Use Of Finite Difference Method Research Articles

Development of Two-Dimensional Thermal Analysis Code for the Analysis of 3D Printed PLA Parts

Fused Deposition Modelling is one of the main 3D printing methods to manufacture plastic parts. The strength of the printed part by FDM is dependent on polymer diffusion between printed layers. The polymer diffusion between two neighboring layers occur not only during the extrusion of the hot top layer, but also during the production of consecutive layers due to thermal conduction. The heat diffusion from upper layers enhances the curing of polymers, which consequently affects the strength of the part. Therefore, the history of the temperature variations - curing time and curing temperature - should be analyzed to predict the strength of the part. The goal of this study is to develop a two-dimensional transient thermal analysis solver for the investigation of time-dependent thermal changes during the printing process. This solver is developed with the use of finite difference method employed under implicit scheme. The transient temperature pattern is qualitatively compatible to the experimental results in literature. The solver can be utilized for further thermal analyses to correlate temperature, polymer diffusion and strength with the inclusion of deposition path in the third dimension.

Acoustic and thermal numerical modelling of transcranial ultrasound thermal therapy

Ultrasound has long been recognized as a therapeutic tool with potential to treat disorders throughout the brain. However, the attenuation and distortion of acoustic waves caused by the skull has been a significant challenge to the development and clinical application of transcranial ultrasound therapies. Numerical models, such as the use of finite difference method, can be prohibitively expensive in required computational resources. While some numerical models can be computationally prohibitive, the ray-tracing modelling method used in the CIVA Healthcare simulation platform has been shown to be an efficient and computationally inexpensive method of modelling transcranial focused ultrasound, and can be coupled to Pennes’ bioheat transfer equation to model temperature in biological tissue. The present work quantifies the sensitivity of transcranial ultrasound to clinically relevant uncertainties in the physical properties of the skull (i.e., shear conversion, wave speed), and combines the ray-tracing acoustic model with a numerical thermal model to provide a versatile and unified software platform for use in clinical ultrasound treatment planning. Results are compared with experimentally measured thermal rise from transcranial focused ultrasound using a clinical Insightec ExAblate 650 kHz 1000-element system, and shown to exhibit a high degree of accuracy. [This work is supported by the Focused Ultrasound Foundation.]

Optimal design of high-performance aerostatic equipment based on finite difference method with flux error feedback

This paper proposes an aerostatic lubrication model with the use of finite difference method, and the solution of the model is designed with the combination of flux error feedback and grid parameter optimization. The model and solution are validated by the performance test on the slider with the diameter of orifice at 50 and 200 μm. Finally, the proposed model is applied to the optimal design of the aerostatic turntable, and the type, parameter and configuration of the restriction are determined by the optimal results. To guarantee the accuracy of key parts, the ultra-precision turning technology is applied to manufacture of turntable, and the runout of the end face meets the design requirements through the verification. The proposed model and solution are significant to the analysis and design of high-performance aerostatic equipments.

Prototyping a numerical model coupled with remote sensing for tracking harmful algal blooms in shallow lakes

In this paper, we present the design, development and test cases of LakeM, a software package for modeling the transport pathways of harmful algal blooms in shallow lakes. LakeM integrates implementation of numerical methods and data visualization for the purpose of facilitating the communication between modelers and decision makers. By making use of Finite Difference Method, LakeM is able to numerically solve the Shallow Water Equations on a staggered grid. The core of LakeM consists of two modules, the hydrodynamic and tracer module, which are run separately. The hydrodynamic module simulates the surface level and water movement, and the tracer module simulates the movement of harmful algal blooms subject to advection and dispersion processes. Most of the LakeM programs are developed based on the Python language in an object-oriented fashion, which can be readily adapted to changing requirements. Computationally intensive programs are written in Fortran routines for interoperating with Python. Model validation is performed by comparing model predictions to analytical solutions. Additionally, the capability and effectiveness of LakeM are illustrated by a case study of modeling algal bloom transport in Taihu Lake. The results have shown good agreements with analytical solutions and available remote sensing images. The model is efficient, robust, and can be used to aid environmental managers in developing emergency response actions for HABs. Besides that, the development of LakeM offers a design pattern for future model development efforts in tracking HABs.

