Closed-form Asymptotic Solution Research Articles

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

Recently, some researchers have revisited the analysis of chlorine transportation in cylindrical pipes by deploying a coupling between the Laplace transform method and the complex analysis’ residue approach for inverting complex integrals. This method yielded interesting results after the incorporation of root-finding numerical schemes. Thus, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of a bi-layered pipe using the hybrid of the Laplace transform method and the asymptotic approximations method. The need for asymptotic approximations for the modified Bessel functions, which arise in the reduced ordinary differential equations, necessitates the quest for closed-form analytical solutions, which are largely considered benchmark solutions for numerical investigations. Moreover, the obtained closed-form asymptotic solutions have been examined graphically; where it was observed that both the radial diffusion coefficient η and the spatial radial variable are contributory in the transport of chorine concentration in the media.

A New Analytical Method for Calculating Subsidence Resulting by Fluid Withdrawal from Disk-Shaped Confined Aquifers

This work presents the derivation of analytical solutions concerning the radial subsidence distribution ensuing from fluid extraction from a disk-shaped confined aquifer in homogeneous formations. The study draws upon methodologies developed in petroleum geomechanics of deep reservoirs to estimate surface uplift due to CO2 injection using Hankel-transformed thin plate theory. These methods yield simplified expressions as compared to previous results derived using the superposition principle on surface uplift from a uniform pressure field. Hence, closed-form formulas for the subsidence at the well location are re-derived, while the formulas for the subsidence field are deducted by both methods and the mathematical relation between the two methodologies is discussed. Additionally, innovative closed-form asymptotic solutions for radial subsidence distribution are deduced for scenarios involving deep aquifers. These solutions demonstrate exceptional accuracy when aquifer depth exceeds aquifer diameter, exhibiting independence from formation permeability and fluid viscosity. The study explores the influence of physical parameters on the subsidence field.

Alpha dose modeling in diffusing alpha-emitters radiation therapy-Part I: single-seed calculations in one and two dimensions.

Diffusing alpha-emitters Radiation Therapy ("DaRT") is a new method, presently in clinical trials, which allows treating solid tumors by alpha particles. DaRT relies on interstitial seeds carrying μCi-level 224 Ra activity below their surface, which release a chain of short-lived alpha emitters that spread throughout the tumor volume primarily by diffusion. Alpha dose calculations in DaRT are based on describing the transport of alpha emitting atoms, requiring new modelingtechniques. A previous study introduced a simplified framework, the "Diffusion-Leakage (DL) model", for DaRT alpha dose calculations, and employed it to a point source, as a basic building block of arbitrary configurations of line sources. The aim of this work, which is divided into two parts, is to extend the model to realistic seed geometries (in Part I), and to employ single-seed calculations to study the properties of DaRT seed lattices (Part II). Such calculations can serve as a pragmatic guide for treatment planning in future clinicaltrials. We derive a closed-form asymptotic solution for an infinitely long cylindrical source, and extend it to an approximate time-dependent expression that assumes a uniform temporal profile at all radial distances from the source. We then develop a finite-element one-dimensional numerical scheme for a complete time-dependent solution of this geometry and validate it against the closed-form expressions. Finally, we discuss a two-dimensional axisymmetric scheme for a complete time-dependent solution for a seed of finite diameter and length. Different solutions are compared over the relevant parameter space, providing guidelines on their usability and limitations. We show that approximating the seed as a finite line source comprised of point-like segments significantly underestimates the correct alpha dose, as predicted by the full two-dimensional calculation. The time-dependent one-dimensional solution is shown to coincide to sub-percent-level with the two-dimensional calculation in the seed midplane, and maintains an accuracy of a few percent up to ∼2mm from the seed edge. For actual treatment plans, the full two-dimensional solution should be used to generate dose lookup tables, similarly to the TG-43 format employed in conventional brachytherapy. Given the accuracy of the one-dimensional solution up to ∼2mm from the seed edge it can be used for efficient parametric studies of DaRT seedlattices.

Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients

For the purpose of solving a second-order singularly perturbed problem (SPP) with variable coefficients, a mth-order asymptotic-numerical method was developed, which decomposes the solutions into two independent sub-problems: a reduced first-order linear problem with a left-end boundary condition; and a linear second-order problem with the boundary conditions given at two ends. These are coupled through a left-end boundary condition. Traditionally, the asymptotic solution within the boundary layer is carried out in the stretched coordinates by either analytic or numerical method. The present paper executes the mth-order asymptotic series solution in terms of the original coordinates. After introducing 2(m+1) new variables, the outer and inner problems are transformed together to a set of 3(m+1) first-order initial value problems with the given zero initial conditions; then, the Runge–Kutta method is applied to integrate the differential equations to determine the 2(m+1) unknown terminal values of the new variables until they are convergent. The asymptotic-numerical solution exactly satisfies the boundary conditions, which are different from the conventional asymptotic solution. Several examples demonstrated that the newly proposed method can achieve a better asymptotic solution. For all values of the perturbing parameter, the method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy of the solution in the entire domain. We derive the sufficient conditions, which terminate the series of asymptotic solutions for inner and outer problems of the SPP without having the spring term. For a specific case, we can derive a closed-form asymptotic solution, which is also the exact solution of the considered SPP.

Modeling Bistatic Coherent Scattering From Multilayered Rough Surface Using Its Effective Dielectric Constant at P- and L-Bands

This paper proposes a closed-form asymptotic solution for the bistatic coherent scattering of a multilayered rough surface structure based upon its effective dielectric constant (EDC) in specular direction. The EDC is modeled by establishing an equivalence of the coherent scattering between the multilayered rough surface structure and a half-space homogeneous medium. The scattering solution is then solved using the scalar Kirchhoff approximation (SKA) method. This new method is referred to as the SKA-EDC method, and it is applied to analyze the sensitivity of the EDC and coherent reflectivity to bare soils with realistic parameters at P- and L- bands. The result indicates that EDC can give different responses to the soil moisture variations with respect to the coherent reflectivity, enabling the potentials of root-zone soil moisture retrieval. At incidence angle smaller than 35°, EDC gives the same value for both polarizations, and the coherent reflectivity can show a significant response to soil at depths < 15-50 cm at 0.80 GHz and <5-15 cm at 1.57 GHz, depending on soil moisture. The simulation also demonstrates that subsurface roughness has a trivial effect on the EDC and coherent reflectivity for three-rough-surfaces separated by continuously dielectric profiles. Subsurface can thus be assumed to be flat for reducing uncertainty in soil moisture inversion algorithms when the subsurface roughness is unknown.

Optimal Retirement Under Partial Information

The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure nonnegativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial-information settings.

Estimating indirect parental genetic effects on offspring phenotypes using virtual parental genotypes derived from sibling and half sibling pairs.

Indirect parental genetic effects may be defined as the influence of parental genotypes on offspring phenotypes over and above that which results from the transmission of genes from parents to their children. However, given the relative paucity of large-scale family-based cohorts around the world, it is difficult to demonstrate parental genetic effects on human traits, particularly at individual loci. In this manuscript, we illustrate how parental genetic effects on offspring phenotypes, including late onset conditions, can be estimated at individual loci in principle using large-scale genome-wide association study (GWAS) data, even in the absence of parental genotypes. Our strategy involves creating “virtual” mothers and fathers by estimating the genotypic dosages of parental genotypes using physically genotyped data from relative pairs. We then utilize the expected dosages of the parents, and the actual genotypes of the offspring relative pairs, to perform conditional genetic association analyses to obtain asymptotically unbiased estimates of maternal, paternal and offspring genetic effects. We apply our approach to 19066 sibling pairs from the UK Biobank and show that a polygenic score consisting of imputed parental educational attainment SNP dosages is strongly related to offspring educational attainment even after correcting for offspring genotype at the same loci. We develop a freely available web application that quantifies the power of our approach using closed form asymptotic solutions. We implement our methods in a user-friendly software package IMPISH (IMputing Parental genotypes In Siblings and Half Siblings) which allows users to quickly and efficiently impute parental genotypes across the genome in large genome-wide datasets, and then use these estimated dosages in downstream linear mixed model association analyses. We conclude that imputing parental genotypes from relative pairs may provide a useful adjunct to existing large-scale genetic studies of parents and their offspring.

