Finite-difference Calculations Research Articles

Two-dimensional time-domain finite-difference modeling for viscoelastic seismic wave propagation

Real Earth media are not perfectly elastic. Instead, they attenuate propagating mechanical waves. This anelastic phenomenon in wave propagation can be modeled by a viscoelastic mechanical model consisting of several standard linear solids. Using this viscoelastic model, we approximate a constant Q over a frequency band of interest. We use a four-element viscoelastic model with a trade-off between accuracy and computational costs to incorporate Q into 2-D time-domain first-order velocity–stress wave equations. To improve the computational efficiency, we limit the Q in the model to a list of discrete values between 2 and 1000. The related stress and strain relaxation times that characterize the viscoelastic model are pre-calculated and stored in a database for use by the finite-difference calculation. A viscoelastic finite-difference scheme that is second order in time and fourth order in space is developed based on the MacCormack algorithm. The new method is validated by comparing the numerical result with analytical solutions that are calculated using the generalized reflection/transmission coefficient method. The synthetic seismograms exhibit greater than 95 per cent consistency in a two-layer viscoelastic model. The dispersion generated from the simulation is consistent with the Kolsky–Futterman dispersion relationship.

Intensity correlations in metal films with periodic-on-average random nanohole arrays

We report detailed numerical studies based on three-dimensional finite-difference time domain computations of the intensity–intensity correlations in deliberately randomized, periodic-on-average systems. Correlation analyses are carried out in plasmonic thin films with nanohole arrays as a function of strength of disorder. We find that the intensity at certain uncharacteristic wavelengths remains strongly correlated with that in the periodic system, and these wavelengths do not match the global maxima of the periodic transmission spectrum. The study indicates that the strength of correlations is related to the pinning of the intensity to the holes. Since the intensity pinning is special characteristic of metals, the effect is only applicable in plasmonic systems.

Influence of the sputtering flow regime on the structural properties and magnetic behavior of Fe-Ga thin films (Ga ∼ 30 at.%)

In this paper we analyze the structure of Fe-Ga layers with a Ga content of \ensuremath{\sim}30 at.% deposited by the sputtering technique under two different regimes. We also studied the correlation between the structure and magnetic behavior of the samples. Keeping the Ar pressure fixed, we modified the flow regime from ballistic to diffusive by increasing the distance between the target and the substrate. X-ray diffraction measurements have shown a lower structural quality when growing in the diffusive flow. We investigated the impact of the growth regime by means of x-ray absorption fine structure (XAFS) measurements and obtained signs of its influence on the local atomic order. Full multiple scattering and finite difference calculations based on XAFS measurements point to a more relevant presence of a disordered $A2$ phase and of orthorhombic Ga clusters on the Fe-Ga alloy deposited under a diffusive regime; however, in the ballistic sample, a higher presence of $D{0}_{3}/B2$ phases is evidenced. Structural characteristics, from local to long range, seem to determine the magnetic behavior of the layers. Whereas a clear in-plane magnetic anisotropy is observed in the film deposited under ballistic flow, the diffusive sample is magnetically isotropic. Therefore, our experimental results provide evidence of a correlation between flow regime and structural properties and its impact on the magnetic behavior of a rather unexplored compositional region of Fe-Ga compounds.

Overflow-assisted laser machining of titanium alloy: surface characteristics and temperature field modeling

Underwater laser machining process is a promising method to cut materials with less thermal damage. A variation of underwater technique is overflow-assisted laser ablation. This process can introduce a higher thermal convection and more uniform water layer than the typical underwater method. Such characteristics can encourage the damage-free fabrication and also stabilize the laser ablation in water. In this study, cut profile and temperature distribution of workpiece induced by the overflow technique were investigated. Titanium alloy (Ti-6Al-4V) used as a work sample was grooved by a nanosecond pulse laser under different overflow conditions. The effects of laser power, laser repetition rate, and water flow velocity were experimentally and numerically examined. A clean and smooth cut surface can be fabricated when the overflow technique was used. Microcracks and porosities found on the laser-ablated area were also addressed in this study. The temperature field of titanium alloy under the different ablation conditions was simulated by using the finite difference computation. The transient heat conduction model was implemented together with the enthalpy method and temperature-dependent material properties. By using the developed model, the groove depths obtained from the experiment and simulation were in a good agreement.

