scholarly journals Analytic and numerical solutions to the seismic wave equation in continuous media

2020 ◽  
Vol 476 (2243) ◽  
pp. 20200636
Author(s):  
S. J. Walters ◽  
L. K. Forbes ◽  
A. M. Reading
Keyword(s):  
Numerical Scheme ◽  
Wave Speed ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Initial Condition ◽  
Seismic Structure ◽  
Absorbing Boundary ◽  
Alternative Approach ◽  
Isotropic Media

This paper presents two approaches to mathematical modelling of a synthetic seismic pulse, and a comparison between them. First, a new analytical model is developed in two-dimensional Cartesian coordinates. Combined with an initial condition of sufficient symmetry, this provides a valuable check for the validity of the numerical method that follows. A particular initial condition is found which allows for a new closed-form solution. A numerical scheme is then presented which combines a spectral (Fourier) representation for displacement components and wave-speed parameters, a fourth-order Runge–Kutta integration method, and an absorbing boundary layer. The resulting large system of differential equations is solved in parallel on suitable enhanced performance desktop hardware in a new software implementation. This provides an alternative approach to forward modelling of waves within isotropic media which is efficient, and tailored to rapid and flexible developments in modelling seismic structure, for example, shallow depth environmental applications. Visual comparisons of the analytic solution and the numerical scheme are presented.

Numerical Solutions of Generalized Oscillator Equations

2013 ◽  
Author(s):  
Yufeng Xu ◽  
Om P. Agrawal
Keyword(s):  
Closed Form ◽  
Numerical Scheme ◽  
Fractional Integral ◽  
Numerical Solutions ◽  
Lagrange Equation ◽  
Closed Form Solution ◽  
Fractional Power ◽  
Form Solution ◽  
Harmonic Oscillators ◽  
General Nature

Harmonic oscillators play a fundamental role in many areas of science and engineering, such as classical mechanics, electronics, quantum physics, and others. As a result, harmonic oscillators have been studied extensively. Classical harmonic oscillators are defined using integer order derivatives. In recent years, fractional derivatives have been used to model the behaviors of damped systems more accurately. In this paper, we use three operators called K-, A- and B-operators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K-, A- and B-operators allow the kernel to be arbitrary. In the case when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K-, A- and B-operators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler-Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A numerical scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution. It is demonstrated that the numerical scheme is convergent, and the order of convergence is 2. For a special kernel, this scheme reduces to a scheme presented recently in the literature.

Models and Numerical Solutions of Generalized Oscillator Equations

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Yufeng Xu ◽  
Om P. Agrawal
Keyword(s):  
Closed Form ◽  
Numerical Scheme ◽  
Fractional Integral ◽  
Numerical Solutions ◽  
Lagrange Equation ◽  
Closed Form Solution ◽  
Fractional Power ◽  
Form Solution ◽  
Harmonic Oscillators ◽  
General Nature

In this paper, we use three operators called K-, A-, and B-operators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K-, A-, and B-operators allow the kernel to be arbitrary. In the case, when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K-, A-, and B-operators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler–Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A finite difference scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution, which demonstrates that the numerical scheme is convergent.

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

Modeling Controlled Articulated Flexible Systems Using Assumed Modes: Part I — Theory

2001 ◽  
Author(s):  
Theodore G. Mordfin ◽  
Sivakumar S. K. Tadikonda
Keyword(s):  
Closed Form ◽  
Structural Model ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Flexible Multibody Dynamics ◽  
Companion Paper ◽  
Form Solution ◽  
Control Laws ◽  
Comparison Functions ◽  
Flexible Behavior

Abstract Guidelines are sought for generating component body models for use in controlled, articulated, flexible multibody dynamics system simulations. In support of this effort, exact closed-form and numerical solutions are developed for the small elastic motions of a planar, flexible, single link system, in which the link is represented as an Euler-Bernoulli bar in transverse vibration. The link is connected to ground by a pin joint, and the articulation is controlled by proportional and proprotional/derivative (PD) feedback control laws. The characteristics of the closed-form solution are shown to consist of combinations of the characteristic expressions associated with classical end conditions. A large-articulation flexible body model of a controlled-articulation flexible link is then developed and linearized about an arbitrary reference angle. This model uses the method of assumed modes to represent the flexible behavior of the link. It is shown the model is analytically equivalent to a purely structural model which uses a hybrid set of assumed modes, and that numerical convergence can be investigated in terms of admissible functions and quasi-comparison functions. Numerical evaluation of the use of various types of assumed modes is presented in a companion paper.

Three-dimensional stress state in inclined backfilled stopes obtained from numerical simulations and new closed-form solution

2018 ◽  
Vol 55 (6) ◽  
pp. 810-828 ◽  
Author(s):  
Abtin Jahanbakhshzadeh ◽  
Michel Aubertin ◽  
Li Li
Keyword(s):  
Stress State ◽  
Closed Form ◽  
Numerical Solutions ◽  
Hanging Wall ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Solid Waste Disposal ◽  
Inclination Angles ◽  
Backfilled Stopes

Backfill is commonly used world-wide in underground mines to improve ground stability and reduce solid waste disposal on the surface. Practical solutions are required to assess the stress state in the backfilled stopes, as the stress state is influenced by the fill settlement that produces a stress transfer to the adjacent rock walls. The majority of existing analytical and numerical solutions for the stresses in backfilled openings were developed for two-dimensional (plane strain) conditions. In reality, mine stopes have a limited extension in the horizontal plane so the stresses are influenced by the four walls. This paper presents recent three-dimensional (3D) simulations results and a new 3D closed-form solution for the vertical and horizontal stresses in inclined backfilled stopes with parallel walls. This solution takes into account the variation of the stresses along the opening width and height, for various inclination angles and fills properties. The numerical results are used to validate the analytical solution and illustrate how the stress state varies along the opening height, length, and width, for different opening sizes and inclination angles of the footwall and hanging wall. Experimental results are also used to assess the validity of the proposed solution.

