New protocols for solving geometric calculation problems incorporating dynamic geometry and computer algebra software

The ready availability today of both software for geometric construction, measurement and calculation, and software for numerical calculation and symbolic analysis, when taken together allow new approaches to the solution of geometric problems. These computer-aided graphical, numerical and algebraic methods of solution are illustrated and discussed by means of examples, using an appropriate choice of software tools, and provide an indication of the possible future when fully integrated software is available.     

Technique and software for implementing interactive intelligent modules in geometric modeling systems

We consider methods of constructing intelligent systems of geometric modeling and propose a technique and software for implementing these methods. The results obtained can be used in computer-aided design. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Modeling and homogenization of textile reinforced composites based on the isogeometric analysis

In this contribution, the isogeometric analysis is used to compute the effective material properties of textile reinforced composites. The isogeometric analysis based on non-uniform rational B-splines (NURBS) provides an efficient approach for numerical modeling because there is no need for a mesh generation. There are further advantages such as the availability of a geometry representation based on NURBS in computer-aided design software and the possibility to apply different refinement methods which do not change the geometry of the numerical model. These properties motivate the combination of the isogeometric analysis with the homogenization method. Therefor, the unit cell model representing the inner architecture of a textile reinforced composite is defined using NURBS. In order to compute the effective mechanical properties of the heterogeneous material, the homogenization method with periodic boundary conditions is applied. Finally, two examples demonstrate the advantages of this approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of non-linear thermal transient problems by a new adaptive time-step method in quenching process

Quenching process is a thermo-elastic-plasticity problem with a high material non-linearity. The numerical oscillation is likely caused in the simulation of quenching process. In order to avoid the numerical oscillation and improve the calculation accuracy of temperature and phase-transformation fields in the quenching process, a new self-adaptive time-step size method is presented. The method can adjust the time-step size according to the maximum and minimum differences of temperature fields between the previous simulation step and the current simulation step. FEM software for evaluating the temperature, stress/strain and phase-transformation is also developed. A cooling example with numerical analytical results and a quenching example with experiment results are used to verify the calculation accuracy of this software. Five methods including the method in this paper, two constant time-step sizes and two geometric proportion time-step sizes are applied to simulate the quenching process of a 40Cr steel cylinder, respectively. A comparison of the simulation results shows that, the method presented in this paper can effectively avoid the numerical oscillation, ensure the calculation accuracy and cost less calculation time.     

About Tracing Problems in Dynamic Geometry

Dynamic Geometry is the field of interactively performing geometric construction on a computer. In addition to simulating ruler-and-compass constructions we allow a drag mode. This drag mode allows to move geometric objects that have at least one degree of freedom. The remaining part of the construction should adjust automatically. Thus, during the motion, we have to trace the resulting paths of all geometric objects. This path tracking problem is known as the Tracing Problem from Dynamic Geometry. It combines the step-by-step procedure of doing geometric constructions with the continuous concept of motions. This study is based on the model for Dynamic Geometry used in the interactive geometry software Cinderella. We give a numerical solution to the Tracing Problem based on continuation methods and a reliable algorithm based on real and complex interval arithmetic. Degenerate situations like the intersection of two identical lines lead to critical points in the configuration space and are treated separately.     

Adaptive Numerical Treatment of Elliptic Systems on Manifolds

Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established in [55] are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented; more detailed examples using MC for this application may be found in [26].     

Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations

Periodic travelling waves (wavetrains) are an important solution type for many partial differential equations. In this paper I review the use of numerical continuation for studying these solutions. I discuss the calculation of the form and stability of a given periodic travelling wave, and the calculation of boundaries in a two-dimensional parameter plane for wave existence and stability. I also describe the automated implementation of these numerical continuation procedures via the software package wavetrain (http://www.ma.hw.ac.uk/wavetrain). I conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves.     

Fast and accurate evaluation of dual Bernstein polynomials

Numerical Algorithms - Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis, and computer-aided geometric design. In this...     

Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations

A variety of numerical methods are available for determining the stability of a given solution of a partial differential equation. However for a family of solutions, calculation of boundaries in parameter space between stable and unstable solutions remains a major challenge. This paper describes an algorithm for the calculation of such stability boundaries, for the case of periodic travelling wave solutions of spatially extended local dynamical systems. The algorithm is based on numerical continuation of the spectrum. It is implemented in a fully automated way by the software package wavetrain, and two examples of its use are presented. One example is the Klausmeier model for banded vegetation in semi-arid environments, for which the change in stability is of Eckhaus (sideband) type; the other is the two-component Oregonator model for the photosensitive Belousov–Zhabotinskii reaction, for which the change in stability is of Hopf type.     