Necking evolution in dynamically stretched bars: New experimental and computational insights

This paper presents new results on dynamic neck evolution in steel bars of varying diameters. Dynamic tensile tests were carried out in a Kolsky apparatus using cylindrical steel specimens with various cross-section diameters ranging from 1.5mm to 4mm. A high speed digital camera was used to record the deformation of the specimen during the loading process. Video recording of the tests enabled accurate experimental measurements of the necking evolution, specifically its growth rate as a function of the diameter. The experiments show that increasing the specimen cross-section slows down the neck development. This behavior has been further investigated using two different kinds of numerical calculations: (1) axisymmetric finite element simulations and (2) one-dimensional finite difference computations. While the finite difference model only considers the normal stress along the longitudinal direction of the bar, the finite element model does not entail any simplification on the stress state of the specimen during the loading process. In agreement with the experiments, the finite element calculations show a decrease of the necking growth rate with the increase in the cross-section of the sample. On the contrary, the damping effect of the specimen cross-section on the necking evolution is not captured by the finite difference computations. We postulate that this result comes from the one-dimensional nature of the finite difference model. This work uncovers, by means of combined experiments and modelling, the key role played by stress multiaxiality in the growth rate of dynamic necks.

Design Equations for Concrete Bridge Decks with FRP Stay-in-Place Structural Forms

AbstractBridge deck construction using stay-in-place (SIP) fiber-reinforced polymer (FRP) structural forms has shown promise as an efficient, rapid, and low-waste alternative to conventional methods. Several studies have shown that the load capacity of this system is typically governed by punching shear strength, although no design-oriented equations have yet been developed to predict failure load or deflection at service. In this paper, two simple design equations are proposed for the ultimate load and stiffness of concrete deck slabs with SIP FRP forms, derived from the results of a comprehensive parametric study using a rigorous finite-difference computer model that is readily available. The equations were then validated by using experimental results from a database of 52 tests reported in the literature including FRP SIP forms of several shapes, sizes, surface treatments, spliced connections, boundary conditions, environmental exposures, and loading protocols. The equations showed an average predicted...

Finite-Difference–Integral Method for Computing Low-Frequency Electromagnetic Fields in a Nonhomogeneous Medium

We derive integral relationships for electromagnetic fields, integral recalculation formulas for anomalous electric and magnetic fields, and integral boundary conditions. The efficiency of the finite-difference---integral method is checked by computing the apparent resistance curves for the graben model. Apparent resistance curves computed from finite-difference formulas and from formulas with integral smoothing are compared. Finite-difference graphs are polygonal curves that for small wavelengths lie below the smoothed curves, because the finite-difference computation of the derivative as if reduces the impact of the surface skin effect on the apparent resistance. As the wavelength increases, the surface skin effect diminishes and the finite-difference curves approach the integral curves.

Temperature field analysis of a cold-region railway tunnel considering mechanical and train-induced ventilation effects

In accordance with the unsteady-state finite-difference equations for heat transfer and heat convection in a cold-region railway tunnel, an unsteady-state finite-difference computer model was developed to study the heat convection between the air and the tunnel wall as well as the heat transfer in the surrounding rock and at interfaces between different materials in the structure of the tunnel. The wind speed and wind direction of actual mechanical ventilation wind produced during operation as well as the train-induced wind and natural wind in the tunnel were considered in the analysis of the temperature field distribution of a cold-region railway tunnel under actual periodic variations in entrance wind temperature. Good agreement was observed between the calculated frozen length and the frozen length observed in situ at the entrance of the railway tunnel. The results show that mechanical ventilation winds during operation and train-induced winds significantly influence the temperature field distribution in the tunnel and that the variables associated with mechanical ventilation winds and train-induced winds should be considered in the engineering design of cold-region tunnels. In this way, the temperature field of a railway tunnel can be correctly predicted via finite differencing.

