Semi-analytical Solution Research Articles

An Efficient Parallel Iterative Solver for Controlled-source Electromagnetic 3-D Adaptive Forward Modeling in General Anisotropic Media

Controlled-source electromagnetic (CSEM) surveys are moving increasingly toward realistic and complicated scenarios where the survey region may contain undulated topography, complex geometries, and electrical anisotropy media. In this paper, the adaptive finite element numerical method is employed to discrete the total electric field equation, which can provide precise electromagnetic (EM) responses even with a coarse initial mesh. The unstructured tetrahedral grid is employed to effectively address arbitrary irregular geometry and mountain terrain. Then, the flexible generalized minimum residual solver (FGMRES), auxiliary-space Maxwell pre-conditioner, and grid division technique were used to solve the large-scale linear system of equations, which can stably solve ill-conditioned problems using fewer computing resources. Finally, we conducted a numerical experiment via our newly proposed forward modeling scheme on a synthetic model with multi-contrast electrical anisotropy. It validated that accurate EM fields could be obtained against the semi-analytic solutions, and this iterative solver has good robustness for various anisotropic media. As a result, we have developed a state-of-the-art 3D CSEM anisotropic forward modeling engine, which can quickly and accurately deal with large-scale and complex geo-models.

Fast and Accurate Semi-Analytical Solution to Skin and Proximity Effects in Shielded Cable Systems

Fast and Accurate Semi-Analytical Solution to Skin and Proximity Effects in Shielded Cable Systems

A size-dependent 3D solution of functionally graded shallow nanoshells

Abstract An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen’s nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid for spherical and cylindrical shells, and rectangular plates. The stresses and displacements are assumed in terms of the Navier method which is applicable for simply supported structures. The derivatives in terms of thickness are approximated by the differential quadrature method (DQM). The thickness domain is discretized by the Chebyshev–Gauss–Lobatto grid distribution. Lagrange interpolation polynomials are considered as the basis function for DQM. The correct free surface boundary condition for out-of-plane stresses is considered. Several problems of isotropic and functionally graded shells subjected to different types of loads are analyzed. The results are compared with other three-dimensional solutions and higher-order theories. It is important to emphasize that the radii of curvature are crucial at nanoscale, so it should be considered in the design of nanodevices.

Elastoplastic solution of a circular tunnel in surrounding rock with any nonlinear yield criteria and plastic flow envelopes

Besides the (semi-)linear Mohr-Coulomb and Hoek-Brown criteria, other nonlinear functions are also used to describe the plastic response of rock mass. For a nonlinear yield and plastic flow function in the form of τ=fσn and/or σ1=fσ3, the equivalent cohesive strength and frictional angle corresponding to the Mohr-Coulomb yield criterion were derived. The normal stress was further related to the minimum principal stress using the geometrical relationship of a Mohr circle. The equivalent dilatancy angle was also derived using the same method. According to the solution of a circular tunnel in Mohr-Coulomb rock mass, this paper derived a semi-analytical solution of a circular tunnel in any nonlinear yield and plastic flow rock masses with a form of τ=fσn and/or σ1=fσ3. A numerical procedure was developed for the stresses and displacement using a reserve method. The proposed solution was verified by the results of a circular tunnel in elasto-perfectly(brittle-)-plastic and/or strain-softening rock mass. The strain-softening analysis of a circular tunnel in the assumed two nonlinear yield criteria and/or plastic flow rock masses was studied. For a linear deteriorated material parameters leads to a nonlinear evolution of equivalent cohesive strength, frictional angle and dilatancy angle in the plastic region.

Efficient prediction of residual stress in additive manufacturing based on semi-analytical solution

Efficient prediction of residual stress in additive manufacturing based on semi-analytical solution

Well-Testing Model for Dual-Porosity Reservoir considering Stress-Sensitivity and Elastic Outer Boundary Condition

