3D Analytical Solution Research Articles

Identification of bored pile defects utilizing torsional low strain integrity test: Theoretical basis and numerical analysis

The torsional low strain integrity test (TLSIT), known for its advantages such as a smaller detection blind zone, improved identification of shallowly buried defects, stable phase velocity for signal interpretation, and better adaptability for existing pile testing. However, it lacks a comprehensive understanding of the authentic three-dimensional (3D) strain wave propagation mechanism, particularly wave reflection and transmission at defects. To address this gap, a novel 3D theoretical framework is introduced in this context to model the authentic 3D wave propagation during the TLSIT. The proposed approach is validated by comparing its results with those obtained from 3D finite element method (FEM) simulations and simplified 1D (one-dimensional) and 3D analytical solutions. Additionally, a parametric study is conducted to enhance insights into the formation mechanism of high-frequency interference observed during the TLSIT. Finally, a defect identification study is performed to provide guidance for interpreting the wave spectrum in terms of defect characteristics.

Analytical model for Joule-Thomson cooling under heat exchange during CO2 storage

This paper discusses axi-symmetric flow during CO2 injection into a non-adiabatic reservoir accounting for Joule-Thomson cooling and steady-state heat exchange between the reservoir and the adjacent layers by Newton's law. An exact solution for this 1D problem is derived and a new method for model validation by comparison with quasi 2D analytical heat-conductivity solution is developed. The temperature profile obtained by the analytical solution shows a temperature decrease to a minimum value, followed by a sharp increase to initial reservoir temperature on the temperature front. The temperature distribution head of the front is determined by the initial reservoir temperature, while the solution behind the front is determined by the temperature of injected CO2. The analytical model exhibits stabilisation of the temperature profile and the cooled zone. The explicit formula for temperature distributions allows determining the maximum injection rate that avoids hydrate formation.

The Navier-Stokes system with temperature and salinity for free surface flows. Numerical scheme and validation

In this paper, we propose and study a numerical scheme for the Navier-Stokes-Fourier system derived and studied by the authors in [12]. This system models hydrostatic free surface flows with density variations depending on temperature or salinity. We show that the finite volume/finite element scheme presented – based on a layer averaged formulation of the model – is well-balanced with regards to the steady state of the lake at rest and preserves the nonnegativity of the water height. A maximum principle on the density is also proved as well as a discrete entropy inequality (when the thermal and viscous effects are neglected). Some numerical validations are finally shown with comparisons to 3D analytical solutions and experiments.

3D Multi‐Fluid MHD Simulation of the Early Time Behavior and Cross‐Field Propagation of an Artificial Plasma Cloud in the Bottom Side Ionosphere

AbstractThe relative motion of an artificial plasma stream in an ambient plasma background transverse to the geomagnetic field produces a polarization current and an drift in the direction of the injected photoionized neutral cloud. The polarization current couples the ion cloud momentum to the ambient plasma causing the stream to brake or “skid” before coming to rest. A multi‐fluid five‐moment resistive magnetohydrodynamic (MHD) model has been previously developed and is used in this study to simulate an artificial barium cloud released at 8 km/s in the lower ionosphere transverse to the magnetic field. The MHD model’s governing equations are used to derive 2D analytical solutions to the equations of motion that capture oscillatory behavior relating the cloud ion cyclotron frequency, artificial and ambient plasma density, initial release velocity, and photoionization rate. The analytical solutions are compared to the MHD results for time t < 1 s and previous 1D momentum coupling models. It is shown that motion in the cloud release direction is consistent with previous models while motion in the polarization direction is consistent with numerical results.

3D elasticity solutions for stress field around square hole in functionally graded plate

Since the square holes are widely used in many structures and machines, the purpose of this work is to obtain analytical solutions for the three-dimensional (3D) elastic stress field of transversely isotropic functionally graded (FG) infinite plates containing a single square hole with small round corners subjected to far-field loads or on the edges of the holes. Based on the generalized England-Spencer plate theory, material parameters can vary continuously and arbitrarily along the thickness direction of the plate, the solutions are obtained by using the conformal mapping technology combined with the Cauchy integral method. Concentration factors of internal forces Kij around the square hole are explicitly expressed. The obtained explicit expressions of solutions can be validated against the classical two-dimensional (2D) solutions when the material degenerates into an isotropic and homogeneous material. The detailed numerical examples are given to further investigate the influence of several parameters such as the material gradient factor and load conditions on the stress field around the hole. The present 3D analytical solutions can not only provide an in-depth understanding of the 3D stress field of FG plates with holes, but also be utilized as a benchmark to verify the numerical solutions of the same problems.

