The Cumulative Method for Multiplication and Division

This paper provides a method of the process of computation called the cumulative method, it is based upon repeated cumulative process. The cumulative method is being adapted to the purposes of computation, particularly multiplication and division. The operations of multiplication and division are represented by algebraic formulas. An advantage of the method is that the cumulative process can be performed on decimal numbers. The present paper aims to establish a basic and useful formula valid for the two fundamental arithmetic operations of multiplication and division. The new cumulative method proved to be more flexible and made it possible to extend the multiplication and division based on repeated addition/subtraction to decimal numbers.

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method,particle methods,and other spatial methods on a single body sub-dividedintomultiple subdomains.This is in conjunctionwithimplementing thewell known Generalized Single Step Single Solve(GS4)family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework.In the current state of technology,the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as theNewmarkmethods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier.However,the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses,overcomes such barriers and allows a wide variety of arbitrary implicit-implicit,implicit-explicit,and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement,velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains.By selecting an appropriate spatialmethod and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body,the proposed work has the potential to better capture the physics of a given simulation.The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility.The accuracy and efficacy of the present work have not yet been seen in the current field,and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams

This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved.A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams.The proposed algorithm,named as two-parameter multiple shooting method,is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions.The impact of the axial force and additional point masses is also taken into account.Due to the fact that the method is based directly on the fourth-order ordinary differential equation,the structures do not have to be divided into many small elements to obtain an accurate enough solution,even though the geometry is very complex.To verify the proposed method,three different examples are considered,i.e.,a three-segment non-prismatic beam,a prismatic column subject to non-uniformly distributed compressive loads,and a two-segment beam with an additional point mass.Numerical analyses are carried out with the software MATHEMATICA.The results are compared with the solutions computed by the commercial finite element program SOFiSTiK.Good agreement is achieved,which confirms the correctness and high effectiveness of the formulated algorithm.

The improved artificial bee colony algorithm for mixed additive and multiplicative random error model and the bootstrap method for its precision estimation

To solve the complex weight matrix derivative problem when using the weighted least squares method to estimate the parameters of the mixed additive and multiplicative random error model(MAM error model),we use an improved artificial bee colony algorithm without derivative and the bootstrap method to estimate the parameters and evaluate the accuracy of MAM error model.The improved artificial bee colony algorithm can update individuals in multiple dimensions and improve the cooperation ability between individuals by constructing a new search equation based on the idea of quasi-affine transformation.The experimental results show that based on the weighted least squares criterion,the algorithm can get the results consistent with the weighted least squares method without multiple formula derivation.The parameter estimation and accuracy evaluation method based on the bootstrap method can get better parameter estimation and more reasonable accuracy information than existing methods,which provides a new idea for the theory of parameter estimation and accuracy evaluation of the MAM error model.

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.

THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs

In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.

Determination of the natural frequencies of axially moving beams by the method of multiple scales

The natural frequencies of an axially moving beam were determined by using the method of multiple scales. The method of second-order multiple scales could be directly applied to the governing equation if the axial motion of the beam is assumed to be small. It can be concluded that the natural frequencies affected by the axial motion are proportional to the square of the velocity of the axially moving beam. The results obtained by the perturbation method were compared with those given with a numerical method and the comparison shows the correctness of the multiple-scale method if the velocity is rather small.

Thermoelastic analysis of multiple defects with the extended finite element method

In this paper, the extended finite element method (XFEM) is adopted to analyze the interaction between a single macroscopic inclusion and a single macroscopic crack as well as that between multiple macroscopic or microscopic defects under thermal/mechanical load. The effects of different shapes of multiple inclusions on the material thermomechanical response are investigated, and the level set method is coupled with XFEM to analyze the interaction of multiple defects. Further, the discretized extended finite element approximations in relation to thermoelastic problems of multiple defects under displacement or temperature field are given. Also, the interfaces of cracks or materials are represented by level set functions, which allow the mesh assignment not to conform to crack or material interfaces. Moreover, stress intensity factors of cracks are obtained by the interaction integral method or the M-integral method, and the stress/strain/stiffness fields are simulated in the case of multiple cracks or multiple inclusions. Finally, some numerical examples are provided to demonstrate the accuracy of our proposed method.

