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Published byNigel Rogers Modified over 9 years ago
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MECH345 Introduction to Finite Element Methods Chapter 1 Numerical Methods - Introduction
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Numerical Methods - Introduction
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Definition: - Methods that seek quantitative approximations to the solutions of mathematic problems. Applications: - structural and stress analysis - thermal analysis - dynamic analysis - acoustic analysis - manufacturing process modeling - fluid mechanics - ………….
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Numerical Integration Calculate: Newton – Cotes integration Trapezoidal rule – 1 st order Newton-Cotes integration Trapezoidal rule – multiple application
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Numerical Integration Calculate: Newton – Cotes integration Simpson 1/3 rule – 2 nd order Newton-Cotes integration
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Numerical Integration Calculate: Gauss Quadrature Trapezoidal Rule : Gauss Quadrature : Chooseaccording to certain criteria
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Numerical Integration Calculate: Gauss Quadrature 2pt Gauss Quadrature 3pt Gauss Quadrature Let:
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Numerical Integration - Example Calculate: Trapezoidal rule Simpson 1/3 rule 2pt Gauss quadrature Exact solution
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Linear System Solver Solve: Gauss Elimination: forward elimination + back substitution Example:
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Linear System Solver Solve: Gauss Elimination: forward elimination Example:
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Linear System Solver Solve: Gauss Elimination: back substitution Example: Computer sum
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Linear System Solver Solve: Gauss Elimination: forward elimination + back substitution Pseudo code: Forward elimination: Back substitution: Do k = 1, n-1 Do i = k+1,n Do j = k+1, n Do ii = 1, n-1 i = n – ii sum = 0 Do j = i+1, n sum = sum +
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Finite Difference Method Example 1: t u Find 0 1 Selection of :
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Finite Difference Method Example 2: Find
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Finite Difference Method Example 3: Find 0 1
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