Finite-Difference Methods

In this paper, a physics-based model for a snare drum will be discussed, along with its finite difference simulation. The interactions between a mallet and the membrane and between the snares and the membrane will be described as perfectly elastic collisions. A novel numerical scheme for the implementation of collisions will be presented, which allows a complete energy analysis for the whole system. Viscothermal losses will be added to the equation for the 3D wave propagation. Results from simulations and sound examples will be presented.

Nesse trabalho é estudado um problema de valor de contorno fuzzy (FPVC) que envolve uma equação diferencial fuzzy com termo fonte considerado como uma função fuzzy e condições de contorno modeladas por números fuzzy. ´ E apresentada uma estrutura matricial para obter uma solução numérica a partir do método de diferenças finitas combinado ao uso de derivadas fuzzy. A solução fuzzy é obtida a partir de seus α-níveis, que são intervalos da reta real. É apresentado um exemplo de resolução para um FPVC que modela a equação de Bessel de ordem 1/2, a partir de esquemas de diferenças finitas centradas, comparando o uso da derivada de Hukuhara (H-diferenciabilidade) e da derivada generalizada de Hukuhara (gH-diferenciabilidade).

Palavras-chave. Problema de valor de contorno, Números fuzzy, Função fuzzy, Método de diferenças finitas, Equação de Bessel, Derivada de Hukuhara, Derivada generalizada de Hukuhara.

The current project is a study of the Laplace equation that is used nowadays in many applications related to the electrical, magnetic, gravitational potentials, steady state temperatures, and in hydrodynamics.. The system corresponds to an elliptic second order partial differential equation and is widely used in the study of the inviscid incompressible potential flow in fluid mechanics. To solve this equation, numerical methods are used and different schemes are introduced to better control the solution through stability, accuracy and consistency. In what follows, a numerical study and investigation of the 2D Laplace Equation.

Sound synthesis based on physical models of musical instruments is, ultimately, an exercise in numerical simulation. As such, for complex systems of the type seen in musical acoustics, simulation can be a computationally costly
undertaking, particularly if simplifying hypotheses, such
as those of traveling wave or mode decompositions are not
employed. In this paper, large scale time stepping methods, such as the finite difference time domain and finite
volume time domain methods are explored for a variety of
systems of interest in musical acoustics, including brass instruments, percussion instruments based on thin plate and
shell vibration, and also their embeddings in 3D acoustic
spaces. Attention is paid here to implementation issues,
particularly on parallel hardware, which is well-suited to
time stepping methods operating over regular grids. Sound
examples are presented.

In this paper, a simple finite difference scheme for a rect-angular dynamic nonlinear plate, under free boundary con-ditions is presented. The algorithm is straightforward to program, and is capable of reproducing, to a first approx-imation, the behaviour of various percussion instruments whose timbre depends crucially on nonlinear effects (due to high-speed strikes), including transient pitch glides and the buildup of high-frequency energy. Though computa-tionally intensive, algorithms such as that presented here promise more faithful sound synthesis and, as with all phys-ical model inspired synthesis algorithms, require the specifi-cation of only a few, physically meaningful parameters. Full details of the algorithm, including the setting of boundary conditions and computational demands are provided. Nu-merical simulation results are presented.

The following paper investigates the effect of damping on a vibrating string. Using the finite difference method of approximating derivatives, an ordinary partial differential equation is solved to describe the motion of a string. This equation is modeled through the use of MATLAB with which an animation is created to show the motion in the x and y direction of the string with respect to time. There are four initial shapes used. Three shapes are in the form of free vibration. The shapes are described using piecewise equations. The final shape is in the form of forced vibration where the initial equation is equal to zero. From modeling the motion of the string using MATLAB, it is shown that increasing the damping on all four strings increases the maximum amplitude to the point where it rips the string and decreases the frequency of oscillations as well as the number of oscillations, while increasing the frequency on the forced vibration string increases the wave speed and amplitude.

The present work named «Finite difference method for the resolution of some partial differential equations», is focused on the resolution of partial differential equation of the second degree.
To establish this work we have first present and classify the partial differential equations. And follow we had present and describe the finite difference method.
After we had applied those methods for the numerical resolution of some partial differential equations
The implementation was done using Matlab language. The result shows that the finite difference method is very efficient for the resolution of partial differential equations, the reason why it is useful for scientists such us engineers, physician...
Key words: Partial differential equations, finite difference method, forward difference, backward difference, centre difference, Euler method, explicit method, implicit method, and Crank-Nicolson method.

