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Analysis of numerical methods for the simulation of deformable models

VISUAL COMPUTER 2003年第7-8期19卷 581-600页

作者： Hauth, M Etzmuss, O Strasser, W Univ Tubingen WSI GRIS Tubingen Germany

Simulating deformable objects based on physical laws has become the most popular technique for modeling textiles, skin, or volumetric soft objects like human tissue. The physical model leads to an ordinary differential equation. Recently, several approaches to fast algorithms have been proposed. In this work, more profound numerical background about numerical stiffness is provided. Stiff equations impose stability restrictions on a numerical integrator. Some one-step and multistep methods with adequate stability properties are presented. For an efficient implementation, the inexact Newton method is discussed. Applications to 2D and 3D elasticity problems show that the discussed methods are faster and give higher-quality solutions than the commonly used linearized Euler method.

关键词： deformable objects cloth modeling numerical integration methods differential equations

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numerical integration using local Taylor expansions in nodes

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期192卷 332-336页

作者： Hashemiparast, S. M. KN Toosi Univ Technol Dept Math Tehran Iran

In this paper we introduce a new formula for approximating the integration based on locally Taylor expansions in each one of the nodes and reduce the results in respect of difference equation. Finally some examples are given to illustrate the application of the formulae for comparison the results with other related formulae. (C) 2007 Elsevier Inc. All rights reserved.

关键词： equal coefficient quadrature rules numerical integration methods precision the method of undetermined coefficient the method of solving non-linear system

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Modular simulation in multibody system dynamics

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MULTIBODY SYSTEM DYNAMICS 2000年第2-3期4卷 107-127页

作者： Kübler, R Schiehlen, W Univ Stuttgart Inst Mech B D-70550 Stuttgart Germany

The dynamic analysis of complex engineering systems like automobiles is often relieved by a modular approach to make it treatable by a team of engineers. The modular decomposition is based on engineering intuition of corresponding engineering disciplines. In this paper, a modular formulation of multibody systems is proposed which is based on the block representation of a multibody system with corresponding input and output quantities. Advantages of this modular approach range from independent and parallel modeling of subsystems over the easy exchange of the resulting modules to the use of different software for each module. However, the modular simulation of the global system by coupling of simulators may result in an unstable integration if an algebraic loop exists between the subsystems. This numerical phenomenon is analyzed and a method of simulator coupling which guarantees stability for general systems including algebraic loops is introduced. numerical results of the modular simulation of a slider-crank mechanism are presented.

关键词： mechatronics block representation algebraic loops modular modeling simulator coupling zero-stability numerical integration methods

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The first kind Chebyshev-Lobatto quadrature rule and its numerical improvement

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期171卷 1104-1118页

作者： Masjed-Jamei, M Hashemiparast, SM Eslahchi, MR Dehghan, M Sanjesh Org Minist Sci Res & Technol Ctr Res & Studies Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran Management & Planning Org Tehran Iran

One of the integration methods is the first kind Chebyshev-Lobatto quadrature rule, denoted by integral(1)(-1) (f(x))(root 1-x2) dx similar or equal to (pi)(n+1) (r=1)Sigma(n)f (cos(((2k-1)pi)(2n))) + (pi)(2(n+1)) f(-1) + (pi)(2(n+1)) f(1). According to this rule, the precision degree of above formula is the highest, i.e. 2n + 1. Hence, it is not possible to increase the precision degree of Chebyshev-Lobatto integration formulas anymore, we will present a matrix proof for this matter. But, on the other hand, we claim that one can improve the above formula numerically. To do this, we consider the integral bounds as two new unknown variables. This causes to numerically be extended the monomial space f(x) = x(j) from j = 0, 1,...,2n + 1 to j = 0, 1,...,2n + 3 (two monomials more than the first kind Chebyshev-Lobatto integration method). In other words, we present an approximate formula as integral(b)(a) (f(x))(root 1-x2) dx similar or equal to (i=1)Sigma(n) w(i)f(x(i)) + nu(1)f(a) + nu(2)f(b), in which a, b and w(1),w(2),..,w(n) and x(1),x(2),...,x(n) are all unknowns and the formula is almost exact for the monomial basis f(x) = x(j), j = 0,1,..., 2n + 3. Several tests are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual first kind Chebyshev-Lobatto nodes and weights. Let us add that in this part we have also some wonderful 5-point formulas that are comparable with 100002-point formulas of the first kind Chebyshev Lobatto quadrature rules in average. (c) 2005 Published by Elsevier Inc.

