Rigorous Solution Research Articles

External gravitational field of a homogeneous ellipsoidal shell: a reference for testing gravity modelling software

There are numerous applications in geodesy and other geo-sciences in which the gravitational potential effect or other functions of the potential are computed by forward modelling from a given mass distribution. Different volume discretisations, e.g. prisms, tesseroids or mass layers are used. In order to control the numerical realisation of the forward calculation in the practical application, e.g. in reduction tasks, these evaluation programs should be verified against rigorous analytical solutions. In this contribution, a closed analytical solution for the potential of an ellipsoidal shell as a test body is presented. Furthermore, we derive the respective closed formulae for the gravity vector and the gravity gradient tensor. Program implementations of the tesseroid approach are compared on the basis of this ellipsoidal mass arrangement. For the practical usage, fast-converging expansions in spherical harmonics are provided in addition. The derivation of the formulae is based on a closed solution of the potential of a homogeneous ellipsoid for computation points situated on the rotation axis, which then is extended to the external space.

Tunnel Face Stability Considering the Influence of Excess Slurry Pressure

With excess slurry pressures exerted on the tunnel face, slurry particles tend to infiltrate into the soil in front of the tunnel. There will be excess pore pressure ahead of the tunnel in the case of infiltration, leading to an impairment in the supporting effect contributed by the excess slurry pressure. Corresponding to three slurry infiltration scenarios distinguished by the forms of the filter cake, different pressure transfer models are employed to describe the pore pressure distribution. Using the kinematic approach of limit analysis and the numerically simulated seepage field, the study of tunnel face stability under different slurry infiltration cases is extended by employing a 3D discretization-based failure mechanism. In addition, two simple empirical formulas describing the pore pressure distributions above the tunnel and in advance of the tunnel are established and verified. Combined with the dichotomy method and strength reduction method, the safety factors yielding rigorous upper-bound solutions are obtained by optimization. The proposed method is validated by a comparative analysis. The developed framework allows considering the influence of excess pore pressure on the whole failure mechanism and the three-dimensional characteristics of seepage. A parameter analysis is performed to study the effect of the excess slurry pressure, hydraulic conditions, soil strength properties, and pressure drop coefficient. The results show that the steady-state flow model leads to much more conservative results than the full-membrane model. The safety factor increases with the increasing excess slurry pressure and the decreasing pressure drop coefficient. The present work provides an effective framework to quickly assess the face stability of tunnels under excess slurry pressure considering different filter cake scenarios.

Efficient and accurate intensity diffraction tomography of multiple-scattering samples.

Optical Diffraction Tomography (ODT) is a label-free method to quantitatively estimate the 3D refractive index (RI) distributions of microscopic samples. Recently, significant efforts were directed towards methods to model multiple-scattering objects. The fidelity of reconstructions rely on accurately modelling light-matter interactions, but the efficient simulation of light propagation through high-RI structures over a large range of illumination angles is still challenging. Here we present a solution dealing with these problems, proposing a method that allows one to efficiently model the tomographic image formation for strongly scattering objects illuminated over a wide range of angles. Instead of propagating tilted plane waves we apply rotations on the illuminated object and optical field and formulate a new and robust multi-slice model suitable for high-RI contrast structures. We test reconstructions made by our approach against simulations and experiments, using rigorous solutions to Maxwell's equations as ground truth. We find the proposed method to produce reconstructions of higher fidelity compared to conventional multi-slice methods, especially for the challenging case of strongly scattering samples where conventional reconstruction methods fail.

