Infinite Number Of Possible Solutions Research Articles

Sampling‐based modified ant colony optimization method for high‐speed rail alignment development

AbstractHigh‐speed railway (HSR) alignment development is a complex and tedious problem due to an infinite number of possible solutions, the existence of non‐linear costs and impacts, and complex location and geometric design constraints. In this study, a low‐discrepancy point sampling‐based modified ant colony optimization (SMACO) algorithm for obtaining horizontal alignments with optimized HSR‐specific cost and impact, including noise and vibration impacts, is proposed. The low‐discrepancy sampling approach is used to identify the potential, from which appropriate intermediate s are selected to develop feasible alignments using the SMACO algorithm. It effectively avoids restricted land parcels and satisfies HSR‐related geometric design requirements. A real‐world case study demonstrated that the HSR alignment obtained using the proposed method was marginally better than the path planner method‐based alignment and the constructed alignment. The sensitivity analysis highlighted the impact of two key parameters, that is, the right of way widths and noise and vibration screening distances on the HSR alignment development. This study advances the alignment development automation, particularly the HSR horizontal alignment for design speeds over 180 km/h. It facilitates extensive search space exploration independent of infeasible regions, identifies and selects without being constrained to prespecified locations and a user‐defined number, and proposes a suitably modified ACO algorithm for the HSR alignment development.

Development and validation of a genetic L-System programming framework for topology optimization of multifunctional structures

When considering the complex problem of developing new multifunctional structures, it is essential to narrow the vast design space and reduce the infinite number of possible solutions to a finite subset of feasible designs. Nature provides examples of ramified, or branched, topologies that form non-intuitive solutions to various structural design problems. This work focuses on the development of a bio-inspired topology optimization framework that couples genetic algorithms with a parallel rewriting system known as a Lindenmayer System (or L-System), which acts as an analogy to the evolutionary process and formalizes the encoding of a 2-D structure. Example design problems and the solutions determined using this novel framework are presented and compared to ideal solutions, where it is shown that a family of branched solutions allowed to evolve over generations can eventually arrive at effective multifunctional structures. Select designs from each example problem are also thickened into 3-D bodies, which are assessed experimentally via fully functional prototypes, demonstrating that the L-System framework is capable of generating realistic solutions for multifunctional structure development.

Improved Raven Roosting Optimization algorithm (IRRO)

Abstract Optimization means finding the best solution from among an infinite number of possible solutions to a complex problem. Several methods are generally used for solving complex problems, such as meta-heuristic algorithms that are inspired from living organisms. Raven Roosting Optimization (RRO) is an algorithm inspired by the mimicking behavior of ravens but it has the problem of premature convergence. This paper provides an expansion of RRO algorithm namely IRRO to resolve the problem. It focuses on a population that follows the leader and divides the population into groups of weak ravens and greedy ravens. It also uses a parameter to control the amount of food remaining for the ravens. This is then extended by comparison of the proposed algorithm to RRO, Particle Swarm Optimization (PSO) approach Bat algorithm (BA), Chicken Swarm Optimization (CSO), Grey Wolf Optimization (GWO), Whale Optimization Algorithm (WOA). The experimental results on 30 standard benchmark functions how the improvement of the proposed algorithm compared to RRO, PSO, BA, CSO, GWO and WOA algorithms.

Strength design optimization of structural steel members according to Eurocode 3

In order to design a steel member subjected to a bending moment and an axial load, there are an infinite number of possible solutions of I- or H‐steel cross-sections, the doubly-symmetric solution being just one of them. This paper presents a procedure to obtain the optimal steel cross-section from the infinite number of possible solutions. The process is based on the Reinforcement Sizing Diagrams employed in reinforced concrete strength design. The procedure looks for any type of solution regarding compact or non-compact steel sections. All aspects related to local instabilities are taken into account, as well as some special considerations in order to address the global instabilities associated with the slenderness of the steel element.

A Customized Particle Swarm Method to Solve Highway Alignment Optimization Problem

Abstract: Optimizing highway alignment requires a versatile set of cost functions and an efficient search method to achieve the best design. Because of numerous highway design considerations, this issue is classified as a constrained problem. Moreover, because of the infinite number of possible solutions for the problem and the continuous search space, highway alignment optimization is a complex problem.In this study, a customized particle swarm optimization algorithm was used to search for a near‐optimal highway alignment, which is a compound of several tangents, consisting of circular (for horizontal design) and parabolic (for vertical alignment) curves. The selected highway alignment should meet the constraints of highway design while minimizing total cost as the objective function. The model uses geographical information system (GIS) maps as an efficient and fast way to calculate right‐of‐way costs, earthwork costs, and any other spatial information and constraints that should be implemented in the design process. The efficiency of the algorithm was verified through a case study using an artificial map as the study region. Finally, we applied the algorithm to a real‐world example and the results were compared with the alignment found by traditional methods.

