Occurrence Of Multiple Solutions Research Articles

Geometric Interpretation of the Multi-solution Phenomenon in the P3P Problem

It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its three control points and the position of the optical center. Since in any real applications, the knowledge on the exact number of possible solutions is a prerequisite for selecting the right one among all the possible solutions, and the study on the phenomenon of multiple solutions in the P3P problem has been an active topic since its very inception. In this work, we provide some new geometric interpretations on the multi-solution phenomenon in the P3P problem, and our main results include: (1) the necessary and sufficient condition for the P3P problem to have a pair of side-sharing solutions is the two optical centers of the solutions both lie on one of the three vertical planes to the base plane of control points; (2) the necessary and sufficient condition for the P3P problem to have a pair of point-sharing solutions is the two optical centers of the solutions both lie on one of the three so-called skewed danger cylinders;(3) if the P3P problem has other solutions in addition to a pair of side-sharing (point-sharing) solutions, these remaining solutions must be a point-sharing (side-sharing ) pair. In a sense, the side-sharing pair and the point-sharing pair are companion pairs; (4) there indeed exist such P3P problems that have four completely distinct solutions, i.e., the solutions sharing neither a side nor a point, closing a long guessing issue in the literature. In sum, our results provide some new insights into the nature of the multi-solution phenomenon in the P3P problem, and in addition to their academic value, they could also be used as some theoretical guidance for practitioners in real applications to avoid occurrence of multiple solutions by properly arranging the control points.

Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

AbstractThe aim of this paper is analyzing the positive solutions of the quasilinear problem-(u′/1+(u′)2)′=λ⁢a⁢(x)⁢f⁢(u) in ⁢(0,1),u′⁢(0)=0,u′⁢(1)=0,-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,whereλ∈ℝ{\lambda\in\mathbb{R}}is a parameter,a∈L∞⁢(0,1){a\in L^{\infty}(0,1)}changes sign once in(0,1){(0,1)}and satisfies∫01a⁢(x)⁢𝑑x<0{\int_{0}^{1}a(x)\,dx<0}, andf∈𝒞1⁢(ℝ){f\in\mathcal{C}^{1}(\mathbb{R})}is positive and increasing in(0,+∞){(0,+\infty)}with a potential,F⁢(s)=∫0sf⁢(t)⁢𝑑t{F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at+∞{+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions,𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from(λ,0){(\lambda,0)}at someλ0>0{\lambda_{0}>0}and from(λ,∞){(\lambda,\infty)}at someλ∞>0{\lambda_{\infty}>0}. It also establishes that𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}consists of regular solutions if and only if∫0z(∫xza(t)dt)-1/2dx=+∞ or ∫z1(∫xza(t)dt)-1/2dx=+∞.\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty.Equivalently, the small positive regular solutions of𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}become singular as they are sufficiently large if and only if(∫xza(t)dt)-1/2∈L1(0,z) and (∫xza(t)dt)-1/2∈L1(z,1).\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1).This is achieved by providing a very sharp description of the asymptotic profile, asλ→λ∞{\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions ofλ0{\lambda_{0}}andλ∞{\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.

Finding a Common Weight Vector of Data Envelopment Analysis Based upon Bargaining Game

Data Envelopment Analysis (DEA) is a mathematical programming method for measuring the relative efficiency of Decision Making Units (DMUs) by evaluating their outputs and inputs. In the history of DEA, the cross-efficiency of jth DMU is widely used as an efficiency measure of a given DMUo among researchers. The approach always utilizes weights related to inputs and outputs in the assessment. Unfortunately, the weights are not always uniquely determined in the cross-efficiency measurement because DEA always suffers from an occurrence of multiple solutions, so indicating an occurrence of multiple weights. To overcome such a difficulty, this paper proposes a new approach for determining a common weight vector of DEA based on bargaining game.