Rigorous solution for drained analysis of spherical cavity expansion in soils of finite radial extent

This paper revisits the expansion problem of a spherical cavity in an elastic–plastic soil mass with a finite radial extent under drained conditions, which proves to be non-self-similar. Rigorous semi-analytical solutions for the analysis of this problem are developed adopting several well-known soil models. With the small strain assumption, analytical expressions for both elastic stresses and displacements are obtained considering the stress dependency of soil moduli and influences of the outer boundary. In the plastic zone, the non-self-similar expansion process is formulated into a set of first-order partial differential equations (PDEs) based on the innovative combination use of the Lagrangian and Eulerian descriptions, and an efficient algorithm for solving these equations is developed. The solution method is quite general to be extended to many other constitutive models and is validated by comparing with existing solutions in some special cases. Then the cavity expansion behaviours in both clay and sand are discussed with a particular focus on the boundary effect. It is shown that, when the ratio of the outer to inner radius is small, the boundary effect may significantly influence the cavity expansion response. The present solution may provide a useful theoretical method for the validation of relevant finite element calculations and interpretation of some typical geotechnical problems such as cone penetration tests in small-size chambers, considering the boundary effect.

Vertical kinematic response of an end-bearing pipe pile in fractional viscoelastic unsaturated soil under vertically-incident P-waves

This work investigates the kinematic response of a pipe pile embedded in a fractional-order viscoelastic unsaturated soil stratum over a rigid base subjected to vertically-propagating seismic P-waves based on a continuum model. The fractional standard line solid model is introduced to the governing equation of unsaturated soil to refine the frequency-dependent viscoelastic behavior of the soil skeleton. The Rayleigh-Love rod theory is utilized to characterize the vertical motion of the pipe pile. A rigorous series form solution for the vertical kinematic response of the end-bearing pipe pile under vertically incident P-waves is yielded theoretically with boundary conditions of pile-soil system. The proposed solution is then verified by comparing with the numerical result of the reported solution. Based on the given rigorous analyzed solution, the effects of physical parameters in pile-soil system on the vertical kinematic response of the end-bearing pipe pile under harmonic P-waves are studied in both the frequency and time domains through calculation example and parametric analysis. Finally, relevant meaningful conclusions are included and indicate that soil parameters and pipe pile dimensions have significant effect on the vertical kinematic response of the pipe pile.

Fluid–structure interaction simulation of a flapping flag in a laminar jet

The flapping flag is a classical and important Fluid–Structure Interaction (FSI) problem with many applications in nature, industry, and biological systems. However, the interaction between the flow field and the flag response is not fully understood due to the complex and nonlinear nature of the problem. Although theoretical models qualitatively predict the flapping frequency and the onset velocity, there are still quantitative discrepancies between the theoretical predictions and the experimental data. In this study, numerical FSI simulations of a standard flag in a viscous uniform laminar flow are performed to characterize the flow field around the flapping flag and correlate the flow field with the flag response for six Reynolds numbers (Re= 4.5 ×104, Re= 5.3 ×104, Re= 5.7 ×104, Re= 5.9 ×104, Re= 6.6 ×104, and Re= 7.5 ×104). Two-way coupled FSI simulations are conducted using commercial software ANSYS®. Rigorous solution verification is conducted first on three Re to ensure that solutions are independent of mesh and time-step refinement. Then validation is achieved by comparing the simulation predictions with experimental data. Overall, results showed a good agreement with an error range between 0.7%–4.7% for oscillation amplitude (Am), 1.3%–6.41% for drag coefficient (Cd), and 1.3%–5.3% for frequency (f). Moreover, a comparison between inviscid and viscous models is performed to evaluate the accuracy of using inviscid theoretical models for the same flow. It is observed that the inviscid model accurately predicted f but underpredicted Am and Cd, and overpredicted lift coefficient Cl. In the case of the viscous model, it is observed that a flow separation bubble and a thin flow circulation region exist on the suction side of the flag during the limit cycle oscillation (LCO). Finally, an instantaneous negative drag is observed for a short time during the LCO, suggesting that flag reconfiguration and flow separation significantly affects the drag. These observations suggest that viscous models may be needed to improve the theoretical predictions.