SU-E-T-386: Gamma Analysis of Normalized and Un-Normalized Dose Distributions.

The gamma index method, as currently implemented in all commercial QA software, calls for selection of a normalization point to evaluate agreement between two dose distributions. The implication of this is that there is an infinite number of possible solutions! Which one to pick? A unique and more relevant solution is obtained only if no normalization point is used. The set of test cases suggested by the AAPM TG1 19 were planned using Pinnacle 8.0m and delivered on a Varian 21EX linac for 6 and 18 MV photons. The recommended point and planar dose measurements were obtained using a Pinpoint ion chamber, EDR2 film and MatriXX. The gamma index method using typical 3%, 3 mm criteria with and without a normalization point was used to assess the agreement between calculated and delivered planar dose distributions. The analysis was extended to a set of data for clinically treated patients. The comparison with the TG119 benchmark data showed that all point dose and planar measurements for 6 MV were within the published range. Similar results, although without published data to compare with, were obtained for 18 MV as well. For all complex tests, the percentage of points passing the gamma criteria of 3%, 3 mm was (95.8±1.6)% and (95.6±1.0)% for 6 MV and 18 MV, respectively. Without a normalization point, however, the same gamma analysis fell to (20.7±6.7)% and (13.9±4.0)% for 6 MV and 18 MV, respectively. The clinical data set showed the same trend, with the gamma passing rate declining from (98.9±0.7)% to (33.4±13.1)%. The gamma index method provides a unique answer for gamma passing rate only without normalizing dose distributions to any particular point. The common gamma criteria of 3%, 3 mm, however, is a very poor metric in that case.

D-MORPH regression: application to modeling with unknown parameters more than observation data

Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) is a model exploration method, originally developed for differential equations. We extend D-MORPH to regression treatment of a model described as a linear superposition of basis functions with unknown parameters being the expansion coefficients. The goal of D-MORPH regression is to improve prediction accuracy without sacrificing fitting accuracy. When there are more unknown parameters than observation data, the corresponding linear algebraic equation system is generally consistent, and has an infinite number of solutions exactly fitting the data. In this case, the solutions given by standard regression techniques can significantly deviate from the true system structure, and consequently provide large prediction errors for the model. D-MORPH regression is a practical systematic means to search over system structure within the infinite number of possible solutions while preserving fitting accuracy. An explicit expression is provided by D-MORPH regression relating the data to the expansion coefficients in the linear model. The expansion coefficients obtained by D-MORPH regression are particular linear combinations of those obtained by least-squares regression. The resultant prediction accuracy provided by D-MORPH regression is shown to be significantly improved in several model illustrations.

On the convergence of the LMS algorithm with a rank-deficient input autocorrelation matrix

In all books and papers on adaptive filtering, the input autocorrelation matrix R xx is always considered positive definite and hence the theoretical Wiener–Hopf normal equations ( R xx h= r xd ) have a unique solution h= h opt (“ there is only a single global optimum”, [B. Widrow, S. Stearns, Adaptive Signal Processing, Prentice-Hall, 1985, p. 21]) due to the invertibility of R xx (i.e., it is full-rank). But what if R xx is positive semi-definite and not full-rank? In this case the Wiener–Hopf normal equations are still consistent but with an infinite number of possible solutions. Now it is well known that the filter coefficients of the least mean square (LMS), stochastic gradient algorithm, converge (in the mean) to the unique Wiener–Hopf solution ( h opt) when R xx is full-rank. In this paper, we will show that even when R xx is not full-rank it is still possible to predict the (convergence) behaviour of the LMS algorithm based upon knowledge of R xx , r xd and the initial conditions of the filter coefficients.

