Finite Difference Numerical Method Research Articles

3D dynamic simulations of spontaneous rupture propagation governed by different constitutive laws with rake rotation allowed

In this work we present a 3D Finite Difference numerical method to model the dynamic spontaneous propagation of an earthquake rupture on planar faults in an elastic half-space. We implement the Traction-at-Split-Nodes fault boundary condition for a system of faults, either vertical or oblique, using different constitutive laws. We can adopt both a slip-weakening law to prescribe the traction evolution within the breakdown zone or rate- and state-dependent friction laws, which involve the choice of an evolution relation for the state variable. Our numerical procedure allows the use of oblique and heterogeneous distribution of initial stress and allows the rake rotation. This implies that the two components of slip velocity and total dynamic traction are coupled together to satisfy, in norm, the adopted constitutive law. The simulations presented in this study show that the rupture acceleration to super-shear crack speeds occurs along the direction of the imposed initial stress; the rupture front velocity along the perpendicular direction is slower than that along the pre-stress direction. Depending on the position on the fault plane the orientation of instantaneous total dynamic traction can change with time with respect to the imposed initial stress direction. These temporal rake rotations depend on the amplitude of initial stress and on its distribution on the fault plane. They also depend on the curvature and direction of the rupture front with respect to the imposed initial stress direction: this explains why rake rotations are mostly located near the rupture front and within the cohesive zone.

Modeling the dynamic rupture propagation on heterogeneous faults with rate- and state-dependent friction

We investigate the effects of non-uniform distribution of constitutive parameters on the dynamic propagation of an earthquake rupture. We use a 2D finite difference numerical method and we assume that the dynamic rupture propagation is governed by a rate- and state-dependent constitutive law. We first discuss the results of several numerical experiments performed with different values of the constitutive parameters a (to account for the direct effect of friction), b (controlling the friction evolution) and L (the characteristic length-scale parameter) to simulate the dynamic rupture propagation on homogeneous faults. Spontaneous dynamic ruptures can be simulated on velocity weakening (a b) can arrest a dynamic rupture, but can be driven to an instability if suddenly loaded by the dynamic rupture front. Our simulations provide a picture of the complex interactions between fault patches having different frictional properties and illustrate how the traction and slip velocity evolutions are modified during the propagation on heterogeneous faults. These results involve interesting implications for slip duration and fracture energy.

Study of the influence of heat exchangers' thermal resistances on a thermoelectric generation system

Abstract In this paper, a computational study of the influence of the heat exchangers' thermal resistances (in both the hot and cold side) on the efficiency of a thermoelectric generation device has been carried out. For this purpose, a computational model has been developed. This model uses the numerical method of finite differences to simulate the performance of the thermoelectric generation system, including the heat exchangers, the heat source and the heat sink. The accuracy of this computational model was experimentally verified, by constructing and testing a prototype. It was obtained that the maximum error between experimental and simulated values of electric power generated is lower than 5%. The generation of thermoelectric power, using as heat source the heat of the smoke from a paper mill's combustion boiler, has been studied too. The results demonstrated that it is possible to generate about 1 kW per meter of chimney height, that is, about 300 W/m 2 . Therefore, it can be stated that this device has good prospects for the future.

“Extended Black” term structure models

This paper examines “Extended Black” term structure models (EBTSM), which are multi-factor extensions of the one-factor Black model (Black, F., 1995. Interest rates as options. Journal of Finance 50, 1371–1376). EBTSM are not affected by the admissibility restrictions that plague canonical affine models. EBTSM encompass quadratic models, but unlike in quadratic models bond yields are sufficient statistics to infer the latent factors driving the short interest rate. EBTSM are amenable to econometric estimation despite the need to solve bond pricing equations through finite difference numerical methods. Estimation through the Iterated Extended Kalman filter reveals that a two-factor EBTSM fit well the observed cross section and time series of Japanese Government bond yields. A three-factor EBTSM is also proposed.