An algorithm for hyperspectral remote sensing of aerosols: 1. Development of theoretical framework

This paper describes the first part of a series of investigations to develop algorithms for simultaneous retrieval of aerosol parameters and surface reflectance from a newly developed hyperspectral instrument, the GEOstationary Trace gas and Aerosol Sensor Optimization (GEO-TASO), by taking full advantage of available hyperspectral measurement information in the visible bands. We describe the theoretical framework of an inversion algorithm for the hyperspectral remote sensing of the aerosol optical properties, in which major principal components (PCs) for surface reflectance is assumed known, and the spectrally dependent aerosol refractive indices are assumed to follow a power-law approximation with four unknown parameters (two for real and two for imaginary part of refractive index). New capabilities for computing the Jacobians of four Stokes parameters of reflected solar radiation at the top of the atmosphere with respect to these unknown aerosol parameters and the weighting coefficients for each PC of surface reflectance are added into the UNified Linearized Vector Radiative Transfer Model (UNL-VRTM), which in turn facilitates the optimization in the inversion process. Theoretical derivations of the formulas for these new capabilities are provided, and the analytical solutions of Jacobians are validated against the finite-difference calculations with relative error less than 0.2%. Finally, self-consistency check of the inversion algorithm is conducted for the idealized green-vegetation and rangeland surfaces that were spectrally characterized by the U.S. Geological Survey digital spectral library. It shows that the first six PCs can yield the reconstruction of spectral surface reflectance with errors less than 1%. Assuming that aerosol properties can be accurately characterized, the inversion yields a retrieval of hyperspectral surface reflectance with an uncertainty of 2% (and root-mean-square error of less than 0.003), which suggests self-consistency in the inversion framework. The next step of using this framework to study the aerosol information content in GEO-TASO measurements is also discussed.

An overview of the computer-integrated systems and manufacturing technologies of pistons of internal combustion engines

The technological features of manufacture of pistons of internal combustion engines (ICE) taking into account the operational characteristics that form the functional requirements of the piston material are considered. It is shown that the chill casting of pistons is recommended for alloys with a low coefficient of linear expansion, and the liquid forging of pistons is preferable to the hot forging since it allows to obtain defect-free, dense billets with the fine structure of the piston material. For the heavy-loaded diesel engine pistons, chill casting is acceptable, provided the maximum possible porosity prevention in massive parts of castings.For the simulation of the casting processes of ICE pistons, it is advisable to use the LVMFlow package since it is focused on the finite-difference computation algorithms, ensuring stable operation of the computational station at a relatively low consumption of its hardware resources, ease of interface, computation speed and low cost of the software package. For the engineering simulation of the thermal and the stress-strain state of the ICE cast pistons, the ANSYS complex that has a high interaction with existing CAD and CAE systems having full interaction with WorkbenchProducts and well-known ANSYS, as well as a large number of mathematical solvers, allowing to quickly and accurately conduct computations in the WorkbenchProducts can be recommended.

Analytical GGA exchange–correlation kernel calculation in auxiliary density functional theory

ABSTRACTThe general working equations for the analytical calculation of the kernel of gradient corrected functionals within auxiliary density functional theory (ADFT) are presented. This formulation improves even further the computational performance for the calculation of second-order energy derivatives already achieved within the ADFT framework. To show this, we present benchmark polarisability calculations of alkane chains and fullerenes. As a specific example, for a gradient corrected functional, the PW86-P86 functional was used. In comparison to finite-difference kernel calculations, we observe speed-ups of at least a factor of 2 with the new analytical kernel evaluation. The here presented implementation will be of particular importance for indirect spin–spin coupling constant and vibrational frequency calculations in the framework of ADFT.

Explicit GPU Based Second-Order Finite-Difference Modeling on a High Resolution Surface, Feather River, California

An explicit second-order finite-difference computer model was developed and optimized for solution of the Shallow Water Equations. The model was applied to the Feather River below the Oroville Dam and Thermalito Afterbay near Gridley, California. Two versions of the computer model were constructed to run on either Central Processing Units or Graphical Processing Units, utilizing Fortran, C, C++, and the NVIDIA Compute Unified Device Architecture (CUDA) parallel computing platform. The underlying algorithm utilizes a structured grid and is capable of handling the wetting and drying of cells. It was developed with view to maximizing stability while maintaining accuracy, and allowing for flexibility of the computational domain. Comparisons with analytical and observed results showed the proposed methodology to be robust, accurate, and efficient. The models were applied to a section of the Feather River where observations of flow depths and volumetric flow rates are available for multiple flood events. The domain surface was partially developed using high-resolution photogrammetric data obtained through use of unmanned aerial vehicles. Runtimes and results were compared to the United States Bureau of Reclamations’ implicit finite-volume numerical method and with field observation with generally good correspondence.

