General Solution Technique Research Articles

Analytical technique for hygrothermo-electroelastic field in piezoelectric bodies with D ∞ symmetry

A general solution technique for hygrothermo-electroelastic problems in bodies with D ∞ symmetry is developed. The displacement and electric field are expressed in terms of their respective potential functions, and the hygrothermo-electroelastic field quantities are expressed in terms of elastic and piezoelastic potential functions, each of which satisfies a Laplace equation with respect to the appropriately transformed spatial coordinates, combined with the two hygrothermoelastic displacement potential functions. As an application of this technique, a theoretical analysis of an infinite cylinder subjected to distributions of moisture content, temperature, and stress was performed, and the significance of the hygrothermo-electroelastic analyses was demonstrated.

A similarity solution for spherical cavity drained expansion in overconsolidated soils considering large deformation

This paper presents a novel similarity solution for drained spherical cavity expansion in overconsolidated soils, which creatively incorporates the large strain into similarity solution. The salient feature of the present solution lies in that it properly represents the large deformation associated responses of the overconsolidated soil during cavity expansion. The logarithmic strain components are reasonably incorporated into the similarity solution, which is a general solution technique that has been widely used to solve cavity expansion problems, to creatively account for the large deformation of soils during cavity expansion in such a solution framework. The competent modified unified-hardening (UH) model is employed to properly represent the unique behavior of overconsolidated soils during expansion, where three-dimensional strength characteristics of the soil are taken into account as well. The partial differential governing equations under Eulerian description are transformed into a set of first order ordinary differential equations under Lagrangian description and further solved as an initial value problem by MATLAB. The present method is verified by comparing with the modified Cam-clay model based solutions and an extensive parameter analysis is subsequently conducted for soils with different overconsolidation ratio. The unique expansion behaviors in the overconsolidated soils including the peak strength behavior, strain-hardening/softening and shear dilatancy behaviors are discussed in a comprehensive and in-depth manner.

An Algorithm for Computing Solutions to the Range Limited Routing Problem Using Electrical Trucks

Increasing urban intensification has caused civic planners to devote additional resources to more appropriate logistics planning. Electric vehicles have proved to be both a lower cost alternative and more environmentally friendly than the more ubiquitous internal combustion engine vehicles. However, prevailing decision-making formulations employed by municipalities and businesses are not necessarily computationally conducive for the evaluation and optimization of urban transportation systems using electric vehicles. An innovative computational approach, the range limited routing problem, is introduced that enables urban planners to more readily evaluate the contributions of electric vehicles to the city logistics decision-making process. While there is no generalized solution technique for solving this new formulation, this paper employs the Firefly Algorithm (FA) metaheuristic to solve the range limited routing problem using electric trucks.

Nonlinear flexure coupling elements for precision control of multibody systems

Conventional multibody systems used in robotics and automated machinery contain bearing components that exhibit complex and uncertain tribological characteristics. These limit fundamentally the precision of the automated motion and also cause wear. Replacing traditional bearing joints with flexure couplings eliminates these tribological effects, together with wear, reducing necessary system maintenance and offering a potential for increased motion precision. A flexure-coupled multibody system is considered and a novel general solution technique is presented. Derivation of a large deflection flexure coupling model is provided and subsequently validated using an experimental facility. A focused study of a unique double-flexure-coupling rigid body system is given; the formulated nonlinear mathematical model can be used for feedforward control. Equivalent control is also applied to a corresponding system with traditional bearing joints. The feasibility of replacing bearing joints by flexure couplings is demonstrated in terms of accurate large displacement control and reduction of high-frequency disturbances.

On a cell division equation with a linear growth rate

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

Solving an incomplete markets model with a large cross-section of agents

This paper shows that perturbation methods can be applied to a DSGE model with incomplete markets and a finite but arbitrarily large number of heterogeneous agents. We develop a simple but general solution technique that handles many state and choice variables for each agent and thus has an extremely high-dimensional state space. The method is based on perturbations around a point at which the solution is known. The novel idea is to exploit the symmetry of the problem to overcome the curse of dimensionality. We use the analysis to demonstrate the impact of heterogeneity on macroeconomic quantities and the pricing of risk. Furthermore, we set our technique apart from standard methods used in the literature.