Squeeze Film Characteristics of Circular Contacts at Constant Squeeze Velocity Motion with Non-Newtonian Lubricants

2011 ◽  
Vol 63-64 ◽  
pp. 147-151
Author(s):  
Li Ming Chu ◽  
Wang Long Li ◽  
Hsiang Chen Hsu
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Hydrodynamic Lubrication ◽  
Closed Form Solution ◽  
Elastohydrodynamic Lubrication ◽  
Form Solution ◽  
Squeeze Film ◽  
High Pressure Stage ◽  
Time Calculation ◽  
Film Characteristics

In this paper, the numerical solutions in pure squeeze motion are explored by using hydrodynamic lubrication (HL) and elastohydrodynamic lubrication (EHL) models at constant squeeze velocity with power law lubricants. This paper also proposes a closed form solution to calculate the relationship between central pressure and central film thickness under HL condition. In order to save time calculation, the present closed form solution can be used as the initial condition for analysis of EHL at the high-pressure stage. In addition, this paper also discussed the HL and EHL squeeze film characteristics.

Exact Analytical Hyperbolic Temperature Profile in a Three-Dimensional Media Under Pulse Surface Heat Flux

2015 ◽  
Vol 32 (3) ◽  
pp. 339-347 ◽  
Author(s):  
M. R. Talaee ◽  
V. Sarafrazi ◽  
S. Bakhshandeh
Keyword(s):  
Heat Flux ◽  
Numerical Solutions ◽  
Rectangular Cross Section ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Biological Tissues ◽  
Form Solution ◽  
Hyperbolic Heat Conduction ◽  
Duhamel Integral

AbstractIn this paper three-dimensional hyperbolic heat conduction equation in a cubic media with rectangular cross-section under a pulsed heat flux on the upper side has been solved analytically using the method of separation of variables and the Duhamel integral. The closed form solution of both Fourier and non-Fourier profiles are introduced with both modes of steady and pulsed fluxes. The results show the considerable difference between the Fourier and Non-Fourier temperature profiles. Then the answer procedure is used for modeling of interaction of a cubical tissue under a short laser pulse heating. The effects of pulse duration and laser intensity are studied analytically. Furthermore the results can be applied as a verification branch for other numerical solutions or laser treatments of biological tissues.

Closed-form and numerical solutions to the laser heating process

1998 ◽  
Vol 212 (2) ◽  
pp. 141-151 ◽  
Author(s):  
B S Yilbas ◽  
M Sami ◽  
A Al-Farayedhi
Keyword(s):  
Closed Form ◽  
Laser Heating ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heat Of Fusion ◽  
Heating Process ◽  
Temperature Profiles ◽  
Depth Analysis ◽  
Analytical Approaches

The laser processing of engineering materials requires an in-depth analysis of the applicable heating mechanism. The modelling of the laser heating process offers improved understanding of the machining mechanism. In the present study, a closed-form solution for a step input laser heating pulse is obtained and a numerical scheme solving a three-dimensional heat transfer equation is introduced. The numerical solution provides a comparison of temperature profiles with those obtained from the analytical approach. To validate the analytical and numerical solutions, an experiment is conducted to measure the surface temperature and evaporating front velocity during the Nd—YAG laser heating process. It is found that the temperature profiles resulting from both theory and experiment are in a good agreement. However, a small discrepancy in temperatures at the upper end of the profiles occurs. This may be due to the assumptions made in both the numerical and the analytical approaches. In addition, the equilibrium time, based on the energy balance among the internal energy gain, conduction losses and latent heat of fusion, is introduced.

Analytical Solutions for Heat Transfer During Cyclic Melting and Freezing of a Phase Change Material Used in Electronic or Electrical Packaging

2003 ◽  
Vol 125 (1) ◽  
pp. 126-133 ◽  
Author(s):  
Suman Chakraborty ◽  
Pradip Dutta
Keyword(s):  
Heat Transfer ◽  
Phase Change ◽  
Phase Change Material ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Electronics Cooling ◽  
Form Solution ◽  
Transfer Model ◽  
Change Material

In this paper, we develop an analytical heat transfer model, which is capable of analyzing cyclic melting and solidification processes of a phase change material used in the context of electronics cooling systems. The model is essentially based on conduction heat transfer, with treatments for convection and radiation embedded inside. The whole solution domain is first divided into two main sub-domains, namely, the melting sub-domain and the solidification sub-domain. Each sub-domain is then analyzed for a number of temporal regimes. Accordingly, analytical solutions for temperature distribution within each sub-domain are formulated either using a semi-infinity consideration, or employing a method of quasi-steady state, depending on the applicability. The solution modules are subsequently united, leading to a closed-form solution for the entire problem. The analytical solutions are then compared with experimental and numerical solutions for a benchmark problem quoted in the literature, and excellent agreements can be observed.

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