Computer-aided design of synchronous generators with comb rotors

A mathematical model of a synchronous generator with comb rotor is implemented in a computer-aided design (CAD) package. The model integrates the differential equations characteristic of an electric machine; tabled data and empirical coefficients; and geometric, electromagnetic, and economic parameters of machine design. The complexity of the model requires that the solution procedure adopt a naive random search methodology, and an expert system is integrated within the package. The expert system is used mainly for guidance, and the priority of initial parameter values is delegated to the designer. An illustrative example is described to demonstrate the methodology of the computer-aided design procedure.     

Principles of verified numerical integration

We describe methods for the numerical calculation of integrals with verified error bounds. The problems range from integration over an interval to integration of parameter-dependent integrands over the whole d-variate space. It is argued, why we use bounds for the integrands in the complex plane as a tool for bounding the error in our own integration software.     

Exponentially fitted variants of Euler's method for ODEs

A new class of Euler's method for the numerical solution of ordinary differential equations is presented in this article. The methods are iterative in nature and admit their geometric derivation from an exponentially fitted osculating straight line. They are single-step methods and do not require evaluation of any derivatives. The accuracy and stability of the proposed methods are considered and their applicability to stiff problems is also discussed.     

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.     

A geometric approach for solving fuzzy linear programming problems

In this paper we first recall some definitions and results of fuzzy plane geometry, and then introduce some definitions in the geometry of two-dimensional fuzzy linear programming (FLP). After defining the optimal solution based on these definitions, we use the geometric approach for obtaining optimal solution(s) and show that the algebraic solutions obtained by Zimmermann method (ZM) and our geometric solutions are the same. Finally, numerical examples are solved by these two methods.     

Symmetric Projection Methods for Differential Equations on Manifolds

Projection methods are a standard approach for the numerical solution of differential equations on manifolds. It is known that geometric properties (such as symplecticity or reversibility) are usually destroyed by such a discretization, even when the basic method is symplectic or symmetric. In this article, we introduce a new kind of projection methods, which allows us to recover the time-reversibility, an important property for long-time integrations.     

Probabilistic modeling of computer system availability

System availability is becoming an increasingly important factor in evaluating the behavior of commercial computer systems. This is due to the increased dependence of enterprises on continuously operating computer systems and to the emphasis on fault-tolerant designs. Thus, we expect availability modeling to be of increasing interest to computer system analysts and for performance models and availability models to be used to evaluate combined performance/availability (performability) measures. Since commercial computer systems are repairable, availability measures are of greater interest than reliability measures. Reliability measures are typically used to evaluate nonrepairable systems such as occur in military and aerospace applications. We will discuss system aspects which should be represented in an availability model; however, our main focus is a state of the art summary of analytical and numerical methods used to solve computer system availability models. We will consider both transient and steady-state availability measures and for transient measures, both expected values and distributions. We are developing a program package for system availability modeling and intend to incorporate the best solution methods.     

Numerical study of learning algorithms on Stiefel manifold

Convex optimization methods are used for many machine learning models such as support vector machine. However, the requirement of a convex formulation can place limitations on machine learning models. In recent years, a number of machine learning methods not requiring convexity have emerged. In this paper, we study non-convex optimization problems on the Stiefel manifold in which the feasible set consists of a set of rectangular matrices with orthonormal column vectors. We present examples of non-convex optimization problems in machine learning and apply three nonlinear optimization methods for finding a local optimal solution; geometric gradient descent method, augmented Lagrangian method of multipliers, and alternating direction method of multipliers. Although the geometric gradient method is often used to solve non-convex optimization problems on the Stiefel manifold, we show that the alternating direction method of multipliers generally produces higher quality numerical solutions within a reasonable computation time.     

Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.     

用Excel软件实现最大似然估计数值计算的方法

本文给出用Excel软件实现最大似然估计数值计算求解的方法。内容包括对似然方程求数值解、直接对似然函数或对数似然函数求极大值以及分组数据最大似然估计的数值计算。