Performance analysis of no-vent fill process for liquid hydrogen tank in terrestrial and on-orbit environments

Two finite difference computer models, aiming at the process predictions of no-vent fill in normal gravity and microgravity environments respectively, are developed to investigate the filling performance in a liquid hydrogen (LH2) tank. In the normal gravity case model, the tank/fluid system is divided into five control volume including ullage, bulk liquid, gas–liquid interface, ullage-adjacent wall, and liquid-adjacent wall. In the microgravity case model, vapor–liquid thermal equilibrium state is maintained throughout the process, and only two nodes representing fluid and wall regions are applied. To capture the liquid–wall heat transfer accurately, a series of heat transfer mechanisms are considered and modeled successively, including film boiling, transition boiling, nucleate boiling and liquid natural convection. The two models are validated by comparing their prediction with experimental data, which shows good agreement. Then the two models are used to investigate the performance of no-vent fill in different conditions and several conclusions are obtained. It shows that in the normal gravity environment the no-vent fill experiences a continuous pressure rise during the whole process and the maximum pressure occurs at the end of the operation, while the maximum pressure of the microgravity case occurs at the beginning stage of the process. Moreover, it seems that increasing inlet mass flux has an apparent influence on the pressure evolution of no-vent fill process in normal gravity but a little influence in microgravity. The larger initial wall temperature brings about more significant liquid evaporation during the filling operation, and then causes higher pressure evolution, no matter the filling process occurs under normal gravity or microgravity conditions. Reducing inlet liquid temperature can improve the filling performance in normal gravity, but cannot significantly reduce the maximum pressure in microgravity. The presented work benefits the understanding of the no-vent fill performance and may guide the design of on-orbit no-vent fill system.

A micromechanical analysis of deformation-induced surface roughening in surface-modified polycrystalline materials

The effect of surface layer modification on the deformation-induced surface roughening in polycrystalline materials is investigated numerically. Three-dimensional constitutive models of as-received and surface-hardened polycrystals are constructed and implemented in finite-difference calculations. The free surface is shown to undergo out-of-plane displacements due to microscale stresses developing in the subsurface layer and acting across the free surface. The surface-hardened layer as thin as half the grain diameter efficiently suppresses grain-scale surface displacements. The thicker is the hardened layer, the smoother is the surface of the material.

Applying full multigroup cell characteristics from MCU code to finite difference calculations of neutron field in VVER core

Abstract Up to now core calculations with Monte Carlo provided only average cross-sections of mesh cells for further use either in finite difference calculations or as benchmark ones for approximate spectral algorithms. Now MCU code is capable to handle functions, which may be interpreted as average diffusion coefficients. Subsequently the results of finite difference calculations with cells characteristic sets obtained in such a way can be compared with Monte Carlo results as benchmarks, giving reliable information on quality of production code under consideration. As an example of such analysis, the results of mesh calculations with 1-, 2-, 4-, 8- and 12 neutron groups of some model VVER fuel assembly are presented in comparison with the exact Monte Carlo solution. As a second example, an analysis is presented of water gap approximate enlargement between fuel assemblies, allowing VVER core region be covered by regular mesh.

Dynamic dip-slip fault rupture in a layered geological medium: Broken symmetry of seismic motion