InSAR-observed surface deformation in New Mexico’s Permian Basin shows threats and opportunities presented by leaky injection wells

Knowledge of aquifer dynamics, including groundwater storage changes, is key to effective groundwater resource and reservoir management. Resolving and accurate modeling of these processes requires knowledge of subsurface poroelastic properties and lateral heterogeneity within units of interest. Computationally demanding methods for determining lateral heterogeneity in poroelastic properties exist but remain difficult to practically employ. The InSAR-based detection of uplift over a New Mexico well with a casing breach provides an opportunity to determine poroelastic properties using a tractable 2D analytical plane strain solution for surface uplift created by a pressurized reservoir with overburden. Using a Bayesian inversion framework, we calculate poroelastic properties under deep (depth of well-screen) and shallow (depth of well-breach) conditions. We find that shallow injection is necessary to produce the observed deformation. However, pressure-varying forward solutions for uplift are required to reproduce the temporal evolution of deformation. For this we use realistic shallow poroelastic properties and well dynamics, which reflect the evolving injection conditions at the well breach as the casing further erodes. Analysis of individual interferograms or InSAR time series may provide insights into shallow subsurface heterogeneity or anomalous injection conditions at operating wells more rapidly than scheduled field inspections.

A serpent-type wave energy converter

The problem of the interaction of waves with a serpent-type wave energy converter was investigated, and a novel 3D analytical solution was derived. The optimal parameters of the energy converter were derived for the first time from wave energy principles. The results show that the wave power captured by the device increases with increasing wave lengths until a maximum and then decreases. The efficiency increases with decreasing stiffness of the device in shallow and intermediate waters. In deep water, the efficiency increases with increasing stiffness until a local maximum and then decreases. Moreover, the efficiency increases with the increasing mass of the device in shallow and intermediate waters. In deep water, the efficiency of a converter decreases with the increasing mass of the device. An original approach to determine the optimal parameters of a device for given wave conditions was derived. The derived analytical formula shows that the top efficiency level of power capturing cannot exceed 50%. This is the theoretical maximum for this type of converter. The power take-off optimization analysis also identifies the spectrum of wave conditions for which the efficiency of the generator is close to the maximum. A series of laboratory experiments were conducted in a hydraulic laboratory to verify the model. The comparisons of the recorded data with the analytical solution show a very good agreement.

A general forward solver for 3D CSEMs with multitype sources and operating environments

To determine the electromagnetic (EM) fields of different three-dimensional (3D) controlled-source electromagnetic methods (CSEMs) using the same parameters of the forward solution, by explicitly considering the commonalities, we present a general 3D forward modeling solver for CSEMs with multitype sources and operating environments. The commonality of the solver is reflected in two aspects. First, the solver is based on a frequency-domain (FD) vector Helmholtz equation for determining the scattered electric field. The different types of sources are imposed on the right-hand term of the equation, expressed as background Green’s function. Second, sources of any CSEM can be composed of electric dipole (ED) or magnetic dipole (MD) superposition. Thus, the focus of the 3D forward modeling of CSEMs is reduced to determining the EM fields of ED or MD sources for the background medium. The quasi-minimal residual (QMR) method is used to solve the large sparse complex linear system. Once the FD EM fields have been calculated, the time-domain (TD) response can be obtained using the cosine/sine transformation. The numerical results show that the relative error is less than 5% between the 3D numerical and analytical solutions, which verifies the accuracy of the solver. We further study the difference between the real (bent) and theoretical (straight) wires. We suggest that the shape of the source must be considered for TD and FD CSEMs with a wire source during data processing and inversion. The last example investigated the characteristics of FD EM fields from a finite-length wire and TD EM fields from a rectangular fixed loop on the same conductive tilted disk model buried in resistive sediments. According to the numerical results, we recommend FD CSEMs with a wire source for detecting deep anomalies.