There are many equations available in the literature using the results of CPT (cone penetration test) and SPT (standard penetration test) measurements to predict the end bearing capacity of drilled shafts in sand. However, there are few equations that use soil parameters, such as friction angle and elastic modulus, as input values. Also, these available equations usually overestimate the end bearing capacity, and at times show conflicting results with respect to the parameters they use. This paper describes a numerical procedure to overcome the above shortcomings. The results obtained in this study were compared with both experimental and numerical results available in the literature. A series of analyses is also conducted to assess the effects of various soil and pile parameters on the magnitude of end bearing capacity of piles embedded in sand. These parameters include diameter and length of pile, friction angle, and Poisson's ratio of soil. Based on the numerical analyses carried out in this study, a new equation is proposed to estimate the end bearing capacity of drilled shafts in sands. The results of the proposed equation are also compared with experimental data available in the literature on end bearing capacity of drilled shafts. The comparison shows better agreement of the suggested end bearing capacity equation with respect to other previously proposed methods.

An analytical study of four phenomena associated with the appearance of the electric arc is conducted: the variation of the temperature, the luminous flux, the sound pressure and the magnetic field. An experimental device has been designed to study experimentally the variations of these phenomena. The electric arc is primed in air and water depending on the supply voltage and the size of the fuse. Variations are captured by a sensor system and visualized in real time on programmed interfaces. We were able to measure the energy delivered by priming in air and water. The experimental results were exposed and compared to those of the analytical study. The simulation of the equations describing each phenomenon has good coherence with the experimental results.

One of the classic approaches to find out reasonable earth pressure applied to the retaining walls for seismic condition is Mononobe-Okabe method. Although this method has a wide range of application in dynamic analysis of the retaining walls in cohesionless material, the method has some limitations for cohesive soils and for dynamic computation with different seismic parameters. In this study an investigation was carried out to evaluate lateral earth pressures for retaining wall backfilled with clayey soils under seismic condition. A numerical model of a cantilever wall was conducted with finite difference method using staged construction and installation of different layers of backfill. A 0.6 m thick cantilever wall with the height of 6 meter and width of 4 meter was used for numerical modelling. Mohr-Coulomb constitutive model was used to simulate foundation and backfill soil material. Furthermore, a bilinear elasto-plastic model was used to characterize interaction of wall and soil medium. A full dynamic analysis was conducted to determine lateral earth pressure during the earthquake. Kobe earthquake (1995) was selected for applying dynamic loading. Various clayey backfill materials were examined for this study. The results were compared to lateral earth pressures estimated by limit-equilibrium Mononobe– Okabe method. A comparison between the results of the two methods indicates that there is an inconsistency between lateral earth pressure estimated by two methods and the reasoning is discussed in detail.

The study of mixed convection evaporation of an inclined wet flat plate was carried out. As the resolution of partial differential equations occupies an important place in the world of research, our study can serve as a reference. We have proposed two methods for understanding and solving the physical phenomena interfering in mixed convection. The first being the semi-analytical resolution allowed us to analyse and to study the preponderance of natural convection and then forced convection. The other is the numerical resolution by the finite difference implicit method which displays results similar to the first, just by varying the Richardson number. The influence of the inclination is also presented in each case. Mass and heat transfers correspond to forced convection and natural convection, respectively. The combination of the two gives rise to mixed convection. In our case, we chose air but the equations can be used for other fluids for heat exchange with a solid surface.

Inordinate localized temperature rise in the power transformer causes subsequent thermal breakdown. To prescribe the limits of short-term and long-term loading capability of a transformer, it is necessary to estimate the hottest spot temperature (HST) of transformer. This paper proposes the steady state temperature distribution of the power transformer windings. Oil in the transformer is assumed nearly incompressible and oil properties such as thermal conductivity, special heat, viscosity, and density vary with temperature. Finite difference method is used for numerical solution. The selected model for simulation is a 50KVA, 20 kV/400V oil natural, and air natural cooling (ONAN) power transformer. A Comparison of the author’s results with those obtained from finite integral transform and experimental test confirms the validity and accuracy of the proposed method.