ON A CELL DIVISION EQUATION WITH A LINEAR GROWTH RATE

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

Analytical technique for thermoelectroelastic field in piezoelectric bodies with D∞ symmetry in cylindrical coordinates

ABSTRACTA general solution technique for non-axisymmetric thermoelectroelastic problems in cylindrical domains with D∞ symmetry is constructed. The displacement and electric field are expressed in terms of the respective potential functions, and the thermoelectroelastic field quantities are expressed in terms of the elastic and piezoelastic potential functions, each of which essentially satisfies a Laplace equation with respect to the spatial coordinates, combined with the two thermoelastic displacement potential functions. As an application of the technique, the theoretical analysis of a solid cylinder subjected to combined mechanical and thermal loading is performed, and the necessity of the thermoelectroelastic analyses is demonstrated.

Analytical technique for thermoelectroelastic field in piezoelectric bodies with D∞ symmetry

ABSTRACTA general solution technique for thermoelectroelastic problems in bodies with D∞ symmetry is constructed. The displacement and electric field are expressed in terms of the respective potential functions, and the thermoelectroelastic field quantities are expressed in terms of the elastic and piezoelastic potential functions, each of which satisfies the Laplace equation with respect to the appropriately transformed spatial coordinates, combined with the two thermoelastic displacement potential functions. As an application of the technique, the theoretical analysis of a semi-infinite body subjected to a temperature distribution is performed, and the significance of the thermoelectroelastic analyses is demonstrated.

A general solution technique for electroelastic fields in piezoelectric bodies with D ∞ symmetry in cylindrical coordinates

To elucidate the electroelastic field in bodies with D ∞ symmetry such as wooden materials, we constructed a general solution technique for electroelastic problems in such bodies in a cylindrical coordinate system. We introduced the displacement and electric potential functions to express the displacement and electric field in cylindrical coordinates; their governing equations were obtained using the fundamental equations for the electroelastic field. The electroelastic field quantities could be expressed in terms of two elastic displacement potential functions and two piezoelastic displacement potential functions, each of which satisfies a Laplace equation with respect to the appropriately transformed cylindrical coordinates. As an application of the technique, we analyzed the problem of an infinitely long cylinder subjected to a non-axisymmetrically distributed electric surface potential. Using numerical calculations, we elucidated the electroelastic field quantities within the cylinder and found unique electroelastic coupling behaviors, which clearly demonstrate the necessity for the analytical technique presented. Moreover, we confirmed the possibility of the nondestructive evaluation (NDE) techniques by use of the piezoelectric effects.

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

The present study takes an analytical approach for solving the free vibration problem of a microstructured beam model, in which transverse displacement springs are added to allow for the transverse shear deformation effect in addition to the rotational springs. The exact vibration frequencies for the discrete microstructured beam model with simply supported ends are obtained via matrix decomposition. In addition, a general solution technique involving the use of Padé approximants for the continualization procedure is proposed in order to obtain the continuous equivalent system for the discrete microstructured beam model. The analytical vibration solutions of the equivalent continuous system are obtained and their accuracy is assessed by using the exact solutions. It is found that the solutions of the equivalent continuous system have a first order accuracy when compared with the exact solutions of their discrete counterpart. The length scale coefficient in the nonlocal Timoshenko beam model is calibrated by using the analytical solutions. Two nonlocal Timoshenko beam models, i.e., the Wang model (without the length scale effect in the shear stress strain relation) and the Reddy model, are evaluated based on their ability to capture the nonlocal effect.

Flexural analysis of cross-ply laminated beams using layerwise trigonometric shear deformation theory

In the present work, a layerwise trigonometric shear deformation theory is used for the analysis of two layered (90/0) cross ply laminated simply supported and fixed beams subjected to sinusoidal load. The displacement field of the present theory consists of trigonometric sine function in terms of thickness coordinate to take into account the effect of transverse shear deformation. Theory satisfies the trans-verse shear stress free boundary conditions at top and bottom surfaces of the beam. This model satisfies the constitutive relationship between shear stress and shear strain in both the layers and the axial displacement compatibility at the interface. Virtual work principle is employed to obtain governing equations and boundary conditions. Closed form solution technique has the limitation of simply supported boundary condition. In the present work general solution technique is developed, which can be used for any type of boundary and loading conditions. The transverse shear stresses are obtained using constitu-tive relation as well from the use of equilibrium equations. The results of displacements and stresses obtained by present theory are compared with the available results in the literature.

