2D Finite Difference Model Research Articles

Numerical modelling of reinforced fill over a void considering rate-dependent stiffness of the reinforcement

The problem of a reinforced fill over a void has been the subject of much research in the geosynthetics literature. Previous studies have mainly focused on finding closed-form solutions to predict the tensile loads and strains in the reinforcement layer once a void develops below the fill. In this paper, a 2D finite difference (FLAC) model that implements the hyperbolic isochronous load-strain model for the reinforcement by Bathurst and Naftchali (2021) is used to investigate the influence of the rate-dependent properties of polymeric geosynthetic reinforcement materials on reinforcement tensile strains and load, and overall system performance including vertical deformation at the reinforcement elevation and at the fill surface. The paper also investigates the influence of fill soil properties and constitutive model type, foundation condition, void geometry and fill height on system performance. The results of numerical modelling are compared to predictions made using the closed-form solution of Giroud et al. (1990) and in the BSI 8006-1 (2010) design code. The results of numerical modelling demonstrate that the choice of fill height to void width and the stiffness of the rate-dependent geosynthetic reinforcement layer are important to ensure that the maximum reinforcement strain, allowable strength and fill surface settlement criteria are not exceeded.

Centrifuge model and numerical studies of strip footing on reinforced transparent soils

This paper presents the results of centrifuge model tests to investigate the deformation behaviour of unreinforced and reinforced transparent soil foundations under strip loading. A digital image analysis technique was employed to obtain the soil displacement field and strain distribution of reinforcements. Two-dimensional (2D) numerical models were developed and verified using the test results. The soil was modelled as a linearly-elastic perfectly-plastic material with Mohr–Coulomb failure criterion. The reinforcement was characterised using a linearly-elastic model considering rupture behaviour. Moreover, a parametric study was conducted to investigate the load–settlement response of foundations, distribution of reinforcement tension and failure sequence of reinforcements. The experimental and numerical studies show that the results obtained from the numerical simulations are in good agreement with the results of the centrifuge model tests. The 2D finite difference model developed using the user-defined functions coded into the FLAC programme can better simulate the progressive failure of the reinforcement layers in the tests. The failure sequence of reinforcement layers is not affected by the modulus and internal friction angle of soil and the reinforcement length, but is closely related to the combined effect of spacing and number of reinforcement layers and the combined effect of reinforcement stiffness and strength.

Method to Estimate Thermal Transients in Reactors and Determine Their Parameter Sensitivities without a Forward Simulation

Thermal response time is an important parameter for the control of fast reactors. Modern thermal hydraulic codes allow for the modeling of transient responses and can also be used to understand the dominant factors that affect them. However, simulations can be computationally expensive, particularly for performing parametric analyses of how thermophysical properties affect transient behavior. Here, we present a method for using linear stability analysis to estimate thermal response time and determine the key parameters that affect transient behavior without performing a forward simulation. The approach can also be used to corroborate simulation results and is tested against simulation results produced with a 2D finite difference model. The results show that this approach produces time-dependent temperature profiles that are within 2 × 10−5–0.1% of the numerical results for a single node perturbation. Changes in temperature have the greatest effect on thermal response time, followed by changes to thermal conductivity.

Effects of Lunar Near‐Surface Geology on Moonquakes Ground Motion Amplification

AbstractLunar seismology has focused on exploring the Moon's seismic activity and its internal structure. However, less attention has been given to site specific, near‐surface geology‐related amplification of seismic waves on the Moon. Here, we quantified ground‐motion amplifications of seismic waves due to local site conditions at the Apollo landing sites. We analyzed Apollo Passive Seismic Experiments seismometer recordings of 30 artificial and natural impacts and estimated the mid‐period (0.1–3 Hz) ground‐motion amplification at each site. We applied techniques commonly used in engineering seismology to understand the local site effects, such as the Horizontal over Vertical Spectral Ratio (HVSR) and the Reference Site Method (RSM) on lunar seismic data. The HVSR and the RSM show convergent spectral ratios specific for each landing site. To correlate these empirical results estimated from lunar artificial and meteoroid impact events with the local site geology, we simulated ground motion amplification using 2D finite difference modeling of lunar geological models and estimated the best geological setting capable of producing a similar frequency response to those observed at each Apollo landing site. Correlation results from the empirical methods and the simulated models show similarities ranging between 79% and 92%, a correlation more than 90% between the RSM and the HVSR results, and similarity exceeding 96% between RSM estimations using different reference sites. These results demonstrate the prominence of a local site geology dependent amplification for frequencies between 0.1 and 3 Hz.

