Some geometrical iteration methods for nonlinear equations

This paper describes geometrical essentials of some iteration methods （e.g. Newton iteration, secant line method, etc.） for solving nonlinear equations and advances some geometrical methods of iteration that are flexible and efficient.

Two-dimensional parasitic capacitance extraction for integrated circuit with dual discrete geometric methods

Capacitance extraction is one of the key issues in integrated circuits and also a typical electrostatic prob- lem. The dual discrete geometric method （DGM） is investigated to provide relative solutions in two-dimensional unstructured mesh space. The energy complementary characteristic and quick field energy computation thereof based on it are emphasized. Contrastive analysis between the dual finite element methods and the dual DGMs are presented both from theoretical derivation and through case studies. The DGM, taking the scalar potential as unknown on dual interlocked meshes, with simple form and good accuracy, is expected to be one of the mainstreaming methods in associated areas.

Computation of sensitivities of IC interconnect parasitic capacitances to the process variation with dual discrete geometric methods

Sensitivity analysis methods help to deal with the challenges of process variation in extraction of para- sitic capacitances in an integrated circuit. The dual discrete geometric methods （DGMs）, which have been recently utilized to extract parasitic capacitances, are reviewed. The computation method based on the dual DGMs for sen- sitivities of capacitances with respect to the given process parameters is presented. As the dual DGMs utilize scalar electric potential is unknown, the capacitances are obtained effectively, and then the sensitivities are calculated conveniently.

Geometric Multigrid Method for Isogeometric Analysis

The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.

TOPOLOGY SYNTHESIS OF GEOMETRICALLY NONLINEAR COMPLIANT MECHANISMS USING MESHLESS METHODS

This paper presents a new method for topology optimization of geometrical nonlinear compliant mechanisms using the element-free Galerkin method （EFGM）. The EFGM is employed as an alternative scheme to numerically solve the state equations by fully taking advantage of its capability in dealing with large displacement problems. In the meshless method, the imposition of essential boundary conditions is also addressed. The popularly studied solid isotropic material with the penalization （SIMP） scheme is used to represent the nonlinear dependence between material properties and regularized discrete densities. The output displacement is regarded as the objective function and the adjoint method is applied to finding the sensitivity of the design functions. As a result, the optimization of compliant mechanisms is mathematically established as a nonlinear programming problem, to which the method of moving asymptotes （MMA） belonging to the sequential convex programming can be applied. The availability of the present method is finally demonstrated with several widely investigated numerical examples.

METHOD FOR ADAPTIVE MESH GENERATION BASED ON GEOMETRICAL FEATURES OF 3D SOLID

In order to provide a guidance to specify the element size dynamically during adaptive finite element mesh generation, adaptive criteria are firstly defined according to the relationships between the geometrical features and the elements of 3D solid. Various modes based on different datum geometrical elements, such as vertex, curve, surface, and so on, are then designed for generating local refined mesh. With the guidance of the defmed criteria, different modes are automatically selected to apply on the appropriate datum objects to program the element size in the local special areas. As a result, the control information of element size is successfully programmed covering the entire domain based on the geometrical features of 3D solid. A new algorithm based on Delatmay triangulation is then developed for generating 3D adaptive finite element mesh, in which the element size is dynamically specified to catch the geometrical features and suitable tetrahedron facets are selected to locate interior nodes continuously. As a result, adaptive mesh with good-quality elements is generated. Examples show that the proposed method can be successfully applied to adaptive finite element mesh automatic generation based on the geometrical features of 3D solid.

DIFFERENTIAL GEOMETRICAL METHOD IN ELASTICCOMPOSITE WITH IMPERFECT INTERFACES

A differential geometrical method is for the first time used to calculate the effective moduli of a two-phase elastic composite materials with imperfect interface which the inclusions are assumed to be ellipsoidal of revolutions. All of the interface integral items participating in forming the potential and complementary energy functionals of the composite materials are expressed in terms of intrinsic quantities of the ellipsoidal of revolutions. Based on this, the upper and the lower bound for the effective elastic moduli of the composite materials with inclusions described above have been derived. Under three limiting conditions of sphere, disk and needle shaped inclusions, the results of this paper will return to the bounds obtained by Hashin([6]) (1992).