A Three-Dimensional Analytical Solution of Stress Field in Casing-Cement-Stratum System Considering Initial Stress State

Accurate stress field calculation of the casing-cement-stratum system is crucial for evaluating wellbore integrity. Previous models treated in-situ stress as boundary pressure loads, leading to unrealistic infinite displacements at infinity. This study presents a three-dimensional (3D) analytical solution for the stress field within the casing-cement-stratum system in inclined wells, considering in-situ stress and hydrostatic stress in cement as the initial stress state and taking into account stress components related to the axial direction. Assuming a plane strain condition and superimposing the in-plane plane strain problem, elastic uni-axial stress problem and anti-plane shear problem, a 3D analytical solution is obtained. Comparisons with previous models indicate that the existing model overestimates the absolute values of stress components and failure potential of casing and cement in both 2D and 3D scenarios. The presence of initial stress in cement greatly increases the absolute value of the compressive stress state but decreases the failure potential in cement, which has not been well studied. Additionally, a low Young’s modulus and high initial stress state of the cement benefits the cement’s integrity since the maximum Mises stress significantly decreases. The new 3D analytical solution can provide a benchmark for 3D numerical simulation and quick assessment for wellbore integrity.

Three-dimensional analytical elasticity solution for the mechanical analysis of arbitrarily-supported, cross and angle-ply composite plates under patch loads

Patch-type discontinuous loads lead to highly localized stresses which are three-dimensional (3D) in nature and can initiate failures. Their accurate characterization is of primal importance for composite and sandwich structures. The present contribution elaborates a 3D analytical elasticity solution for the mechanical analysis of cross and angle-ply composite plates under transverse patch loads and arbitrary support conditions. For the solution, the Reissner-type, mixed variational principle is used to derive the governing equations. Analytical solutions are obtained using the multiterm, extended Kantorovich method, by dividing the plate into loaded and unloaded parts. The present analytical solution exactly satisfies inter-segment, inter-layer continuity and edge support conditions, treating stresses and displacements as primal variables. Its accuracy is established by direct comparisons with existing analytical – where available – and 3D finite element results, both for cross-ply and angle-ply designs, for different boundary conditions. Extensive analysis is performed to quantify the effect of thickness, angle-ply and patch load position on the composite mechanical performance. Through thickness stress asymmetries are identified and quantified, along with angle-ply designs that can smoothen stress concentrations in the vicinity of patch loads. The methodology can be used as a reference solution in the design of multilayer anisotropic plates under discontinuous loads.

A three-dimensional (3D) analytical solution for the ultimate side shear resistance of rock-socketed shafts

This paper presents a new analytical solution to the ultimate side shear resistance τsf of rock socketed shafts with consideration of three-dimensional (3D) strength and shaft sidewall roughness. Specifically, the shaft sidewall profile is simplified as triangular asperities and micro-mechanical analysis is performed on the shear behavior of one representative asperity. A 3D version of the Hoek-Brown (HB) criterion for rock mass and the equilibrium equations in a cylindrical coordinate system are first utilized to derive the governing equations. Then the governing equations are solved by using the method of characteristics. Finally, the τsf is obtained by considering the constant normal stiffness condition and the relative displacement of the rock mass due to sliding along the shaft-rock mass interface. Twenty-four (24) model and field test shafts with measured τsf are collected for analysis to verify the proposed solution. The results indicate that the predicted τsf values are in good agreement with the measured τsf values. Extensive parametric studies are also conducted to investigate the effects of rock mass properties, shaft dimensions, and sidewall roughness parameters on the ultimate side shear resistance factor αf (the ratio of τsf to the unconfined compressive strength of intact rock σc) of rock socketed shafts. The results indicate that the αf increases with rock constant mi, geological strength index (GSI) and asperity half chord length λ, but decreases with shaft diameter B, σc and disturbance D. Furthermore, the αf first increases and then decreases with the asperity inclination angle δ, resulting in an optimum asperity inclination angle δOp. Therefore, it is important to reduce the disturbance of rock mass from construction and select a proper asperity inclination angle when designing a rock socketed shaft.