Stress sensitivity and the elastic outer boundary (EOB) condition have a great effect on the analysis of the characteristics of the fluid flow in a reservoir. When researchers analyzed the characteristics of the fluid flow, they have considered the stress sensitivity and the EOB condition separately but have not considered them simultaneously. Therefore, errors are inevitable during the analysis of well testing. The main object of this work is to present a well-testing model for stress-sensitivity dual-porosity reservoir (DPR) with EOB to improve the accuracy of the analysis of well-testing data. To this end, in this paper, we established a well-testing model for the DPR, considering the stress sensitivity and the EOB simultaneously, and presented its semianalytical solution. On the basis of the consideration of the EOB condition and stress sensitivity of permeability (SSP), a seepage model for the DPR with the EOB is built using the continuity equation, motion equation, state equation, and interporosity flow equation between matrix and fracture, which considers the stress sensitivity, wellbore storage, and skin. To solve this model, a nonlinear partial differential equation is changed into a linear form of a partial differential equation by introducing an effective well radius and applying Pedrosa’s transformation and perturbation transformation. Applying the Laplace transformation, an analytical solution in the Laplace space is obtained, and curves of pressure and pressure derivative (PPD) are drawn by numerically inverting them. The model is verified by comparing it with the EOB without consideration of SSP and using case data. The sensitivity of parameters on the curves of PPD is analyzed. This work may be significant for evaluating more accurately the parameters of wells and reservoirs using well testing.

Analytical solution for channel flow of a Giesekus fluid with non-zero solvent viscosity

A semi analytical solution is obtained here for the fully developed plane Poiseuille flow of a Giesekus fluid with a Newtonian solvent. The fluid behaviour is described using the Deborah number, the mobility factor and an appropriate ratio of fluid viscosity to total viscosity as parameters. The given solution shows that the velocity increases significantly with rising the polymer concentration, confirming that dilution of the solution produces the same effect as an increase in resistance. The analysis demonstrates that there are limiting values of Deborah number related to the mobility parameter.

Elastoplastic solutions for deep-buried twin tunnels with arbitrary shapes and various arrangements under biaxial in-situ stress field based on Mohr-Coulomb and generalized Hoek-Brown criteria

Theoretical solutions for elastoplastic analysis of deep-buried twin tunnels with arbitrary shapes and various arrangements are developed under biaxial in-situ stress field based on Mohr-coulomb (M–C) and generalized Hoek-Brown (G-H-B) criteria. The boundary stress transformation formulas adapted for arbitrary stress conditions on non-circular tunnel wall are provided in terms of conformal transformations. Regarding the plastic analysis, analytical expressions for stress characteristic relationships along slip lines under generalized Hoek-Brown criterion are derived. Subsequently, the semi-analytical solutions for plastic stresses of rock surrounding arbitrary-shaped tunnels are constructed based on Mohr-Coulomb and generalized Hoek-Brown criteria applying conformal mapping technique, slip-line method and finite difference-numerical iterative approach. On the basis of the discrete plastic stresses, interpolation functions are generated according to three-dimensional linear interpolation technique. And then, solutions for elastic stress function and plastic radius of twin tunnels with various arrangements are developed with improved mapping functions and newly defined fitness functions through differential evolution method. The accuracy of the semi-analytical solutions is verified by the numerical simulation. Meanwhile, the proposed solutions have the advantages of meshless characteristic, better convergence and higher computing efficiency comparing with the numerical simulation. From the point of engineering, the developed elastoplastic solutions can provide theoretical supports for the stress analysis, plastic radius estimation, safety clearances optimizations of deep-buried twin tunnels.

Traveling wave solutions for a realistic non-Newtonian foam flow model

Due to several environmental and petroleum engineering applications, there is a renewed interest in studying foam flow in porous media, commonly described as shear-thinning non-Newtonian flow. Based on experiments reporting saturation profiles with an unchanging shape displacing through the medium, we investigate the solution of the model describing foam displacement as a traveling wave. For that, we consider a realistic non-Newtonian foam flow model, which successfully described several experimental studies found in the literature. The aimed traveling wave solution is equivalent to the existence of the solution of the corresponding ordinary differential equations connecting initial and injection conditions. We address this problem by proposing a semi-analytical technique for finding such a connection in the phase space using the explicit foam’s apparent viscosity. Our semi-analytical solution is validated with direct numerical simulations and also by comparing it to experimental data from the literature.

