VARIATIONAL AND FINITE ELEMENT METHODS: A SYMBOLIC COMPUTATION APPROACH. Questo articolo non è disponibile.VARIATIONAL AND FINITE ELEMENT METHODS: A SYMBOLIC COMPUTATION APPROACH

The variational approach, including the direct methods and finite elements, is one of the main tools of engineering analysis. However, it is difficult to appreciate not only for seniors but for graduate students too. It is possible to make this subject easier to understand with the help of symbolic manipulation codes (SMC). The easiness with which these codes provide analytical results allow for a student or researcher to focus on the ideas rather than on calculational difficulties. The very process of programming with SMC encourages appreciation of the qualitative aspects of investigations. Saving time and effort, they enable undergraduates to deal with the subjects generally regarded as graduate courses. There is a habitual aspect too. These days it is more convenient for a student (researcher) to work with a keyboard than with a pencil. Moreover, semantic features of the codes may allow for generalizations of the standard techniques, which would be impossible to achieve without the computer's help.

I Symbolic Manipulation Codes.- 1.1 Notions of LISP and Expert Systems.- 1.2 First Sessions.- 1.3 Matrices.- 1.4 Solving Equations.- 1.5 Limits and Expansions.- 1.6 Integration.- 1.7 Some Useful Commands and Options. Pattern Matching.- 1.8 Conditionals, Iterations and Compound Statements.- 1.9 Few Hints.- 1.10 Example: Steady-State Linear Vibrations.- 1.11 Example: Transient Vibrations.- 1.12 Example: Free Nonlinear Vibrations.- 1.13 Example: Forced Nonlinear Vibrations.- II Variational Approach and Equations of Motion.- 2.1 Mechanics of a Particle.- 2.2 System of Particles. Generalized Coordinates.- 2.3 Functional and its Euler-Lagrange Equation.- 2.4 Hamilton’s Principle for Discrete Systems.- 2.5 Constrained Motions.- 2.6 Virtual Work.- 2.7 D’Alembert’s Principle. Nonconservative Systems.- 2.8 Transition to Continuous Systems.- 2.9 Hamilton’s Principle for Continuous Systems, Part I.- 2.10 Hamilton’s Principle for Continuous Systems, Part H.- 2.11 Minimum of Potential Energy. Imposed and Natural Boundary Conditions.- 2.12 Computer-generated Governing Equations.- 2.13 Single Degree of Freedom.- 2.14 Two Degrees of Freedom. Double Nonlinear Pendulum.- 2.15 Dynamic Shock Absorber.- 2.16 Continuous Systems.- 2.17 Automatic Generation, Part I.- 2.18 Automatic Generation, Part II.- 2.19 Second Variation and Nature of Extremum.- 2.20 Legendre’s Condition.- 2.21 Transversality Conditions.- 2.22 Generalizations and Transformations of Variational Problems.- 2.23 Minimum Pressure Drag.- 2.24 Constrained Minimum Pressure Drag.- III Direct Methods.- 3.1 The Philosophy.- 3.2 The Method of Least Squares. Trial Functions.- 3.3 Beam on Elastic Foundation, Part I.- 3.4 Beam on Elastic Foundation, Part II.- 3.5 The Bubnov-Galerkin Method.- 3.6 Beam on Elastic Foundation, Part III.- 3.7 The Rayleigh-Ritz Method.- 3.8 Master Program.- 3.9 Applications.- 3.10 Improved Master Program.- 3.11 Considerations of Accuracy.- 3.12 Plate on Elastic Foundation.- 3.13 Further Investigations of Plates.- 3.14 Other Direct Methods.- 3.15 Shock Absorber, Preliminary Considerations.- 3.16 Shock Absorber, Program and Results.- 3.17 Flow Through a Duct.- 3.18 Temperature Field in a Plate, Part I.- 3.19 Temperature Field in a Plate, Part II.- 3.20 Free Vibrations by the Rayleigh-Ritz Method.- 3.21 Free Vibrations of a Non-uniform Beam.- 3.22 Master Programm.- 3.23 Free Vibrations by gthe Bubnov-Galerkin Method.- 3.24 Nonlinear Vibrations by the Bubnov-Galerkin Method.- 3.25 Mathematical Considerations. Scalar Products of Functions.- 3.26 Operators and Functionals.- 3.27 Symmetric and Positive Definite Operators.- 3.28 Minimum Theorem and Minimizing Sequence.- 3.29 Orthogonal and Linearly Independent Functions.- IV Introduction to the Finite Element Method.- 4.1 Finite Elements. The Element Stiffness Matrix.- 4.2 Energy Analysis of a Finite Element.- 4.3 Truss Element.- 4.4 Physical Meaning of the Element Matrices.- 4.5 Global Reference Systems.- 4.6 Generalizations. Governing Equations of a Structure.- 4.7 Assembling.- 4.8 Formalization of Assembling.- 4.9 Truss.- 4.10 Further Analysis of a Truss.- 4.11 Composite Beam.- 4.12 Particular Cases.- 4.13 Automatic Generation of the Assembly Stiffness Matrix.- 4.14 Optimization.- 4.15 Reduced Stiffness Matrix.- 4.16 Free Vibrations of Beams.- 4.17 Plate Element, Part I.- 4.18 Plate Element, Part II.- 4.19 Particular Cases. Batch Mode.- 4.20 Compatibility and Convergence.- 4.21 Natural Coordinate Systems.- 4.22 The Concept of Isoparametric Elements.- 4.23 Some Plane Elements.- 4.24 Concluding Remarks.- Problems.- Answers.- References.