Flexible Tolerance Method Research Articles

A New Approach to Permeability Inversion of Fractured Rock Masses and Its Engineering Application

This paper presents a new seepage inversion technique to predict the permeability coefficient of the rock mass with developed fracture or fault. With the measured data of flow and water head of boreholes, the permeability coefficient of the rock masses near the boreholes are obtained by inverse calculation on the basis of unsteady seepage tests. Then, a flexible tolerance method is proposed to invert the permeability coefficient of rock masses in different zones of the reservoir area. This comprehensive inversion analysis method is applied in one actual project of the water supply reservoir. The equivalent permeability coefficient of the rock masses in the range of 0 m to 16.0 m below the road surface near the dam axis on the left bank of the mountain is 1.12 × 10−3 cm/s. The root mean squared error and coefficient of determination of the measured and calculated values are 1.33 × 10−4 m3/s and 0.9976 m3/s, respectively. The rock masses in the reservoir site area have high permeability. The groundwater level in the junction area and the mountains on both sides of Shangmo reservoir is low, and the hydraulic gradient is small. The maximum error between the calculated value of the groundwater level and the measured values is −0.41 m, and the relative error is −4.36%. The recommended anti-seepage scheme can effectively solve the problem of large leakage in the reservoir area. The results show that the innovative approach is appropriate for the seepage analysis of the field with the fractured rock masses and more meaningful from an engineering point of view.

Method of solving optimal design problems based on flexible tolerance strategy

In the work on solving problems of optimal design, structures working in aggressive external environments, a modified method is proposed, which is based on the flexible tolerance method. The proposed method allows to control the accuracy of solving systems of differential equations when calculating the constraint functions of an optimisation problem. Based on the information about the degree of closeness of the point of the solution space to the local extremum point, which is received by the neural network controller, its parameters change. For this purpose, various matrices of neural network synapses, trained for different precisions of calculating the functions of constraints, are used. This strategy is used to modify the flexible tolerance method, based on the use of a neural network controller. As a criterion of the flexible tolerance, the error of calculating the constraint functions is used. It is shown that the use of a neural network regulator of the accuracy of calculating the restriction functions in the modified flexible tolerance method allows to significantly increase its efficiency while simultaneously obtaining a solution to the problem with a given accuracy and compensating the computational costs connected with using the α-level generalisation principle.

Nonlinear constrained optimization using the flexible tolerance method hybridized with different unconstrained methods

This paper proposes the use of the flexible tolerance method (FTM) modified with scaling of variables and hybridized with different unconstrained optimization methods to solve real constrained optimization problems. The benchmark problems used to analyze the performance of the methods were taken from G-Suite functions. The original method (FTM) and other four proposed methods: (i) FTM with scaling of variables (FTMS), (ii) FTMS hybridized with BFGS (FTMS-BFGS), (iii) FTMS hybridized with modified Powell's method (FTMS-Powell) and (iv) FTMS hybridized with PSO (FTMS-PSO), were implemented. The success rates of the methods were 80%, 100%, 75%, 95% and 85%, for FTM, FTMS, FTMS-BFGS, FTMS-Powell and FTMS-PSO, respectively. Numerical experiments including real constrained problems indicated that FTMS gave the best performance, followed by FTMS-Powell and FTMS-PSO. Despite the inferior performance compared to FTMS and FTMS-Powell, the FTMS-PSO method presented some advantages since good different initial points could be obtained, which allow exploring different routes through the solution space and to escape from local optima. The proposed methods proved to be an effective way of improving the performance of the original FTM.

An improved flexible tolerance method for solving nonlinear constrained optimization problems: Application in mass integration

This paper proposes the use of the flexible tolerance method (FTM) modified with adaptive Nelder–Mead parameters and barrier to solve constrained optimization problems. The problems used to analyze the performance of the methods were taken from G-Suite functions, and the methods with the best performance were applied in mass integration problems. Four methods were proposed: (1) flexible tolerance method (FTM) using adaptive parameters (FTMA), (2) flexible tolerance method with scaling (FTMS) and with adaptive parameters (FTMAS), (3) FTMS including the barrier modification (MFTMS) and (4) MFTMS hybridized with PSO (MFTMS-PSO). The success rates of these methods were 100% (MFTMS), 85% (MFTMS-PSO), 40% (FTMAS) and 30% (FTMA). Numerical experiments indicated that the MFTMS could efficiently and reliably improve the accuracy of global optima. In mass integration, the method was able, from current process situation, to reach the optimum process configuration that includes integration issues, which was not possible using FTM in its standard formulation. The hybridization of FTMS with PSO (without barrier), FTMS-PSO, was also able to solve mass integration problems efficiently.