Optimal Retirement Problem Under Partial Information

The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure non-negativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning or the Wonham filter. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial information settings.

Optimal Sensors Placement in Dynamic Damage Detection of Beams Using a Statistical Approach

Structural monitoring plays a central role in civil engineering; in particular, optimal sensor positioning is essential for correct monitoring both in terms of usable data and for optimizing the cost of the setup sensors. In this context, we focus our attention on the identification of the dynamic response of beam-like structures with uncertain damages. In particular, the non-localized damage is described using a Gaussian distributed random damage parameter. Furthermore, a procedure for selecting an optimal number of sensor placements has been presented based on the comparison among the probability of damage occurrence and the probability to detect the damage, where the former can be evaluated from the known distribution of the random parameter, whereas the latter is evaluated exploiting the closed-form asymptotic solution provided by a perturbation approach. The presented case study shows the capability and reliability of the proposed procedure for detecting the minimum number of sensors such that the monitoring accuracy (estimated by an error function measuring the differences among the two probabilities) is not greater than a control small value.

Enhancing Secrecy Capacity in FSO Links via MISO Systems Through Turbulence-Induced Fading Channels With Misalignment Errors

It has recently been proved that the free-space optical (FSO) communication links are susceptible to interceptions. Due to this reason, the optics community shows a special interest in studying these high-speed links in greater detail from a physical layer security (PLS) point of view. Therefore, in this paper, we propose, for the first time, enhancing the average secrecy capacity (ASC) in FSO links via multiple-input/single-output (MISO) systems. It is well-known that the fading effects in FSO channels can be significantly mitigated by exploiting spatial diversity techniques at the transmitter end. Thus, we develop a new asymptotic closed-form solution at high signal-to-noise-ratio (SNR) to accurately compute the ASC for MISO based FSO communication systems with equal gain combining (EGC) reception through generalized misalignment and atmospheric turbulence-induced fading channels. As a key feature, we investigate the impact of the eavesdropper's orientation along with its location in the pointing error model. We can conclude that the influence of the eavesdropper on recollecting radiated power is diminished considerably by increasing not only the normalized beam width at the receiver end, but also by increasing the number of laser sources. Numerical results are tested by exact Monte Carlo simulations.

Effect of elastic grading on fretting wear

We consider fretting wear in elastic frictional contact under influence of oscillations of small amplitude and investigate the question, how wear damage can be influenced by the introduction of material gradients. To achieve a general understanding we restrict our consideration to media with a power-law dependency of the elastic modulus on depth. In this case, a complete analytical solution can be found for the final worn shape. In the limiting case of small fretting oscillations we obtain a simple, closed-form asymptotic solution of the problem. We find that the optimum grading depends on the oscillation amplitude: for large amplitudes, the use of materials with a positive exponent decreases the wear volume whilst for very small amplitudes the use of graded materials with slightly negative exponent is beneficial. Especially interesting is the case of the Gibson-medium which may help avoiding both fretting wear and fretting fatigue.

Rayleigh scattering of a cylindrical sound wave by an infinite cylinder.

Rayleigh scattering, in which the wavelength is large compared to the scattering object, is usually studied assuming plane incident waves. However, full Green's functions are required in a number of problems, e.g., when a scatterer is located close to the ocean surface or the seafloor. This paper considers the Green's function of the two-dimensional problem that corresponds to scattering of a cylindrical wave by an infinite cylinder embedded in a homogeneous fluid. Soft, hard, and impedance cylinders are considered. Exact solutions of the problem involve infinite series of products of Bessel functions. Here, simple, closed-form asymptotic solutions are derived, which are valid for arbitrary source and receiver locations outside the cylinder as long as its diameter is small relative to the wavelength. The scattered wave is given by the sum of fields of three linear image sources. The viability of the image source method was anticipated from known solutions of classical electrostatic problems involving a conducting cylinder. The asymptotic acoustic Green's functions are employed to investigate reception of low-frequency sound by sensors mounted on cylindrical bodies.

