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.

Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses:discrete and continuum models

The main objective of this paper is to investigate the influence of inertia of nonlinear springs on the dispersion behavior of discrete monoatomic chains with lumped and distributed masses.The developed model can represent the wave propagation problem in a non-homogeneous material consisting of heavy inclusions embedded in a matrix.The inclusions are idealized by lumped masses,and the matrix between adjacent inclusions is modeled by a nonlinear spring with distributed masses.Additionally,the model is capable of depicting the wave propagation in bi-material bars,wherein the first material is represented by a rigid particle and the second one is represented by a nonlinear spring with distributed masses.The discrete model of the nonlinear monoatomic chain with lumped and distributed masses is first considered,and a closed-form expression of the dispersion relation is obtained by the second-order Lindstedt-Poincare method(LPM).Next,a continuum model for the nonlinear monoatomic chain is derived directly from its discrete lattice model by a suitable continualization technique.The subsequent use of the second-order method of multiple scales(MMS)facilitates the derivation of the corresponding nonlinear dispersion relation in a closed form.The novelties of the present study consist of(i)considering the inertia of nonlinear springs on the dispersion behavior of the discrete mass-spring chains;(ii)developing the second-order LPM for the wave propagation in the discrete chains;and(iii)deriving a continuum model for the nonlinear monoatomic chains with lumped and distributed masses.Finally,a parametric study is conducted to examine the effects of the design parameters and the distributed spring mass on the nonlinear dispersion relations and phase velocities obtained from both the discrete and continuum models.These parameters include the ratio of the spring mass to the lumped mass,the nonlinear stiffness coefficient of the spring,and the wave amplitude.

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.

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.

Protein Flexibility and Multiple Docking in Ligand Docking and Virtual Screening to the BRAF(TypeⅠ1/2)Inhibitors

BRAF has been recognized as a promising target for cancer therapy. A number of crystal structures have been published. Molecular docking is one of the most effective techniques in the field of computer-aided drug design（CADD）. Appropriate protein conformation and docking method are essential for the successful virtual screening experiments. One approach considering protein flexibility and multiple docking methods was proposed in this study. Six DFG-in/αC-helix-out crystal structures of BRAF, three docking programs（Glide, GOLD and Ligand Fit） and 12 scoring functions were applied for the best combination by judging from the results of pose prediction and retrospective virtual screening（VS）. The most accurate results（mean RMSD of about 0.6 A） of pose prediction were obtained with two complex structures（PDB： 3 C4 C and 3 SKC） using Glide SP. From the retrospective VS, the most active compounds were identified by using the complex structure of 3 SKC, indicated by a ROC/AUC score of 0.998 and an EF of 20.6 at 5% of the database screen with Glide-SP. On the whole, PDB 3 SKC could achieve a higher rate of correct reproduction, a better enrichment and more diverse compounds. A comparison of 3 SKC and the other X-ray crystal structures led to a rationale for the docking results. PDB 3 SKC could achieve a broad range of sulfonamide substitutions through an expanded hydrophobic pocket formed by a further shift of the αC-helix. Our study emphasized the necessity and significance of protein flexibility and scoring functions in both ligand docking and virtual screening.

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.

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

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.

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.

New Refinement Relations of Z Specifications for Multiple Viewpoints Oriented Requirements Method

In this paper we develop several new refinement relations of Z for multiple viewpoints oriented requirements method (MVORM). The original motivation is that we found the standard Z refinement relation is not adequate or correct when considering specifications that have temporal relationships of operations. The concept of temporal state variables is introduced into Z. Then new implementation relations are defined and new refinement relations are deduced, mainly for temporal state variables to process temporal relationships of operations. We use state transition systems to abstract the temporal state transitions. A simple example is used to show the procedures of MVORM. Finally some directions of further work are forwarded.

THE METHOD OF MULTIPLE SCALES APPLIED TO THE NONLINEAR STABILITY PROBLEM OF A TRUNCATED SHALLOW SPHERICAL SHELL OF VARIABLE THICKNESS WITH THE LARGE GEOMETRICAL PARAMETER

Using the modified method of multiple scales, the nonlinear stability of a truncated shallow spherical shell of variable thickness with a nondeformable rigid body at the center under compound loads is investigated. When the geometrical parameter k is larger, the uniformly valid asymptotic solutions of this problem are obtained and the remainder terms are estimated.