An effective inverse analysis tool for parameter identification of anisotropic material models

Abstract Effective inverse analysis tools rest on the synergetic combination of even complex experimental and numerical procedures. This is the case of methodologies recently developed for the identification of parameters entering anisotropic elastic–plastic constitutive laws on the basis of data collected from indentation tests. The values of the sought material properties are inferred from a discrepancy minimization procedure, which entails the repeated simulation of the test. In this situation, the overall computing burden can be significantly reduced, without compromising the accuracy of the results, replacing traditional finite element approaches by approximated analytical models, based on the interpolation by radial basis functions of a few numerical results filtered by proper orthogonal decomposition. The effectiveness of this technique is demonstrated in the present study with specific reference to earlier investigated elastic–perfectly-plastic materials and is further verified on the more general case of hardening constitutive models. The accuracy of the identification results and the role of the likely occurrence of multiple solutions are discussed.

The elastic Landau–Levich problem

AbstractWe study the classical Landau–Levich dip-coating problem in the case where the interface has significant elasticity. One aim of this work is to unravel the effect of surface-adsorbed hydrophobic particles on Landau–Levich flow. Motivated by recent findings (Vella, Aussillous & Mahadevan, Europhys. Lett., vol. 68, 2004, pp. 212–218) that a jammed monolayer of adsorbed particles on a fluid interface makes it respond akin to an elastic solid, we use the Helfrich elasticity model to study the effect of interfacial elasticity on Landau–Levich flow. We define an elasticity number, $\mathit{El}$, which represents the relative strength of viscous forces to elasticity. The main assumptions of the theory are that $\mathit{El}$ be small, and that surface tension effects are negligible. The shape of the free surface is formulated as a nonlinear boundary value problem: we develop the solution as an asymptotic expansion in the small parameter ${\mathit{El}}^{1/ 7} $ and use the method of matched asymptotic expansions to determine the film thickness as a function of $\mathit{El}$. The solution to the shape of the static meniscus is not as straightforward as in the classical Landau–Levich problem, as evaluation of higher-order effects is necessary in order to close the problem. A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as ${\mathit{El}}^{4/ 7} $ in qualitative agreement with the experiments of Ouriemi & Homsy (Phys. Fluids, 2013, in press).

DEA congestion and returns to scale under an occurrence of multiple optimal projections

Abstract Economic implications of congestion have been recently discussed in many DEA (data envelopment analysis) studies. In addition, several previous research efforts have explored a theoretical linkage between returns to scale (RTS) and the concept of congestion, because the two economic concepts are closely connected to each other. Tone and Sahoo [Tone, K., Sahoo, B.K., 2004. Degree of scale economies and congestion: A unified DEA approach. European Journal of Operational Research 158, 755–772] have published the theoretical linkage in this journal. All of the previous studies, including their research (2004), assume a unique optimal solution in the investigation on DEA-based congestion. When multiple solutions occur in DEA-based congestion measurement, the economic implications of congestion obtained from the previous research are all problematic from both theoretical and practical perspectives. To deal with the issue, this study explores how to deal with the occurrence of multiple solutions in the DEA-based congestion measurement. This study proposes a new approach for the congestion measurement and theoretically compares the proposed approach with Tone and Sahoo (2004).

Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches

Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.

The measurement of returns to scale under a simultaneous occurrence of multiple solutions in a reference set and a supporting hyperplane

This research theoretically explores the measurement of RTS (Returns to Scale) under a possible occurrence of multiple solutions in DEA (Data Envelopment Analysis). In this study, the occurrence of multiple solutions is classified into Type I and Type II. Type I is an occurrence of multiple solutions in a reference set. Type II is an occurrence of multiple solutions on a supporting hyperplane passing on the reference set. Both Types I and II are very well known among DEA researchers, but previous research has not sufficiently explored a simultaneous occurrence of Type I and Type II in the RTS measurement. The two types of multiple solutions influence a degree of RTS in the DEA measurement. Such a quantitative issue on RTS is examined from the perspective of the Type I and Type II problems. To deal with such difficulties, a new linear programming approach is proposed to identify all efficient DMUs (Decision Making Units) that consist of a reference set, even if multiple solutions occur on the reference set. Based upon the research result, we can identify when multiple solutions of Type I and/or Type II occur on the RTS measurement and how to deal with such difficulties. Our research result makes it possible to measure a degree of scale economies (RTS) under the simultaneous occurrence of Type I and Type II.

