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Bài giảng Phần 2: Mô phỏng máy tính

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Nội dung của bài giảng trình bày ba dạng phương trình đạo hàm riêng cơ bản; bài toán Parabolic (mô hình truyền nhiệt); bài toán Parabolic (mô hình truyền nhiệt tổng quát); bài toán Elliptic; xấp xỉ sai phân...

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Nội dung Text: Bài giảng Phần 2: Mô phỏng máy tính

  1. Phần 2: mô phỏng máy tính Modeling, simulation and optimization for chemical process Instructor: Hoang Ngoc Ha Email: ha.hoang@hcmut.edu.vn Bộ môn QT&TB CuuDuongThanCong.com https://fb.com/tailieudientucntt
  2. Introduction Numerical Computer Analysis Programming TION SIMULA CuuDuongThanCong.com https://fb.com/tailieudientucntt
  3. Computer simulation „ Some simulation techniques for solving some of the systems of equations ‰ Solution of (nonlinear) algebraic equations ‰ Ordinary differential equations (ODEs) ‰ Partial differential equations (PDEs) „ Numerical methods ‰ Iterative methods ‰ Discrete difference methods ‰ Femlab, Fortran, Ansys… Matlab/Simulink CuuDuongThanCong.com https://fb.com/tailieudientucntt
  4. Computer simulation „ Computer programming ‰ Assume that you know some computer programming language ‰ We are not interested in generating the most efficient and elegant code but in solving problems (from point of view of engineers) „ Including extensive comment statements „ Use of symbols (the same ones in the equations describing the systems) „ Debugging (for mistakes in coding and/or in logic) „ … CuuDuongThanCong.com https://fb.com/tailieudientucntt
  5. Computer simulation Example: CuuDuongThanCong.com https://fb.com/tailieudientucntt
  6. Computer simulation CuuDuongThanCong.com https://fb.com/tailieudientucntt
  7. Computer simulation CuuDuongThanCong.com https://fb.com/tailieudientucntt
  8. Computer simulation „ Interval halving (chia đôi khoảng) CuuDuongThanCong.com https://fb.com/tailieudientucntt
  9. Computer simulation „ This problem can be formulated under the following form: f (x) = 0, x ∈ R „ The goal is to find the solution of this nonlinear equations (in ONE VARIABLE) „ Tools (Iterative methods) ‰ Bisection method (phương pháp phân đoạn) ‰ Newton’s (or Newton-Raphson) method CuuDuongThanCong.com https://fb.com/tailieudientucntt
  10. Iterative method „ Intermediate value theorem ‰ If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u If f(a) and f(b) are of opposite sign, there exist a number p in [a, b] with f(p)=0 CuuDuongThanCong.com https://fb.com/tailieudientucntt
  11. Iterative method „ Bisection method Computer programming: Matlab CuuDuongThanCong.com https://fb.com/tailieudientucntt
  12. Iterative method „ Newton’s method Numerical solutions of nonlinear systems of equations (of SEVERAL VARIABLES)Î (See Ref.) CuuDuongThanCong.com https://fb.com/tailieudientucntt
  13. Computer simulation „ Interpolation and polynomial approximation ‰ Interpolation and the Lagrange polynomial ‰ Cubic spline interpolation ‰ … „ Numerical differentiation and intergration ‰ Numerical differentiation ‰ Richardson’s extrapolation ‰ … (See Ref.) CuuDuongThanCong.com https://fb.com/tailieudientucntt
  14. Numerical intergration of Ordinary Differential Equations (ODEs) CuuDuongThanCong.com https://fb.com/tailieudientucntt
  15. Numerical intergration of Ordinary Differential Equations (ODEs) y(t) y(t0 ) x y(t1 ) y(tN ) x x Interpolation t0 t1 tN t CuuDuongThanCong.com https://fb.com/tailieudientucntt
  16. Numerical intergration of Ordinary Differential Equations (ODEs) „ Tools: ‰ Euler’s method ‰ Higher-Order Taylor methods ‰ Runge-Kutta methods ‰ … CuuDuongThanCong.com https://fb.com/tailieudientucntt
  17. Numerical intergration of Ordinary Differential Equations (ODEs) „ Euler’s method CuuDuongThanCong.com https://fb.com/tailieudientucntt
  18. Numerical intergration of Ordinary Differential Equations (ODEs) „ Euler’s method CuuDuongThanCong.com https://fb.com/tailieudientucntt
  19. Numerical intergration of Ordinary Differential Equations (ODEs) „ Example y 0 = y − t2 + 1, t ∈ [0 2] y(0) = 0.5 P/p Euler n=10? Approximate solution? b−a n = 10 ⇒ h = n = 0.2 Exact solution? y(t) = −0.5 exp(t) + (t + 1)2 Computer programming: Matlab CuuDuongThanCong.com https://fb.com/tailieudientucntt
  20. Numerical intergration of Ordinary Differential Equations (ODEs) „ Local truncation error Definition The local truncation error in Euler’s method is O(h) CuuDuongThanCong.com https://fb.com/tailieudientucntt
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