A two-dimensional (2-D) multi-phase fluid flow model is proposed to investigate freak wave impact on a floating body. The model governed by the Navier-Stokes equations with free surface boundary conditions is solved by a Constrained Interpolation Profile (CIP)-based finite-difference method on a fixed Cartesian grid system. The free surface/interface boundary is captured by a Volume of Fluid (VOF)-type scheme, the Tangent of hyperbola for interface capturing/Slope weighting (THINC/SW), which is more accurate than the original THINC scheme. Physical experiments are performed for validation. Series of images of the impact events from high speed camera are recorded; wave elevations along the wave flume and impact pressure have been measured. Fairly good agreements are obtained from the qualitative and quantitative comparisons between numerical results and laboratory data regarding to distorted free surfaces and large amplitude body motions. Numerical solutions for the velocity field and pressure contour in the vicinity of the floating body are also presented for detailed analysis. Some discrepancies were observed for the predicted peak pressure. The effects of grid resolution on body motions and impact pressure are performed for error analysis. The comparison of the numerical results and measured data reveals that the proposed CIP-based model is capable of reproducing the nonlinear dynamics of the floating body for applications.

The digital emulation of analog audio effects and synthesis components, through the simulation of lumped circuit components has seen a large amount of activity in recent years; electromechanical effects have seen rather less, primarily because they employ distributed mechanical components, which are not easily dealt with in a rigorous manner using typical audio processing constructs such as delay lines and digital filters. Spring reverberation is an example of such a system-a spring exhibits complex, highly dispersive behavior, including coupling between different types of wave propagation (longitudinal and transverse). Standard numerical techniques, such as finite difference schemes are a good match to such a problem, but require specialized design and analysis techniques in the context of audio processing. A model of helical spring vibration is introduced, along with a family of finite difference schemes suitable for time domain simulation. Various topics are covered, including numerical stability conditions, tuning of the scheme to the response of the model system, numerical boundary conditions and connection to an excitation and readout, implementation details, as well as computational requirements. Simulation results are presented

Pile foundations are relatively vulnerable to lateral loads. During liquefaction-induced lateral spreading, this vulnerability is particularly conspicuous due to a loss of strength and stiffness in the liquefied soil. A nonlinear effective stress analysis incorporating an elastoplastic constitutive model based on Finite Difference Method (FLAC 2D program) was used to numerically simulate shake table experiment on piles in laterally spreading soils. The soil-pile interaction has been properly considered by using interface elements. The main objective of this paper is to assess the accuracy of a 2D numerical simulation of physical models in predicting the dynamic response of pile foundations and to identify the capability of 2D numerical simulation for 3D effects such as shadow and neighboring effects in pile groups without a pile cap. Results are presented and discussed, in which the obtained response from the simulation is compared to that measured in the test. For the single pile, a fairly good agreement was observed between computed and measured results. It was also found that the shadow and neighboring effects reduced lateral load on the piles by few percent of difference compared with experimental results.

This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.

In spite of Runge-Kutta method is the most used by scientists and engineers, it is not the most powerful method. In this paper, a comparative study between Piece-wise Analytic Method (PAM) and Runge-Kutta Methods is introduced. The result of comparative study shows that PAM is more powerful and gives results better than Runge-Kutta methods. PAM can be considered as a new step in the evolution of solving nonlinear differential equations.

Cet article présente une étude de l'évaporation par convection mixte d'une plaque plane humide inclinée soumise à une densité de flux de chaleur constante. L'écoulement de l'air, les transferts de chaleur et de masse sont régis par les équations de continuité, de mouvement, d'énergie et de diffusion dont les approximations au niveau de la couche limite sont appliquées. La résolution des équations se procède comme suit : adimensionnalisation, application de la méthode implicite des différences finies et programmation sur Matlab. Notre travail se termine avec la présentation des résultats sur l'influence de du nombre de Richardson et de l'inclinaison pour la vitesse, la température, la concentration et les coefficients d'échanges associés à la convection mixte. Mots clés : convection mixte, nombre de Richardson, plaque plane inclinée, transfert de chaleur, transfert de masse, adimensionnalisation, différences finies Abstract This paper presents a study of the evaporation by mixed convection of an inclined wet flat plate subjected to a constant heat flux density. Air flow, heat and mass transfers are governed by the equations of continuity, motion, energy and diffusion to which boundary layer approximations are applied. Adimensionnalisation, implicit method for finite differences and programming on Matlab are used to solve the equations. Our work ends with the presentation of results about the influence of Richardson's number and flat's inclination for velocity, temperature, concentration and coefficients of exchange associated with mixed convection.