Propagation algorithms for attitude-related ambiguity resolution

A new approach for ambiguity resolution is presented and tested by means of Monte-Carlo simulations. This approach applies to both single and double phase differences sampled at a given instant, and to any number of baselines and lines of sight, provided the number of lines of sight exceeds a certain threshold. The algorithms are based on the combined global geometry of the baselines and lines of sight and thereby consider all geometric constraints that are inherent in the problem. The algorithms have been programmed and applied to a large number of different configurations of GPS sight lines, and limited to baselines of approximately 1 m in length and the GPS L1 frequency only. The cause of the occurrence of multiple solutions has been investigated. To check the adequacy and limitations of the new algorithms for practical application, it was assessed in which proportion the algorithms either failed to provide a result, provided a false result, or provided a good result as a function of number of baselines, lines of sight and accuracy of the ambiguous phase difference measurements.

SOLUTION NONUNIQUENESS FOR SEPARATED GAS–LIQUID FLOW IN PIPES AND WELLS. I. OCCURRENCE

ABSTRACT The occurrence of multiple solutions has been investigated and discussed in this paper. Multiple solution regions are displayed by either a U SL–U SG map or a X–Y map. It is found that many parameters impose influence on the multiple solution regions, including fluid properties, pipe ID, pipe inclination angle, wall inflow or outflow, empirical correlations for wall friction and interfacial friction factors, and so on. For example, the multiple solution region could change significantly if different interfacial friction factor correlations are applied. Multiple solutions are only possible for upward flow for separated flow in a pipe without wall mass transfer. However, for wellbore flow case, depending on wall inflow or outflow of fluids and their flow rates, multiple solutions may also exist for horizontal and downward flow. More specifically, for wall inflow case (production well), multiple solutions only exist for upward flow for the gas inflow-dominant case, whereas they may exist for upward, horizontal and downward flow for liquid inflow-dominant situations. The opposite is expected for wall outflow (injection well) case.

Exact analytical solutions of forced convection flow in a porous medium

The forced convection problem in a fluid saturated porous medium is considered. For the self-similar boundary-layer flows past a plane or axisymmetric body with arbitrary shape embedded in this medium and having a power-law surface temperature distribution, analytical solutions are given. In the range λ ≥ 0 of the temperature exponent, the analytical results show an excellent agreement with previous numerical findings. In the range λ < − 1 2 , the existence of a new class of unique solutions, while for − 1 2 < λ < 0 the occurrence of multiple solutions is reported. The heat transfer characteristics and the physical meaning of all these forced convection boundary-layer flows are discussed in detail.

DEA Duality on Returns to Scale (RTS) in Production and Cost Analyses: An Occurrence of Multiple Solutions and Differences Between Production-Based and Cost-Based RTS Estimates

This article discusses the concept of RTS (Returns To Scale) in the framework of DEA (Data Envelopment Analysis) production and cost analyses, focusing on an occurrence of multiple solutions and how to deal with such a difficulty. Dual relationships between production-based and cost-based RTS estimates are also theoretically discussed in this study.

NUMERICAL INVESTIGATION ON THE BIFURCATIVE NATURAL CONVECTION IN A HORIZONTAL CONCENTRIC ANNULUS

Steady state two-dimensional natural convective heat transfer in a horizontal cylindrical annulus was studied by solving the governing equations based on the primitive variables. Emphasis was put on the occurrence of multiple solutions and on the determination of the bifurcation points at which those multiple solutions begin to branch out. The multicellular flow patterns from the results of melting processes in an isothermally heated horizontal cylinder for high Rayleigh numbers were used as initial fields. This approach succeeded in finding new bifurcation points to a tetracellular solution for a set of parameter variables identical to those used in previous studies. Close examination of flow pattern transitions near the bifurcation point was also conducted. It was found that the mechanisms of flow transition are different depending on the critical Rayleigh number of the bifurcation point.

A Galerkin finite-element study of the onset of double-diffusive convection in an inclined porous enclosure

The Darcy model with the Boussinesq approximation is used to study the onset of double-diffusive natural convection in an inclined porous cavity. Transverse gradients of heat and solute are applied on two opposing walls of the cavity, while the other two walls are impermeable and adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing and of equal intensity. The objective of this study is to investigate the critical stability of this system in terms of the inclination angle, the aspect ratio of the cavity and the Lewis number. The subsequent behavior of the convective flow is also discussed in terms of the governing parameters of the problem. Numerical procedures based on the Galerkin and finite element methods are carried out to investigate the onset of double-diffusive convection using the linear stability analysis. It is shown that for values of Lewis number around unit, overstability is possible provided that the normalized porosity of the porous medium ε is made smaller than unity. For supercritical convection, the occurrence of multiple solutions, for a given range of the governing parameters, is demonstrated. The numerical results also indicate the existence of subcritical convective regimes.