Analysis of Steady Groundwater Flow in Confined Aquifer Due to Long-Strip Pit Dewatering with Suspended Cut-Off Wall

This study investigates the steady flow resulting from dewatering by a partially penetrating well in a confined aquifer with a cut-off barrier. By considering flow in both horizontal and vertical directions and incorporating the barrier and pumping well as flow boundary conditions, separate mathematical models are established for the inside and outside of the cut-off barrier. The interaction between these zones is ensured through continuous conditions along the opening of the two zones. A semi-analytical solution is derived for the problem using the finite Fourier cosine transform and boundary transformation methods. The effectiveness of the method is verified by comparing it with the finite element numerical results and pumping test data respectively. Based on the proposed solutions, we proceed to analyze the influence of some relevant factors: the extent to which the cut-off wall is embedded within the confined aquifer, the depth of the partially penetrating well, and the distance to the lateral head boundary. Results indicate that a greater depth of the cut-off wall leads to a reduced pumping rate requirement for achieving a desired drawdown of the confined water level within the excavation. According to the presented solution, placing pumping wells near the top of the confined aquifer in excavation dewatering projects can facilitate a faster reduction of the confined water head at the excavation bottom. Additionally, proximity of lateral head boundary could significantly impact dewatering, with closer boundaries reducing dewatering effectiveness due to improved aquifer recharge. Finally, the use of the Fourier method showcases impressive convergence properties in the approach presented in this study. The computed results maintain a high level of approximation quality, even with extremely coarse discretization.

Smart Elastic Material with a Moving Local Stiffness Zone for Reducing the Effects of Impact Loading

The elastic material properties which change momentarily and locally under the high deformation rate due to the movement of a wavefront are presented. The work contains mathematical formulation, semianalytical results, numerical formulations, and simulation results demonstrating the effectiveness of modifying the rheological properties of the elastic material upon shock load or contact with a rigid obstacle. While the semianalytical solutions can be obtained in a narrow time interval, numerical solutions allow us to track the process of wavefront reflections from edges. The effectiveness of reducing the physical quantities significant for impact in the presented examples reaches 30–70% of forces or accelerations, depending on the adopted criteria.

Semi-analytical solution of Graetz–Brinkman problem combined with non-Newtonian Ellis fluid flow in a passive channel

Semi-analytical solution of Graetz–Brinkman problem combined with non-Newtonian Ellis fluid flow in a passive channel

Grounded source transient electromagnetic 3D forward modeling with the spectral-element method and its application in hydraulic fracturing monitoring

A long wire with large current source transient electromagnetic (TEM) monitoring, with a large detection depth, low cost, safety, and environmental protection, has unique advantages in the testing and identification of unconventional reservoir fluid and the evaluation of stimulated reservoir volume. So, the TEM 3D forward modeling method has become a research hotspot. Although the finite-element method (FEM) is a type of numerical algorithm that has been widely applied in three-dimensional (3D) electromagnetic field forward modeling, the efficiency and accuracy of FEM require further improvement in order to meet the demand of fast 3D inversion. By increasing the order of the basis function and adjusting the principle of mesh discretization, the precision of the mixed-order spectral-element (SEM) result will be increased. The backward Euler scheme is an unconditionally stable technique which can ignore the impact of the scale of the time step. To achieve a better description of the nonlinear electromagnetic (EM) response of the grounded source TEM method and to optimize the efficiency and accuracy/precision of the 3D TEM forward modeling method significantly, we proposed the use of 3D TEM forward modeling based on the mixed-order SEM and the backward Euler scheme, which can obtain more accurate EM results with fewer degrees of freedom. To check its accuracy and efficiency, the 1D and 3D layered models are applied to compare the SEM results with the semi-analytical and FEM solutions. In addition, we analyzed the accuracy and efficiency of the SEM method for different types of order basis functions. Finally, we calculated the long-wire source TEM response for a practical 3D earth model of a shale gas reservoir for fracturing monitoring and tested the feasibility of the TEM method in a hydraulic fracturing monitoring area to further demonstrate the flexibility of the SEM method.

