Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

Fuzzy Inventory Model under Selling Price Dependent Demand and Variable Deterioration with Fully Backlogged Shortages

The objective is to develop a model considering demand dependent on selling price and deterioration occurs after a certain period of time, which follows two-parameter Weibull distribution. Shortages are allowed and fully backlogged. Fuzzy optimal solution is obtained by considering hexagonal fuzzy numbers and for defuzzification Graded Mean Integration Representation Method. A numerical example is provided for the illustration of crisp and fuzzy, both models. To observe the effect of changes in parameters, sensitivity analysis is carried out.

Measurement of mouse liver glutathione S-transferase activity by the integrated method

Objective: The integrated method was investigated to measure Vm/Km of mouse liver glutathione S-transfer-ase (GST) activity on GSH and 7-Cl-4-nitrobenzofurazozan. Methods: Presetting concentration of one substrate twenty-fold above the other's and taking maximum product absorbance Am as parameter while Km as constant, Vm/Km was obtained by nonlinear fitting of GST reaction curve to the integrated Michaelis-Menten equation In [Am/(Am -Ai)] + Ai/ ( ξ× Km ) = ( Vm/Km )×ti (1). Results: Vm/Km for GST showed slight dependence on initial substrate concentration and data range, but it was resistant to background absorbance, error in reaction origin and small deviation in presetting Km. Vm/Km was proportional to the amount of GST with upper limit higher than that by initial rate. There was close correlation between Vm/Km and initial rate of the same GST. Consistent results were obtained by this integrated method and classical initial rate method for the measurement of mouse liver GST. Conclusion: With the concentration of one substrate twenty-fold above the other's, this integrated method was reliable to measure the activity of enzyme on two substrates , and substrate concentration of the lower one close to its apparent Km was able to be used.

Evaluation of integration methods for hybrid simulation of complex structural systems through collapse

This study examines the performance of integration methods for hybrid simulation of large and complex structural systems in the context of structural collapse due to seismic excitations. The target application is not necessarily for real-time testing, but rather for models that involve large-scale physical sub-structures and highly nonlinear numerical models. Four case studies are presented and discussed. In the first case study, the accuracy of integration schemes including two widely used methods, namely, modified version of the implicit Newmark with fixed-number of iteration （iterative） and the operator-splitting （non-iterative） is examined through pure numerical simulations. The second case study presents the results of 10 hybrid simulations repeated with the two aforementioned integration methods considering various time steps and fixed-number of iterations for the iterative integration method. The physical sub-structure in these tests consists of a single-degree-of-freedom （SDOF） cantilever column with replaceable steel coupons that provides repeatable highly- nonlinear behavior including fracture-type strength and stiffness degradations. In case study three, the implicit Newmark with fixed-number of iterations is applied for hybrid simulations of a 1：2 scale steel moment frame that includes a relatively complex nonlinear numerical substructure. Lastly, a more complex numerical substructure is considered by constructing a nonlinear computational model of a moment frame coupled to a hybrid model ofa 1：2 scale steel gravity frame. The last two case studies are conducted on the same porotype structure and the selection of time steps and fixed number of iterations are closely examined in pre-test simulations. The generated unbalance forces is used as an index to track the equilibrium error and predict the accuracy and stability of the simulations.

Analytical mechanics methods for solving Whittaker equations

The purpose of this paper is to study the solution of the celebrated Whittaker equations by using analytical mechanics methods, including the Lagrange-Noether method, Hamilton-Poisson method and potential integral method.

Comparison of capability of time integration methods in capturing dynamic loading

Numerical properties of the time integration method proposed by the first author of this paper in 2007 are the same as those of the constant average acceleration method （AAM） for linear elastic systems, except that the capability to capture dynamic loading was not explored. It was found that there were different quadrature equations to predict the next step displacement increment. A modified quadrature equation of this method was derived so that the equation to determine the next step displacement was numerically equivalent to the equation used in the constant AAM. It was verified that the original form of this method, in general, had a better capability to capture dynamic loadings than the constant AAM. This excellent property, in addition to computational efficiency, will help to make this method competitive with general secondorder accurate integration methods.

