Assessing the persistence of a contaminant plume generated by linear aquifer source depletion and back diffusion from an aquitard.

This study presents analytical solutions for describing contaminant storage and release from an aquitard with linear source depletion (LSD) boundary conditions. We investigated three scenarios for trichloroethylene (TCE) mass exchange before and after the LSD period in an aquifer bounded by an adjacent aquitard based on the LSD dynamics, a resistance coefficient, and the aquitard thickness. The developed analytical solutions showed good agreement with measured profiles and breakthrough curves from a previous study. In three scenarios, the factors delaying the onset of TCE release into the aquifer were a decrease in the resistance coefficient, an increase in LSD period and aquitard thickness. The changes in the duration, mass, and rate of TCE storage in the aquitard during LSD loading process affected the equilibrium of the aquifer-aquitard concentration gradient. After TCE loading, the period maintained above the maximum contaminant level was directly related to the three variables; the longest plume persistence occurred when TCE penetration distance at transition point from storage to release coincided with the aquitard thickness. Overall, the developed analytical solution aids in evaluating the risk of plume persistence, enhancing site management efficiency, and reducing remediation costs.

Analytical solution for surrounding rock temperature of cold-region tunnel considering phase change.

The temperature of the surrounding rock in cold-region tunnels is crucial for antifreeze design, and the water-ice phase transition is essential to addressing the temperature field. This paper proposes a refined method that equates the latent heat of the ice-water phase transition to heat capacity and establishes a one-dimensional radial heat transfer model considering phase change. By defining an average thermal diffusivity coefficient through the concept of equal accumulated temperature, this method overcomes the limitations of classical heat transfer theory in directly solving the temperature field of three zone (unfrozen zone, freezing zone and frozen zone). Additionally, by employing the variable separation method and Fourier integral transformation method, the analytical formula for the transient temperature field considering phase change is derived. Then, the analytical solution was verified based on the field data. The results calculated using this method exhibit greater consistency with field temperature data and outperform the modified Stephan formula in determining the maximum frozen depth of the surrounding rock. Finally, the simplified form of the established analytical solution was further discussed. The research results can provide a theoretical basis for the analysis of the temperature field of the surrounding rock of the tunnel in cold regions and its antifreeze design.

Analytical solution of nano-concrete-epoxy interaction area considering static equilibrium.

This research proposes an analytical solution of the nano-concrete-epoxy interaction area within nano crack region of the reinforced concrete beam by applying Newton's third law in static equilibrium. For deriving the governing equation, the imaginary beam with free ends (no support) is considered within nano crack region. This imaginary beam is acted along the imaginary line of concrete-epoxy interface. Newton's third law is applicable for deriving the governing equation because of assuming the absence of frictional and other external forces. The parametric study is performed for implementing the proposed formula of nano interactive area considering variable nano crack depths and thicknesses. The nano interactive area is increased gradually with the increment of depths and thicknesses based on the parametric study because of linear functionality of interactive area and geometry of nano crack region. The maximum interactive area is found to be 314 nm2 at 0.6 ratio of depths and thicknesses of the nano crack. The incremental differences in interactive area between the crack depth or thickness ratios of 0.1 and 0.6 are found to be 25.4% and 1.6% for variations of the crack depth and thickness ratios, respectively. So, the crack depth shows higher impact on the interaction area compared to the thickness of the crack. However, there is a scope for enhancing this research in future by deriving closed-formed analytical formulations to consider appropriate boundary conditions.

An improved analytical solution on viscous dissipation effect in extended Stokes' second problem in microchannel with isothermal boundaries.

In this analytical study, the fluid motion within a microchannel is induced by the oscillation of one surface parallel to the other stationary surface, termed the extended Stokes' problem. The novelty and research gap are acquiring the thermal effect of such motion due to the viscous dissipation or fluid friction, subject to symmetric isothermal boundary conditions. The study may shed light on the role of viscous dissipation in temperature rise in the synovial fluid of an artificial hip joint, or in the fluid layer of a mechanical bearing. The full exact analytical temperature field, until now, has been unsolved, as it involves unsteady flow with manipulation of a complicated velocity field. The assumptions in the model are one-dimensional, incompressible, laminar, Newtonian flow with constant properties in a microchannel. Through the methodology of partial differential equation analysis, the temperature field is obtained in terms of Brinkman number, Prandtl number and a dimensionless angular frequency, and results are verified with a reported numerical solution, for specified range of the variables. Results complement recent approximate solutions which are valid only for the limited condition of the dimensionless angular frequency being less than or equal to unity, whereby suggesting a new Stokes number.