Minimum structural fire resistance imposed by prescriptive design methods is usually of lesser value than the actual performance of an individual building under fire. The behaviour of the entire structure as a whole is not considered. Real fire conditions are not taken into account. Full-scale fire tests are also proven not to be economically feasible as standard procedure.
The work presented in this dissertation follows a performance-based approach, with the purpose of arriving at a numerical model capable of simulating the structural response of a building under fire conditions.
A field model for a compartment fire was created with computational fluid dynamics software Fire Dynamics Simulator. A ten-storey previously designed steel-concrete composite building was modelled using commercially available finite element analysis software ABAQUS. The effects of fire from the field model were simulated with a thermal-stress analysis.

This work is concerned with the numerical simulation of percussion instruments based on physical principles. Three novel modular environments for sound synthesis are presented: a system composed of various plates vibrating under nonlinear conditions, a model for a nonlinear double membrane drum and a snare drum. All are embedded in a 3D acoustic environment. The approach adopted is based on the finite difference method, and extends recent results in the field. Starting from simple models, the modular instruments can be created by combining different components in order to obtain virtual environments with increasing complexity. The resulting numerical codes can be used by composers and musicians to create music by specifying the parameters and a score for the systems. Stability is a major concern in numerical simulation. In this work, energy techniques are employed in order to guarantee the stability of the numerical schemes for the virtual instruments, by imposing suitable coupling conditions between the various components of the system.

Before presenting the virtual instruments, the various components are individually analysed. Plates are the main elements of the multiple plate system, and they represent the first approximation to the simulation of gongs and cymbals. Similarly to plates, membranes are important in the simulation of drums. Linear and nonlinear plate/membrane vibration is thus the starting point of this work. An important aspect of percussion instruments is the modelling of collisions. A novel approach based on penalty methods is adopted here to describe lumped collisions with a mallet and distributed collisions with a string in the case of a membrane. Another point discussed in the present work is the coupling between 2D structures like plates and membranes with the 3D acoustic field, in order to obtain an integrated system. It is demonstrated how the air coupling can be implemented when nonlinearities and collisions are present. Finally, some attention is devoted to the experimental validation of the numerical simulation in the case of tom tom drums. Preliminary results comparing different types of nonlinear models for membrane vibration are presented.

Brass instrument modelling and synthesis poses many challenges, regardless of the type of numerical techniques employed; this is particularly true of the instrument bore, when boundary layer losses and nonlinear effects are included. Because the system is well modelled in 1D, finite difference time domain (FDTD) methods are a good match, and allow direct modelling for general instrument bores without the need for specialized filter structures, or calibration from measured impedance curves. The framework is well-suited for an extension to the case of fully nonlinear wave propagation, which is the ultimate goal of this work.
In this paper, a simple FDTD scheme is applied to the brass instrument bore, in the context of sound synthesis, including viscous losses, and acoustic radiation. The extension to the case of fully nonlinear propagation is also discussed. Sound synthesis examples will be presented. .

In this study, the idea of randomly generated grids presented for solving two-dimensional partial differential equations using the finite difference method. The finite difference method is based on meshes usually called grids. There is no rule for generating meshes. In this paper, we are using MATLAB code for generating random grids. The numerical solution of the engineering model using the finite difference method over the uniform, randomly generated grids are compared and analyzed. It is concluded that the novel method that is "randomly generated grids" is converging solution-wise, iteration wise and computational time-wise. This research contributes method, which is random meshes to solve the partial differential equation.

