstream. The third application regards the flow development in a rotating axial compressor blade row in which the complexity of the flow field increases by flow effects that are due to centrifugal and Coriolis forces. The comparisons between experimental and numerical results show good agreements for all applications.

Axisymmetric flow near the critical point on the moving boundary

2010 ◽  
Vol 7 ◽  
pp. 182-190
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh. Nasibullaeva
Keyword(s):  
Fluid Flow ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Axial Velocity ◽  
Moving Boundary ◽  
Axisymmetric Flow ◽  
Boundary Motion ◽  
Newton Raphson ◽  
Raphson Method

In this paper the investigation of the axisymmetric flow of a liquid with a boundary perpendicular to the flow is considered. Analytical equations are derived for the radial and axial velocity and pressure components of fluid flow in a pipe of finite length with a movable right boundary, and boundary conditions on the moving boundary are also defined. A numerical solution of the problem on a finite-difference grid by the iterative Newton-Raphson method for various velocities of the boundary motion is obtained.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

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