Algorithm of solving corroding construction optimization problems based on flexible tolerance method

In current paper authors formulated a new problem of hinged-rod constructions optimal design, which considers physicochemical processes in construction elements that cause reduction of their bearing capacity and assumes that solution is obtained with a given accuracy. The search for an optimal solution is made on discrete non-metrical space of varied parameters. Actuality of the problem of corroding constructions optimal design is determined by the requirements for their high reliability and minimal consumption of materials. Utilization of existing algorithms for optimal design of such constructions allows to achieve required solution accuracy only with high computational cost . The paper describes creation of effective algorithm based on flexible tolerance method which allows to obtain solution with given accuracy and, therefore, to ensure required level of reliability of designed construction. Optimization algorithm uses neural network module of computational error control, which allows to change parameters of numerical solution of differential equation system modeling the influence of aggressive environment while solving the optimization problem. It allows to reduce computational cost at initial stages of search for the solution and to ensure required solution accuracy in the vicinity of the extremum. Analysis of results of numerical experiments allows to make a conclusion about high performance of optimization algorithm while ensuring given accuracy of problem solution. Utilization of developed algorithm will allow to solve the problems of corroding hinged-rod constructions optimal design.

A flexible tolerance genetic algorithm for optimal problems with nonlinear equality constraints

A hybrid method called a flexible tolerance genetic algorithm (FTGA) is proposed in this paper to solve nonlinear, multimodal and multi-constraint optimization problems. This method provides a new hybrid strategy that organically merges a flexible tolerance method (FTM) into an adaptive genetic algorithm (AGA). AGA is to generate an initial population and locate the “best” individual. FTM, serving as one of the AGA operators, exploits the promising neighborhood individual by a search mechanism and minimizes a constraint violation of an objective function by a flexible tolerance criterion for near-feasible points. To evaluate the efficiency of the hybrid method, we apply FTGA to optimize four complex functions subject to nonlinear inequality and/or equality constraints, and compare these results with the results supplied by AGA. Numerical experiments indicate that FTGA can efficiently and reliably achieve more accurate global optima of complex, nonlinear, high-dimension and multimodal optimization problems subject to nonlinear constraints. Finally, FTGA is successfully implemented for the optimization design of a crank-toggle mechanism, which demonstrates that FTGA is applicable to solve real-world problems.

THE FLEXIBLE TOLERANCE METHOD FOR ESTIMATING HYDROLOGIC PARAMETERS IN THE ROOT ZONE

: In this study, a constrained minimization method, the flexible tolerance method, was used to solve the optimization problems for determining hydrologic parameters in the root zone: water uptake rate, spatial root distribution, infiltration rate, and evaporation. Synthetic soil moisture data were first generated using the Richards' equation and its associated initial and boundary conditions, and these data were then used for the inverse analyses. The results of inverse simulation indicate the following. If the soil moisture data contain no noise, the rate of estimated water uptake and spatial root distribution parameters are equal to the true values without using constraints. If there is noise in the observed data, constraints must be used to improve the quality of the estimate results. In the estimation of rainfall infiltration and surface evaporation, interpolation methods should be used to reduce the number of unknowns. A fewer number of variables can improve the quality of inversely estimated parameters. Simultaneous estimation of spatial root distribution and water uptake rate or estimation of evaporation and water uptake rate is possible. The method was used to estimate the water uptake rate, spatial root distribution, infiltration rate, and evaporation using long-term soil moisture data collected from Nebraska's Sand Hills.