Un elemento finito lineal con sección transversal constante puede adoptar cualquier orientación en el plano y sus extremos o nodos lo ligan al resto de los elementos. La energ a cinética (T ) y potencial (V ) de un elemento elástico dinámico son el basamento en la implementación del principio de Hamilton para la definición de un elemento finito. La definición de la energía cinética y potencial es el primer paso para la formulación variacional preliminar a la enunciación por elementos finitos que se utiliza para resolver, dígase, los problemas de mecanismos que se mueven en el plano utilizando la ecuación de Hamilton. El objetivo general consistió en definir la ecuación del movimiento de un elemento finito lineal plano elástico dinámico utilizando la ecuación de Hamilton, a partir de la lagrangiana (T –V ) obtenida con el uso de un polinomio de quinto y uno de primer grados, con ocho grados de libertad, cuatro en cada nodo, que representaron las deformaciones: axial (u(x)), transversal (w(x)), pendiente ((dw(x)/dx)) y curvatura ((d2w(x)/dx2)). La deformación debido al cizalleo transversal, insignificante comparado con la deformación flexional y la axial, la inercia rotatoria y las fuerzas friccionales en las uniones,fueron desestimadas con el fin de producir un elemento amigo. Los objetivos específicos fueron producir: (a) la matriz de masa de traslación [MD], (b) la matriz giroscópica de traslación [AD], (c) la matriz de rigidez total de traslación [KD], y (d) el vector de deformación (S). Como resultado se forjó la ecuación del movimiento de un elemento finito lineal plano elástico dinámico [MD]( ¨ S) − 2¨_[AD]( ˙S ) + {[K] − _˙2[MD] − ¨_[AD]}(S) = (Q) . Se concluyó que la ecuación obtenida variacionalmente con la aplicación del principio de Hamilton es un modelo cuyo procedimiento puede ser utilizado cuando se requiera aumentar el número de grados de libertad del modelo.

The present paper models the fundamental problems of fluid flow
using a discretely improved finite difference method on a staggered
computational grid. The developed finite difference formulation is applied to
well-established benchmark problems, namely, the lid-driven cavity flow, the
developing laminar flow in a straight rectangular duct and the backward-facing
step flow. Excellent agreements have been found for all cases. Also, this
approach has successfully handled the pressure of the flow that has been long
considered as one of the main problems in using the finite difference method.

Stress distribution in anisotropic rock medium especially shales are very complex. This anisotropic behavior has a dramatic effect on wellbore stability during fracturing operation to extract oil. To overcome a high quality drilling in the medium, understanding of stress distribution and deformation regime is required. Different parameters of rock medium such as pore pressure, rock strength, and in-situ stresses make a crucial role in evaluating the wellbore stability. The Bakken Formation in Williston Basin, North Dakota, has different layers of shales with anisotropic geomechanical parameters such as Poisson's ratio, Young's modulus, shear modulus, and rock strengths based on the laboratory experiments. Although vertical drilling is the most frequent drilling method, it is not an efficient method for oil production; thus inclined wells are inevitable. In this study, mechanical performance of the inclined drilling method was investigated to fill the current lack of knowledge about its effect on wellbore stability in this anisotropic formation. The elastic anisotropic geomechanical model based on Mohr-Coulomb failure criteria was applied to analyze inclined wellbore stability in Bakken Formation shale layers using Finite Difference Method. To study the mechanical response of drilling in the anisotropic medium, stress distribution, deformation and plastic zones of the rock adjacent to the hole were examined. In addition, the effects of different inclination angles of the wellbore versus depth were studied. Results indicate a correlation between wellbore deformations as the drilling inclination angle changes from vertical to inclined orientation. Stress contours and maximum principle stress distribution indicate an extended zone of higher stresses around the symmetric line of wellbore.

For interface-tracking simulation of incompressible two-phase fluids with high density ratios, a new numerical method was proposed by combining Navier-Stokes equations with a phase-field model based on a van der Waals-Cahn-Hilliard free-energy theory. The method was applied to several benchmark problems. Major findings are as follows: (1) The volume flux derived from a local chemical potential gradient in the Cahn-Hilliard equation leads to accurate volume conservation, autonomic reconstruction of gas-liquid interface, and reduction of numerical diffusion and oscillation. (2) The proposed method gave good predictions of pressure increase inside a bubble caused by the surface tension force. (3) A single liquid drop falling in stagnant gas and merging into a stagnant liquid film was successfully simulated.