An update of the 3D analytical solution for the design of barricades made of waste rocks

Stope backfilling is increasingly used in underground mines. Barricades are necessary to retain slurried backfill poured in mine stopes . In Canada, waste rock barricades (WRB) have been becoming more and more popular to retain paste backfill in stopes. A few solutions have been developed to size the WRB. Among them, a solution developed by Yang and coworkers in 2017 is particularly relevant. However, this solution was calibrated and validated by 2D numerical modeling . Its validity in 3D conditions remains unknown. In addition, the local stability analysis was made by considering a horizontal sliding plane while their numerical modeling clearly showed an inclined plane. In this study, the validity of the Yang et al. solution is first evaluated against a few numerical results obtained with FLAC3D. The subjectivity in evaluating the instability onset of a structure in numerical modeling is further reduced by considering the first occurrence between displacement jump and coalescence of current yield zones passing through a WRB structure from the downstream to upstream slopes. The results show that both the global and local stability analysis equations of the Yang et al. solution need to be updated. For the global stability analysis, the value of the earth pressure coefficient needs to be modified while an inclined sliding plane needs to be considered for the local stability analysis. These considerations result in a considerable improvement because the updated solution does not contain any empirical calibration coefficient. Good agreements are obtained between the results obtained with numerical simulations and those predicted by the updated 3D analytical solution. The proposed solution is then further validated by an experimental result recently available in the literature. It can be used to design WRB to retain fluid-like paste backfill at the preliminary stage of a project.

Viscoelastic free vibration analysis of in-plane functionally graded orthotropic plates integrated with piezoelectric sensors: Time-dependent 3D analytical solutions

This paper proposes an accurate three-dimensional framework for elastic and viscoelastic free vibration investigation of in-plane functionally graded (IPFG) orthotropic rectangular plates integrated with piezoelectric sensory layers. The developed analytical framework is capable of considering layer-wise unidirectional linear functional gradation in both stiffness and density of the orthotropic composite layers. 3D piezoelasticity-based governing equations of motion are formulated in mixed form by employing Hamilton’s principle, and further solved analytically for Levy-type support conditions using the power-series-based extended Kantorovich method (EKM) jointly with Fourier series. The displacements, stresses, and electrical variables (electric field and electric potential) are solved as the primary variables that ensure the point-wise interlayer continuity and electro-mechanical support conditions. The viscoelastic property of the orthotropic interlayer is defined by employing Biot model, which is similar to the standard linear viscoelastic model. The correctness and efficacy of the present mathematical model are established by comparing the present numerical results with published literature and 3D finite element results, obtained by utilizing user material subroutine in the commercial FE software ABAQUS. An extensive numerical study is performed for various configurations and thickness ratios to investigate the influences of in-plane gradation, viscoelasticity and their coupled effects on the free-vibration response of hybrid laminated plates. It is found that in-plane gradation of stiffness and density remarkably alters the flexural frequencies and corresponding mode shapes of the hybrid intelligent rectangular plates. The flexural frequencies and stresses in the plate can be modified by selecting suitable grading indexes. Another interesting observation is that the in-plane gradation shows a considerably less effect on the electrical response of piezoelectric layers, which can play a vital role in the design of sensors and actuators for dynamic applications. Further, the numerical study demonstrates a potential time-dependent structural behaviour based on the present viscoelastic modelling. The consideration of viscoelasticity could be crucial for analysing the mechanical behaviour of a wide range of polymer composites more realistically and for prospective temporal programming in smart structural systems by exploiting the viscoelastic effect. Although the present analytical solution has been proposed for the free-vibration investigation of smart in-plane functionally graded (IPFG) viscoelastic plates, it can also be utilized directly to analyze the symmetric and asymmetric laminated piezoelectric smart plates with constant properties.

Powerful laser-produced quasi-half-cycle THz pulses.