Fabry–Perot resonance in slit and grooves to enhance the transmission through a single subwavelength slit

The enhanced transmission through a single subwavelength slit surrounded with periodic grooves has been simulated by a numerical finite-difference time-domain method. We simulated the transmission as functions of slit depth (film thickness) and groove depth, and also the field distributions in the slit and grooves. The transmission varies periodically as functions of slit depth or groove depth, just like the field distribution in the slit or in the groove. These results imply that the resonance mechanisms in the slit and the grooves are the same and we employ a Fabry–Perot like resonance effect to explain it.

Analysis of Laminar Falling Film Condensation Over a Vertical Plate With an Accelerating Vapor Flow

Laminar falling film condensations over a vertical plate with an accelerating vapor flow is analyzed in this work in the presence of condensate suction or slip effects at the plate surface. The following assumptions are made: (i) laminar condensate flow having constant properties, (ii) pure vapor with a uniform saturation temperature in the vapor region, and (iii) the shear stress at the liquid/vapor interface is negligible. The appropriate fundamental governing partial differential equations for the condensate and vapor flows (continuity, momentum, and energy equations) for the above case are identified, nondimensionalized, and transformed using nonsimilarity transformation. The transformed equations were solved using numerical, iterative, and implicit finite-difference methods. It is shown that the freestream striking angle has insignificant influence on the condensation mass and heat transfer rates, except when slip condition is present and at relatively small Grl/Re2 values. Moreover, it is shown that increasing the values of the dimensionless suction parameter (VS) results to an increase in dimensionless mass of condensate (Γ(L)/(μl Re)) and Nusselt number (Nu(L)/Re1/2). Thus, it results in an increase in condensation mass and heat transfer rates. Finally, it is found that the condensation and heat transfer rates increase as Jakob number, slip parameter, and saturation temperature increase. Finally, the results of this work not only enrich the literature of condensation but also provide additional methods for saving thermal energy.

Investigating the influence of geometric parameters on gas bearing static stiffness with experimental and numerical methods

High‐speed spindle systems designed with gas bearings are widely used in electronic industries for printed circuit board drilling and semiconductor wafer cutting. This study focuses on investigating the stiffness of the gas bearing when the spindle itself is suspended under the supplied air pressure without rotating. The investigation utilized both the experimental ball impact test method and the numerical finite difference method to solve the modeling Reynolds equation for the stiffness evaluation respectively. The most frequently used condition of the 6 bar supply pressure is investigated experimentally for stiffness at the beginning of the study. It is then compared with the numerical results as derived from the theoretical calculated pressure distribution of the gas bearing. The validated numerical scheme is applied further to calculate the stiffness at other supply pressure and geometric variations. Moreover, it is also investigated for influences on the gas bearing stiffness by making a series of changes in the gas bearing design such as different orifice diameters, different length/ diameter (L/D) ratios, and varying types of gas restrictors including the pocketed orifice and inherent orifice.

High-order derivatives of Green’s functions in magneto-electro-elastic materials

Three-dimensional Green’s functions and their arbitrary order derivatives in general anisotropic magneto-electro-elastic materials are derived by using Fourier transform. They are analytical solutions expressed in line integral forms, and can be evaluated by a standard numerical integration method. With this method, we can obtain results with high accuracy. Besides, a numerical finite difference method is also given to evaluate the second-order derivatives quickly. When setting the appropriate material coefficients to zero, the piezoelectric, piezomagnetic, and purely anisotropic elastic Green’s functions and their derivatives can all be obtained from the current solutions.

Development of a hybrid refrigerator combining thermoelectric and vapor compression technologies

A domestic refrigerator with three compartments has been developed: refrigerator compartment, at 4 °C (vapor compression cooling system); freezer compartment, at −22 °C (vapor compression cooling system); and a new super-conservation compartment, at 0 °C (thermoelectric cooling system). The thermoelectric system designed for the super-conservation compartment eliminates the oscillation of its temperature due to the start and stop compressor cycles, obtaining a constant temperature and thus, a better preservation of the food. For the design and optimization of this application, a computational model, based in the numerical method of finite differences, has been developed. This model allows to simulate the complete hybrid refrigerator (vapor compression–thermoelectricity). The accuracy of the model has been experimentally checked, with a maximum error of 1.2 °C for temperature values, and 8% for electric power consumption. By simulations with the computational model, the design of the refrigerator has been optimized, obtaining a final prototype highly competitive, by the features on food preservation and power consumption: 1.15 kW h per day (48.1 W) for an ambient temperature of 25 °C. According to European rules, this power consumption value means that this new refrigerator could be included on energy efficiency class B.