Benefit allocation model of distributed photovoltaic power generation vehicle shed and energy storage charging pile based on integrated weighting-Shapley method

In this study,to develop a benefit-allocation model,in-depth analysis of a distributed photovoltaic-powergeneration carport and energy-storage charging-pile project was performed;the model was developed using Shapley integrated-empowerment benefit-distribution method.First,through literature survey and expert interview to identify the risk factors at various stages of the project,a dynamic risk-factor indicator system is developed.Second,to obtain a more meaningful risk-calculation result,the subjective and objective weights are combined,the weights of the risk factors at each stage are determined by the expert scoring method and entropy weight method,and the interest distribution model based on multi-dimensional risk factors is established.Finally,an example is used to verify the rationality of the method for the benefit distribution of the charging-pile project.The results of the example indicate that the limitations of the Shapley method can be reasonably avoided,and the applicability of the model for the benefit distribution of the charging-pile project is verified.

Precise integration methods based on the Chebyshev polynomial of the first kind

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.

A Survey of Mathematical Tools in Topology and Performance Integrated Modeling and Design of Robotic Mechanism

Topology and performance are the two main topics dealt in the development of robotic mechanisms.However,it is still a challenge to connect them by integrating the modeling and design process of both parts in a unified frame.As the properties associated with topology and performance,finite motion and instantaneous motion of the robot play key roles in the procedure.On the purpose of providing a fundamental preparation for integrated modeling and design,this paper carries out a review on the existing unified mathematic frameworks for motion description and computation,involving matrix Lie group and Lie algebra,dual quaternion and pure dual quaternion,finite screw and instantaneous screw.Besides the application in robotics,the review of the work from these mathematicians concentrates on the description,composition and intersection operations of the finite and instantaneous motions,especially on the exponential-differential maps which connect the two sides.Furthermore,an in-depth discussion is worked out by investigating the algebraical relationship among these methods and their further progress in integrated robotic development.The presented review offers insightful investigation to the motion description and computation,and therefore would help designers to choose appropriate mathematical tool in the integrated design and modeling and design of mechanisms and robots.

Resolving Domain Integral Issues in Isogeometric Boundary Element Methods via Radial Integration:A Study of Thermoelastic Analysis

The paper applied the isogeometric boundary element method(IGABEM)to thermoelastic problems.The Non-Uniform Rational B-splines(NURBS)used to construct geometric models are employed to discretize the boundary integral formulation of the governing equation.Due to the existence of thermal stress,the domain integral term appears in the boundary integral equation.We resolve this problem by incorporating radial integration method into IGABEM which converts the domain integral to the boundary integral.In this way,IGABEM can maintain its advantages in dimensionality reduction and more importantly,seamless integration of CAD and numerical analysis based on boundary representation.The algorithm is verified by numerical examples.

Compact implicit integration factor methods for some complex-valued nonlinear equations

The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.

The efficiency of direct integration methods in elastic contact-impact problems

A comparison of direct integration methods is madeand their efficiency is investigated for impact problems.New-mark,Wilson-θ,Central Difference and Houbolt Methodsare used as direct integration methods.Impact analysisincludes that of elastic and large deformation based uponupdated Lagrangian including buckling check.The resultsshow that the direct integration methods give differentresults in different contact-impact cases.

Integrating object-oriented methods and formal methods for requirement engineering

High quality software requirement specification is crucial for a software development. Although much efforts and research works have been done to address the problem, the errors in user requirement are still prevent us from developing high quality software. To address the problem, this paper proposes integrating graphical specification technique UML with formal specification technique to construct user requirement specification. We also present a prototype tool to perform the automatic translation from UML specification into Object-Z specification.

YIG Thin Film for RF Integrated Inductor

The yttrium iron garnet（YIG） thin films prepared by the sol-gel method and rapid thermal annealing（RTA） process for integrated inductor are investigated. The X-ray diffraction（XRD） results indicate that the YIG film annealed above 650 ℃ is poly-crystalline with single-phase garnet structure. Moreover, it can be found that the initial permeability μi, saturation magnetization MS and coercivity Hc of these YIG films increase with increasing RTA temperature. Low temperature annealing after crystallization can further improve the magnetic properties of YIG film. Thereby, a planar integrated inductor in the presence of Si substrate/SiO2 layer/Y2.8Bi0.2Fe5O12 thin film/Cu spiral coil structure is fabricated successfully by the standard IC processes. Due to the magnetic enhancement originated from YIG film, the inductance L and quality factor Q of the inductor with YIG film are improved in a certain frequency range.

