One-dimensional Analytical Solution Research Articles

Exploring ice melting dynamics in beverageware

The ice-melting process inside water-filled beverageware is studied experimentally, theoretically, and numerically. An ice body is immersed in warm fresh water in different-shaped glass cups. The ice body eventually melts due to heat transfer from the warmer fluid to the ice. As the ice melts and cools the surrounding fluid, it promotes complex time-dependent laminar flows characterized by a vertical downward jet beneath the ice body and rotating convective cells in the fluid domain. Experimental results obtained through dye visualization, particle image velocity, and local temperature measurements show that the size and number of the rotating cells, jet velocity, and melting time of the ice depend significantly on the shape of the beverageware. A one-dimensional analytical solution reproduces qualitatively the hydrodynamical and thermal boundary layers close to the ice body. Further, a two-dimensional numerical model correctly captures the main features of the experimental observations.

Analytical Solution for Contaminant Transport through a Soil–Bentonite (SB)/Geosynthetic Clay Liner (GCL)/Soil–Bentonite (SB) Composite Cutoff Wall and an Aquifer

This study develops a one-dimensional analytical solution for contaminant advection, diffusion and adsorption through a soil–bentonite (SB)/geosynthetic clay liner (GCL)/SB–aquifer composite cutoff wall (CCW) system. The solution agrees well with an existing double-layer model. Adopting toluene as a representative contaminant, using the present solution, the analysis systematically investigates the impact of hydraulic gradient (i) and the hydraulic conductivities of GCL (kgcl) and SB (ksb). The results show the following: (1) Increasing i from 0.1 to 1 reduces the concentration breakthrough time (tcb) from 20 to 11 years and mass flux breakthrough time (tfb) from infinite to 11 years, indicating lower i significantly extend both tcb and tfb, which is crucial for optimizing CCW barrier performance; (2) lowering kgcl from 5.0 × 10−11 m/s to 1 × 10−12 m/s and reducing ksb from 1.0 × 10−9 m/s to 1.0 × 10−11 m/s, would increase the tcb by 36% and 100%, respectively. It demonstrates that reducing kgcl and ksb could enhance barrier performance. (3) To achieve equivalent barrier performance, soil–bentonite cutoff wall (SBCW) requires greater thickness compared to SB/GCL/SB CCW, indicating that GCL reduces the required amount of bentonite; and (4) CCWs can use SB with lower adsorption capacity to achieve equivalent performance, further reducing bentonite requirements. The present solution can aid in the design and optimization of GCL-enhanced CCWs.

Image analysis technique for quantifying fluorescein concentration profiles in clays

The image analysis technique approach links the dye concentration and color intensity, enabling quantification of spatial and temporal dye dynamics in clay media. In this study, batch adsorption tests and two-dimensional diffusion experiments with three types of clays (kaolinite, montmorillonite, and bentonite) were conducted for quantifying the vertical distribution of fluorescein dye tracer to evaluate the feasibility of the image analysis technique. In clay domains, color intensities were measured using images obtained during diffusion experiments, and these intensities were converted to aqueous fluorescein concentrations. Concentration profiles simulated from a one-dimensional analytical solution agreed well with those measured by the LRV technique with a single-channel application for kaolinite (blue channel, EB = 0.94 and 0.96) and montmorillonite (green channel, EG = 0.99 and 0.91) in the early (31.1, 54.6 PVs) and later (75.7, 93.9 PVs) diffusion times, respectively. The single-channel application for the bentonite was appropriate for concentration ranges of 0 − 400 mg/L for the blue channel and 400 − 1800 mg/L for the green channel. For quantifying the bentonite concentration profile with a wide range of dye concentrations (0 − 1800 mg/L), multi-channel application (blue + green channels) was required. Measured LRV concentration profiles with the multi-channel application showed better agreement (EB+G = 0.94) with simulated concentration profiles from the analytical solution than those with the single-channel application (EB = 0.56). The results of this study offer guidance on the application of the LRV approach as a non-destructive optical dye measurement method to diffusion problems between high and low permeability zones.