Rigorous plane-stress solution and buckling analysis of rectangular functionally graded plates subjected to non-uniform edge loads

In this article, analytical solutions for the buckling analysis of simply supported rectangular functionally graded (FG) plates subjected to various forms of non-uniform in-plane loads are presented. Both perfect and porous FG plates with power-law material property variation are considered. The analytical solution is obtained by adopting a two-step approach: First, a rigorous plane-stress solution is developed based on the superposition of Airy’s stress functions, and later, these stress solutions are used in the buckling load computation of the FG plate based on Galerkin’s technique. One key aspect of the proposed solution is that it can consider different non-uniform waveforms of the in-plane loads at two opposite sides of the plate. Results show that the nature of the edge load has a profound effect on the stress solution of the FG plates. Several tables and graphs are presented to obtain a qualitative and quantitative understanding of the buckling behavior of FG plates influenced by various plate design parameters which include plate aspect ratios, power-law exponent, and porosity index. Comparative studies are also presented to highlight some erroneous results published in the open literature. It can be emphasized that the analytical results reported here can be used as a reference to judge the accuracy of other numerical solutions such as the finite element method.

Nonlinear description of active earth pressure for finite backfill based on principal stress trajectory method subjected to steady unsaturated seepage

To simplify calculations, soil cohesion, horizontal shear stress, and unsaturated properties have usually been neglected in previous earth pressure analytical methods. This negligence has certain theoretical flaws and does not conform to the actual situation. Especially for finite backfill, which often occurs, it is worth continuing to expand research in this field to seek a more rigorous solution. This work presents an analytical framework for estimating the active earth pressure of finite backfill on inclined retaining walls under steady unsaturated seepage conditions. The extended Mohr–Coulomb failure criterion expresses soil shear strength, and a unified effective stress method considering the suction stress is adopted. For a classic failure model, a calculation method using curved differential elements is established based on the principal stress trajectory. The proposed method fully considers soil cohesion, arching effect, and unsaturated properties. Relative to the existing test and theoretical results, the rationality of this method is verified. The sensitivities of relevant parameters are discussed based on different assumed soil types. The results show that seepage significantly affects the lateral earth pressure. The earth pressure distributions vary greatly under different unsaturated types for various soils. The results can provide a reference for the selection of engineering backfill.

Prediction of Seismic Bearing Capacity Considering Nonlinearity and Dilatancy by Sequential Quadratic Programming

Most of the published literature regarding bearing capacity are often focused on linear and associative soils. Concerning the intrinsic strength nonlinearity in dilatancy soils, this study investigates the problem of the seismic bearing capacity in the framework of the kinematic theorem of limit analysis. The conventional linear Mohr–Coulomb criterion is substituted with a nonlinear power law criterion to depict the nonlinearity of the soil strength. The non-associative feature of soil materials is considered by defining a nonlinear dilatancy coefficient. A generalized tangential technique is accordingly introduced to linearize the strength envelope for making the nonlinear criterion tractable in the analysis. A non-symmetrical translational failure mechanism that is comprised of several rigid wedges is used to characterize the failure of the foundation at the limit state. Moreover, the seismic action is considered by the classic pseudo-static method. Based upon the energy equilibrium theory of the upper-bound limit analysis, new analytical solutions are derived from the work-balanced equation with nonlinearity and dilatancy. This rigorous upper-bound solution is formulated as a multivariate optimization problem and is readily addressed by sequential quadratic programming (SQP). To verify the reliability of the new expressions, the present results are compared with already posted solutions and the original pseudo-dynamic solutions. The comparative results show a good agreement with previous works, and the correctness and rationality of the new analytical solutions are validated. The detailed parametric study reveals that, in the non-associative flow soils, the ultimate bearing capacity is significantly decreased with a reduction in the dilatancy coefficient. Particularly in the linear condition, namely m = 1, the larger the internal friction angle is, the more obvious the influence of the non-associative feature on the bearing capacity is.