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

In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.

MULTIPLE-SCATTERING STUDIES OF ADSORPTION STRUCTURE OF C_2D_2/Si(111)7×7

The multiple scattering cluster (MSC) method has been employed to perform a theoretical analysis on carbon is near edge X-ray absorption fine structure of the deuteron acetylene (C2 D2) adsorbed on Si(111)7× 7 at room temperature. From the MSC study. it is confirmed that the (22D2 molecule is bonded to a pair of adjacent Si adatom and Si restatom with C-Si bond length about 0.18nm. The carbon-deuteron bond is bent away front the surface and the CCD bond angle is about 120°. The molecule plane tilt slightly away from the surface normal. Compared with C2D2 in gas phase, the C-C bond and C-D bond are elongated by about 0.03nm and 0.02nm respectively when acetylene was adsorbed on the subtrate. Keyowrds: adsorption of deuteron acetylene on Si(111)7×7. near edge X- ray absorption fine structure. multiple scattering cluster method

Two Generic Frameworks of Multiple Viewpoints Oriented Requirements Method and Their Comparison

Traditional requirements method has some problems when it is used for large distributed systems. Multiple viewpoints oriented requirements method (MVORM) is a new method for resolving these problems. This paper develops two generic formal frameworks of MVORM, framework based on refinement relation (FBRR) and framework based on implementation relation (FBIR). They are generic, because no assumptions are made about the development process or the formal description languages to be used. Three kinds of specification relations and three kinds of specification transformations are discussed over FBIR and FBRR. This paper also compares the equivalence between FBIR and FBRR. We point out that an equivalent FBIR can be found for any FBRR, but reverse transformation is not always possible. We think FBIR is better than FBRR on most cases.

COMPUTER COMPUTATION OF THE METHOD OF MULTIPLE SCALES-DIRICHLET PROBLEM FOR A CLASS OF SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .

Multiple solutions and stability of the steady transonic small-disturbance equation

Numerical solutions of the steady transonic small-disturbance（TSD） potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.

Second-order sensitivity of eigenpairs in multiple parameter structures

This paper presents methods for computing a second-order sensitivity matrix and the Hessian matrix of eigenvalues and eigenvectors of multiple parameter structures. Second-order perturbations of eigenvalues and eigenvectors are transformed into multiple parameter forms,and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed.With these formulations,the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate changes of eigenvalues and eigenvectors when the design parameters are changed. The presented method avoids direct differential operation,and thus reduces difficulty for computing the second-order sensitivity matrices of eigenpairs.A numerical example is given to demonstrate application and accuracy of the proposed method.

Multiple mobile-obstacle avoidance algorithm for redundant manipulator

In order to overcome the shortcomings of the previous obstacle avoidance algorithms,an obstacle avoidance algorithm applicable to multiple mobile obstacles was proposed.The minimum prediction distance between obstacles and a manipulator was obtained according to the states of obstacles and transformed to escape velocity of the corresponding link of the manipulator.The escape velocity was introduced to the gradient projection method to obtain the joint velocity of the manipulator so as to complete the obstacle avoidance trajectory planning.A7-DOF manipulator was used in the simulation,and the results verified the effectiveness of the algorithm.

Multiple scales method for analyzing a forced damped rotational pendulum oscillatorwithgallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces.The multiple scales method(MSM)is applied to solve the proposed problem.Several types of rotational pendulum oscillators are studied and talked about in detail.These include the forced damped rotating pendulum oscillator with gallows,the damped standard simple pendulum oscillator,and the damped rotating pendulum oscillator without gallows.The MSM first-order approximations for all the cases mentioned are derived in detail.The obtained results are illustrated with concrete numerical examples.The first-order MSM approximations are compared to the fourth-order Runge-Kutta(RK4)numerical approximations.Additionally,the maximum error is estimated for the first-order approximations obtained through the MSM,compared to the numerical approximations obtained by the RK4 method.Furthermore,we conducted a comparative analysis of the outcomes obtained by the used method(MSM)and He-MSM to ascertain their respective levels of precision.The proposed method can be applied to analyze many strong nonlinear oscillatory equations.