Impact of heat generation/absorption on the unsteady magnetohydrodynamic stagnation point flow and heat transfer of nanofluids

Purpose The purpose of this study is to describe the unsteady three-dimensional magnetohydrodynamic stagnation point flow of nanofluids with heat generation/absorption. Design/methodology/approach The comprehensive numerical simulations in this study accommodate a physical insight into the heat transfer and flow problem. The use of finite difference method through the bvp4c function in Matlab provides the numerical results and graphical illustrations for the heat transfer rate and shear stress. Findings Dual solutions are discovered in this study. Thus, stability analysis is implemented and the first solution complies the stability behavior. Silver nanoparticles dominate the highest thermal conductivity. Accretion of the rate of heat transfer is obtained with an increment in the magnitude of heat absorption, suction parameter and nanoparticle volume fraction. A stronger magnetic field and larger unsteadiness parameter contribute to the increase of the surface shear stress. Practical implications Many practical fluid mechanics problems involve the time-dependent element. Practically, an unsteady flow of nanofluid can be implemented in the micro-manufacturing, periodic heat exchanges process, nano drug delivery system and nuclear reactors. Originality/value In spite of numerous studies on the unsteady flow, none of the researchers combined the effect of heat generation/absorption and magnetic field in the nanofluid model. The behavior of the flow and heat transfer have been analyzed thoroughly with the variations in the unsteadiness parameter, heat source/sink and nanoparticle volume fraction. Moreover, the discovery of dual solutions in this model strengthens the novelty of this study. Subsequently, the implementation of stability analysis leads to a remarkable revelation where the first solution is found to be stable.

Natural convection in the annulus bounded by two wavy wall cylinders having a chemically reacting fluid

We study the flow characteristics of natural convection induced by exothermic reaction in an annulus bounded by two wavy wall cylinders. With a view to achieving the target the dimensionless equations are transformed for the present problem using suitable functions. We solve the new system of equations with the use of finite difference method. Numerical results demonstrate that an increase of the number of crests or troughs of the inner or outer walls of the enclosed domain considerably reduces the strength of streamlines and the maximum temperature. Due to the augmentation of the Frank-Kamenetskii number the intensity of the streamlines and the maximum temperature are found to increase but the minimum value of the concentration decreases. When the Rayleigh number for thermal diffusion is increased, the magnitude of streamlines noticeably increases, whereas there is little effect on the isotherms and isolines of concentration. For larger value of the amplitude of the inner or outer cylinder walls the intensity of streamlines considerably decreases, but the isotherms and the isolines of concentration change a little bit. The remarkable findings are that when Le = 0.5 and 1.0, the inner and outer major vortices contain two and three eyes respectively. But the streamlines for Le = 2.0 show distinct characteristics from those for Le = 0.5 and 1.0. In this case, there occur two elongated and two elliptic vortices adjacent to the inner and outer cylinder walls. The inner elongated vortices hold a single eddy whereas the elongated vortices neighboring the outer cylinder consist of two eddies. Contrary to this, in presence of thermal convection or combined convection for thermal and mass diffusion the inner and outer major vortices have two and three eddies respectively. However, the convection due to mass diffusion yields an astonishing result. Two vortices are recognized near the outer wall while a single vortex is observed adjacent to the inner wall. Both of the inner and the outer elongated vortices consist of two eyes. It is worthy of mentioning that the concept of constructing such a geometry by means of simple transformations might encourage the researchers to think for generating a more complex geometry. Moreover, these results could be useful for the advancement in science, engineering and technology.