LOCAL SALINE APPROXIMATION METHODS FOR SINGULAR PRODUCT INTEGRATION

The purpose of this paper is to propose and study local spline approximation methods for singular product integration,for which;i)the precision degree is the highest possible using splint approximation; ii) the nodes fan be assumed equal to arbitrary points,where the integrand function f is known; iii) the number of the requested evaluations of f at the nodes is low,iv) a satisfactory convergence theory can be proved.

Correlation of serum arylesterase activity on phenylacetate estimated by the integrated method to common classical biochemical indexes of liver damage

The correlation of serum arylesterase(PON1) activity on phenylacetate determined by an integrated method to clas-sical biochemical indexes of liver damage was investigated for the use of PON1 activity to evaluate liver damage.PON1 reaction curve as absorbance at 270 nm for 0.20 mmol/L phenylacetate hydrolysis was analyzed by the integrated method to determine maximal PON1 reaction rate.Classical biochemical indexes of liver damage were determined routinely.The 95% confidence threshold of PON1 activity in sera from healthy individuals was 2.12 mkat/L [(4.73±1.31) mkat/L,n=105].PON1 activity in clinical sera was closely correlated to serum albumin,total protein and the ratio of albumin to globulins,but was weakly correlated to both direct and total bilirubin in serum.There were no correlations of PON1 activity to γ-glutamyltransferase,alkaline phos-phatase,alanine aminotransferase and aspartate aminotransferase.Among 127 clinical sera with PON1 activity>2.12 mkat/L,there were 92% healthy individuals examined by albumin,90% healthy individuals examined by total protein,88% healthy individuals examined by total bilirubin,86% healthy individuals examined by direct bilirubin and 64% healthy individuals examined by the ratio of albumin to globulins,respectively.In each group of healthy individuals judged by classical biochemical indexes of close correlation to PON1 activity,percentage of healthy individuals examined by PON1 activity was always >80%.These results suggested PON1 activity on phenylacetate estimated by the integrated method was also suitable for the evaluation of liver damage.

An integrated method for matching forest machinery and a weight-value adjustment

Proper matching of forestry machinery is important when raising mechanization levels for forestry production. In the matching process, forestry machinery needs not only expertise, but also improved methods for solving problems. I propose combination of case-based reasoning （CBR） and rule-based reasoning （RBR） by calculating the similarity of quantitative parameters of various forestry machines in an analytical and hierarchical process. I calculated the similarity of machin-ery used in forest industries to enable better selection and matching of equipment. I propose a weight-value adjusting method based on sums of squares of deviations in which the individual parameter weights were modified in the process of application. During the process of system design, I put forward a design method knowledge base and generated a dynamic web reasoning framework to integrate the processes of forest industry machinery selection and weight-value adjustment. This enables expansion of the scope of the complete system and enhancement of the reasoning efficiency. I demonstrate the validity and practicability of this method using a practical example.

Integrated Evaluation and Its Application on Decision-Making of Comprehensive Transportation Projects

The comprehensive transportation project involves with many different transportation methods, such as railway, highway, waterway, airline, and pipeline, which have their special characteristics. To evaluate the transporta- tion project with different transportation methods under the same system, the integrated evaluation method should be built up on the basis of a common index system. This paper suggests a feasible integrated evaluation method including index system establishing, indices transacting, the index weight distributing, and weighted value integrating.

Progress on Therapy of Infertility by Integrated Traditional Chinese and Western Medicine

Infertility is one of the difficult complicated diseases. Many couples suffer from it. The pathogenesis is very complicated. The imbalance or lesson of any link of the reproductive system can cause infertility. This paper summarizes the treatment of female infertility by integrated Traditional Chinese Medicine (TCM) and Western Medicine (WM) which can not only improve the ovulation rate and pregnancy rate, but also decrease the complications. The effects are better than that by TCM or WM only. Therefore, the coupling method is worth to be used widely in clinical practice.