An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations

In recent years, machine learning algorithms, especially deep learning, have shown promising prospects in solving Partial Differential Equations (PDEs). However, as the dimension increases, the relationship and interaction between variables become more complex, and existing methods are difficult to provide fast and interpretable solutions for high-dimensional PDEs. To address this issue, we propose a genetic programming symbolic regression algorithm based on transfer learning and automatic differentiation to solve PDEs. This method uses genetic programming to search for a mathematically understandable expression and combines automatic differentiation to determine whether the search result satisfies the PDE and boundary conditions to be solved. To overcome the problem of slow solution speed caused by large search space, we propose a transfer learning mechanism that transfers the structure of one-dimensional PDE analytical solution to the form of high-dimensional PDE solution. We tested three representative types of PDEs, and the results showed that our proposed method can obtain reliable and human-understandable real solutions or algebraic equivalent solutions of PDEs, and the convergence speed is better than the compared methods. Code of this project is at https://github.com/grassdeerdeer/HD-TLGP.

BDFLUX: A new bounded domain model for quantifying vertical fluxes in streambeds using heat as a tracer

The use of heat as a tracer has become a widely used approach to estimate vertical streambed fluxes (VSFs). We present BDFLUX, a new bounded domain model that can be used as a rapid-screening tool to quantify VSFs and thermal properties of streambed sediments by analyzing on-site temperature–time series. BDFLUX uses a closed-form analytical solution that takes thermal advection and thermal conduction into account. BDFLUX, unlike other one-dimensional closed-form analytical solutions, does not assume a semi-infinite half-space. The initial and boundary conditions are described by arbitrary functions that can be directly determined by fitting the measured temperature–time series in the streambed at various depths. The particle swarm optimization (PSO) method and Markov Chain Monte Carlo (MCMC) method are used to estimate VSFs using temperature–time series observed at three depths. BDFLUX is validated using synthetic temperature–time series, demonstrating that it can perform well when the top and bottom boundary conditions are determined directly by the measured data. The capabilities of BDFLUX are further evaluated using laboratory and field data. The simulation results show that BDFLUX performed well in terms of interpreting laboratory experiments and field data, with a root mean square error as low as 0.059. The simulation of laboratory experiments shows that the MCMC method has a lower computational efficiency than PSO method. The magnitude of nonvertical flow components at a field site were quantified by analyzing the results of estimated VSFs in different vertical subdomains. Overall, BDFLUX improves one-dimensional heat analytical models in terms of assessing the magnitude of the nonvertical flow component in heterogeneous streambed sediments.

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.

Analytical study on the seepage field of EPBS and preliminary discrimination of spewing

Considering the two-dimensional seepage effect of a soil chamber, the spewing problem of an earth pressure balance shield (EPBS) is analysed by an analytical method. In this paper, an explicit analytical solution of the two-dimensional shield seepage field considering the water level distribution at the cutter surface is derived by using the separation of variables method and the superposition principle, and the unknowns in the analytical solution are solved by constructing a non-homogeneous system of equations. The correctness of the solution is verified by using the experimental and numerical simulation results, and a parameter analysis was conducted to investigate the effect of shield geometry and water head at the center of the cutter surface. The results show that water head on the cutter surface, screw conveyor angle, screw conveyor length and screw conveyor in the water flow and water pressure with an approximate linear relationship, reducing the screw conveyor diameter is most conducive to prevent spewing. On the basis of previous studies, the analytical solution can calculate the critical permeability coefficient for the occurrence of a spewing, and use the undisturbed soil or predicted permeability coefficient to preliminarily discriminate whether a spewing will occur. By comparing with the shield spewing example and with the shield one-dimensional analytical solution, it is proven that spewing discrimination based on this analytical solution is effective and more accurate. Since the analytical solution considers the water level distribution on the cutter surface, it can be used to discriminate the spewing of shields of any size.

A particle-size dependent smoothing scheme for polydisperse Euler-Lagrange simulations

In Euler-Lagrange models for particulate systems, the information exchange of multiphase coupling terms, i.e. drag force, heat transfer, or mass transfer, is often smoothed from particles' location to nearby computational cells when the cell and particle sizes are comparable. The diffusion-based smoothing is among the most popular approach. However, when it comes to polydisperse systems, the state-of-the-art constant diffusivity approach does not consider variation in particle sizes, resulting in an even smoothing of exchanged information across all particles with different sizes.In this paper, a particle-size dependent smoothing scheme named “non-constant diffusivity approach” is proposed. This approach distributes the exchanged information based on the local Sauter diameter d32 such that the area of influence per particle varies by size. The one-dimensional analytical solution of the non-constant diffusivity approach is first derived and two control parameters L⁎ and S are identified, where L⁎ determines the characteristic length of the diffusivity and S determines the magnitude of the diffusivity. Together they control the characteristic length scale ℓ˜c of the smoothing operation. In the one-dimensional study, it is found that the proposed smoothing method is most suitable for the dilute particulate regime. The investigation is extended to a three-dimensional dilute particle flow based on the experiment of the CRIEPI (Central Research Institute of Electric Power Industry) pulverized coal burner. It is shown that using the non-constant diffusivity smoothing, the prediction of the O2 concentration is more accurate than when using the constant diffusivity smoothing approach, which assumes a universal smoothing length for all sizes of particles. The findings suggest that when it comes to polydisperse particulate systems, the area of influence per particle on the fluid depends on the particle size. The proposed non-constant diffusivity approach can describe that with the benefits of cheap computational cost, conservation of exchanged quantities, and easy-to-implement.