Analytical Solution for Dynamic Response of a Reinforced Concrete Beam with Viscoelastic Bearings Subjected to Moving Loads.

To provide a theoretical basis for eliminating resonance and optimizing the design of viscoelastically supported bridges, this paper investigates the analytical solutions of train-induced vibrations in railway bridges with low-stiffness and high-damping rubber bearings. First, the shape function of the viscoelastic bearing reinforced concrete (RC) beam is derived for the dynamic response of the viscoelastic bearing RC beam subjected to a single moving load. Furthermore, based on the simplified shape function, the dynamic response of the viscoelastic bearing RC beam under equidistant moving loads is studied. The results show that the stiffness and damping effect on the dynamic response of the supports cannot be neglected. The support stiffness might adversely increase the dynamic response. Further, due to the effect of support damping, the free vibration response of RC beams in resonance may be significantly suppressed. Finally, when the moving loads leave the bridge, the displacement amplitude of the viscoelastic support beam in free vibration is significantly larger than that of the rigid support beam.

Novel analytical solutions for convolution in compartmental pharmacokinetic models and application to non-bioequivalent formulations.

Deconvolution and convolution are powerful tools that allow decomposition and reconstruction, respectively, of plasma versus time profiles from input and impulse functions. While deconvolution have commonly used compartmental approaches (e.g., Wagner-Nelson or Loo-Riegelman), convolution most typically used the convolution integral which can be solved with numerical methods. In 2005, an analytical solution for one-compartment pharmacokinetic was proposed and has been widely used ever since. However, to the best of our knowledge, analytical solutions for drugs distributed in more than one compartment have not been reported yet. In this paper, analytical solutions for compartmental convolution from both original and exact Loo-Riegelman approaches were developed and evaluated for different scenarios. While convolution from original approach was slightly more precise than that from the exact Loo-Riegelman, both methods were extremely accurate for reconstruction of plasma profiles after respective deconvolutions. Nonetheless, convolution from exact Loo-Riegelman was easier to interpret and to be manipulated mathematically. In fact, convolution solutions for three and more compartments can be easily written with this approach. Finally, our convolution analytical solution was applied to predict the failure in bioequivalence for levonorgestrel, demonstrating that equations in this paper may be useful tools for pharmaceutical scientists.

Analytical study of water infiltration and contaminant transport in barrier systems.

An analytical model was developed to assess the service time of the barrier system consisting of a two-layer cover system and a cut-off wall. The recursive method is used to evaluate the influence of the variable head loss boundary condition caused by the water infiltration. The impact of the types of cover systems and cut-off walls on the barrier system performance is assessed. The results show that cover system types are more likely to influence the long-time performance of barrier systems. Contaminant concentrations with H1* = 0.5 m and H2* = 0.3 m when t = 40 and 100 years are 1.17 and 1.42 times larger than those with H1* = H2* = 0.5 m, respectively. The decrease in hydraulic conductivity of cut-off wall and the increase in the thicknesses and retardation factors of it can also significantly improve the performance of barrier systems. Among all of the parameters, the cut-off wall thickness poses the most significant influence on the contaminant cumulative concentrations, followed by the retardation factor of the cut-off wall, the thickness and hydraulic conductivity of the lower cover layer, the hydraulic conductivity of the cut-off wall, and the thickness and hydraulic conductivity of the upper cover layer. Additionally, the proposed solution is used for the barrier system design of a mine legacy site. The minimum design thicknesses of the cut-off walls for three different cover system types and service times are obtained.

Impact of Multi-Scattered LiDAR Returns in Fog.

