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Báo cáo môn Mô phỏng và tối ưu hóa các quá trình hóa học - Phần 2: Mô phỏng máy tính - Hoàng Ngọc Hà

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Phần 2: mô phỏng máy tính  
Modeling, simulation and optimization for chemical process  
Instructor: Hoang Ngoc Ha  
Email: [email protected].vn  
Bộ môn QT&TB  
Introduction  
Numerical  
Analysis  
Computer  
Programming  
SIMULATION  
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  
Matlab/Simulink  
‰ Femlab, Fortran, Ansys…  
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)  
„ …  
Computer simulation  
Example:  
Computer simulation  
Computer simulation  
Computer simulation  
„ Interval halving (chia đôi khoảng)  
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  
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  
c ∈ [a, b]  
f(b), then there is a  
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  
Iterative method  
„ Bisection method  
Computer programming: Matlab  
Iterative method  
„ Newton’s method  
Numerical solutions of nonlinear systems of equations (of  
SEVERAL VARIABLES)Î (See Ref.)  
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.)  
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
y(t )  
y(t)  
0
y(t1)  
x
y(tN )  
x
x
Interpolation  
t0 t1  
tN  
t
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
„ Tools:  
‰ Euler’s method  
‰ Higher-Order Taylor methods  
‰ Runge-Kutta methods  
‰ …  
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
„ Euler’s method  
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
„ Euler’s method  
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
„ Example  
y0 = y − t2 + 1, t ∈ [0 2]  
y(0) = 0.5  
P/p Euler n=10?  
Approximate solution?  
b−a  
n = 10 ⇒ h = = 0.2  
n
Exact solution?  
y(t) = −0.5 exp(t) + (t + 1)2  
Computer programming: Matlab  
Numerical intergration of Ordinary  
Differential Equations (ODEs)  
„ Local truncation error  
Definition  
The local truncation error in Euler’s method is  
O(h)  
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