A length scale insensitive phase field model based on geometric function for brittle materials

The commonly used phase field model has a broad application in static and dynamic scenarios with brittle materials. Following the assumption of irreversibility of fracture, monotonically increasing phase and tensile strain energy are applied in the commonly used phase field model. By doing so, a clear stiffness reduction, however, is observed in stress–strain curve before reaching fracture strength, which conflicts with the linear elastic behavior of brittle materials. The limitation has also been mentioned in publications when length scale is the only parameter to adjust fracture strength. To solve these problems, a generalized quadratic geometric function is proposed to build a length scale insensitive phase field model. Governing equations and one-dimensional analytical solutions (homogeneous and nonhomogeneous solutions) of this length scale insensitive phase field model are presented. By adjusting the extra parameter in the generalized quadratic geometric function and zeroing negative phase, different phase profile (ranged from exponential profile to linear profile) and fracture strength can be obtained, which could be more suitable to cover different types of fractures. Meanwhile, a narrower crack band and linear elastic behavior can be guaranteed. LS-DYNA user-defined element and user-defined material subroutines are used to implement the proposed length scale insensitive phase field model. A tensile test successfully verifies the properties of a narrower crack band and linear elastic behavior. Three representative numerical examples show the feasibility of this model to simulate fracture propagation, including a Mode Ⅰ failure of wedge splitting test, a mixed-mode failure of l-shaped panel test and a mixed-mode failure of notched beam test. From the crack band and force–displacement curve, the proposed model can handle fracture propagation better for brittle materials.

Theoretical analysis for thermal consolidation of marine sediments with depth variability subjected to time-dependent loading and heating

Because the spatial distribution of energy is restricted, many energies geotechnics would build on marine sediments and the resulting thermal consolidation would pose an inevitable threat to the safety and stability of the project. In this study, a new governing equation for thermal consolidation of saturated marine sediments is proposed, by considering the depth variability of the marine sediment layer and the time-dependent external loading and temperature. The corresponding one-dimensional (1D) analytical solution for thermal consolidation of saturated marine sediments is derived. The average degree of consolidation (Ua) and the normalized excess pore water pressure (u/u0) in the saturated marine sediment layer at different depths and time durations are calculated and compared with the typical loading case. The results show that the loading rate of the external force only affects the amplitude of the excess pore water pressure u and does not affect the proportion of u with depth; the depth variability of bulk modulus has a greater effect on the distribution of Ua relative to the depth variability of permeability; the depth variability of permeability has a greater effect on the distribution of u/u0 with depth relative to the depth variability of bulk modulus when Ua = 50%; the assumption of instant thermal loading will lead to an over-assessment of Ua and u/u0. This study provides useful insights for energy geotechnical engineering design and practice.

In-process nip point temperature estimation in automated tape placement based on analytical solution and remote thermal measurements

The optimisation and control of automated tape placement (ATP) requires fast analysis tools able to utilise process data for predictions and monitoring. In this study, a strategy for in-process estimation of nip point temperatures is proposed. The method is based on a combination of two one-dimensional analytical solutions of heat transfer in ATP using temperature data measured on the tool surface, combined with an inverse solution for the estimation of power delivered by the heating device on the composite surface. The performance of the method is examined against a validated finite element model. Approximations of nip point temperature show good correlation for different tool materials, with an average error of 15°C and a maximum of 50°C which is satisfactory for the processing of high-temperature thermoplastic materials. The analytical scheme offers real-time estimations of the nip point temperature with the potential to be used for process control of ATP.