In the context of autonomous driving, the augmentation of existing data through simulations provides an elegant solution to the challenge of capturing the full range of adverse weather conditions in training datasets. However, existing physics-based augmentation models typically rely on single scattering approximations to predict light propagation under unfavorable conditions, such as fog. This can prevent the reproduction of important signal characteristics encountered in a real-world environment. Consequently, in this work, Monte Carlo simulations are employed to assess the relevance of multiple-scattered light to the detected LiDAR signal in different types of fog, with scattering phase functions calculated from Mie theory considering real particle size distributions. Bidirectional path tracing is used within the self-developed GPU-accelerated Monte Carlo software to compensate for the unfavorable photon statistics associated with the limited detection aperture of the LiDAR geometry. To validate the Monte Carlo software, an analytical solution of the radiative transfer equation for the time-resolved radiance in terms of scattering orders is derived, thereby providing an explicit representation of the double-scattered contributions. The results of the simulations demonstrate that the shape of the detected signal can be significantly impacted by multiple-scattered light, depending on LiDAR geometry and visibility. In particular, double-scattered light can dominate the overall signal at low visibilities. This indicates that considering higher scattering orders is essential for improving AI-based perception models.

A Theoretical Study on Static Gas Pressure Measurement via Circular Non-Touch Mode Capacitive Pressure Sensor.

A circular non-touch mode capacitive pressure sensor can operate in both transverse and normal uniform loading modes, but the elastic behavior of its movable electrode plate is different under the two different loading modes, making its input-output analytical relationships between pressure and capacitance different. This suggests that when such a sensor operates, respectively, in transverse and normal uniform loading modes, the theory of its numerical design and calibration is different, in other words, the theory for the transverse uniform loading mode (available in the literature) cannot be used as the theory for the normal uniform loading mode (not yet available in the literature). In this paper, a circular non-touch mode capacitive pressure sensor operating in normal uniform loading mode is considered. The elastic behavior of the movable electrode plate of the sensor under normal uniform loading is analytically solved with the improved governing equations, and the improved analytical solution obtained can be used to mathematically describe the movable electrode plate with larger elastic deflections, in comparison with the existing two analytical solutions in the literature. This provides a larger technical space for developing the circular non-touch mode capacitive pressure sensors used for measuring the static gas pressure (belonging to normal uniform loading).

Analytical solution of the Bloch-McConnell equations for steady-state CEST Z-spectra.

PURPOSE: To derive an analytic expression for the steady-state Chemical Exchange Saturation Transfer (CEST) Z-spectra of a two-pool proton-exchanging system, facilitating simulations and expedited fitting of steady-state Z-spectra. METHOD: The analytical expression is derived by directly solving the set of Bloch-McConnell differential equations in matrix form for a two-pool exchanging system, determining water magnetization under steady-state saturation across the entire Z-spectrum. The analytic solution is compared and validated against the numerical solution of Bloch-McConnell equations under prolonged saturation. The study also explores the line shape of a CEST peak, interpolating under-sampled Z-spectra, and Z-spectral fitting in the presence of noise. RESULTS: The derived analytic solution accurately reproduces spectra obtained through numerical solutions. Direct fitting of simulated CEST spectra with the analytical solution yields the physical parameters of the exchanging system. The study shows that the analytical solution enables the reproduction of fully sampled spectra from sparsely sampled Z-spectra. Additionally, it confirms the approximation of the CEST spectrum of a single exchanging proton species with a Lorentzian function. Monte Carlo simulations reveal that the accuracy and precision of Z-spectral fittings for physical parameters are significantly influenced by data noise. The study also derives and discusses the analytical solution for three-pool Z-spectra. CONCLUSION: The derived analytic solution for steady state Z-spectra can be utilized for simulations and Z-spectrum fitting, significantly reducing fitting times compared to numerical methods employed for fitting CEST Z-spectra.

An Improved Theory for Designing and Numerically Calibrating Circular Touch Mode Capacitive Pressure Sensors.