As recorded recently on 22 November 2014 during the Nagano-ken Hokubu (Kamishiro Fault), Japan, earthquake, one important observation in dynamic rupture (fracture) related to a shallow dip-slip earthquake (usually assumed as mode-II shear crack propagation from depth towards a free surface) is the broken symmetry of seismic motion in the vicinity of the fracturing fault plane. Generally, the strong motion (particle motion) on the hanging wall is much larger than that on the footwall, but the physical mechanisms causing this asymmetry have not been fully clarified yet. Here, utilising the techniques of finite difference calculations and dynamic photoelasticity in conjunction with high speed cinematography, we investigate the fracture dynamics of a dip-slip fault plane situated near a free surface and try to explain the mechanics behind the asymmetry, numerically as well as experimentally. In our two-dimensional crack-like rupture models, we prepare a flat fault plane, which dips either vertically or at an angle, in a monolithic (scenario (1)) or layered (scenario (2)) linear elastic medium (representing rocks). In the basic scenario (1), when the primary fault rupture initiated at some depth approaches the free surface, four Rayleigh-type waves may be induced: two of them propagate along the free surface as Rayleigh surface waves into the opposite directions to the far field, and the other two travel back downwards along the fractured fault plane as interface waves into depth. If the fault plane is inclined, in the hanging wall, the interface and Rayleigh waves may interact with each other and a shear wave carrying concentrated energy (corner wave) can be produced to cause stronger disturbances. The corner waves, generated by primary fault rupture, may exist only when the fault plane is inclined, i.e. only when the geometry considered is asymmetric. In the scenario (2), however, we indicate that (anti-)symmetry of mode-II seismic motion can be easily broken even in geometrically symmetric models if the secondary fracture is allowed at an interface between layers. If primary vertical dip-slip fault rupture in an (anti-)symmetric model moves from depth and interacts with a horizontal interface that follows, for example, a tensile fracture criterion, basically only the interface segments where the primary rupture induces dynamic tension (in the relatively subsiding footwall) may be fractured and the segments in compression (in the rising hanging wall) may remain unbroken. In this case, in the hanging wall, the dynamic stresses in the upper layer above the interface become relatively large because the compressive parts of the primary rupture-induced wave (rupture front wave) can propagate from the lower layer across the still bonded interface into the upper layer, with less reflection back to the lower one. On the contrary, in the footwall, much of the energy carried by the rupture front wave in the lower layer is reflected at the broken interface and dynamic disturbances in the upper layer tend to become smaller. Thus, the strong motion on the hanging wall is expected to become larger than that on the footwall not only in geometrically asymmetric cases but also in symmetric ones.

Finite difference calculations of permeability in large domains in a wide porosity range

Determining effective hydraulic, thermal, mechanical and electrical properties of porous materials by means of classical physical experiments is often time-consuming and expensive. Thus, accurate numerical calculations of material properties are of increasing interest in geophysical, manufacturing, bio-mechanical and environmental applications, among other fields. Characteristic material properties (e.g. intrinsic permeability, thermal conductivity and elastic moduli) depend on morphological details on the porescale such as shape and size of pores and pore throats or cracks. To obtain reliable predictions of these properties it is necessary to perform numerical analyses of sufficiently large unit cells. Such representative volume elements require optimized numerical simulation techniques. Current state-of-the-art simulation tools to calculate effective permeabilities of porous materials are based on various methods, e.g. lattice Boltzmann, finite volumes or explicit jump Stokes methods. All approaches still have limitations in the maximum size of the simulation domain. In response to these deficits of the well-established methods we propose an efficient and reliable numerical method which allows to calculate intrinsic permeabilities directly from voxel-based data obtained from 3D imaging techniques like X-ray microtomography. We present a modelling framework based on a parallel finite differences solver, allowing the calculation of large domains with relative low computing requirements (i.e. desktop computers). The presented method is validated in a diverse selection of materials, obtaining accurate results for a large range of porosities, wider than the ranges previously reported. Ongoing work includes the estimation of other effective properties of porous media.

Optical field emission from resonant gold nanorods driven by femtosecond mid-infrared pulses

We demonstrate strong-field photoelectron emission from gold nanorods driven by femtosecond mid-infrared optical pulses. The maximum photoelectron yield is reached at the localized surface plasmon resonance, indicating that the photoemission is governed by the resonantly-enhanced optical near-field. The wavelength- and field-dependent photoemission yield allows for a noninvasive determination of local field enhancements, and we obtain intensity enhancement factors close to 1300, in good agreement with finite-difference time domain computations.

Efficient finite-difference time-domain computation of resonant frequencies of rectangular Lamé mode resonators via a combination of the symmetry boundary conditions and the Padé approximation

Resonant frequency analysis of the fundamental and higher-order modes of Lamé mode resonators on a lossless isotropic solid is carried out by the finite-difference time-domain (FD-TD) method with the staggered grid with collocated grid points of velocities (SGCV). The symmetry boundary condition is implemented to reduce the size of the computational domain for the FD-TD method with the SGCV. One spectrum estimation technique based on the Padé approximation is employed to effectively extract the resonant frequencies from a spectrum transformed from the time series data calculated by the FD-TD method. Numerical results show the validity and efficiency of these techniques.

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-called primitive regime, the rest belong to a later category of the modified type, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.