A study on the effects of piled-raft foundations on the seismic response of a high rise building resting on clayey soil

Piled-raft foundations are often used to transfer the structural load to the depth of the soil and tall structures can be constructed on medium to soft soils by utilizing a group of piles. The static effects of a group of piles are quite evident while their dynamic effects are quite unknown and therefore, not considered in the design of the superstructure.In the present research, the performance of pile-raft foundations on the seismic response of a 20-story benchmark building resting on soft clayey soil has been investigated using a 2D finite difference model, namely FLAC. The study has been carried out by analyzing the seismic response of the structure with and without pile-raft foundations, on the surface and inside an excavation.The results show that when the structure is constructed on the surface, piled-raft foundations have a significant beneficial effect on the seismic response of the superstructure, for instance a 40% decrease in the base shear. In the case where the structure is located inside an excavation the use of piled-raft foundations are not noteworthy and do not have a considerable effect.The results of this study give an insight on the effects of pile-raft foundations on the seismic response of a superstructure to overcome the current shortage of information in this area and to better predict the behavior of tall structures in soft soil situations.

Rapid drawdown on earth dam stability after a strong earthquake

The paper focuses on the rapid drawdown of earth dams in the hypothesis that a prompt reservoir lowering is needed soon after a strong earthquake. The interest on such a topic comes from the fact that in Italy and worldwide there are many large dams placed very close to active faults. In this case, the dam may be asked to face the earthquake first and the rapid drawdown later to face an emergency state. The problem has been numerically investigated with reference to a zoned earth dam (Campolattaro Dam) placed in a highly seismic area of Southern Italy. To solve the overall boundary value problem, encompassing several stages of the dam lifetime, a 2D finite difference model has been solved through the FLAC2D code, implementing a coupled transient seepage formulation, which accounts for partial saturation of the soil during the drawdown stage. Unlike past literature studies, in which the drawdown stage was modelled without accounting for the past loading history of the dam, in the performed simulation the drawdown stage was placed at the end of a quite real sequence of events, encompassing the embankment construction, the reservoir impounding, and earthquake scenarios compatible with the dam site seismic hazard. The performed analyses pointed out that a seismic stage previously experienced by the dam could further contribute to decreasing dam stability during the drawdown especially during the first stages of reservoir lowering. In addition, the faster the drawdown, the smaller the dam safety factor against stability (FOS) with more prominent effects of the initial (pre-drawdown) soil conditions.

Effect of Tidal Fluctuations on Contaminant Transport in Coastal Unconfined Aquifers

The environment of the coastal aquifers becomes worse and worse, due to the rapid development of economy, especially for the unconfined aquifers which direct connects with the contaminates from the land surface. In this study, a 2D finite-difference model describing variably saturated flow and solute transport (VS2DT) was modified to simulate solute transport in unconfined coastal aquifers with periodic tidal boundary conditions. In order to improve the understanding of tidal effects on contaminant transport, simulations with a simplified seaward boundary condition at a fixed position were compared to simulations with periodic moving boundary condition. Tidal fluctuations were modeled respectively as diurnal tide, semi-diurnal tide, and multi-periods semi-diurnal tide. In comparison with a fixed seaward boundary condition, a periodic moving boundary condition induced a larger water table over height and causes a lower transport velocity of the contaminant plume. Therefore, neglecting the horizontal movements of the seaward boundary underestimated the migration rate of the contaminant plume towards the sea and incorrectly describes the temporal and spatial distribution of contaminant plumes in costal aquifers. The results found with three different kinds of tidal fluctuations revealed that the multi-periods of tide also affected the movement of contaminant plume. There were lower migration rates during neap tides and faster transportation rates during spring tides. Our results indicated that it was necessary to simulate both the vertical tidal fluctuation and the horizontal movement of the seaward boundary condition to predict the movements of contaminant plumes through unconfined coastal aquifers.

Surface to sewer flow exchange through circular inlets during urban flood conditions

Abstract Accurately quantifying the capacity of sewer inlets (such as manhole lids and gullies) to transfer water is important for many hydraulic flood modelling tools. The large range of inlet types and grate designs used in practice makes the representation of flow through and around such inlets challenging. This study uses a physical scale model to quantify flow conditions through a circular inlet during shallow steady state surface flow conditions. Ten different inlet grate designs have been tested over a range of surface flow depths. The resulting datasets have been used (i) to quantify weir and orifice discharge coefficients for commonly used flood modelling surface–sewer linking equations and (ii) to validate a 2D finite difference model in terms of simulated water depths around the inlet. Calibrated weir and orifice coefficients were observed to be in the range 0.115–0.372 and 0.349–2.038, respectively, and a relationship with grate geometrical parameters was observed. The results show an agreement between experimentally observed and numerically modelled flow depths but with larger discrepancies at higher flow exchange rates. Despite some discrepancies, the results provide improved confidence regarding the reliability of the numerical method to model surface to sewer flow under steady state hydraulic conditions.