The design, especially the numerical calibration, of a circular touch mode capacitive pressure sensor is highly dependent on the accuracy of the analytical solution of the contact problem between the circular conductive membrane and the rigid plate of the sensor. In this paper, the plate/membrane contact problem is reformulated using a more accurate in-plane equilibrium equation, and a new and more accurate analytical solution is presented. On this basis, the design and numerical calibration theory for circular touch mode capacitive pressure sensors has been greatly improved and perfected. The analytical relationships of pressure and capacitance are numerically calculated using the new and previous analytical solutions, and the gradually increasing difference between the two numerical calculation results with the gradual increase in the applied pressure is graphically shown. How to use analytical solutions and analytical relationships to design and numerically calibrate a circular touch mode capacitive pressure sensor with a specified pressure detecting range is illustrated in detail. The effect of changing design parameters on capacitance-pressure analytical relationships is comprehensively investigated; thus, the direction of changing design parameters to meet the required or desired range of pressure or capacitance is clarified.

A Hybrid Convolutional and Recurrent Neural Network for Multi-Sensor Pile Damage Detection with Time Series.

Machine learning (ML) algorithms are increasingly applied to structure health monitoring (SHM) problems. However, their application to pile damage detection (PDD) is hindered by the complexity of the problem. A novel multi-sensor pile damage detection (MSPDD) method is proposed in this paper to extend the application of ML algorithms in the automatic identification of PDD. The time-series signals collected by multiple sensors during the pile integrity test are first processed by the traveling wave decomposition (TWD) theory and are then input into a hybrid one-dimensional (1D) convolutional and recurrent neural network. The hybrid neural network can achieve the automatic multi-task identification of pile damage detection based on the time series of MSPDD results. Finally, the analytical solution-based sample set is utilized to evaluate the performance of the proposed hybrid model. The outputs of the multi-task learning framework can provide a detailed description of the actual pile quality and provide strong support for the classification of pile quality as well.

Flexoelectric and size-dependent effects on hygro-thermal vibration of variable thickness fluid-infiltrated porous metal foam nanoplates.

The Galerkin-Vlasov approach based on the improved first-order shear deformation theory (i-FSDT) and nonlocal elasticity theory are proposed to investigate the free vibration response of variable-thickness fluid-infiltrated porous metal foam (FPMF) nanoplates with flexoelectricity effect resting on Pasternak elastic foundation in the hygro-thermal environment. The FPMF nanoplate thickness varies according to both the length and width directions. The novelty of the present work is to consider the influence of the nonlocal's spatial variation and flexoelectric coefficients on the free vibration behavior of the nanoplates. Based on Hamilton's principle, the governing equation of FPMF nanoplate is established. The accuracy of the proposed method is checked by comparing the obtained results with those of available work in the literature. The effects of the parameters such as the flexoelectric coefficient, nonlocal coefficient, porosity coefficient, Skempton factor, temperature and moisture, thickness variation, and various boundary conditions on the natural frequency of the nanoplate are examined.

A general analytical solution for fluid flow and heat convection through arbitrary-shaped triangular ducts: A variational analysis.

This paper presents an analytical solution for fluid flow and heat transfer inside arbitrarily-shaped triangular ducts for the first time. The former analytical solutions are limited to the special case of isosceles triangular ducts. The literature has no report about the analytical solution for the general case of arbitrarily-shaped triangular ducts. Due to the significant role of fluid flow through non-circular channels in industry and the large number of triangular shapes, a method for solving the heat transfer problem for all triangular shapes is needed. The heat transfer of a fluid flow through a channel with an arbitrary triangular cross-section for the case of constant heat flux at the walls is solved in this work for the first time, considering viscous dissipation. Here, the functionals of flow and heat transfer equations are derived, and the resulting Euler-Lagrange equations are solved using the Ritz method. The effect of the duct geometry on the velocity profile and friction coefficient is studied in detail. The effect of the Brinkman number on the temperature distribution and Nusselt number is investigated for both cooling and heating cases. The results reveal that the critical Brinkman Number distinguishes between the cooling and heating cases and represents the critical point at which the Nusselt number approaches infinity. The value of the Nusselt number decreases with the increase of the Brinkman number in both the wall cooling and heating modes. It is also found that the equilateral triangle exhibits the minimum friction coefficient and the maximum value of the Poiseuille number.

