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Numerical Methods Applied to Chemical Engineering >> Content Detail



Study Materials



Readings

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Unless otherwise noted, the readings are from the required text:

Amazon logo Beers, Kenneth J. Numerical Methods for Chemical Engineering: Applications in MATLAB®. New York, NY: Cambridge University Press, November 2006. ISBN: 9780521859714.

In general, the readings were assigned ahead of the corresponding lecture topic.

The calendar below provides information on the course's lecture (L) and quiz (Q) sessions.


SES #TOPICSREADINGSREADINGS TOPICS
L1Using MATLAB® to evaluate and plot expressions

pp. 1-25.

MATLAB® Tutorial



Linear Algebra


Linear Systems of Algebraic Equations

Review of Scalar, Vector, and Matrix Operations

Elimination Methods for Solving Linear Systems

Existence and Uniqueness of Solutions

L2Solving systems of linear equationspp. 25-32 and 36-56.

Linear Algebra


Existence and Uniqueness of Solutions

Matrix Inversion

Matrix Factorization

Matrix Norm and Rank

Submatricies and Matrix Partitions

Example. Modeling a Separation System

Sparse and Banded Matricies

L3

Matrix factorization

Modularization



Condition Number


Amazon logo Heath, Michael T. Scientific Computing: An Introductory Survey. 2nd ed. New York, NY: McGraw-Hill Companies, Inc., 2002, pp. 5-6 and 52-65. ISBN: 9780072399103.

Amazon logo Recktenwald, Gerald W. Introduction to Numerical Methods with MATLAB®: Implementations and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000, pp. 402-410. ISBN: 9780201308600.

L4When algorithms run into problems: Numerical error, ill-conditioning, and tolerancespp. 61-77.

Nonlinear Algebraic Systems


Existence and Uniqueness of Solutions to a Nonlinear Algebraic Equation

Iterative Methods and Use of Taylor Series

Newton's Method for a Single Equation

The Secant Method

Bracketing and Bisection Methods

Finding Complex Solutions

Systems of Multiple Nonlinear Algebraic Equations

Newton's Method for Multiple Nonlinear Equations

L5Introduction to systems of nonlinear equationspp. 77-85 and 88-99.

Nonlinear Algebraic Equations


Estimating the Jacobian and Quasi-Newton Methods

Robust Reduced-step Newton's Method

The Trust - Region Newton Method

Solving Nonlinear Algebraic Systems in MATLAB®

Homotopy

Example. Steady-state Modeling of a Condensation Polymerization Reactor

Bifurcation Analysis

L6Modern methods for solving nonlinear equationspp. 104-113.

Matrix Eigenvalue Analysis


Orthogonal Matrices

Eigenvalues and Eigenvectors Defined

Eigenvalues / Eigenvectors of a 2×2 Real Matrix

Multiplicity and Formulas for the Trace and Determinant

Eigenvalues and the Existence/uniqueness Properties of Linear Systems

Estimating Eigenvalues; Gershgorin's Theorem

L7Introduction to eigenvalues and eigenvectorspp. 117-123 and 148-149.

Matrix Eigenvalue Analysis


Eigenvector Matrix Decomposition and Basis Sets

Computing Roots of a Polynomial

L8Constructing and using the eigenvector basispp. 123-126 and 137-141.

Matrix Eigenvalue Analysis


Numerical Calculation of Eigenvalues and Eigenvectors in MATLAB®

Eigenvalue Problems in Quantum Mechanics

L9Function space vs. real space methods for partial differential equations (PDEs)pp. 141-149.

Matrix Eigenvalue Analysis


Singular Value Decomposition

Computing the Roots of a Polynomial

L10Function spacepp. 126-134.

Matrix Eigenvalue Analysis


Computing Extremal Eigenvalues

The QR Method for Computing all Eigenvalues

L11

Numerical calculation of eigenvalues and eigenvectors

Singular value decomposition (SVD)



Initial Value Problems


Initial Value Problems of Ordinary Differential Equations (ODE-IVPs)

Polynomial Interpolation

Newton-cotes Integration

Linear ODE Systems and Dynamic Stability

Q1Quiz 1pp. 154-163 and 169-176.

Initial Value Problems


Initial Value Problems of Ordinary Differential Equations (ODE-IVPs)

Polynomial Interpolation

Newton-cotes Integration

Linear ODE Systems and Dynamic Stability

L12Ordinary differential equation - initial value problems (ODE-IVP) and numerical integrationpp. 176-194.

Initial Value Problems


Overview of ODE-IVP Solvers in MATLAB®

Accuracy and Stability of Single-step Methods

Stiff Stability of BDF Methods

L13Stiffness. MATLAB® ordinary differential equation (ODE) solverspp. 195-203.