Semi-analytical solution for internal forces of tunnel lining with multiple longitudinal cracks

Longitudinal cracks on the tunnel lining significantly influence the performance of tunnels in operation.In this study,we propose a semi-analytical method that provides a simple and effective way to calculate the internal forces of tunnel linings with multiple cracks.The semi-analytical solution is obtained using structural analysis considering the flexural rigidity for the cracked longitudinal section of the tunnel lining.Then the proposed solution is verified numerically.Using the proposed method,the influences of the crack depth and the number of cracks on the bending moment and modified crack tip stress are investigated.With the increase in crack depth,the bending moment of lining scetion adjacent to the crack decreases,while the bending moment of lining scetion far away from the crack increases slightly.The more the number of cracks in a tunnel lining,the easier the new cracks initiated.

Multiple Suppression Using T-A Dual Polynomial Fitting Method

Principles of polynomial fitting zero offset profile are introduced, and a new polynomial fitting method, tbe time-amplitude dual fitting method, is developed. The method can be used to purify seismic waves and suppress multiples. The effect of suppressing multiples is compared with other multiple suppression methods.

Reliability analysis for aeroengine turbine disc fatigue life with multiple random variables based on distributed collaborative response surface method

The fatigue life of aeroengine turbine disc presents great dispersion due to the randomness of the basic variables,such as applied load,working temperature,geometrical dimensions and material properties.In order to ameliorate reliability analysis efficiency without loss of reliability,the distributed collaborative response surface method(DCRSM) was proposed,and its basic theories were established in this work.Considering the failure dependency among the failure modes,the distributed response surface was constructed to establish the relationship between the failure mode and the relevant random variables.Then,the failure modes were considered as the random variables of system response to obtain the distributed collaborative response surface model based on structure failure criterion.Finally,the given turbine disc structure was employed to illustrate the feasibility and validity of the presented method.Through the comparison of DCRSM,Monte Carlo method(MCM) and the traditional response surface method(RSM),the results show that the computational precision for DCRSM is more consistent with MCM than RSM,while DCRSM needs far less computing time than MCM and RSM under the same simulation conditions.Thus,DCRSM is demonstrated to be a feasible and valid approach for improving the computational efficiency of reliability analysis for aeroengine turbine disc fatigue life with multiple random variables,and has great potential value for the complicated mechanical structure with multi-component and multi-failure mode.

BLOCK BIDIAGONALIZATION METHODS FOR MULTIPLE NONSYMMETRIC LINEAR SYSTEMS

The symmetric linear system gives us many simplifications and a possibility to adapt the computations to the computer at hand in order to achieve better performance. The aim of this paper is to consider the block bidiagonalization methods derived from a symmetric augmented multiple linear systems and make a comparison with the block GMRES and block biconjugate gradient methods.

ON DECOMPOSITION METHOD FOR ACOUSTIC WAVE SCATTERING BY MULTIPLE OBSTACLES

Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared with the standard scattering problem for one obstacle, the difficulty of such a new problem is the interaction of scattered wave by different obstacles. A decomposition method for solving this multiple scattering problem is developed. Using the boundary integral equation method, we decompose the total scattered field into a sum of contributions by separated obstacles. Each contribution corresponds to scattering problem of single obstacle. However, all the single scattering problems are coupled via the boundary conditions, representing the physical interaction of scattered wave by different obstacles. We prove the feasibility of such a decomposition. To compute these contributions efficiently, an iteration algorithm of Jacobi type is proposed, decoupling the interaction of scattered wave from the numerical points of view. Under the well-separation assumptions on multiple obstacles, we prove the convergence of iteration sequence generated by the Jacobi algorithm, and give the error estimate between exact scattered wave and the iteration solution in terms of the obstacle size and the minimal distance of multiple obstacles. Such a quantitative description reveals the essences of wave scattering by multiple obstacles. Numerical examples showing the accuracy and convergence of our method are presented.

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multi- relaxation-time （MRT） lattice Boltzmann flux solver （LBFS） is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations （LBEs）. The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

Fail-Safe Topology Optimization of Continuum Structures with Multiple Constraints Based on ICM Method

Traditional topology optimization methods may lead to a great reduction in the redundancy of the optimized structure due to unexpected material removal at the critical components.The local failure in critical components can instantly cause the overall failure in the structure.More and more scholars have taken the fail-safe design into consideration when conducting topology optimization.A lot of good designs have been obtained in their research,though limited regarding minimizing structural compliance(maximizing stiffness)with given amount of material.In terms of practical engineering applications considering fail-safe design,it is more meaningful to seek for the lightweight structure with enough stiffness to resist various component failures and/or to meet multiple design requirements,than the stiffest structure only.Thus,this paper presents a fail-safe topology optimization model for minimizing structural weight with respect to stress and displacement constraints.The optimization problem is solved by utilizing the independent continuous mapping(ICM)method combined with the dual sequence quadratic programming(DSQP)algorithm.Special treatments are applied to the constraints,including converting local stress constraints into a global structural strain energy constraint and expressing the displacement constraint explicitly with approximations.All of the constraints are nondimensionalized to avoid numerical instability caused by great differences in constraint magnitudes.The optimized results exhibit more complex topological configurations and higher redundancy to resist local failures than the traditional optimization designs.This paper also shows how to find the worst failure region,which can be a good reference for designers in engineering.