Analytical and numerical (FEA) solution for steady-state heat transfer in generic FGM cylinder coated with two layers of isotropic material under convective-radiative boundary conditions.

An investigation was carried out in order to develop an accurate analytical solution and a numerical (FEA) solution for steady-state heat transfer in a circular sandwich structure incorporated with convective-radiative boundary conditions. The dimensional governing equations and boundary conditions were developed in the form of a 4th order algebraic equation, and then the solution was obtained using Ferrari's method. By solving for the roots of the quartic equation, we were able to determine the dimensionless temperature fields of the FG sandwich composite. The findings obtained utilizing the exact analytical solution for the FG sandwich composite under thermal loads were satisfactorily validated against those data obtained using the Galerkin finite element approximation. The impact of geometric and thermo-physical characteristics, such as Biot number (Bii=1,2), Inner and outer surface thickness ratio (ri=1,2Ro), ambient temperature ratio (θd), radiation-conduction parameter (Nr), and thermal conductivity ratio (λ3λ1) on the efficiency of heat transfer, has also been studied. This study reveals the distinct effect of Biot number on the inner and outer layers of the composite cylinder. It shows that Bi1 has a negligent effect on temperature distribution; on the other hand, the outer surface (Bi2≤1) minimizes temperature variation. However, for design consideration, a thicker inner face sheet is not recommended in high thermal load, as Nr>4 has an insignificant impact on inner surface thickness on top surface temperature. Moreover, the outer surface temperature appears to be more sensitive to θd than the radiation-convection side. Furthermore, the given analytical solution is adequately verified against the proposed FEA method, having an error of less than 1.5%.

Interaction between an edge dislocation and a circular elastic inhomogeneity with Steigmann-Ogden interface.

We propose an effective method for the solution of the plane problem of an edge dislocation in the vicinity of a circular inhomogeneity with Steigmann-Ogden interface. Using analytic continuation, the pair of analytic functions defined in the infinite matrix surrounding the inhomogeneity can be expressed in terms of the pair of analytic functions defined inside the circular inhomogeneity. Once the two analytic functions defined in the circular inhomogeneity are expanded in Taylor series with unknown complex coefficients, the Steigmann-Ogden interface condition can be written explicitly in complex form. Consequently, all of the complex coefficients appearing in the Taylor series can be uniquely determined so that the two pairs of analytic functions are then completely determined. An explicit and general expression of the image force acting on the edge dislocation is derived using the Peach-Koehler formula.

An Analytical Thermal Buckling Model for Semiconductor Chips on a Substrate.

Semiconductor chips on a substrate have a wide range of applications in electronic devices. However, environmental temperature changes may cause mechanical buckling of the chips, resulting in an urgent demand to develop analytical models to study this issue with high efficiency and accuracy such that safety designs can be sought. In this paper, the thermal buckling of chips on a substrate is considered as that of plates on a Winkler elastic foundation and is studied by the symplectic superposition method (SSM) within the symplectic space-based Hamiltonian system. The solution procedure starts by converting the original problem into two subproblems, which are solved by using the separation of variables and the symplectic eigenvector expansion. Through the equivalence between the original problem and the superposition of subproblems, the final analytical thermal buckling solutions are obtained. The SSM does not require any assumptions of solution forms, which is a distinctive advantage compared with traditional analytical methods. Comprehensive numerical results by the SSM for both buckling temperatures and mode shapes are presented and are well validated through comparison with those using the finite element method. With the solutions obtained, the effects of the moduli of elastic foundations and geometric parameters on critical buckling temperatures and buckling mode shapes are investigated.

Poisson representation: a bridge between discrete and continuous models of stochastic gene regulatory networks.