Proportioning of steel beam–column members based on RSD optimization methodology

Due to the cost and energy embodied in steel production, a reduction in the cross-sectional area of a steel structural member could imply a significant saving of money and reduction in greenhouse gas emissions. Hernandez-Montes et al. [Hernández-Montes E, Aschheim M, Gil-Martin LM. The impact of optimal longitudinal reinforcement on the curvature ductility capacity of reinforced concrete column sections. Mag Concrete Res 2004;56(9):499–512; Hernández-Montes E, Gil-Martín LM, Aschheim M. The design of concrete members subjected to uniaxial bending and compression using reinforcement sizing diagrams. ACI Struct J 2005;102(1):150–8] proposed an analytical approach for reducing the amount of longitudinal reinforcement in reinforced concrete members subjected to the action of axial force and bending moment acting about a principal axis of the cross-section. The approach, which makes use of Reinforcement Sizing Diagrams (RSD), makes use of a graphical representation of all possible reinforcement solutions for a particular concrete cross-section subjected to a combined loading consisting of bending moment and axial load ( M , N ) . The common symmetric solution is recognized as just one of the infinite number of possible solutions. The RSD methodology used in that approach is extended in the present paper to steel sections having at least one axis of symmetry, subjected to axial force, bending moment about the strong axis, and shear acting in the plane of the bending moment. Special considerations are introduced to address the instabilities associated with slender steel elements. The methodology is developed following the Eurocode 3 provisions for compact steel members.

Load Extraction from Photoelastic Images Using Neural Networks

Photoelastic materials develop colored fringes under white light when subjected to mechanical stresses, which can be viewed through a polariscope. This technique has traditionally been used for stress analysis of loaded components, however, this can also be potentially used in sensing applications where the requirement may be measurement of the stimulating forces causing the generation of fringes. This leads to inverse photoelastic problem where the developed image can be analyzed for the input forces. However, there could be infinite number of possible solutions which cannot be determined by conventional techniques. This paper presents neural networks based approach to solve this problem. Experiments conducted to prove the principle have been verified with theoretical results and finite element analysis of loaded specimens. The developed technique, if generalized, can be implemented for whole-field analysis of the stress patterns involving complex fringes under different loading conditions. This can also provide direct visualization of the stress field, which may find application in a variety of specialized areas including biomedical engineering and robotics.

Mechanical models of trishear-like folds

Previous workers have formulated velocity descriptions of the trishear kinematic model of fault propagation-folds, which are inherently non-unique. We present two complete mechanical models of fault-related folding and assess the validity of the assumptions used in the assignment of velocities in the trishear description and to eliminate the untenable situation of an infinite number of possible solutions for velocity fields. The mechanical model of forced-folding, Forced Fold, based on viscous folding theory, is used to derive the velocity fields in an anisotropic sedimentary cover overlying faulted and displaced rigid basement blocks. The solution of displacements around a stress-free fault in an elastic body is used to model fault-arrest folds that form around a fault imbedded in a deformed medium. The results indicate that the velocity fields assumed in the trishear model more closely resemble the velocities derived in the mechanical Forced Fold model than the mechanical model of fault-arrest folding. The Forced Fold model produces a triangular region of concentrated deformation similar to the trishear region assumed in the kinematic models, while the deformation produced by the fault-arrest model is not concentrated within a triangular zone.

Phase-conjugate picture of a wave-packet interference design for arbitrary target states

Target states are specifically desired states of systems that allow the control of processes or the investigation of unique phenomena. A time-domain scheme for the excitation of such states by the interference of multiple wave packets is developed. The scheme leads to analytical results for the optical field required to excite an arbitrary target state. The scheme suggests a clear physical picture for the design process that is applicable to many systems. The design has strong similarities to the phase-conjugation process in optics. A strong point of the scheme is that it picks a simple set of solutions from the infinite number of possible solutions for a particular target state. This characteristic may be attributed to the choice of a classical wave-packet basis as the appropriate basis for this time-domain scheme. We show that a small number of pulses are sufficient to produce a wide range of target states (both classical and nonclassical target states are designed).

Minimum Muscle-Tension Change Trajectories Predicted by Using a 17-Muscle Model of the Monkey's Arm.

Four computational problems to be solved for visually guided reaching movements, hand path, and trajectory formations, coordinate transformation, and calculations of muscle tensions are ill-posed in redundant biological control systems. These problems are ill-posed in the sense that there exist an infinite number of possible solutions. In this article, it is shown that the nervous system can solve those problems simultaneously by imposing a single global constraint: finding the smoothest muscle- tension trajectory that satisfies the desired final hand position, velocity, and acceleration. Horizontal trajectories were simulated by using a l7-muscle model of the monkey's arm as the controlled object. The simulations predicted gently curved hand paths for lateral hand movements and for movements from the side of the body to the front, and a roughly straight hand path for anterioposterior movements. The tangential hand velocities were roughly bell shaped. The simulated results were in agreement with the actual biological movements.