Corrigendum: Shallow Compaction Modeling and Upscaling: A One-Dimensional Analytical Solution and Upscaling

Natural depositional processes frequently give rise to the heterogeneous multi-layer system, which is often overlooked but essential for simulation of a geological process. The sediments undergo the large-strain process in shallow depth and small-strain process in deep depth. With the transform matrix and Laplace transformation, a new method of solving multilayer small-strain (Terzaghi) and large-strain (Gibson) consolidations is proposed. Results from this work matches the numerical results and other analytical solutions well. According to the method of transform matrix which can consider the integral properties of multilayer consolidation, relevant upscaling method is developed. This method showed to be more effective than the normally-used weighted average method. Correspondingly, the upscaling results indicate that the upscaled properties of a multilayer system vary in consolidation process.

Shallow Compaction Modeling and Upscaling: A One-Dimensional Analytical Solution and Upscaling

Natural depositional processes frequently give rise to the heterogeneous multilayer system, which is often overlooked but essential for the simulation of a geological process. The sediments undergo the large-strain process in shallow depth and the small-strain process in deep depth. With the transform matrix and Laplace transformation, a new method of solving multilayer small-strain (Terzaghi) and large-strain (Gibson) consolidations is proposed. The results from this work match the numerical results and other analytical solutions well. According to the method of transform matrix which can consider the integral properties of multilayer consolidation, a relevant upscaling method is developed. This method is more effective than the normally used weighted average method. Correspondingly, the upscaling results indicate that the upscaled properties of a multilayer system vary in the consolidation process.

3D edge-based and nodal finite element modeling of magnetotelluric in general anisotropic media

Research on the magnetotelluric (MT) forward modeling response in anisotropic media has been important since the discovery of anisotropic electric fields in the Earth's interior. In this study, we presented an edge-based and nodal finite-element method (FEM) for three-dimensional (3D) MT forward modeling problems in general anisotropic media. Partial differential equations are derived using the Coulomb-gauged approach from Maxwell's equation and discretized by the edge-based and nodal FEM. The magnetic vector A was discretized by an edge-based FEM and the electric scalar ψ was discretized by nodal FEM. The computational domain was refined using a hexahedral mesh. The Galerkin variant of the weighted residuals method was used to obtain a sparse linear system for magnetic vector A and electric scalar ψ. Three models were designed to validate and analyze our forward modeling code by comparing it with a one-dimensional analytical solution and with the results from codes developed by other scholars. A direct solver and iterative solvers with preprocessors were used to evaluate the performance of our code. Numerical experiments showed that our forward modeling code is accurate and robust and converges faster. In addition, the stiffness matrix assembled by our algorithm was smaller than that assembled by the nodal FEM, which further demonstrates the advantage of our algorithm in terms of computational memory.

Pulse compression favorable frequency modulated thermal wave imaging for non-destructive testing and evaluation: an analytical study

InfraRed Thermography (IRT) is one of the non-destructive testing and evaluation (NDT&E) approach widely used for testing and evaluation of wide verity of materials such as metals, semiconductors and composites. Among the widely used Thermal NDT&E (TNDT&E) approaches for better depth resolution and sensitivity for detection of defects located at different depths inside the test specimen recently proposed correlation based approach gained importance due to its enhanced defect detection capabilities. The present paper introduces a novel one-dimensional analytical solution for the frequency modulated excitation scheme under adiabatic boundary conditions for detection of flat bottom holes as defects in a mild steel sample. The performance of the Pulse Thermography (PT), Lock-in Thermography (LT) and Frequency Modulated Thermal Wave Imaging (FMTWI) methods are highlighted their defect detection capabilities have been compared by adopting the recently introduced correlation based post-processing approach. Finally, the proposed analytical method has been validated with the results obtained from the commercially available finite element based software.

Spreadsheet Tools for Quantifying Seepage Flux Across the GW-SW Interface.

Identifying the spatial distribution and magnitude of seepage flux across the groundwater-surface water (GW-SW) interface is critical for assessing potential impairments and restoration alternatives for water bodies adjacent to sites with groundwater contamination. Measurement of the vertical distribution and time-varying characteristics of temperature in sediments provides an indirect way to map out spatial and temporal patterns of seepage flux into surface water. Two spreadsheet-based calculation tools are introduced that implement four one-dimensional analytical solutions to calculate the magnitude and direction of seepage flux based on measurement of steady-state vertical temperature profiles or transient diel temperature signals at two depths within sediment. Performance of these calculation tools is demonstrated for a pond receiving contaminated groundwater discharge from an adjacent landfill. Transient versus steady-state model performance is compared, and limitations of transient modelsare illustrated for a situation with unfavorable sediment characteristics and inadequate sensor spacing. The availability of a range of analytical solutions implemented within Microsoft Excel® is intended to encourage practitioners to explore use of this seepage flux characterization method and develop greater insight into best practices for model selection and use.