Stochastic gene expression dynamics can be modelled either discretely or continuously. Previous studies have shown that the mRNA or protein number distributions of some simple discrete and continuous gene expression models are related by Gardiner's Poisson representation. Here, we systematically investigate the Poisson representation in complex stochastic gene regulatory networks. We show that when the gene of interest is unregulated, the discrete and continuous descriptions of stochastic gene expression are always related by the Poisson representation, no matter how complex the model is. This generalizes the results obtained in Dattani & Barahona (Dattani & Barahona 2017 J. R. Soc. Interface 14, 20160833 (doi:10.1098/rsif.2016.0833)). In addition, using a simple counter-example, we find that the Poisson representation in general fails to link the two descriptions when the gene is regulated. However, for a general stochastic gene regulatory network, we demonstrate that the discrete and continuous models are approximately related by the Poisson representation in the limit of large protein numbers. These theoretical results are further applied to analytically solve many complex gene expression models whose exact distributions are previously unknown.

One-dimensional consolidation of highly saturated abandoned-soil under time-dependent stress: An analytical study.

A thorough knowledge of the consolidation behavior of highly saturated soil under time-dependent stress is essential for the design and construction of abandoned-soil dump sites in the soft soil regions of China. In this study, one-dimensional consolidation analytical solutions are derived for such soil under one-way and two-way drainage conditions, accommodating the time-dependent stress created by various dumping protocols. Representative soil samples are obtained, and consolidation tests are conducted with various saturation degrees (one-way drainage) and loading protocols (two-way drainage), to verify the consolidation equation and determine its range of applicability to various saturation degrees. The effects of layer thickness, dumping type, and compaction degree on the consolidation behaviors of highly saturated abandoned-soil dumps are investigated. The one-dimensional consolidation equation is applicable to soil with saturation degree not lower than 75% under instantaneous stress, stepped stress, and linear stress. The pore pressure distribution with depth is not symmetrical; the eccentric distance of consolidation degree increases with increasing layer thickness in the stress application stage and is approximately zero in the stress keeping stage. The pore pressure at middle of the soil layer increases with increasing layer thickness and decreases with increasing dumping rate from the completion of soil dumping. With increasing compaction degree, the middle pore pressure increases, while the surface settlement decreases. In the premise of the stability of an abandoned-soil dump, where the goals are to reduce post-construction settlement and to shorten the consolidation process of the entire soil layer, the important factors are smaller layer thickness, higher dumping rate, and larger compaction degree.

Radiostrontium determination by combination of automated Sr isolation and "on-column" Cherenkov detection using chromatographic model.

A novel analytical solution of non-linear chromatography in case of parabolic isotherm for frontal analysis was obtained by combination of Cole-Hopf and Laplace transform. It was used for simulation of strontium capturing on chromatographic column with aim to improve quantitative determination of low-level 90Sr activities. From the experimentally determined breakthrough curves, the retention factor and the number of theoretical plates were calculated using the Glueckauf and Wenzel relations and by fitting the breakthrough curves for the linear isotherm using MatLab. These were used to simulate the breakthrough curves using a parabolic isotherm solution where the non-linear term of the isotherm was taken as a small negative deviation of the retention factor. On the base of theoretical prediction and experimental data, procedure for automated capturing of strontium on chromatographic column with specific dimension and off line "on-column" Cherenkov detection on commercial ultra low-level liquid scintillation counter was developed. It was shown that analytical solution for parabolic isotherm in comparison with solution for linear isotherm gives better prediction of mass of captured Sr on column filled with small amount of Sr resin and SuperLig®620 in case of elevated Sr concentration, even when non-linear effect is not obvious. The solution also makes it possible to predict the mass of resin required for strontium isolation at 100% yield under given conditions. Considering the limited dimensions of the column, and consequently small mass of the resin in them, it resulted in the low efficiency of the columns, which, however, did not affect the yield in real conditions of isolation. The results have shown that the yields achieved after isolation on SuperLig®620 from real samples are 100%. In addition, it is shown that captured 90Sr can be detected through 90Y ingrowth, on column filled with strontium specific resin, with Cherenkov detection efficiency of at least 50%. The efficiency may be enhanced to 60%, depending on parameters which can affect detection efficiency change (type of column, resin type, surrounding solution, etc.). The developed procedures enable quantitative determination of 90Sr in natural water samples with MDAC below 12 mBq l-1 and solid (soil and vegetation) samples with MDAC below 6 Bq kg-1 within 2-3 days. The proposed solution may easily be implemented in radiochemical laboratories where this type of analysis is routinely done within environmental monitoring or other purposes.