High-frequency response of a calculation methodology for gas dynamics based on Independent Time Discretisation

A calculation methodology to solve the one-dimensional governing equations system is presented. This calculation methodology is based on the Independent Time Discretisation (ITD) of the ducts composing the system. The purpose is the improvement of the trade-off between the accuracy and the computational cost that the current 1D gas dynamic models can yield. The ITD methodology is applied to the specific problem of noise prediction in internal combustion engines in order to evaluate its performance in the frequency domain. The application of the ITD methodology to the well-known acoustic configurations which are representative of the main attenuation mechanisms in commercial mufflers shows its ability. The potential is evaluated in terms of reduction of the computational cost and the accuracy and robustness provided by the results as a function of the spatial mesh size and the family of finite difference numerical method applied.

Analysis of transient heat transfer in multilayer thin films with nonlinear thermal boundary resistance

Transient energy transport in thin-layer films with a nonlinear thermal boundary resistance is analyzed theoretically within the framework of the dual-phase-lag heat conduction model. An iterative finite difference numerical method is used and is verified using a derived semi-analytical solution of the problem. Effects of the thermo-physical properties on energy transport when a two-layer film is exposed to a thermal pulse of certain duration and strength are presented. The thermal boundary resistance, the heat flux and temperature gradient phase lags and the thermal conductivities and heat capacities all are important factors that characterize energy transport through the interface and the temperature distribution in the two layers. The maximum interfacial temperature difference that takes place in the transient process of thermal pulse propagation is found to be the proper choice to measure the perfect-ness of the interface with a finite thermal boundary resistance. The results show that even with high values of the thermal boundary resistance the maximum interfacial temperature difference can be very small when the thermal pulse propagates from a high-thermal conductivity and heat capacity layer to a low-thermal conductivity and heat capacity layer. For a certain range of the thermal conductivities and heat capacities, the maximum interfacial temperature difference approaches zero even with high values of the thermal boundary resistance. Thermal conductivities and heat capacities are much more important in characterizing transient heat transfer through the imperfect interface than the phase lags of the heat flux and temperature gradient.

Numerical Study of Reinforced Soil Segmental Walls Using Three Different Constitutive Soil Models

A numerical finite-difference method (FLAC) model was used to investigate the influence of constitutive soil model on predicted response of two full-scale reinforced soil walls during construction and surcharge loading. One wall was reinforced with a relatively extensible polymeric geogrid and the other with a relatively stiff welded wire mesh. The backfill sand was modeled using three different constitutive soil models varying as follows with respect to increasing complexity: linear elastic-plastic Mohr-Coulomb, modified Duncan-Chang hyperbolic model, and Lade's single hardening model. Calculated results were compared against toe footing loads, foundation pressures, facing displacements, connection loads, and reinforcement strains. In general, predictions were within measurement accuracy for the end-of-construction and surcharge load levels corresponding to working stress conditions. However, the modified Duncan-Chang model which explicitly considers plane strain boundary conditions is a good compromise between prediction accuracy and availability of parameters from conventional triaxial compression testing. The results of this investigation give confidence that numerical FLAC models using this simple soil constitutive model are adequate to predict the performance of reinforced soil walls under typical operational conditions provided that the soil reinforcement, interfaces, boundaries, construction sequence, and soil compaction are modeled correctly. Further improvement of predictions using more sophisticated soil models is not guaranteed.