Parameter identification for improved viscoplastic model considering dynamic recrystallization

The successful application of viscoplastic model considering dynamic recrystallization depends on how well the parameters are identified. However, it is difficult to obtain satisfactory parameters using conventional parameter identification methods. The reasons are due to difficulties in obtaining homogeneous deformation, high complexity of physical process described by the model and large number of parameters. In this paper, the material parameters are identified by inverse analysis. Global information on objective function is firstly studied by an improved uniform random sampling method; secondly, a hybrid global optimization method, which combines the genetic algorithm, the Levenberg–Marquardt algorithm, the augmented Gauss–Newton algorithm and the flexible tolerance method, is constructed and an inverse analysis numerical procedure, which combines the proposed optimization method with the finite element analysis, is proposed; at last, a set of satisfactory material parameters for 26Cr2Ni4MoV is obtained by the proposed inverse analysis numerical procedure.

Smooth and time-optimal trajectory planning for industrial manipulators along specified paths

This article presents a method for determining smooth and time-optimal path constrained trajectories for robotic manipulators and investigates the performance of these trajectories both through simulations and experiments. The desired smoothness of the trajectory is imposed through limits on the torque rates. The third derivative of the path parameter with respect to time, the pseudo-jerk, is the controlled input. The limits on the actuator torques translate into state-dependent limits on the pseudo-acceleration. The time-optimal control objective is cast as an optimization problem by using cubic splines to parametrize the state space trajectory. The optimization problem is solved using the flexible tolerance method. The experimental results presented show that the planned smooth trajectories provide superior feasible time-optimal motion. © 2000 John Wiley & Sons, Inc.

A new upper-bound elemental technique approach

A new upper-bound elemental method is proposed to improve the ineffectiveness of the upper bound elemental technique (UBET) for solving forging problems that are geometrically complex or need a forming simulation for predicting the profile of the free boundary. This method combines the advantages of stream function and the finite element method (FEM). Specifically, the curve fitting property of FEM and the fluid incompressibility of the stream function. The formulated optimal design problems with constrained conditions are solved by the flexible tolerance method. Three forming problems, ring upsetting, closed-die, and backward-extrusion forging are used to illustrate this method: the results of ring upsetting show a good ability of simulating the forming profile of the free boundary, the closed-die forging results show lower upper-bound solutions than UBET and backward-extrusion forging shows the flexible curve fitting property for a complex geometric boundary.

Analysis of tube-spinning processes by the upper-bound stream-function method

Trapezoidal and spherical velocity fields using the stream function are suggested in tube spinning. In this paper the basic mechanisms of tube spinning are explained. A search is made for the optimum roller oblique angle in terms of process parameters, such as the feed rate, the ratio of the wall thickness reduction and the friction factor. The acquired total power consumption is optimized by the Flexible Tolerance Method which is used for minimizing the multi-variable non-linear function with constraint conditions. Thus the total power consumption required in deformation and the related tangential forces are obtained using the upper-bound stream-function method. The results show that the trapezoid velocity field using the stream function provides the best agreement with the results of model-material experiment.

The load analysis of the plane—strain forging processes using the upper-bound stream-function elemental technique

The upper-bound stream-function elemental technique (UBST) for the load analysis of the plane—strain forging processes is developed. This technique improves the ineffectiveness of UBET for solving forging problems that are geometrically complex or need a forming simulation for predicting the profile of the free boundary. This method combines the advantages of the stream function and the finite-element method (FEM); specifically, the curve-fitting property of the FEM and the fluid incompressibility of the stream-function. The formulated optimal design problems are solved by the flexible tolerance method. Two forming problems, strip upsetting and cosine-die extrusion, are used to illustrate the method, the reasonable results obtained showing that this technique can provide a powerful tool for industrial applications.

A new upper-bound elemental technique approach to axisymmetric metal forming processes

A new upper-bound elemental technique (UBET) is proposed to improve the ineffectiveness of UBET for solving forging problems that are geometrically complex or need a forming simulation for predicting the profile of free boundary. This method combines the advantages of the stream function and the finite element method (FEM); specifically, the curve fitting property of FEM and the fluid incompressibility of the stream function. The formulated optimal design problems with constrained conditions are solved by the flexible tolerance method. Three forming problems (ring upsetting, closed-die, and backward-extrusion forging) are used to illustrate this method: the results of ring upsetting show a good ability of simulating for predicting the forming profile of a free boundary; the closed-die forging produces a lower upper-bound solution than UBET; and backward-extrusion forging demonstrates the flexible curve fitting property for a complex geometric boundary.