Variational interface element model for 2D and 3D hydraulic fracturing simulations

This paper presents an extension of the variational interface fracture model proposed in Khisamitov and Meschke (2018) to model fluid driven fracture in porous materials. The fluid saturated material is described by the theory of poroelasticity and the effective stress concept, while the coupling between fluid flow and fracture initiation and propagation is accomplished via a variational interface element formulation, using a damage variable c as an additional degree of freedom in conjunction with linear momentum and mass balance equations along the interface zones. Introducing the damage variable c, fracture propagates according to the values of the minimizers of the total potential energy expressed in terms of effective stresses. Hence, neither local crack propagation criteria nor tracking algorithms are required. Taking into account that a fracture grows quasi-statically, the discretized system of PDEs is solved by the backward Euler-scheme, ignoring the contribution from inertia forces. In the Newton–Raphson iterative solution procedure, an operator splitting algorithm is employed to solve first the poroelastic equations, and updating subsequently the damage variable c. The proposed model for hydraulic fracturing is validated first by means of one-dimensional analytical solutions in toughness as well as in viscosity dominated regimes. The performance and the applicability of the model to simulate interactions between propagating and existing fractures is demonstrated by means of 2D and 3D numerical examples.

Meshless modeling of coupled transdermal pharmacokinetics with analytical validation

An efficient localized collocation meshless computational approach is applied to the study of transdermal drug delivery (TDD). To validate the numerical technique, a one-dimensional analytical solution is first derived for the set of coupled differential equations that govern the pharmacokinetic process in the two-compartment compound diffusion model. These equations consist of one partial differential equation (PDE) and one ordinary differential equation (ODE). These equations are solved in one-dimensional space via superposition and separation of variables to arrive at the field solution for the compound concentration in the tissue. The convergence of the numerical result to the analytical result indicates the viability of the localized collocation meshless method (LCMM) as a valid approach to the solution of TDD problems. The analytical solution is used to benchmark the numerical technique for several cases with differing pharmacokinetic coefficients, demonstrating the robustness and accuracy of the meshless numerical technique. Furthermore, the highly efficient LCCM implementation is extended to address multi-dimensional TDD cases achieving stable and physically consistent results.

A two-dimensional analytical model for organic contaminants transport in a transition layer-cutoff wall-aquifer system

Cutoff walls are widely used as in-situ vertical barriers to control pollution plumes migration in aquifer. A two-dimensional analytical model for the transport of organic contaminants through a layered subsurface system with a cutoff wall is developed. The model considers a coupled advection-dispersion-adsorption-degradation process in the whole system and a non-uniform concentration distribution of a contaminant source with respect to depth. The solution was effectively validated against a one-dimensional analytical solution and a two-dimensional numerical solution. The migration behavior of contaminants in the triple-layer system and the service performance of a cutoff wall are investigated. It is demonstrated that the previous one-dimensional contaminant migration solution would underestimate the polluted ranges but overestimate the migration distance. The consideration of a non-uniformly distributed contamination source is of great importance, especially for narrowly contaminated aquifer. The biodegradation in a cutoff wall has a significant influence on the transport of an organic contaminant through the layered system when its half-life in the cutoff wall is less than 5 years, and this effect would be enhanced when the biodegradation in natural soil is also considered. The proposed model is a practical tool for performing a comprehensive design of the location and thickness of a cutoff wall.

Uncoupling Electrokinetic Flow Solutions

The continuum-scale electrokinetic porous-media flow and excess charge redistribution equations are uncoupled using eigenvalue decomposition. The uncoupling results in a pair of independent diffusion equations for "intermediate" potentials subject to modified material properties and boundary conditions. The fluid pressure and electrostatic potential are then found by recombining the solutions to the two intermediate uncoupled problems in a matrix-vector multiply. Expressions for the material properties or source terms in the intermediate uncoupled problem may require extended precision or careful re-writing to avoid numerical cancellation, but the solutions themselves can be computed in typical double precision. The approach works with analytical or gridded numerical solutions and is illustrated through two examples. The solution for flow to a pumping well is manipulated to predict streaming potential and electroosmosis, and a periodic one-dimensional analytical solution is derived and used to predict electroosmosis and streaming potential in a laboratory flow cell subjected to low frequency alternating current and pressure excitation. The examples illustrate the utility of the eigenvalue decoupling approach, repurposing existing analytical solutions and leveraging simpler-to-derive solutions or numerical models for coupled physics.