A Wavelet-Optimized Adaptive Grid Method for Finite-Difference Simulation of Wave Propagation

To accurately and efficiently model the complex spatiotemporal localization of wave propagation in heterogeneous media, we present a wavelet-optimized gridding scheme for the staggered-grid finite-difference (FD) numerical method. An average-interpolating (AI) wavelet transform with its wavelet coefficients obtained from linear combinations of samples, rather than from integrals, enables the gridding scheme constructed in the physical space to determine where to refine/coarsen the grid adapted to the complex spatial localization of problems and the relatively irregular sampling of seismic signals. The AI wavelet is a perfect transform with its wavelet coefficients decaying rapidly with increasing resolution. The wavelet scale parameter is related directly to the level of meshes. We can reconstruct the solution with good accuracy using a finite number of terms of wavelet expansion under some threshold sets for refinement/coarsening in a low- or high- degree of regularity. The adaptive meshes, in response to the heterogeneities of media and the resolution scales of wave fields, lead not only to a great saving of computation time and memory but also to an enabling method for solving problems with high accuracy in some local complex zones. Numerical examples demonstrate that the proposed wavelet-optimized adaptive FD scheme is efficient and accurate for wave propagation simulation.

The Visualization of Three Dimensional Brain Tumors’ Growth on Distributed Parallel Computer Systems

The main objective of this study is to visualize the brain tumors’ growth in three-dimensional and implement the algorithm on distributed parallel computer systems. The Partial Differential Equations (PDE) to solve the mathematical problem will be discussed in this study. The growth of the brain tumor through angiogenic process is described as parabolic model in partial differential equations. The discretization of the three-dimensional parabolic equations for the brain tumor’s growth mathematical model using a numerical finite-difference method will be implemented from the previous study of two dimensional model and thus a parallelization of algorithm simulation to computational resources based on high-performance computing systems will be used to generate the growth of the brain tumor in three dimensional. The study also includes an observation of the behaviour of the cells graphically and Parallel Virtual Machine (PVM) is used to communicate the platforms involved in the computational clusters. A comparison of sequential and parallel algorithm will be discussed and this study will address the major issues of the parallel computers performance in terms of efficiency, effectiveness, speedup, and temporality.

Optimisation of Plate Thickness Using Finite Difference Method

A finite difference numerical method of solving biharmonic equation is presented. The biharmonic equation and plate theory are used to solve a classical engineering problem involving the optimisation of plate thickness to minimise deformations and stresses in the plate. Matlab routines were developed to solve the resulting finite difference equations. The results from the finite difference method were compared with results obtained using ANSYS finite element formulation. Using the finite difference method, a plate thickness of 277 mm was obtained with a mesh size of 3 m and a plate thickness of 271 mm was obtained with a mesh size of 1 m., whiles using ANSYS finite element formulation, a plate thickness of 270 mm was obtained. The significance of these results is that, by using off-the-shelf general application tool and without resorting to expensive dedicated application tool, simple engineering problems could be solved. Keywords: Finite Difference (FD), Biharmonic Equation, Finite Element (FE), Plate Theory Journal of Science & Technology (Ghana) Vol. 28 (3) 2008: pp. 135-151

Introduction to computation and modeling for differential equations

An introduction to scientific computing for differential equations Introduction to Computation and Modeling for Differential Equations provides a unified and integrated view of numerical analysis, mathematical modeling in applications, and programming to solve differential equations, which is essential in problem-solving across many disciplines, such as engineering, physics, and economics. This book successfully introduces readers to the subject through a unique Five-M approach: Modeling, Mathematics, Methods, MATLAB, and Multiphysics. This approach facilitates a thorough understanding of how models are created and preprocessed mathematically with scaling, classification, and approximation, and it also illustrates how a problem is solved numerically using the appropriate mathematical methods. The book's approach of solving a problem with mathematical, numerical, and programming tools is unique and covers a wide array of topics, from mathematical modeling to implementing a working computer program. The author utilizes the principles and applications of scientific computing to solve problems involving: Ordinary differential equations Numerical methods for Initial Value Problems (IVPs) Numerical methods for Boundary Value Problems (BVPs) Partial Differential Equations (PDEs) Numerical methods for parabolic, elliptic, and hyperbolic PDEs Mathematical modeling with differential equations Numerical solution Finite difference and finite element methods Real-world examples from scientific and engineering applications including mechanics, fluid dynamics, solid mechanics, chemical engineering, electromagnetic field theory, and control theory are solved through the use of MATLAB and the interactive scientific computing program Comsol Multiphysics. Numerous illustrations aid in the visualization of the solutions, and a related Web site features demonstrations, solutions to problems, MATLAB programs, and additional data. Introduction to Computation and Modeling for Differential Equations is an ideal text for courses in differential equations, ordinary differential equations, partial differential equations, and numerical methods at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for researchers and practitioners in the fields of mathematics, engineering, and computer science who would like to refresh and revive their knowledge of the mathematical and numerical aspects as well as the applications of scientific computation.