The Maxwell equations-based 3D-analytical solution for the terahertz (THz) half-cycle electromagnetic wave transition radiation pulse has been found. This solution describes generation and propagation of transition radiation into free space from laser-produced relativistic electron bunch which crosses a target-vacuum interface as a result of ultrashort laser pulse interaction with a thin high-conductivity target. The analytical solution found complements the theory of laser initiated transition radiation. It describes the THz wave half-cycle pulse at the arbitrary distance from a target surface including near-field zone rather than its standard far-field characterization. The analytical research has also been supplemented with the 3D simulations using the finite-difference time-domain method, which makes it possible for description of much wider spatial domain as compared to that from the particle-in-cell approach. The presented result sheds light fundamentally on the interference of the electron bunch field and the generated THz field of broadband transition radiation from laser-plasma interaction. The latter is studied for a long time in the experiments with solid density plasma and the theory developed may inspire to targeted measurements and investigations of unique super intense half-cycle THz radiation waves near the laser target.

Joint inversion of tectonic stress and magma pressures using dyke trajectories

AbstractIn volcano-tectonic regions, dyke propagation from shallow magmatic chambers is often controlled by the interaction of the local and regional stress fields. The variations of the stress fields result from a combination of factors including the regional tectonic stress, the geometry of pressurized magma chambers, the layering and the pre-existing discontinuities (e.g. fractures). In this contribution, we describe and apply a new multiparametric inversion technique based on geomechanics that can invert for both the far field stress attributes and the internal pressure of magma chambers or stocks, constrained by observed dyke or eruptive fissure orientations. This technique is based on the superposition principle and uses linear elastic models that can be solved using many types of numerical methods. For practical reasons, we chose a 3D boundary element method (BEM) for a heterogeneous elastic half-space, where magma chambers are modelled as pressurized cavities. To verify this approach, the BEM solution has been validated against the known 3D analytical solution of a pressurized cylindrical cavity. Then the effectiveness of this technique and its practical use is demonstrated through application to natural examples of dyke network development around two different volcanic systems, the Spanish Peaks (USA) and the Galapagos Islands (Ecuador). Results demonstrate that regional stress characteristics as well as the internal pressure of magma chambers can be estimated from observed radial and circumferential dyke patterns and some knowledge of magma chamber geometry.

Analytical Solution for the Three-Dimensional Vibration of a Rectangular Functionally Graded Material Plate with Two Simply Supported Opposite Faces

Analytical solutions based on three-dimensional (3D) elasticity for the vibrations of functionally graded material (FGM) plates are valuable for assessing the validity and accuracy of various plate theories and numerical approaches. Few benchmark 3D analytical solutions for the vibrations of FGM plates are available in the literature. In this study, analytical solutions based on Fourier series and 3D elasticity were developed for the first time for the vibrations of FGM rectangular plates with two simply supported opposite edge faces. The distributions of the properties of FGMs through the thickness follow a simple power law. The proposed solutions were validated by conducting comprehensive convergence studies on the vibration frequencies of square plates with different thickness-to-side ratios and boundary conditions as well as comparisons with published results. The benchmark nondimensional frequencies were tabulated for plates with free boundary conditions on the top and bottom faces and six combinations of boundary conditions on the other two faces. Moreover, the effects of aspect ratio and gradient index on the vibration frequencies of FGM plates were investigated. The influence of the thickness ratio of the FGM layer to the homogenous layer on the vibration frequencies of sandwich plates with FGM face sheets and a homogeneous core was also studied.

Dynamic torsional impedance of large-diameter pipe pile for offshore engineering: 3D analytical solution

Wind-induced or current-induced torsional vibration is a long-neglected issue for offshore engineering (e.g., offshore wind turbines, offshore oil platforms, etc.). As the dimension of the pipe pile implemented offshore increases dramatically, the one-dimensional rod theory can no longer precisely compute the dynamic torsional impedance. This paper proposes a novel 3D torsional soil-pile interaction model and derives the corresponding analytical solution for the 3D dynamic torsional impedance at the pile cross-section with the adoption of the Laplace transform, variable separation, and impedance transfer method. The solution is then validated by comparisons against some widely-accepted existing solutions. The main conclusions of this study can be drawn as follows: (1). Classic one-dimensional theory overestimates the impedance of the pipe pile, which could result in severe disasters if the dynamic design is radical; (2). Even for small diameter pipe piles, the impedance would vary in the radial direction at the pile cross-section. Usually, the impedance at the outer radius would be slightly larger than that at the inner radius; (3). For large-diameter pipe piles or pipe piles with high-strength soil plugs, the most significant impedance could appear at the inner radius, indicating the resistance provided by the soil plugs can not be overlooked for large-diameter pipe piles.