Semi-Analytical Solution Based on Strip Method for Buckling and Vibration of Isotropic Plate

The present paper achieves a semianalytical solution for the buckling and vibration of isotropic rectangular plates. Two opposite edges of plate are simply supported and others are either free, simply supported, or clamped restrained against rotation. The general Levy type solution and strip technique are employed with transition matrix method to develop a semianalytical approach for analyzing the buckling and vibration of rectangular plates. The present analytical approach depends on reducing the strips number of the decomposed domain of plate without escaping the results accuracy. For this target, the transition matrix is expressed analytically as a series with sufficient truncation numbers. The effect of the uni-axial and bi-axial in-plane forces on the natural frequency parameters and mode shapes of restrained plate is studied. The critical buckling of rectangular plate under compressive in-plane forces is also examined. Analytical results of buckling loads and vibration frequencies are obtained for various types of boundary conditions. The influences of the aspect ratios, buckling forces, and coefficients of restraint on the buckling and vibration behavior of rectangular plates are investigated. The presented analytical results may serve as benchmark solutions for such plates. The convergence and efficiency of the present technique are demonstrated by several numerical examples compared with those available in the published literature. The results show fast convergence and stability in good agreement with compressions.

Large-Scale Bi-Level Strain Design Approaches and Mixed-Integer Programming Solution Techniques

The use of computational models in metabolic engineering has been increasing as more genome-scale metabolic models and computational approaches become available. Various computational approaches have been developed to predict how genetic perturbations affect metabolic behavior at a systems level, and have been successfully used to engineer microbial strains with improved primary or secondary metabolite production. However, identification of metabolic engineering strategies involving a large number of perturbations is currently limited by computational resources due to the size of genome-scale models and the combinatorial nature of the problem. In this study, we present (i) two new bi-level strain design approaches using mixed-integer programming (MIP), and (ii) general solution techniques that improve the performance of MIP-based bi-level approaches. The first approach (SimOptStrain) simultaneously considers gene deletion and non-native reaction addition, while the second approach (BiMOMA) uses minimization of metabolic adjustment to predict knockout behavior in a MIP-based bi-level problem for the first time. Our general MIP solution techniques significantly reduced the CPU times needed to find optimal strategies when applied to an existing strain design approach (OptORF) (e.g., from ∼10 days to ∼5 minutes for metabolic engineering strategies with 4 gene deletions), and identified strategies for producing compounds where previous studies could not (e.g., malate and serine). Additionally, we found novel strategies using SimOptStrain with higher predicted production levels (for succinate and glycerol) than could have been found using an existing approach that considers network additions and deletions in sequential steps rather than simultaneously. Finally, using BiMOMA we found novel strategies involving large numbers of modifications (for pyruvate and glutamate), which sequential search and genetic algorithms were unable to find. The approaches and solution techniques developed here will facilitate the strain design process and extend the scope of its application to metabolic engineering.

Equilibrium Existence and Approximation for Incomplete Market Models with Substantial Heterogeneity

This paper contains an analysis of incomplete market models with finitely but arbitrarily many heterogeneous agents. We discuss the mathematical foundation for equilibrium conditions which leads to two findings. First, we establish existence of recursive equilibria for small and large risks. Second, we develop a simple but general solution technique which handles many state and choice variables for each agent and thus an extremely high-dimensional state space. The method is based on perturbations around a point at which the solution is known. The novel idea is to exploit the symmetry of the problem to overcome the curse of dimensionality. We use the analysis to demonstrate the impact of heterogeneity on macroeconomic quantities and the pricing of risk. Furthermore, we set our technique apart from the standard method used in the literature.

