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A Modification and Application of Parametric Continuation Method to Variety of Nonlinear Boundary Value Problems in Applied Mechanics.

机译：参数连续法在应用力学中各种非线性边值问题的改进和应用。

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摘要

In the field of engineering, researches often come across strong nonlinear boundary value problems which cannot be solved easily. Numerical convergence for many problems, typically solved by the Newton-Raphson linearization algorithm, is sensitive to the initial approach, relaxation parameters and differential topology. Emphasis in the present work is placed on the alternative approach, the so called parametric imbedding of a particular problem into the family of problems. While this may appear to complicate rather than to simplify the problem, its justification lies in the fact that a relation between infinitesimally close neighboring processes results in a simple Cauchy problem with respect to the introduced parameter.;Many problems in applied mechanics are reduced to the solutions of systems of nonlinear algebraic, transcendental, differential or integral-differential equations containing an explicit parameter. These are problems in the areas of thermo-fluids, gas dynamics, deformable solids, heat transfer, biomechanics, analytical dynamics, catastrophe theory, optimal control and others. A parameter found in these models is not unique, and may be easily identified as a load which could be geometric, structural, and physical or it could be introduced artificially. An important aspect of these problems is a question of the variation of the solution when parameter is incrementally changed.;The growing interest in nonlinear problems in engineering has been intensified by the use of digital computers. This paved a way in development of the solution procedures which can be applied to a large class of nonlinear problems containing a parameter. An important aspect of these problems is the variation of the solution of with the parameter. Hence, method of continuing the solution with respect to the parameter is a natural and universal tool for the 4 analysis. It was originally introduced by Ambarzumian and Chandrasekar, and intensively studied by Bellman, Kalaba and others. Different problems of applied mechanics and physics with dominant nonlinearities due to convective phenomena, constituent models, finite deformation, bifurcation and others are analyzed and solved in the present work. The choice of the optimal continuation parameter, which ensures the best conditioning of the corresponding system of nonlinear equations, is discussed. Some modifications for stiff systems of ordinary nonlinear differential equations are suggested and applied. Effectiveness of the continuation method is demonstrated by comparing the results with the stiff boundary value problem numerical solvers implemented using commercial softwares. The objective of the research is to investigate applicability of the method as a universal approach to the wide range of nonlinear boundary value problems in different areas of mechanics: nonlinear mechanics of solids, bifurcation problems, Newtonian and Non-Newtonian fluids, thermo-fluids, gas-dynamics, control, inverse problems.
机译：在工程领域，研究经常遇到难以解决的强烈的非线性边值问题。通常通过Newton-Raphson线性化算法解决的许多问题的数值收敛对初始方法，松弛参数和微分拓扑敏感。当前工作的重点放在替代方法上，即将特定问题参数化地嵌入到一系列问题中。尽管这似乎使问题变得复杂而不是简化，但其合理性在于以下事实，即无限接近的相邻过程之间的关系导致相对于引入参数的简单柯西问题。;应用力学中的许多问题都简化为包含显式参数的非线性代数，超越，微分或积分微分方程组的解。这些是热流体，气体动力学，可变形固体，传热，生物力学，分析动力学，突变理论，最优控制等领域的问题。在这些模型中找到的参数不是唯一的，并且可以很容易地识别为可以是几何，结构和物理负载，也可以人为引入的负载。这些问题的一个重要方面是当参数增量更改时解决方案的变化问题。；通过使用数字计算机，人们对工程非线性问题的兴趣日益浓厚。这为解决程序的开发铺平了道路，可以将其应用于包含参数的一大类非线性问题。这些问题的重要方面是随参数的解的变化。因此，针对参数继续求解的方法是进行4分析的自然而通用的工具。它最初由Ambarzumian和Chandrasekar提出，并由Bellman，Kalaba等人进行了深入研究。在本工作中，分析和解决了由于对流现象，组成模型，有限变形，分叉等导致应用非线性的主要应用力学问题。讨论了最佳连续参数的选择，该参数可确保相应非线性方程组的最佳条件。建议并应用了对普通非线性微分方程的刚性系统的一些修改。通过将结果与使用商业软件实现的刚性边界值问题数值求解器进行比较，证明了延续方法的有效性。研究的目的是研究该方法作为通用方法在各种不同力学领域中广泛应用的非线性边界值问题的适用性：固体的非线性力学，分叉问题，牛顿和非牛顿流体，热流体，气体动力学，控制，反问题。

著录项

  • 作者

    Patil, Akshay.;

  • 作者单位

    Rochester Institute of Technology.;

  • 授予单位 Rochester Institute of Technology.;
  • 学科 Mechanical engineering.
  • 学位 M.S.
  • 年度 2016
  • 页码 91 p.
  • 总页数 91
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 公共建筑;
  • 关键词

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