Fluid Penetration in a Deformable Permeable Web Moving Past a Stationary Rigid Solid Cylinder

We present an analysis for the process of fluid infiltration into a deformable, thin and permeable web that moves in close proximity over a rigid and stationary solid cylinder. While this is a process of significant interest in a range of coating, printing and composites pultrusion processes, its hydrodynamics have received limited attention in the open literature. The flow in the film separating the web from the cylinder is described by lubrication theory, while fluid transfer into the web is governed by Darcy’s law. The deformation of the web at each position is a linear function of the local gap pressure; this is consistent with the assumption of a thin and rigidly supported web. Our results indicate that the web/fluid interface is forced away from the cylinder surface as it approaches it and bounces back downstream from the minimum separation point. This behavior produces a non-symmetric gap between the adjacent surfaces, and this is shown to have critical influence on the final amount of penetrating fluid. The extent of fluid penetration is also found to be affected by the web elasticity (expressed by the dimensionless Ne number) and permeability (expressed in dimensionless form via \(\hat{{K}})\) where under a specific Ne and \(\hat{{K}}\) combination a maximum penetration depth is obtained. Finally, we derive a closed-form asymptotic solution for the final infiltration depth in the limit of Ne \(<<\) 1 and \(\hat{{K}}<<\)1 and test its predictions against the above-mentioned numerical results.

Rolling behavior of a micro-cylinder in adhesional contact.

Understanding the rolling behavior of a micro-object is essential to establish the techniques of micro-manipulation and micro-assembly by mechanical means. Using a combined theoretical/computational approach, we studied the critical conditions of rolling resistance of an elastic cylindrical micro-object in adhesional contact with a rigid surface. Closed-form dimensionless expressions for the critical rolling moment, the initial rolling contact area, and the initial rolling angle were extracted after a systematic parametric study using finite element method (FEM) simulations. The total energy of this system is defined as the sum of three terms: the elastic energy stored in the deformed micro-cylinder, the interfacial energy within the contact area, and the mechanical potential energy that depends on the external moment applied to the cylindrical micro-object. A careful examination of the energy balance of the system surprisingly revealed that the rolling resistance per unit cylindrical length can be simply expressed by “work of adhesion times cylindrical radius” independent of the Young’s modulus. In addition, extending a linear elastic fracture mechanics based approach in the literature, we obtained the exact closed-form asymptotic solutions for the critical conditions for initial rolling; these asymptotic solutions were found in excellent agreement with the full-field FEM results.

Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

In a continuous-time economy, we investigate the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation between the two risky assets. The portfolio selection that is robust to the uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation. Thus, it becomes a maximin problem. We solve the problem under the Black--Scholes model for risky assets with an ambiguous correlation using the theory of $G$-Brownian motions. We then extend the problem to stochastic volatility models for risky assets with an ambiguous correlation between risky asset returns. An asymptotic closed-form solution is derived for a general class of utility functions, including constant relative risk aversion and constant absolute risk aversion utilities, when stochastic volatilities are fast mean reverting. We propose a practical trading strategy that combines information from the option implied volatility surfaces of risky assets through the ambiguous correlation.

Capacity Analysis for Correlated Multi-Hop MIMO Channels under Colored Noise

A capacity analysis for generally correlated wireless multi-hop multi-input multi-output (MIMO) channels is presented in this paper. The channel at each hop is spatially correlated, the source symbols are mutually correlated, and the additive Gaussian noises are colored. First, by invoking Karush-Kuhn-Tucker condition for the optimality of convex programming, we derive the optimal source symbol covariance for the maximum mutual information between the channel input and the channel output when having the full knowledge of channel at the transmitter. Secondly, we formulate the average mutual information maximization problem when having only the channel statistics at the transmitter. Since this problem is almost impossible to be solved analytically, the numerical interior-point-method is employed to obtain the optimal solution. Furthermore, to reduce the computational complexity, an asymptotic closed-form solution is derived by maximizing an upper bound of the objective function. Simulation results show that the average mutual information obtained by the asymptotic design is very closed to that obtained by the optimal design, while saving a huge computational complexity.