A Stock Dependent Economic Order Quantity Model for Perishable Items Under Inflationary Conditions

Problem statement: In most of the earlier inventory models, effect of inflation has been ignored, which is playing pilot role in present environment. In this article, we have proposed an economic order quantity model for deteriorating items having stock dependent demand (whose demand varies with the stock) under the effect of inflation. Approach: Firstly, problem is framed in the form of linear differential equation model and then this model has been solved with general solution technique of linear differential equations. Using the solution the expression for total cost of inventory has been obtained. Results: Using total cost equation an optimal order quantity is obtained and convexity of various costs is also studied. This model helps retailer to decide his economic order quantity for deteriorating items having stock dependent demand. The consideration of effect of inflation makes this model more relevant for present business environment. Conclusion: The convexity of various costs proves the validity of this model. The effect of inflation with two different rates is also studied in tables below considering both constant as well as stock dependent demand. The graph of these tables shows that the optimal inventory cost increases with increase in number of cycles it increases rapidly in starting but becomes stable for large number of cycles. The proposed model can be extended in several ways. For instance, we may generalize the model to allow for shortages, quantity discounts, time value of money, finite replenishment rate, permissible delay in payments and others.

Feedback Dual Controller Design and Its Application to Monocular Vision-Based Docking

A novel stochastic feedback control approach, which is capable of considering the presence of uncertainty during the controller design phase, is developed and applied to the automatic docking problem for an unmanned vehicle with a monocular vision sensor. A typical procedure for linear stochastic control employs a two-step approach that separates estimation and control: estimate the system state using noise corrupted measurements, and control the system based on the estimated state. However, for nonlinear stochastic control systems and adaptive control systems with uncertain parameters, control and estimation are often not separable because control inputs can affect not only the system state but also the quality of the state estimation. Dynamic programming and search-based methods are general solution techniques for solving these types of control problems, however, those solution techniques are often infeasible for real-time applications due to the huge computational requirements even for small problems. In this paper, a new systematic, suboptimal control algorithm is presented, which can be implemented as an online feedback controller for a stochastic control problem of moderate dimension. For an application example, automated docking of a nonholonomic vehicle with a monocular vision sensor is considered. The results from a series of numerical simulations and experimental tests are presented.

An Optimal Inventory Policy for Items having Linear Demand and Variable Deterioration Rate with Trade Credit

Problem statement: Demand considered in most of the classical inventory models is constant, while in most of the practical cases the demand changes with time. In this study model has been framed to study the items whose demand changes with time and deterioration rate increases with time. The effect of permissible delay is also incorporated in this study. The objective of this research is to develop an inventory model for perishable items whose perish-ability rate as well as demand increases with time Approach: Firstly, problem is framed in the form of linear differential equation model and this model had been solved using general solution techniques of linear differential equations. The solution obtained gives the inventory level at any particular time of the cycle period. With the help of this inventory level, total as well as average inventory cost has been obtained. Results: This study developed a model to determine an optimal order quantity by using calculus technique of maxima and minima. Thus it helps retailer to decide its optimal ordering quantity under the constraints of variable deterioration rate and linear pattern of demand. Conclusion: Numerical solution of the suggested model had also been proposed, the above model can be converted into constant demand model, or for items having no deterioration. This study can further be extended for items having some other demand pattern, also time value of money and inflation can be incorporated in this model to make it more realistic and present business environment suited.

Accurate solution to scattering by a semi-circular groove

Wave scattering by a semi-circular groove in an otherwise flat surface is considered. A mode-matching technique is employed in which the inner region is circular, the lower half of which corresponds to the groove. A Galerkin approximation is then made in which the solution (or its derivative, depending on the boundary conditions) on the matching semi-circle is expanded in a set of basis functions involving ultraspherical Gegenbauer polynomials which incorporate the known singularity at the groove edges. In this way a highly accurate and robust scheme is developed which can be used as a benchmark against which to test more general solution techniques.

Electroelastic Analysis of Functionally Graded Piezoelectric Material Beams

A general two-dimensional solution is proposed for functionally graded piezoelectric material beams with arbitrary graded material properties along the beam thickness direction. The beam is subjected to normal and shear tractions of polynomial form on the top and bottom surfaces, while the end boundary conditions can be cantilever, simply supported or rigidly clamped. The Airy stress function and the electric potential are expressed in a finite power series, and the corresponding recurrence-governing equations are obtained. Explicit expressions of analytical solution to a specific example for a cantilever beam are obtained to demonstrate the usefulness of the proposed general solution technique. The proposed solution method will be useful in analyzing functionally graded piezoelectric structures with arbitrary variations of material properties, and the obtained results can serve as a basis for establishing simplified theories and numerical methodology of functionally graded piezoelectric material beams.