Estimate of Throughput Improvement Due to a Metasurface Reflector in 5G Millimeter-Wave Links

Rigorous solution of Maxwell's equations is used to show that electromagnetic field scattered by a metasurface (MS) reflector based on engineered phase gradient closely resembles the scattering pattern of a properly irradiated metal reflector of the same size. Approximate analytical expressions for the spatial distribution of the scattered signal are derived to estimate an increase in the achievable bit rate in a 5G-NR link due to an MS reflector. A figure-of-merit (FoM) based on the wireless link throughput is introduced. The FoM analysis shows that to affect a 5G wireless link, the MS reflector should exceed some critical size, which depends on the propagation distances. Furthermore, performance of a larger reflector initially increases linearly with its size, followed by a slower sublinear growth.

Алгоритм газодинамического расчета асимметричных конусов методом локальных клиньев и конусов

Studies related to alteration in the shape of a body moving in the dense layers of the Earth’s atmosphere at high speed are associated with the need to consider an entire set of problems having no strict physical and mathematical description. To solve problems of this kind, it becomes necessary to use semi-empirical approaches tested on fundamental experimental data obtained in a fairly wide range of determining factors on the models characterized, as a rule, by an extremely simple pattern of the gas flow. Introduction of this mathematical description in real operating conditions is characterized by the need to study the processes of gas dynamics and heat transfer in the three-dimensional formulation combined with the complex nature of their course. These processes determine alteration in the shape of a body causing serious problems associated with rigorous solution of this complex mutually conjugate problem. As a result, there appeared a large number of publications devoted to approximate methods for calculating the spatial flow around bodies by the gas flow used to calculate intensity of the convective heat transfer. This article describes in detail the algorithm for solving the problem of spatial flow around blunt cones by the method of local wedges and cones

Обобщенная задача динамического уравновешивания и перспективные направления ее применения

The paper considers the generalized problem of the machines and mechanisms dynamic balance in terms of ensuring the given laws of altering reactions in the selected links. Representation of equations of the mechanical systems dynamics in the form of differential algebraic equations was used making it possible to obtain mathematical models of the nonlinear mechanical systems dynamics with the arbitrary structure of kinematic and force connections. With this approach, the constraint reactions are determined by algebraic equations from the system coordinates. The problem solution is based on changing the bonds selected reactions due to the impact on reactions in the other selected bonds by adding non-stationary terms to the selected bonds equation. Conditions for rigorous solution of the optimal control problem for a mechanical system are shown for the integral quality criterion not explicitly containing the control functions. The method is aimed at numerical models of the mechanical systems widely used in the programs for dynamic analysis of the coupled systems of bodies. Test examples are provided for manipulator, anthropomorphic robot and controlled suspension of a transport vehicle. The method was realized in the software package for simulating the controlled motion dynamics of the coupled systems of bodies.

Axial kinematic response of single floating pipe piles to vertically propagating P waves

This paper presents a rigorous analytical solution for predicting the axial kinematic response of single floating pipe piles subjected to vertically-propagating seismic P waves, by means of a refined Tajimi-type continuum elastodynamic model. The outer and inner soil motion are decomposed as the sum of free-field motion and the scattered soil motion due to the existence of the pipe pile, and their solutions are derived from the governing equations separately. Closed-form expressions for the responses of the pipe pile and soil are obtained by utilizing the continuity conditions at the interfaces between outer and inner soil and the pile shaft as well as its underlying soil column. The effects of key problem parameters on the seismic responses of the pipe pile and soil are discussed.

Light transport through a magneto-optical medium: simple theory revealing fruitful phenomena.