The problem of redundancy in movement control: the adaptive model theory approach.

The problem of redundancy in movement control is encountered when one attempts to answer the question: How does the central nervous system (CNS) determine the pattern of neural activity required in some 5,000,000 descending motor fibres to control only 100-150 biomechanical degrees of freedom of movement? Mathematically this is equivalent to solving a set of simultaneous equations with many more unknowns than equations. This system of equations is redundant because it has an infinite number of possible solutions. The problem is solved by the neuronal circuitry hypothesized in Adaptive Model Theory (AMT). According to AMT, the CNS includes neuronal circuitry able to compute and maintain adaptively the accuracy of internal models of the reciprocal multivariable relationships between outgoing motor commands and their resulting sensory consequences. To identify these input-output relationships by means of regression analysis, correlations between the input signals have to be taken into account. For example, if the inputs are perfectly correlated, the model reduces to a virtual one-input system. In general, the number of inputs modelled equals the number of degrees of freedom encoded by the signals; that is, the number of independently varying (orthogonal) signals. The adaptive modelling circuitry proposed in AMT automatically tunes itself to extract independently varying sensory and motor signals before computing the dynamic relationships between them. Inverse models are employed during response execution to translate movements pre-planned as desired trajectories of these high-level sensory-feature signals into appropriately co-ordinated motor commands to send to the muscles. Since movement is pre-planned in terms of a number of orthogonalized sensory-feature signals equal to the number of degrees of freedom in the desired response, the problem of redundancy is solved and the correlation or co-ordination between motor-command signals is automatically introduced by the adaptive models.

Solving Underdetermined Tracer Inverse Problems by Spatial Smoothing and Cross Validation

Tracer conservation equations may be inverted to determine the flow field and macroscopic diffusion coefficients from known tracer distributions. An underdetermined system leads to an infinite number of possible solutions. The solution that is selected is the one that is as smooth as possible while still reproducing the tracer observations. The procedure suggested here is to define a penalty function that balances solution smoothnness, based on spatial derivatives of the solution, against residuals in the conservation equations. The ratio of detail in the solution to equation error is controlled by one or more smoothing parameters, which will not usually be known prior to the inversion. A parameter estimation technique known as generalized cross-validation is used to determine the degree of smoothing based on optimizing the prediction of withheld information. The method is tested for the case of steady flow containing a range of spatial scales in a two-dimensional channel with a spatially varying diffusion coefficient. It is shown that the correct flow field and diffusivity may be reproduced relatively accurately from a knowledge of the distribution of two tracers for a variety of flow configurations. The impact on the solution of errors in the equations and errors in the tracer data is studied. It is found that relatively large (correlated) errors in the equations due to numerical truncation error have the same effect as relatively small random errors in the data. A useful qualitative diagnostic measure of the value of an inverse solution is introduced. It is a measure of the loss of independent information due to smoothing the solution and is related to the data resolution matrix of classical discrete inverse theory.

Surface‐consistent deconvolution in the log/Fourier domain

In the surface‐consistent hypothesis, a seismic trace is the convolution of a source operator, a receiver operator, a reflectivity operator (representing the subsurface structure) and an offset‐related operator. In the log/Fourier domain, convolutions become sums and the log of signal amplitude at a given frequency is the sum of source, receiver, structural, and offset‐related terms. Recovering the amplitude of the reflectivity for a given frequency is then a linear problem (very similar to a surface‐consistent static correction problem). However, this linear system is underconstrained. Thus, among the infinite number of possible solutions, a particular one must be selected. Studies with real data support the choice of a spatially band‐limited solution. The surface‐consistent operators can then be calculated very efficiently using an inverse Hessian method. Applications to real seismic data show improvement compared with previous techniques. Surface‐consistent deconvolution is robust and fast in the log/Fourier domain. It allows the use of long operators, improves statics estimation, and removes the amplitude variations due to source or receiver coupling.

The Dehoop‐Knopoff representation theorem as a linear inverse problem

The deHoop‐Knopoff representation theorem, which relates observed seismic waves to a displacement discontinuity s defined on a surface S, is posed so that seismograms may be directly inverted for estimates of s and of the spatial and temporal resolution of s. Solutions s can be constructed either by parametrizing the fault surface as a number of point double couples or by representing s as an expansion of orthogonal functions. Of the infinite number of possible solutions satisfying a single seismic‐data set, methods for constructing particular solutions (e.g., best fitting, or closest to a desired solution) are given. Application of inverse theory to the deHoop‐Knopoff representation theorem leads to a convenient way to include various types of seismic data—e.g., long‐ and short‐period teleseismic, near field accelerogram, and geodetic—into a single inversion.