Erratum to: Development and Testing of 2D Finite Difference Model in Open Channels

Erratum to: Development and Testing of 2D Finite Difference Model in Open Channels

Development and Testing of 2D Finite Difference Model in Open Channels

This study develops a depth-averaged two-dimensional (2D) numerical model using a finite difference method (FDM) on a staggered grid. The governing equations were solved using the Marker and Cell method that was developed at the Los Alamos laboratories by Harlow and Welch in 1965. In the paper, an explicit FDM was used to solve the governing equations. A first-order approximation was used for the temporal derivative. Second-order central difference approximations were used for space discretization. The time step is limited by the Courant–Friedrichs–Lewy (CFL) condition. The time step used in this study depends on the grid spacing and velocity components in the x- and y-directions. The study is divided into two steps: the first step is to develop a depth-averaged 2D numerical model to simulate the flow process. The second constructs a module to calculate the bed load transport and simulate the river morphology in the areas that have steep slopes, torrents, and mountain rivers. Developed model was applied to the artificial channel and a flood event in the Asungjun River section of the mountainous Yangyang Namdae River (South Korea). General simulation results showed that the developed model was in good agreement with the observed data.

Probabilistic analysis of a coal mine roadway including correlation control between model input parameters

Probabilistic analysis and numerical modelling techniques have been combined to analyse a deep coal mine roadway. Using the Monte Carlo method, a correlation control algorithm and a FLAC 2D finite difference model, a probability distribution of roof displacements has been calculated and compared to a set of measurements from an actual mine roadway. The importance of correlation between input parameters is also considered. The results show that the analysis performs relatively well, but does tend to over predict the magnitude of displacements. Correlation between parameters is shown to be very important, particularly between the three model stresses.

A mathematical model for understanding fluid flow through engineered tissues containing microvessels.

Knowledge is limited about fluid flow in tissues containing engineered microvessels, which can be substantially different in topology than native capillary networks. A need exists for a computational model that allows for flow through tissues dense in nonpercolating and possibly nonperfusable microvessels to be efficiently evaluated. A finite difference (FD) model based on Poiseuille flow through a distribution of straight tubes acting as point sources and sinks, and Darcy flow through the interstitium, was developed to describe fluid flow through a tissue containing engineered microvessels. Accuracy of the FD model was assessed by comparison to a finite element (FE) model for the case of a single tube. Because the case of interest is a tissue with microvessels aligned with the flow, accuracy was also assessed in depth for a corresponding 2D FD model. The potential utility of the 2D FD model was then explored by correlating metrics of flow through the model tissue to microvessel morphometric properties. The results indicate that the model can predict the density of perfused microvessels based on parameters that can be easily measured.

Numerical Modeling of Vertical Geothermal Heat Exchangers Using Finite Difference and Finite Element Techniques

This paper presents the development of a 2D finite difference modelling approach and a 3D finite element numerical model for simulating vertical geothermal heat exchangers (GHEs), explaining the theory governing the thermal processes, element discretization and the selection of the appropriate boundary conditions. Both of these models provide fully coupled solutions for the fluid flow in the circulation pipes and the thermal processes between the fluid and solid domains (pipes, grout and soil). The numerical models are verified with a field test and subsequently they are utilized to simulate the thermal performance of a borehole heat exchanger integrated with a single U-tube. Two different thermal operation cases are analyzed; a constant rate heat injection and a fluid injection at a constant temperature. A model validation study is also carried out for the constant rate heat injection case by comparing the numerical results with the available analytical solution for a finite line source. Furthermore, effective thermal conductivity of the ground back-calculated from the results of the numerical analyses is compared with the value used in the numerical models. Comparison of the results obtained from both numerical models and validating model predictions with the analytical solution confirms that both FE and FD models can accurately simulate the heat transfer mechanisms governing the thermal performance of GHE systems.

A new model for the description of the heat transfer for plane thermo-active geotechnical systems based on thermal resistances

In this paper, a thermal resistance model for an energy wall using the example of thermo-active seal panels is presented. In the developed model, the resistances of the pipes as well as the resistance of the structure itself are considered. The resistance model is transferred to a 2D finite difference model, which itself is implemented into the general 3D subsurface heat and flow transport code SHEMAT-Suite. This coupling of a semi-analytical model with a numerical code avoids a complete discretisation of the model domain and thus enables fast computing times. This new approach has been verified by pure finite element simulations and by laboratory tests.