Dynamics of reactive chromatographic columns of inert core/hollow/film coated spherical packing: An analytical solution and applications

The transient behavior of reactive-diffusive fixed-bed (chromatographic columns or reactors) packed with inert core/shell core/hollow/thin film coated monodisperse spherical rigid particles is investigated. We modeled the reactive sorption system with linear mass exchange law between the flowing and the stationary phases. The coupled governing partial differential equations were integrated analytically using Laplace transformations. The analytical closed form solution involves two infinite integrals. The closed form solution was integrated numerically and the breakthrough curves of the system were studied under the influence of the system parameters. The limiting cases are discussed, a simple asymptotic closed form solution for fixed-bed desorbers with monodisperse spherical particles is obtained, and results are generalized for similar kinetics of reactive chromatographic columns (fixed-beds) for diffusion/absorption into the inert core spherical packing. Further, potential applications of the mathematical model and results in processes such as liquid vapor extraction, various forms of chromatography, reactive sorption, multifunctional reactors, process intensification, and regeneration of ion exchange columns are briefly discussed.

Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

In a continuous-time economy, we investigate the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation between the two risky assets. The portfolio selection that is robust to the uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation. Thus, it becomes a maximin problem. We solve the problem under the Black-Scholes model for risky assets with an ambiguous correlation using the theory of G-Brownian motions. We then extend the problem to stochastic volatility models for risky assets with an ambiguous correlation between risky asset returns. An asymptotic closed-form solution is derived for a general class of utility functions, including CRRA and CARA utilities, when stochastic volatilities are fast mean-reverting. We propose a practical trading strategy that combines information from the option implied volatility surfaces of risky assets through the ambiguous correlation.

Reflection of an acoustic line source by an impedance surface with uniform flow

An exact analytic solution is derived for the 2D acoustic pressure field generated by a time-harmonic line mass source located above an impedance surface with uniform grazing flow. Closed-form asymptotic solutions in the far field are also provided. The analysis is valid for both locally-reacting and nonlocally-reacting impedances, as is demonstrated by analyzing a nonlocally reacting effective impedance representing the presence of a thin boundary layer over the surface. The analytic solution may be written in a form suggesting a generalization of the method of images to account for the impedance surface. The line source is found to excite surface waves on the impedance surface, some of which may be leaky waves which contradict the assumption of decay away from the surface predicted in previous analyses of surface waves with flow. The surface waves may be treated either (correctly) as unstable waves or (artificially) as stable waves, enabling comparison with previous numerical or mathematical studies which make either of these assumptions.The computer code for evaluating the analytic solution and far-field asymptotics is provided in the supplementary material. It is hoped this work will provide a useful benchmark solution for validating 2D numerical acoustic codes.

Multiscale approach to micro/macro fatigue crack growth in 2024-T3 aluminum panel

When two contacting solid surfaces are tightly closed and invisible to the naked eye, the discontinuity is said to be microscopic regardless of whether its length is short or long. By this definition, it is not sufficient to distinguish the difference between a micro- and macro-crack by using the length parameter. Microcracks in high strength metal alloys have been known to be several centimeters or longer. Considered in this work is a dual scale fatigue crack growth model where the main crack can be micro or macro but there prevails an inherent microscopic tip region that is damaged depending on the irregularities of the microstructure. This region is referred to as the “micro-tip” and can be simulated by a sharp wedge with different angles in addition to mixed boundary conditions. The combination is sufficient to model microscopic entities in the form of voids, inclusions, precipitations, interfaces, in addition to subgrain imperfections, or cluster of dislocations. This is accomplished by using the method of “singularity representation” such that closed form asymptotic solutions can be obtained for the development of fatigue crack growth rate relations with three parameters. They include: (1) the crack surface tightness σ* represented by σo/σ∞ = 0.3–0.5 for short cracks in region I, and 0.1–0.2 for long cracks in region II, (2) the micro/macro material properties reflected by the shear modulus ratio µ* (=µmicro/µmacro varying between 2 and 5) and (3) the most sensitive parameter d* being the micro-tip characteristic length d* (=d/do) whose magnitude decreases in the direction of region I→II. The existing fatigue crack growth data for 2024-T3 and 7075-T6 aluminum sheets are used to reinterpret the two-parameter da/dN=C(ΔK)n relation where ΔK has now been re-derived for a microcrack with surfaces tightly in contact. The contact force will depend on the mean stress σm or mean stress ratio R as the primary parameter and on the stress amplitude σa as the secondary parameter.