Electromagnetic wave transmission in a magneto-optical (MO) medium is a basic and old topic but has raised new interest in recent years, because MO medium plays a vital role in optical isolator, topological optics, electromagnetic field regulation, microwave engineering, and many other technological applications. Here, we describe several fascinating physical images and classical physical variables in MO medium by using a simple and rigorous electromagnetic field solution approach. We can easily obtain explicit formulations for all relevant physical quantities, such as the electromagnetic field distribution, energy flux, reflection/transmission phase, reflection/transmission coefficients, and Goos-Hänchen (GH) shift in MO medium. This theory can help to deepen and broaden our physical understanding of basic electromagnetics, optics, and electrodynamics in application to gyromagnetic and MO homogeneous medium and microstructures, and might help to disclose and develop new ways and routes to high technologies in optics and microwave.

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

This work reports new analytic free in-plane vibration solutions for orthotropic non-Lévy-type rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM), which has never been applied to in-plane elasticity problems in any existing works. Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge in mechanical analysis of plates. The clamped and free plates are considered as two most representative cases of non-Lévy-type plates. The SSM is implemented in the Hamiltonian system-based symplectic space, where the separation of variables and the symplectic eigen expansion prove to be well-grounded. These two mathematical treatments are adopted to first gain the analytic solutions of two elementary problems. The final analytic free in-plane vibration solutions are obtained by superposition of the two elementary problems. Comprehensive new natural frequencies and vibration modes are studied and validated by reference solutions from the finite element method or other approaches. The rigorous solution procedure, fast convergence, and highly accurate results render the present framework capable of serving as benchmarks for future comparison and applicable to analytic investigation of more plate problems.

Automatic segmentation and parallel phased least squares for highly efficient geodetic network parameter estimation

This paper describes a novel automatic segmentation algorithm and parallel phased least squares strategy. These developments have been motivated by a requirement to repeatedly estimate extremely large parameter sets from massive and constantly changing geodetic networks in a highly efficient manner. The challenge of solving parameter sets efficiently, together with the inverse of the normal equations to obtain the full matrix of parameter estimate precisions, may be addressed by using highly optimised parallel matrix factorisation libraries. While offering notable reductions in time on multi-core CPU architectures, the simultaneous solution of extremely large measurement sets still requires excessive amounts of time and computational resources (if attainable). Historically, the challenge has been addressed by geodesists through a range of Helmert blocking or matrix partitioning techniques and sequential least squares models. While capable of producing rigorous parameter estimates, certain methods suffer from computational inefficiency; implementation complexity; inflexibility in re-forming the parameterised expressions upon the introduction of new measurements or non-rigorous variance matrices. Tienstra's phased least squares method is able to overcome all of the aforementioned problems, provided an efficient and flexible process is available for rigorously subdividing the parameters and measurements. In this contribution, we present an automatic segmentation algorithm and a strategy for parallelising Tienstra's phased least squares method using a dynamic scheduling queue. These developments offer a breakthrough in geodetic methodology, in that for the first time they allow for the rigorous solution of both parameters and variances from massive systems of observation equations subject to continual change in an adaptive and highly efficient manner.

Distinguishing Heterogeneous and Homogeneous CE Mechanisms: Theoretical Insights into Square-Wave Voltammetry

A theoretical study of an electrode mechanism where the electrode reaction (E) is preceded by a chemical reaction that takes place solely at the electrode surface (Chet) is presented under conditions of square-wave voltammetry (SWV). Rigorous mathematical solutions in the form of integral equations derived by means of Laplace transforms are presented for the surface concentration of all species involved in the electrode mechanism, yielding explicit recurrent formulas for the simulation of the voltammetric response. The theory approximately predicts that the chemical reaction starts at the beginning of the voltammetric experiment, disregarding its occurrence in the short time period between inserting the electrode in the solution until starting the voltammetric experiment. It is demonstrated that SWV can differentiate between the common CE mechanism, where C is a homogeneous chemical reaction taking place in the vicinity of the electrode, and the current ChetE mechanism, which is relevant for plethora of electrocatalytic processes at electrodes modified with catalytically active enzymes and/or noble-metal nanoparticles, as well as for electrocatalytic processes of fundamental importance such as CO2, N2, O2, and H+ reductions.