Parameter Identification in an Inhomogeneous Medium With the Finite-Element Method

Abstract This paper studies the development of a systematic procedure for solving the problem of parameter identification. The parameter function to parameter identification. The parameter function to be identified is the permeability embedded in a nonlinear, partial differential equation of parabolic type. Finite elements are used to represent the unknown function parametrically in terms of nodal values over a suitable discretization of a flow region. These nodal values are then determined by a least-squares fitting between the observed and calculated flow potentials, subject to linear inequality constraints imposed upon the parameters to be identified. To handle such constraints, Rosen's gradient projection technique is combined with the Gauss-Newton method. Numerical procedures are presented for the solution of the procedures are presented for the solution of the partial differential equation using the Galerkin partial differential equation using the Galerkin method and a predictor-corrector approach. Simultaneous identification of 25 parameters embedded in a two-dimensional, nonlinear, diffusion-type equation is demonstrated by two examples. Introduction Mathematical models have been used as powerful tools in analyzing groundwater or oil reservoir systems. The application of such models to real field problems requires the determination of parameters embedded in partial differential parameters embedded in partial differential equations. These parameters are not simply measurable from die physical point of view. Recently, considerable attention has been directed toward developing analytical procedures for solving the parameter identification problem. Proposed methods may be classified into two Proposed methods may be classified into two groups: the direct approach and the indirect approach. The direct approach treats the parameters as dependent variables in the form of a formal boundary-value problem. If the flow potentials are known over the entire flow region, the original governing equation becomes a linear, first-order, partial differential equation of the hyperbolic type partial differential equation of the hyperbolic type in terms of the unknown permeability function. With the aid of boundary conditions and flow data, a unique solution of the equation can be obtained. Nelson and Nelson and McCollum proposed the energy dissipation method for solving the unknown parameters. Frind and Pinder used a Galerkin parameters. Frind and Pinder used a Galerkin method for steady flow only. Another direct-method approach that specially takes account of unsteady flow problems is the minimization of the residual error involved in the differential equation. When the error functional is defined as the sum or maximum norm, the minimization problem can be solved by the linear programming technique. The time component in a linear differential equation may be also eliminated by an integration with respect to time. This was studied by Nutbrown. When the flow data and necessary boundary conditions are lacking, the inverse problem yields an infinite number of possible solutions. The inverse problem, even in the case where its solution is guaranteed to be unique, is not generally well posed. In order to obtain a meaningful solution to the inverse problem, it is necessary to impose certain prior problem, it is necessary to impose certain prior restrictions on the admissible function space of parameters. The methods that have been proposed parameters. The methods that have been proposed in the literature include the use of a flatness criterion by Emsellem and de Marsilly and a multiple objective decision process by Neuman. For the direct approach a difficulty arises in the data requirement. In field practice, observation wells are sparsely distributed in an arbitrary fashion rather than regularly distributed; and only a limited number of wells are available. An attempt to obtain space and/or time derivative approximations of flow potentials from the insufficient data by finite-difference or finite-element methods would be a multisource of errors in the results of parameter identification. Sagar et al. suggested the method of spline interpolation to fit the discrete data of observations; but the method still requires a sufficient number of the observation points to adequately approximate the whole flow potential surface in the region under consideration. The indirect approach is an optimization procedure in which the algorithm starts from a set of initial estimates of the parameters and improves it in an iterative manner until the model response is sufficiently close to that of the field observation. SPEJ P. 217

Mathematical formulation of viscous-inviscid interaction problems insupersonic flow.

Abstract : The problem area of the interaction between the viscous and inviscid flowfields around a body is investigated. The study is based on ascertaining a flow model which has the ability of determining interaction effects wityout the possibility of having to choose between an infinite number of possible solutions. The importance of the inclusion of a two-family characteristic net for the outer supersonic flow and the normal momentum for the inner viscous flow is discussed and the necessity of their inclusion in order to have a properly posed model is presented. A result of the numerical analysis based on the flow model is presented and the uniqueness of such results is discussed.

Comments on the Eichenbaum-Hernqvist Model of a Low-Pressure Diode

Using a special model of a collisionless plasma diode, Eichenbaum and Hernqvist have shown the interesting circumstance that two different interelectrode space-potential distributions may exist for the same boundary conditions and the same output voltage and current. It is shown here, using the same boundary conditions, that there are at least several additional and probably an infinite number of possible solutions.