Inconsistency in the Structure of Analyses Proposed for Shear Banding.

Shear banding is considered as localized necking where multiple solutions can exist on the necking plane comprising the boundary between the shear band and its surroundings. Therefore, shear banding is identical with localized necking though it is investigated under plane strain. On the necking plane the discontinuity of velocity gradient can be permitted. Hill and Hutchinson [J. Mech. Phys. Solids, 23 (1975), 239-264] investigated shear band without considering the discontinuity, but their method is methodically equivalent to that which takes into account the discontinuity. In previously published analyses of shear banding, it is assumed that the regions separated by the necking plane keep contact each other during the onset of shear banding, and the equilibrium of forces is maintained on the plane. However, it is proven that this assumption cannot provide useful information for determining the deformation when shear banding begins to occur. Furthermore, the assumption does not allow the formation of a common boundary condition imposed on both regions separated by the necking plane. When shear banding occurs, the spin rate of the regions differs because of multiple solutions. The criterion for determining the occurrence of shear banding can be deduced by imposing a rigid spin to make the spin rates identical. It is found that this criterion agrees with one of the universal criteria obtained from considering the occurrence of multiple solutions under the common boundary condition on the surface of a body.

The stability of activated sludge reactors with substrate inhibition kinetics and solids recycle

The steady-state simulation of activated sludge reactors with settler and recycle of the concentrated biomass was considered for the case of substrate inhibition kinetics. Analytical relationships were developed for the continuous stirred tank reactor; numerical simulation was performed for other reactors. Occurrence of multiple solutions and hysteresis behavior are examined as a function of space time θ, recycle ratio R, recycle solids concentration X R and sludge age θ c. The sensitivity of the outlet substrate conversion with respect to the degree of substrate inhibition and of hydraulic mixing is discussed; criteria are suggested in order to operate such bioreactors and to prevent washout occurrence. All results are presented in dimensionless form.

Loss of stability at turning points in nonlinear elliptic variational inequalities

A CONSIDERABLE amount of work has been devoted, in recent years, to stationary free boundary problems. In particular, obstacle problems have been the subject of extensive work concerning existence, estimates and regularity of the solution or of the coincidence set, and occurrence of multiple solutions. In [6, 8, 161 the authors consider obstacle problems with nonlinear forcing terms, leading to multiple solutions, hence to turning points when a real parameter appears in the forcing term. A local study of the branches of solutions is set up; the results obtained are the extension, in a weaker form, of those well known for equations [ll, 12). However, some important classical questions remain open for obstacle problems:

The occurrence of multiple solutions for the TSD-Euler equation

In a recent paper the TSD-Euler equation for transonic flow has been derived. This equation is similar in some respects to the TSD equation but has both entropy and vorticity terms retained. In this paper the existence of multiple solutions for the TSD-Euler equation is examined and it is found that such solutions exist for a small range of Mach numbers and airfoil thickness. It is found also that the addition of a vorticity flux on the airfoil surface can enhance the appearance of multiple solutions.

The effect of entropy production on the occurrence of multiple solutions in transonic flow

In recent years it has become apparent that the transonic potential equation does not give a unique solution when shock waves are present in the flow. A previous paper has examined the causes of the phenomenon. The present paper is concerned with the extension of the previous analysis to include a simplified form of the Euler equations. Multiple solutions for the Euler equations can occur when the average of the velocity on the upper and lower surfaces of the airfoil exceeds sonic values.

Observations on the occurrence of multiple solutions in transonic potential theory

In recent years it has become apparent that the transonic potential equation does not give a unique solution when shock waves are present in the flow. Most of the studies of this problem are computational in nature. This paper is concerned with an analytic examination of these nonunique solutions. It is concluded that multiple solutions occur when the flow has a supersonic region and that all of the solutions satisfy the usual boundary conditions.