Boundary-Layer Separations of Mixed Convection Flow Past an Isothermal Circular Cylinder

This study reveals the boundary-layer separations of unsteady mixed convection flow of incompressible fluid past a circular cylinder which is kept at a constant temperature. The governing equations are made dimensionless using an appropriate transformation. These equations are solved with the use of finite difference method. We also solve the problem using perturbation method for small time and local non-similarity method for long time. Results show that numerical solutions agree well with the series solutions obtained by perturbation method for small time and asymptotic method for long time. For an increase of Richardson number and Prandtl number, the local friction factor and the rate of heat transfer increases. Besides, the streamlines and isotherms reveal that when the Richardson number and the Prandtl number are increased, the point of separation of boundary-layer reduces and the size of vortex decreases. Moreover, the larger values of Richardson number thicken the thermal boundary layer whereas the reverse characteristic is observed for higher Prandtl numbers.

Inverse estimation of boundary conditions with forced convection heat transfer in two dimensional channels

In this paper, the inverse estimation of boundary conditions in a parallel plates channel with forced convection heat transfer is studied. The direct problem involves solving energy equation which is solved by use of finite difference method. A hydrodynamically fully developed and thermally developing laminar flow is considered. To minimize the objective function, heuristic optimization algorithms includes the particle swarm optimization and the gravitational search algorithms are used. The objective function is expressed by the sum of square residuals between measured and estimated temperatures. At the end, the results of solving the direct problem are validated by previous research. Then, the ability of optimization algorithms mentioned in inverse estimation of boundary conditions is discussed. The results indicate that the particle swarm optimization algorithm has a high accuracy in estimating boundary conditions, but in the gravitational search algorithm, the convergence rate becomes constant after the first few iterations. Furthermore, the effect of sensors number in the boundary conditions estimation indicates that the increase in the number of sensors improves the accuracy of estimations.

Numerical study of the transient processes in a long pipeline caused by its depressurization

A mathematical model for mass and momentum transfer in a long gas pipeline under its depressurization has been constructed. Nonlinear problems for transient processes in the pipeline under a local depressurization have been formulated. With the use of finite difference method, the nonstationary distributions of pressure and flow rate in the pipeline have been studied for various regimes of compressors operation. The obtained results can be used for modeling gas-dynamic processes in main gas pipelines during their accidental depressurization, forecasting of possible losses, and decision making on how to cure the accident and minimize its consequences.

Tuning framework for stencil computation in heterogeneous parallel platforms

Image processing and computer vision applications are usually complex in terms of the large amount of processed data and high computation loads. To cope with this, optimization techniques and high-performance hardware platforms are required. Since these applications present many opportunities for parallelism, heterogeneous parallel platforms (HPPs) are an interesting choice, offering a good balance between high computation capabilities and flexibility to handle a large spectrum of application features. Applications such as image filtering and edge detection make extensive use of finite difference method to solve partial derivative equations, which computational pattern is called stencil computation. Stencil computations are known as memory-bound, so that reducing high-latency memory access becomes the biggest challenge to reach high performance. In this paper, we present our methodology as a basis of a performance tuning framework to optimize the implementation of multiple stencil computations on HPPs. Results show that our approach outperforms SDK-based methodologies, improving performance. Moreover, using the proposed approach, the developer has the ability of investigating efficiently the performance of the stencil computations before implementing actual code on the target platforms.

Effect of Radiation on Unsteady MHD Flow of a Chemically Reacting Fluid Past a Hot Vertical Porous Plate: A Finite Difference Approach

The paper deals with the effect of radiation on unsteady free convention flow of a viscous, incompressible, electrically conducting fluid past an infinite hot vertical porous plate embedded in porous medium. A magnetic field of uniform strength is applied normal to the fluid flow. Chemical reaction, viscous dissipation and Heat generation/absorption effects are included. Temperature of the plate is assumed to be span wise cosinusoidally fluctuating with time. In order to establish a mathematical convenience of converging the solution at a finite point ( 1 → η ), the governing equations of the problem are transformed to a new system of co-ordinates and then resulting equations are reduced to coupled non-linear ordinary differential equations of zeroth and first order, using appropriate perturbation technique. These zeroth and first order equations are solved numerically by making use of finite difference method and the simulation is carried out by coding in C-Program. Graphical results for velocity, temperature and concentration fields are presented and Original Research Article

Analysis of the buckling of rectangular nanoplates by use of finite-difference method