关键词： first kind Chebyshev-Lobatto quadrature formula numerical integration methods precision degree the method of undetermined coefficient the method of solving nonlinear systems

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Higher-order numeric Wazwaz-El-Sayed modified Adomian decomposition algorithms

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2012年第11期63卷 1557-1568页

作者： Duan, Jun-Sheng Rach, Randolph Shanghai Inst Technol Coll Sci Shanghai 201418 Peoples R China

In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz-El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge-Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Adomian decomposition method (ADM) Adomian polynomials Nonlinear differential equations numerical integration methods

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Stability preserving integration of index-1 DAEs

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APPLIED numerical MATHEMATICS 2003年第2-3期45卷 175-200页

作者： Higueras, I März, R Tischendorf, C Humboldt Univ Math Inst D-10099 Berlin Germany Univ Publ Navarra Dept Matemat & Informat Pamplona 31006 Spain

For index-1 DAEs with properly stated leading term, we characterize dissipative and contractive flows and study how the qualitative properties of the DAE solutions are reflected by numerical approximations. The best situation occurs when the discretization and the decoupling procedure commute. It turns out that this is the case if the relevant part of the inherent regular ODE has a constant state space. Different kinds of reformulations are studied to obtain this property. Those reformulations might be expensive, hence, in order to avoid them, criteria ensuring the given DAE to be numerically equivalent to a numerically qualified representation are proved. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

关键词： differential algebraic equations numerical integration methods global stability

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New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

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APPLIED MATHEMATICS AND COMPUTATION 2011年第6期218卷 2810-2828页

作者： Duan, Jun-Sheng Rach, Randolph Shanghai Inst Technol Coll Sci Shanghai 201418 Peoples R China

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Pade approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian's representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order a

关键词： Adomian decomposition method (ADM) Modified decomposition method (MDM) Adomian-Rach Theorem Adomian polynomials Nonlinear differential equations numerical integration methods

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RECENT PROGRESS IN THE THEORY AND APPLICATION OF SYMPLECTIC INTEGRATORS

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CELESTIAL MECHANICS & DYNAMICAL ASTRONOMY 1993年第1-2期56卷 27-43页

作者： Yoshida, Haruo Natl Astron Observ Tokyo 181 Japan

In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the form H = T(p) + V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets.

关键词： numerical integration methods long time evolution symplectic mapping

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THE INFLUENCE OF TEMPERATURE-GRADIENTS ON THE KINETIC BOUNDARY-LAYER PROBLEM FOR A CONDENSING DROPLET

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JOURNAL OF STATISTICAL PHYSICS 1992年第1-2期67卷 347-367页

作者： WIDDER, ME TITULAER, UM 1. Institut für Theoretische Physik Johannes Kepler Universit?t Linz A-4040 Linz Austria

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in a spatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state;the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermo-diffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.

关键词： CONDENSATION KINETIC BOUNDARY LAYERS BROWNIAN MOTION THERMODIFFUSION numerical integration methods

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On the calculation of convolutions with Gaussian kernels

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APPLIED MATHEMATICS AND COMPUTATION 2006年第1期176卷 383-387页

作者： Bailon, Martin L. Horntrop, David J. New Jersey Inst Technol Dept Math Sci Newark NJ 07102 USA

The calculation of convolutions with Gaussian kernels arises in many applications. The accuracy of a family of recently developed numerical integration rules is studied in this paper. In spite of the attractive implementation properties of the method, the poor convergence properties greatly restrict the situations in which the method should be used. (c) 2005 Elsevier Inc. All rights reserved.

关键词： quadrature methods numerical integration methods convolutions

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