Inversion of the Auroral Electrojets from Magnetometer Chain Data Based on the Flexible Tolerance Method.

An improved method is developed to invert a one-dimensional east-west electrojet from the H component of the geomagnetic disturbances measured along a meridian chain of magnetometer stations. This nonlinear inversion method is based on the flexible tolerance method. The main difference between this new method and the previous methods is that the height, the width and the location of the auroral electrojet current are determined together from the data by the flexible tolerance method, rather than assumed or calculated independently as in the previous methods. It should be noted also that the method presented in this paper is fast enough to allow a real-time display of the inverted electrojet current distribution by a workstation.

Application of Three Non-Linear Programming Techniques in Optimizing Machining Conditions

Mathematical models used for determining optimal machining conditions are non-linear functions subject to non-linear constraints. In this paper models for unit production cost and unit production time for multi-pass milling operations are developed, taking cutting speed, depth of cut, feed and number of passes as design variables. The problem is then solved by three non-linear programming methods, namely the generalized reduced gradient (GRG) method, the sequential unconstrained minimization technique (SUMT) and the flexible tolerance method. The efficiency of the methods in obtaining the solution is discussed. It is shown through computational experience that their usefulness differs considerably, some parameters in the computer codes used requiring careful selection for the models in question.

A MODIFIED FLEXIBLE TOLERANCE METHOD FOR NONLINEAR PROGRAMMING

A modified flexible tolerance method for nonlinear optimization is presented. The modification incorporates a random search technique with the flexible tolerance method, making the flexible tolerance method more efficient and effective. Test problems of nonlinear programming algorithms presented by previous investigators are included to illustrate the advantages of the modified method

Reactive power control for real power-loss minimization

An accurate and fast technique for real-time control of system voltage and reactive power of large-scale power systems is presented. The one-bus-away method has been modified to obtain accurate results with moderate computer requirements for online application to large-scale power networks. The approach was used to obtain a constrained minimum of the objective function for each subsystem in a deterministic number of steps, in most cases using the special geometric features of the problem. For the few cases where the optimal solution was not obtained from direct steps, good initial starting conditions for the iterative nonlinear optimal technique were obtained. The technique used was the flexible tolerance method, which works directly on objective functions and on inequality constraints without evaluating derivatives or gradients. It was selected because of its robustness, ease of implementation, and fast convergence properties for a wide range of nonlinear small-size problems. The required CPU time was small enough for significant online applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multistep methods with optimal stability regions

In order to optimize the stability regions we construct families of k multi-step methods with free parameters and of order k+1. We exhibit parameters for which the real interval of stability is arbitrarly large and using the Flexible Tolerance Method we show how these parameters optimize - under a predefined criterion - the stability region. Computational experience, while limited, has indicated that the procedure is efficient

Computer-aided optimization of saturated electromagnetic systems

A method for determining the optimum correlations between the basic geometric parameters of C-shaped electromagnetic system is described, when the saturation and the leakage can not be neglected. A non-linear programming problem is formulated on the basis of an equivalent magnetic circuit of the electromagnetic system. Flexible tolerance method program is used to find the optimum solution of the problem. As an illustration, the aplication of the method for optimum design of an electromagnet is shown.

A Weighted Flexible Tolerance Algorithm for Constrained Minimization

A direct search algorithm based upon the flexible tolerance method developed by Paviani and Himmelblau [1] for minimization of a functional subject to nonlinear equality and inequality constraints is proposed and evaluated. The modification incorporates a weighting on the direction of search in which the flexible polyhedron formed orients itself toward the gradient of the objective function (and clings to all active constraints). This is accomplished by weighting the centroid, in the sense of minimization, of the polyhedron with respect to the objective function. A number of problems have been solved satisfactorily by this algorithm, generally with more rapid convergence.