Maximizing deviation method for multiple attribute decision making under q-rung orthopair fuzzy environment

Because of the uncertainty and subjectivity of decision makers in the complex decision-making environment,the evaluation information of alternatives given by decision makers is often fuzzy and uncertain.As a generalization of intuitionistic fuzzy set(IFSs)and Pythagoras fuzzy set(PFSs),q-rung orthopair fuzzy set(q-ROFS)is more suitable for expressing fuzzy and uncertain information.But,in actual multiple attribute decision making(MADM)problems,the weights of DMs and attributes are always completely unknown or partly known,to date,the maximizing deviation method is a good tool to deal with such issues.Thus,combine the q-ROFS and conventional maximizing deviation method,we will study the maximizing deviation method under q-ROFSs and q-RIVOFSs in this paper.Firstly,we briefly introduce the basic concept of q-rung orthopair fuzzy sets(q-ROFSs)and q-rung interval-valued orthopair fuzzy sets(q-RIVOFSs).Then,combine the maximizing deviation method with q-rung orthopair fuzzy information,we establish two new decision making models.On this basis,the proposed models are applied to MADM problems with q-rung orthopair fuzzy information.Compared with existing methods,the effectiveness and superiority of the new model are analyzed.This method can effectively solve the MADM problem whose decision information is represented by q-rung orthopair fuzzy numbers(q-ROFNs)and whose attributes are incomplete.

Description of Mechanism Function about Dehydration of Cobalt Oxalate Dihydrate by Multiple Rates Isotemperature Method

A new method of the multiple rates isotemperature is proposed to define the most probable mechanismg(α) of thermal anlaysis; the iterative isoconversional procedure has been employed to estimate apparent activation energyE; the pre-exponential factorA is obtained on the basis ofE andg(α). By this new method, the thermal analysis kinetics triplet of dehydration of cobalt oxalate dihydrate is determined, apparent activation energyE is 99.84 kJ·mol?1; pre-exponential factorA is 3.427×109–3.872×109 s?1 and the most probable mechanism belongs to nucleation and growth,A m model, the range ofm is from 1.50 to 1.70. Key words multiple rates isotemperature method - isoconversional method - cobalt oxalate dihydrate - accomodation function - differential scanning calorimetry (DSC) CLC number O 636.1 Foundation item: Supported by the Key Foundation of the Science and Technology Committee of Hubei Province (2001ABA009)Biography: Li Li-qing (1977-), female, Master candidate, research direction: material synthesize and thermal analysis kinetics.

A Multiple Nested Lattice Boltzmann Method and Its Application

The standard lattice Boltzmann method utilizes uniform grids to maintain a compact computational procedure. However, it is often less efficient to perform hydrodynamic and aerodynamic flow simulations when there is a need for high resolution. To resolve these difficulties, a multiple nested lattice Boltzmann method(MNLBM) was developed, which contains several overlapped layers with different resolutions in the computational domain. The data transference of flow field on two layers is accomplished by a Filippova procedure which is proved to satisfy the continuity of mass, momentum, and stresses across the interface. The proposed method is based on the standard lattice Boltzmann method, so it is easily performed.By numerical investigation, the result of present method has been agreed with that of literature, but the computation efficiency is higher than the standard lattice Boltzmann method.

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.

MULTIPLE RECIPROCITY METHOD WITH TWO SERIES OF SEQUENCES OF HIGH-ORDER FUNDAMENTAL SOLUTION FOR THIN PLATE BENDING

The boundary value problem of plate bending problem on two_parameter foundation was discussed.Using two series of the high_order fundamental solution sequences, namely, the fundamental solution sequences for the multi_harmonic operator and Laplace operator, applying the multiple reciprocity method(MRM), the MRM boundary integral equation for plate bending problem was constructed. It proves that the boundary integral equation derived from MRM is essentially identical to the conventional boundary integral equation. Hence the convergence analysis of MRM for plate bending problem can be obtained by the error estimation for the conventional boundary integral equation. In addition, this method can extend to the case of more series of the high_order fundamental solution sequences.