Initial Value Problems


Differential-Algebraic Equation (DAE) Systems

L14

Implicit ordinary differential equation (ODE) solvers

Shooting

pp. 212-231.

Numerical Optimization


Local Methods for Unconstrained Optimization Problems

The Simplex Method

Gradient Methods

Newton Line Search Methods

Trust-region Newton Method

Newton Methods for Large Problems

Unconstrained Minimizer fminunc in MATLAB®

Example. Fitting a Kinetic Rate Law to Time-dependent Data

L15

Differential algebraic equations (DAEs)

Introduction: Optimization

pp. 231-246.

Numerical Optimization


Lagrangian Methods for Constrained Optimization

Constrained Minimizer fmincon in MATLAB®

L16Unconstrained optimizationpp. 258-270.

Boundary Value Problems (BVPs)


BVPs from Conservation Principles

Real-space vs. Function-space BVP Methods

The Finite Difference Method Applied to a 2-D BVP

Extending the Finite Difference Method

Chemical Reaction and Diffusion in a Spherical Catalyst Pellet

L17Constrained optimizationpp. 270-279.

Boundary Value Problems


Finite Differences for a Convection/diffusion Equation

L18

Optimization

Sensitivity analysis

Introduction: Boundary value problems (BVPs)

pp. 282-299.

Boundary Value Problems


Numerical Issues for Discretized PDEs with More Than Two Spatial Dimensions

The MATLAB® 1-D Parabolic and Elliptic Solver pdepe

Finite Differences in Complex Geometries

The Finite Volume Method

L19Boundary value problems (BVPs) lecture 2pp. 299-311.

Boundary Value Problems


The Finite Element Method (FEM)

FEM in MATLAB®

Further Study in the Numerical Solution of BVPs

L20Boundary value problems (BVPs) lecture 3: Finite differences, method of lines, and finite elements
L21TA tutorial on BVPs, FEMLAB®
L22Introduction: Models vs. Datapp. 372-389 and 325-338.

Bayesian Statistics and Parameter Estimation


General Problem Formulation

Example. Fitting Kinetic Parameters of a Chemical Reaction

Single-response Linear Regression

The Bayesian View of Statistical Inference

The Least Squares Method Reconsidered



Probability Theory and Stochastic Simulation


Important Probability Distributions

  • Bernoulli Trials
  • The Random Walk Problem
  • The Binomial Distribution
  • The Gaussian (Normal) Distribution
  • The Central Limit Theorem of Statistics
  • The Gaussian Distribution With Non-zero Mean
  • The Poisson Distribution

Random Vectors and Multivariate Distributions

  • The Boltzmann Distribution
  • The Maxwell Distribution
L23Models vs. Data lecture 2: Bayesian viewpp. 389-403.

Bayesian Statistics and Parameter Estimation


Selecting a Prior for Single-response Data

Confidence Intervals From the Approximate Posterior Density

L24Uncertainties in model predictionspp. 403-431.

Bayesian Statistics and Parameter Estimation


MCMC Techniques in Bayesian Analysis

MCMC Computation of Posterior Predictions

Applying Eigenvalue Analysis to Experimental Design

Bayesian Multi Response Regression

Analysis of Composite Data Sets

Bayesian Testing and Model Criticism

L25Conclude models vs. data
L26TA led review
Q2Quiz 2 (lectures 1 - 21)

Probability Theory and Stochastic Simulation


Markov Chains and Processes; Monte Carlo Methods

Markov Chains

Monte Carlo Simulation in Statistical Mechanics

Monte Carlo Integration

Simulated Annealing

L27

Models vs. Data recapitulation

Monte carlo and molecular dynamics

L28Guest lecture on Monte Carlo / molecular dynamics: Frederick Bernardinpp. 363-364.

Probability Theory and Stochastic Simulation


Genetic Programming

L29

Global optimization

Multiple minima

L30

Modeling intrinsically stochastic processes

multiscale modeling

pp. 338-353.

Probability Theory and Stochastic Simulation


Brownian Dynamics and Stochastic Differential Equations (SDEs)

L31Fluctuation-dissipation theorem
L32Kinetic Monte Carlo and turbulence modeling
L33

Operator splitting

Strang splitting



Strang Splitting


Schwer, Douglas A., Pisi Lu, William H. Green, Jr., and Viriato Semião. "A Consistent-splitting Approach to Computing Stiff Steady-state Reacting Flows With Adaptive Chemistry." Combust Theory Modelling 7 (2003): 383-399.

L34

Fourier transforms

Fast fourier transform (FFT)

pp. 436-452.

Fourier Analysis


Fourier Series and Transforms in One Dimension

1-D Fourier Transforms in MATLAB®

Convolution and Correlation

Fourier Transforms in Multiple Dimensions

L35Summary: Problem solving
L36TA led final review

 








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