In recent years nanostructures have been widely used in industry, for example in nanoelectromechanical systems (NEMS); knowledge of the mechanical behavior of nanostructured materials is therefore important. In the work discussed in this paper, the non-dimensional buckling load of rectangular nano-plates was determined for general boundary conditions. Non-local theory was used to derive the governing equation, and this equation was then solved, by use of the finite-difference method, by applying different combinations of boundary conditions. To verify the proposed method, the non-dimensional buckling load determined for a simply supported plate was compared with results obtained by use of local theory and with results reported in the literature. When the method was used to calculate the buckling load of nano-beams, results were in good agreement with literature results. As a novel contribution of the work, non-symmetric boundary conditions were also studied. The non-dimensional buckling load was obtained for several values of aspect ratio, non-local variables, and different types of boundary condition. For better understanding, mode shapes are also depicted. The finite-difference method could be a powerful means of determination of the mechanical behavior of nanostructures, with little computational effort, and the results could be as reliable as those obtained by use of other methods. The ability to deal with a combination of boundary conditions illustrates the advantages of this method compared with other methods.

Correlational Research of Bionics Design of Dry Gas Face Seal Groove

Based on statistical method and gas lubrication theory, the correlation mathematical models of outline constructional feature of a bird wing and face groove feature of a dry gas seal(DGS) are established, and the film pressure controlling equation of a DGS is set up. The membership grades and their calculation expressions used to define face groove geometrical feature of a DGS are presented based on fuzzy mathematics, and DGSs are classified according to the membership grades. The gas film pressure control equations are resolved by use of finite difference method, and the gas film stiffness of a DGS is obtained. The correlation between the constructional feature parameters of bird wing outlines and flight speed and the correlation between face groove geometric parameters and operational stability of a DGS at different speeds are comparatively analyzed. Results show that the correlations between the geometric constructional feature parameters of bird wing outlines or face grooves and their flight speeds or operational speeds are in moderate degree. That is to say long wings give a bird stability, and long narrow wings give a bird more stability at high flight speeds, and long grooves give a DGS stability, and long narrow grooves give a DGS more stability at high operational speeds, correspondingly. Results also reveal that the relationship between a bird biological model and a DGS engineering model exists exactly. The feasibility of bionic design of face groove based on the geometric characteristics of bird wing outlines is demonstrated. The important theory basis is provided for the bionic design of a DGS.

Analysis on Aerodynamic Performance of Horizontal Axis Wind Turbine Based on Nonlinear Wake Model

A computational method based on nonlinear wake model was established for horizontal axis wind turbines aerodynamic performance prediction. This method makes use of finite difference method to solve the integral differential equation of the model, the induced velocity of wake vortex can be calculated from equations and compared with the induced velocity of wake vortex in linear model. The comparison between the calculated results of wind turbine under axis flow condition, including tip vortex geometry and aerodynamic performance, and available experimental data shows that this method is suitable for wind turbine aerodynamic performance analysis. Finally, a series of numerical calculations were made to investigate the change of wake geometry and aerodynamic performance of the wind turbine when yawing and pitch angle increasing, which provide foundations for aerodynamic optimization design of horizontal axis wind turbines.

The Temperature Field Caused by Sphere Inclusion in Dielectric Irradiated by Single Pulse Laser

The thermal effect arisen from absorbing inclusions is the main factor which causes the damage of optical materials or component irradiated by the longer pulse duration laser. The unsteady heat conduction depends markedly on both the thermal properties of inclusions and the parameters of laser. Based on the differential equation of heat conduction, the temperature distribution caused by single absorbing inclusion is solved by use of finite difference method. The effect of the laser intensity and the pulse duration on temperature field is analyzed in detail. The result demonstrates that the smaller size inclusion and the smaller pulse duration cause relative safe thermal effect, consequently, the less probability to be damaged by thermal effect.

The Temperature Field Caused by Sphere Inclusion in Dielectric Irradiated by Multi-Pulse Laser

The thermal effect caused by absorbing inclusions irradiated by multi-pulse laser is different from that of single pulse laser. The temperature field induced by multi-pulse laser depends markedly on both inclusion size and pulse duration, and repetition interval of pulse. Based on the differential equation of heat conduction, the temperature field caused by single absorbing inclusion is solved by use of finite difference method. The effect of inclusion size, pulse duration and repetition interval of pulse on the evolution of temperature field at the center of inclusion and interface between inclusion and dielectric are discussed qualitatively.