General Semianalytical Solution for Axisymmetric Consolidation of Unsaturated Soil with Unified Boundary under Electroosmotic and Surcharge Preloading

General Semianalytical Solution for Axisymmetric Consolidation of Unsaturated Soil with Unified Boundary under Electroosmotic and Surcharge Preloading

Homotopy Analysis Fractional Sumudu Transform Method for Semi-analytic Solution of 2-D Time-Fractional Black–Scholes Equations

Homotopy Analysis Fractional Sumudu Transform Method for Semi-analytic Solution of 2-D Time-Fractional Black–Scholes Equations

Semianalytical Solution for the Uplift Bearing Capacity of Spread Foundations in Sand

Semianalytical Solution for the Uplift Bearing Capacity of Spread Foundations in Sand

Modeling the impregnation of roselle antioxidants into papaya cubes

The increasing demand for healthy foods has recently encouraged research focused on incorporating antioxidant compounds from bioactive-rich sources into other foodstuffs; however, the studies aimed at modeling the mass transfer occurring during the enrichment of food matrices with bioactive substances remain scarce. This work investigated the modeling of water loss, solids gain, impregnation of bioactive compounds - total phenolics (TPC) and total anthocyanins (TAN) - and changes in total antioxidant capacity (TAC) of papaya cubes infused with a roselle extract/sucrose solution (RESS). The RESS was used to impregnate 15 mm papaya cubes in the mass ratio of 15:1 at 45, 55 and 65 °C for different immersion times (0, 10, 15, 30, 60, 90, 120, 180, 240, 300, and 360 min). A novel semi-analytical solution for the unsteady-state diffusion equation considering the product shrinkage was proposed for describing experimental mass transfer data and for obtaining equilibrium points and effective diffusivities of migrating species. After 360 min, RESS-impregnated papaya cubes increased their TPC, TAN, and TAC in about 670–990 mg GAE/100 g fresh weight (FW), 48.8–65.0 mg C3G/100 g FW, and 805–1004 mg TE/100 g FW from 402 ± 53 mg GAE/100 g FW, 0 mg C3G/100 g FW, and 679 ± 62 mg TE/100 g FW. Diffusivities for water, total solids, TPC, TAN, and TAC were in the ranges of 1.76–12.6, 0.90–2.54, 0.65–10.8, 0.36–1.32, and 3.31–11.1 ( × 10−10) m2/s, respectively. These values were overestimated in about 23 ± 6% when papaya shrinkage, accounting up to a 50% volume loss, was not considered in the calculations.

Semi- Analytical Study on Non-Isothermal Steady R-D Equation in a Spherical Catalyst and Biocatalyst

The Lane-Emden Boundary Value Problem as it appears in chemical applications, science, and biochemical applications are employed. Two specific models are solved by applying the Ananthaswamy-Sivasankari method (ASM). The model in first problem is a reaction–diffusion equation of a spherical catalyst and the model in second problem is the reaction–diffusion process of a spherical biocatalyst. Obtain a reliable semi-analytical expression of the effectiveness factors and the concentrations. A graph is constructed for the obtained semi-analytical solutions.The effects of several parameters like dimensionless activation energy, Thiele modulus and dimensionless heat of reaction are shown in graphical representation. Our semi-analytical solution is compared with numerical simulation by using MATLAB and finds good fit in all parameters. The new analytical method ASM is helpful to solve many non-linear problems mainly Reaction-Diffusion equation.

Jacobi Rational Operational Approach for Time-Fractional Sub-Diffusion Equation on a Semi-Infinite Domain

In this study, we employ a rational Jacobi collocation technique to effectively address linear time-fractional subdiffusion and reaction sub-diffusion equations. The semi-analytic approximation solution, in this case, represents the spatial and temporal variables as a series of rational Jacobi polynomials. Subsequently, we apply the operational collocation method to convert the target equations into a system of algebraic equations. A comprehensive investigation into the convergence properties of the dual series expansion employed in this approximation is conducted, demonstrating the robustness of the numerical method put forth. To illustrate the method's accuracy and practicality, we present several numerical examples. The advantages of this method are: high accuracy, efficiency, applicability, and high rate of convergence.

Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials

Approximate solutions for a family of nonlinear fractional-order differential equations are introduced in this work. The fractional-order operator of the derivative are provided in the Caputo sense. The third-kind Chebyshev polynomials are discussed briefly, then operational matrices of fractional and integer-order derivatives for third-kind Chebyshev polynomials are constructed. These obtained matrices are a critical component of the proposed strategy. The created matrices are used in the context of approximation theory to solve the stated problem. The fundamental advantage of this method is that it converts the nonlinear fractional-order problem into a system of algebraic equations that can be numerically solved. The error bound for the suggested technique is computed, and numerical experiments are presented to verify and support the accuracy and efficiency of the proposed method for solving the class of nonlinear multi-term fractional-order differential equations.