An Approximate Analytical Approach to Steady State Simulation of Unglazed Solar Collectors

This paper illustrates the thermal modeling of a flat plate unglazed solar collector with the aim of developing a simplified but accurate method for the collector steady state simulation without requiring a large amount of computational power and time. T he 1D+1D model is derived from the Riesz-Galerkin method applied to the approximate analytical solution of the heat equation. The model allows calculating heat transfer rate, fluid outlet temperature, collector efficiency and plate temperature distributio n. Accuracy and ability in simulating solar collector are checked using a 2D finite-difference model as a benchmark. The results of the simplified model show very good agreement with those provided by the finite-difference model, allowing to state that the simplified model is a valuable tool for fast and reliable collector simulation.

Sea‐bed diffractions and their impact on 4D seismic data

ABSTRACTA sea‐bed reflection is used to estimate changes in water layer velocities between time‐lapse seismic surveys. Such corrections are crucial in order to obtain high‐quality 4D seismic data. This might be a challenge in areas where rough sea‐bed topography creates sea‐bed diffractions that interfere with the sea‐bed reflection, as in several of the northern fields in the Norwegian Sea. These diffractions and diffracted multiples are difficult to attenuate during data processing and become a source of noise in time‐lapse data.In this work we study how cross‐correlation analysis of time‐lapse seismic data at a sea‐bed reflection may be perturbed by the presence of diffractions. The sea‐bed topography from the Norne field is used in 2D finite difference modelling to explain some of the observed variation in a water layer time‐shift in field data. We find that a rough sea‐bed and the diffracted energy it induces cause residual noise on the 4D data. The variation to the water layer time‐shift increases with sea‐bed complexity and is amplified by interaction with other sources of non‐repeatability like water column variations, mis‐positioning and strength of the ice scours creating the diffracted energy. Time‐shift variations with mis‐positioning and velocity changes between the surveys seem to best explain the observed variation in the time‐shift for time‐lapse seismic field data from Norne.

An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves

Perfectly matched layer (PML) absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling. When modeling Rayleigh waves with the existence of the free surface, the classical PML algorithm becomes unstable when the Poisson’s ratio of the medium is high. Numerical errors can accumulate exponentially and terminate the simulation due to computational overflows. Numerical tests show that the divergence speed of the classical PML has a nonlinear relationship with the Poisson’s ratio. Generally, the higher the Poisson’s ratio, the faster the classical PML diverges. The multiaxial PML (M-PML) attenuates the waves in PMLs using different damping profiles that are proportional to each other in orthogonal directions. The proportion coefficients of the damping profiles usually vary with the specific model settings. If they are set appropriately, the M-PML algorithm is stable for high Poisson’s ratio earth models. Through numerical tests of 40 models with Poisson’s ratios that varied from 0.10 to 0.49, we found that a constant proportion coefficient of 1.0 for the x- and z-directional damping profiles is sufficient to stabilize the M-PML for all 2D isotropic elastic cases. Wavefield simulations indicate that the instability of the classical PML is strongly related to the wave phenomena near the free surface. When applying the multiaxial technique only in the corners of the PML near the free surface, the original M-PML technique can be simplified without losing its stability. The simplified M-PML works efficiently for homogeneous and heterogeneous earth models with high Poisson’s ratios. The analysis in this paper is based on 2D finite difference modeling in the time domain that can easily be extended into the 3D domain with other numerical methods.

Site response effects on an urban overpass

Strong ground motion variability due to rapid changes in subsoil conditions may lead to different site responses, which in turn yields to beneficial or detrimental soil–foundation–structure interaction. This technical note presents the results of a seismic soil–structure interaction analysis conducted using a 2D finite difference model, developed with the program FLAC, of a critical section of a 60 km long strategic urban overpass, which is under construction in Mexico City, for a M w 8.7 earthquake. Initially, the response of the free field was calibrated comparing the values obtained with FLAC, with those gathered using the computer code QUAD4M. Good agreement was observed between the results generated with these programs. Accelerations and displacements were determined at the upper deck and foundation of the urban overpass. Important seismic soil–structure interaction was observed along the overpass at the supports analyzed. This numerical study helps to gain insight regarding the site response ground motion incoherence effects that influence the dynamic behavior of this kind of structures during extreme events.

Implementation and Validation of a New Combined Model for Outdoor to Indoor Radio Coverage Predictions

A new model used to compute the outdoor to indoor signal strength emitted from an outdoor base station is presented. This model is based on the combination of 2 existing models: IRLA (Intelligent Ray Launching), a 3D Ray Optical model especially optimized for outdoor predictions, and MR-FDPF (Multiresolution Frequency Domain ParFlow), a 2D Finite Difference model initially implemented for indoor propagation. The combination of these models implies the conversion of the ray launching paths on the border of the buildings, into virtual source flows that will be used as input for the indoor model. The performance of the new combined model is evaluated via measurements at 2 frequencies (WiMAX and WiFi). This solution appears to be efficient for radio network planning, in term of both accuracy and computational cost.