Experimental and Modeling Investigation of Supercritical Extraction of Mannitol from Olive Leaves

AbstractThe experimental feasibility of mannitol extraction from olive leaves using supercritical carbon dioxide was investigated. The experimental data indicated that increasing the pressure from 200 to 350 bar and decreasing the temperature from 80 to 40 °C resulted in an enhancement of the extraction yield and reduced the partition coefficient significantly. In addition, increasing the extraction time from 10 to 90 min increased the extraction yield, while further increases up to 180 min did not cause any further change. Ethanol was utilized as an entrainer and the maximum extraction yield was obtained using 20 % of ethanol. Moreover, modeling of the supercritical fluid extraction was carried out with the relevant mass transfer mechanisms involved in the supercritical and solid phases, and the appropriate numerical method of finite difference. The numerical results show that the model with three adjustable parameters is capable of predicting the experimental data very well.

Sequential Algorithm of Parabolic Equation in Solving Thermal Control Process on Printed Circuit Board

This paper focuses on the implementation of sequential algorithms for the simulation of parabolic equation in solving the thermal control systems. The platform of the temperature behaviour prediction is based on printed circuit board. The numerical finite-difference method (FDM) is used to design the discretization of these partial differential equations. The results of finite-difference approximation are presented graphically. Numerical methods that are used in solving the problem are the method of Jacobi, Gauss-Seidel, Red-Black Gauss-Seidel, Successive Over Relaxation (SOR) and Red-Black Successive Over Relaxation. The numerical analysis is treated done in terms of time execution, computation complexity, number of iterations, accuracy and convergence rate.

Numerical analysis of pile lateral behavior of pile supported embankment

A finite difference numerical method was adopted to evaluate the pile lateral behavior of pile supported embankment. A published case history was used to verify the proposed methodology. By simulating the case history, the determination of parameters needed were verified. Then three embankments constructed on different ground conditions with different soil-pile relative stiffnesses were analyzed to study pile lateral behaviors including pile deflection and bending moment. The results show that pile deflections and bending moments induced by soil lateral deformation and embankment vertical load are different for piles at different positions under the same embankment. The relative stiffness between pile and soil affected by the properties of different reinforcing piles such as concrete pile and deep mixing method pile exert important effects on the pile lateral behavior and the pile’s failure modes. Consequently, it is necessary to consider the different piles lateral behaviors and possible failure modes at different positions and the different piles proprieties with different reinforcing methods in the embankment stability analysis.

THE MATRIX FORMULATION OF BOUNDARY INTEGRAL MODELING OF ELASTIC WAVE PROPAGATION IN 2D MULTI-LAYERED MEDIA WITH IRREGULAR INTERFACES

A semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces is presented. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave fields between arbitrary shapes of layers can be computed using the indirect boundary element (BEM) method. Like any standard BEM methods, the primary task of the BEM-based propagation matrix method (thereafter called PM–BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial boundaries due to limited model space. To our knowledge, such implementation has not appeared in the literature. Several examples are presented in this paper to demonstrate the effectiveness of this proposed PM–BEM method for modeling elastic waves in media with complex structure.