Diffraction of the Field of a Grounded Cable on an Elongated Dielectric Spheroid in a Conducting Layer

Based on a rigorous solution to the problem, analytical expressions are obtained for calculating the diffraction of the electromagnetic field of a grounded cable on an elongated dielectric spheroid in a conductive layer. The field of a grounded AC cable in a conductive layer is determined by solving the Helmholtz equation for the vector potential by using the method of integral Fourier–Bessel transformations, taking into account the boundary conditions at the bottom and surface of the conductive layer. The process of finding the secondary field of an elongated dielectric spheroid on an alternating current in a conducting layer is divided into two stages. First, we find an exact solution to the problem of an elongated dielectric spheroid at a constant current in a homogeneous field, in free space, decomposing this solution into a Taylor series and retaining the first term, which is a dipole approximation. In the second stage, the resulting field as the sum of the fields of the horizontal and vertical dipoles is analytically continued into the frequency domain. The field of the horizontal and vertical dipoles in the conducting layer is obtained by using the method of integral Fourier–Bessel transformations, taking into account the boundary conditions at the bottom and surface of the conducting layer. The resulting solution is presented in a closed form in elementary functions and has an accuracy level acceptable for the practice. Graphs showing the flow characteristics of an elongated dielectric spheroid modeling a swimmer in a light diving suit are given. The influence of the water–air boundary on the increase in the secondary field of the dielectric spheroid, which leads to an increase in the reliability of object detection, is revealed. The practical implementation of the described device protected by a patent and the experimental data of testing the device layout on the Gulf of Finland are given. A good agreement between the theoretical and experimental flow characteristics of a dielectric object both in shape, amplitude, and phase, is revealed.

Refinement of the Heuristic Solution for the Problem of TE-Polarized Electromagnetic Wave Diffraction on a Half-Plane with Two-Sided Impedance Boundary Conditions

A technique for obtaining refined heuristic solutions for problems with electromagnetic wave diffraction by scatterers with non-ideal boundary conditions is proposed. Based on the method of fundamental components for the problem of TE-polarized wave diffraction on a “semitransparent” half-plane with two-sided impedance boundary conditions, an improved heuristic solution for the scattered field is obtained by replacing the primary heuristic formula. A quantitative and qualitative assessment of the accuracy of the heuristic relationship is carried out as well as a physical interpretation of the rigorous solution based upon them. A classification of heuristic solutions is proposed according to the purpose of their application in practical problems.

Influence of Interpolation Scheme on the Accuracy of Overset Method for Computing Rudder-Propeller Interaction

Abstract The overset method and associated interpolation schemes are usually thoroughly verified only on synthetic or academic test cases for which conclusions might not directly translate to real engineering problems. In the present work, an overset grid method is used to simulate a rudder-propeller flow, for which a comprehensive verification and validation study is performed. Three overset-related interpolation schemes (first order inverse distance, second order nearest cell gradient and third order least squares) are tested to quantify and qualify numerical errors on integral quantities, mass imbalance, flow features and rudder pressure distributions. The performance overhead is also measured to help make accuracy versus performance balance decisions. Rigorous solution verification is performed to estimate time and space Discretization, iterative and statistical uncertainties. Validation of the propeller-rudder flow against experimental data is also done. The results show that, while the choice of interpolation scheme has minimal impact on time-averaged integral quantities (like propeller and rudder forces), they do influence the smoothness of the time signals, with the first order scheme resulting in large intensity high-frequency temporal oscillations. Lower order interpolation methods also produce more interpolation artifacts in fringe cells, which are then convected downstream. Mass imbalance is also affected by the interpolation scheme, with higher order schemes such as the third order least squares approach resulting in an order of magnitude lower flux errors. The limitations of first order schemes do not, however, result in significant lower computational overhead, with the second order nearest cell gradient being even cheaper than the inverse distance scheme in the tested implementation. Lastly, validation shows promising results with rudder forces within 10% of the experiments.