Photoelastic‐numerical treatment of Beltrami‐Michell boundary value problems

AbstractOne of the main challenges of using photoelasticity has always been its complexity to determining all stress components, for it provides an unfinished solution. Indeed, the use of the Beltrami‐Michell Boundary Value Problem remains of practical interest for the study of the stress‐separation, when complemented by the photoelasticity analysis to acquire the Dirichlet conditions. The synergy of both is enhanced with the use of Finite Difference Method to work out a photoelastic‐numerical hybrid method for stress analysis under plane conditions. On the other hand, while for photoelasticity specially, annular disks under diametral compressive load are usually used as standard models allowing to check the performance of any developed method; because theoretical solutions exist. A comprehensive study has been carried out to show the photoelastic‐numerical method in order to investigate the in‐plane stress distribution using isochromatic values only on the boundaries of an annular object subjected to diametral compression. Compared with a reference work, the obtained results are more than concluding. The method is fast in the analysis for a low cost.

Evolution of emmitted light intensity in two-photon pumping random lasers

Based on the diffusive equation of incoherent-feedback random lasers，whose gain is modified by adding a saturation factor，the evolution of the emitted light intensity in two-photon pumping random lasers varying with time is studied by use of finite difference method. Numerical results show how the intensity of emitted light varies from the spontaneous radiation，which undergoes amplification at small-signal gain，to the steady output due to the saturation effect. It is noted that the nonlinear effect，e.g. saturation of gain，in such laser system can be achieved under low intensity conditions due to the disorder of media. Our results give a theoretical support to obtaining the steady output of the two-photon pumping random lasers.

Calculation of DC Corona Wind

Analytical-digital method of calculation the velocities of DC corona wind in wire-to-plain electrode system is discussed. Themethod is based upon an assumption that the electric field of the discharge is independent from the corona wind field, and, vice versa,this field is the function of electric field quantities of the discharge. Differential quantities of the electric field of DC corona dischargeare obtained analytically by using the Deutch‘s-Popkov‘s assumption of coincidence the directions of electrostatic field strength vectorand the field vector of corona discharge. The dependence of the force acting the elementary charge on the coordinates is essential andthe influence of any coordinate to the force must not be denied. The results of analysis of DC corona field differential quantities arechecked indirectly by comparison of theoretical and experimental voltampere characteristics. It‘s misalignment do not exceed 8 %. Theset of equations of corona wind represented by the Navier and Stokes equations and the equation of stream continuity are solved by useof finite difference method. The values of analytically determined components of spatial force acting the charge element in the nodes ofsquare net are being used in the process of iterative computation. Ill.3, bibl. 15 (in Lithuanian; summaries in Lithuanian, English andRussian).

A semi-closed-form solution for chloride diffusion in concrete with time-varying parameters

Reinforcement corrosion in concrete is a worldwide problem and billons of dollars are spent each year on repair and replacement of reinforced concrete structures because of corrosion damage of reinforcement. Chloride induced corrosion is the major type of corrosion encountered in reinforced concrete. The corrosion is initiated in reinforcement when chloride diffuses through the concrete cover and the chloride concentration at the reinforcement reaches a critical threshold level. Fick's second law of diffusion is commonly used for one-dimensional chloride diffusion. A closed-form solution based on error function is commonly used to predict the chloride concentrations, if the diffusion coefficient (D) and surface chloride concentration (Cs) remain constant. This solution is not applicable when D and Cs vary with time, which is the case in real-life situations. The finite difference or finite element methods with specialised software are generally used to solve such problems. This paper reviews existing methods and presents an alternative method based on a general closed-form solution with varying D but constant Cs. To incorporate the effect of varying Cs, an adaptation of the closed-form solution is presented. This solution can be used without the use of finite difference method or special computer software to predict the chloride concentrations.