Characteristics of Multi-Objective Linear Programming Problem and Multi-Objective Linear Fractional Programming Problem Taking Maximum Value of Multi-Objective Functions

In this paper, a new statistical averaging technique is proposed for finding an optimal solution to a multi-objective linear fractional programming problem (MOLFPP) and multi-objective linear programming problem (MOLPP) by using new arithmetic averaging method and new geometric averaging method. It is significantly noticeable same characteristics among all the technique while taking maximum or minimum among all optimized values for multi-objective functions using simplex algorithm. The characteristics provided from the problems are verified by the numerical examples.

INFLUENCES OF EDGING ROLL SHAPE ON THE PLASTIC STRAIN DISTRIBUTION OF SLAB DURING MULTI-PASS V-H ROLLING PROCESS

Multi-pass slab vertical-horizontal （V-H） rolling process with variable edging roll shape have been simulated with explicit dynamic finite element method and updating geometric method. The distributions of plastic strain contour in slab daring rolling process with different edging roll and under different rolling stage have been obtained. The results show that there exist two thin strain assembling zones in slab when the flat edging roll is used, and there just exist one strain assembling zone in slab when the edging roll with groove is used. And compared the deformation equality between flat edging roll and edging roll with groove, the lateris better than the former, which supplies the theory prove to the slab deformation distribution during V-H rolling process and is helpful for predicting the slab texture.

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.

Global geometrical optics method for vector-valued Schrodinger problems

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.

A GEOMETRICAL INTERPRETATION OF THE WITHOUT-EXCEPTION FEASIBILITY OF PL HOMOTOPY METHODS

PL homotopy metheds are effective methods to locate zerces(or fixed points) of highly nonlinearmappirgs. Due to the Jexicographical system, the methods are feasible without exceptions Thispaper presents a geemetrical interpretation of the without-exception feasibility.

A Geometric Space-Time Multigrid Algorithm for the Heat Equation

We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint.The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation’s elliptic operator with a multiscale solution propagation in time.While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded,the holistic approach promises to exhibit a better parallel scalability than classical time stepping,adaptive dynamic refinement in space and time fall naturally into place,as well as the treatment of periodic boundary conditions of steady cycle systems,on-time computational steering is eased as the algorithm delivers guesses for the solution’s long-term behaviour immediately,and,finally,backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.

Geometric Schrdinger-Airy Flows on Khler Manifolds

We define a class of geometric flows on a complete Kahler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equa- tions, derivative nonlinear SchrSdinger equations etc. Furthermore, we consider the existence for these flows from S1 into a complete Kahler manifold and prove some local and global existence results.

The GI/M/1 Queue in a Multi-phase Service Environment with Working Vacations and Bernoulli Vacation Interruption

In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation of random length,causing the system to move to vacation phase 0.During phase 0,the server takes service for the customers at a lower rate rather than stopping completely.When a vacation ends,if the queue is non-empty,the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N.Moreover,we assume Bernoulli vacation interruption can happen.At a service completion instant,if there are customers in a working vacation period,vacation interruption happens with probability p,then the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N,or the server continues the vacation with probability 1−p.Using the matrix geometric solution method,we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs.The waiting time of an arbitrary customer is also derived.Finally,several numerical examples are presented.

Performance analysis of LTE-U coexistence network with WiFi using queueing model

The unforeseen mobile data explosion as well as the scarce of spectrum resource pose a major challenge to the performance of today＇s cellular networks which are in urgent need of novel solutions to handle such voluminous mobile data. Long term evolution-unlicensed （LTE-U）, which extends the LTE standard operating on the unlicensed band, has been proposed to improve system throughput. In LTE-U system, arriving users will contend the unlicensed spectrum resource with wireless fidelity （WiFi） users to transmit data information. Nevertheless, there is no clear consensus as to the benefits of transmission using unlicensed bands for LTE users. To this end, in this paper an analytical model is presented based on a queue system to understand the performance achieved by unlicensed based LTE system taking quality of services （QoS） and LTE-U users＇ behaviors into account. To obtain the stead-state solutions of the queue system, a matrix geometric method is used to solve it. Then, the average delay and utilization of unlicensed band for the LTE-U users is derived by using the queuing model. The performance of LTE-U coexistence is evaluated with WiFi using the proposed model and provide some initial insights as to the advantage of LTE-U in practice.