Numerical investigation on arching effect surrounding deep cylindrical shaft during excavation process

Predicting the inner displacements of deep vertical shafts during the excavation process has been a difficult task considering the geological, structural, and constructional influences. In fact, the two-dimensional (2D) analytical solution based on the retaining wall model remains insufficient for understanding the actual behavior during an excavation. This is because the deformation of vertical shafts becomes complicated due to the unexpected arching effect brought about by the three-dimensional (3D) flexible displacements occurring in the excavation process. Previous analytical solutions only considered the limit equilibrium. Therefore, the present study deals with a 3D soil-structure simulation by considering the displacements of a cylindrical shaft and the mechanical behavior of the surrounding soil as well as the geometry of the cylindrical structure. Moreover, this mechanical behaviors of the surrounding soil and shaft are controlled by the shaft stiffness; hence, the relationships among the shaft stiffness, mechanical behavior of the surrounding soil (in terms of earth pressure coefficient), and shaft displacement were investigated. A cylindrical model, 120 m in depth and 20 m in diameter, was positioned at the center of a sand domain, and each excavation step was performed at an interval depth of 20 m. A 3D finite difference method analysis was applied using the modified Cam-Clay (MCC) model to represent the soil behavior. As a result, the present study provides a new normalized lateral earth pressure theory for excavated shafts by considering the 3D arching effect obtained from parametric studies using various levels of shaft stiffness. From a comparison with the analytical solutions of previous studies (Terzaghi, 1943a; Prater, 1977; Cheng & Hu, 2005), it is found that the previous studies underestimated the earth pressure acting on the cylindrical shaft because they did not consider the accurate arching effect.

A series of elasticity solutions for flexural responses of functionally graded annular sector plates

This paper aims at investigating the flexural responses of transversely isotropic functionally graded (FG) annular sector plates subjected to biharmonic loads, where the material parameters are assumed to vary arbitrarily along the thickness direction and the cylindrical boundaries is simply supported at the two radial edges and arbitrary at the two circumferential edges. Based on the extended England-Spencer plate theory, a series of analytical solutions to FG annular sector plates are derived by virtue of the mid-plane displacements of the plate, which can be expressed by four analytic functions of the complex variable, and the corresponding unknown constants can be determined by the boundary conditions at two circumferential edges. In order to validate the proposed analytical solutions, a finite element (FE) model is built to obtain numerical solutions for comparison, which shows that the obtained deflection and stress components are coherent with those from the FE results. Furthermore, two typical biharmonic loads are taken in the numerical examples to discuss the effects of material gradient index, boundary conditions and thickness-to-radius ratio on the flexural responses of the plate in detail. This work gives exact 3D analytical solutions for the FG annular sector plates, which not only provide an in-depth understanding of their mechanical behaviors and a new theoretical route for the analysis of engineering structures, but also serve as benchmark to check analytical solutions based on any simplified plate theories or numerical solutions to the same problem.

3D Exact Analytical Solutions of Two-fluid Plasma, Magnetohydrodynamics, and Neutral Fluid Equations for the Creation of Ordered Structures as well as Jet-like Flows

Abstract The 3D exact analytical solutions of ideal two-fluid plasma, single-fluid plasma, and neutral fluid equations have been found using physically justifiable assumptions. Surprisingly these solutions satisfy all nonlinearities in the systems. It is pointed out that these solutions explain the fundamental mechanism behind the creation of a vast variety of ordered structures in plasmas and fluids. In the limiting case of 2D dependence of fields, the theoretical model for plasma is applied to explain the formation of spicules in the solar chromosphere. It is pointed out that the main contribution of electron (ion) baroclinic vectors is to produce vorticity in the plasma, and that magnetic field generation is coupled with the flow of both electrons and ions.