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課程來源:MIT
     


16.901航空工程的計算方法(2005春季課程)

16.901 Computational Methods in Aerospace Engineering, Spring 2005

 

譯者:任维

編輯:陳盈、馬景文

Turbine blade.

 

   

馬赫Mach2時圓柱體周圍無粘流場的模擬,來自第一專題。(圖片由Darmofal教授提供。)

 

課程重點

本課程有完整的課堂講稿作業,也有豐富的習資料

 

課程描述

本課程介紹航空工程的計算技術,應用在航空結構力學,空氣動力學,動力學和 控制,以及航空系統;這包括:系統常微分方程的數值積分;有限差分,有限體積和有限元對偏微分方程的離散化;數值線性代數;特徵值問題;約束優化問題。

 

技術需求

要用MATLAB® software運行課程站點的.m.mat文件。

 

教學大綱

課程目標

學生成功完成16.901之後學會:

  1. 對航空系統分析和設計普遍使用的計算方法有概念性理解。
  2. 對計算方法有應用方面的認識,包括為航空工程典型問題建模的經驗。
  3. 分析計算方法行為的理論技術基礎。


定量指標

這科目分成四部份:

  • 常微分方程(ODE)的數值積分
  • 偏微分方程(PDE)的有限體積法和有限差分法
  • 偏微分方程的有限元法
  • 機率模擬技術

每部份的定量指標如下,尤其學生成功完成16.901課程將學會:

 

常微分方程的積分方法

   1.  (a) 描述亞當斯-巴什福思法(Adams-Bashforth)、亞當斯-莫爾頓法(Adams-Moulton)和多步法的後向微分族;
        (b) 描述多階段法龍格-庫塔(Runge-Kutta)族的形式;
        (c) 闡明多步法和多階段法兩者相對的計算代價。
   2. (a) 闡明方程組剛度的概念;以及
       (b) 描述這如何影響選擇數值方法求解方程。
   3. (a) 闡明顯式和隱式格式法對常微分方程組積分方面的差別和相對優勢;和
       (b) 解釋對於非線性方程組來說,牛頓-拉夫遜(Newton-Raphson)法如何用於隱式格式法的解題。
   4. (a) 定義一個收斂方法;
       (b) 定義一個一致性方法;
       (c) 解釋(臨界)穩定性;和
       (d) 通過描述收斂方法、一致性和穩定性之間的關係表明對Dahlquist等價定理的理解。
   5. 確定多步法是否穩定和一致。
   6. (a) 定義常微分方程積分法精度的總體和局部階數,
       (b) 描述總精度和局部精度之間的關係,以及
       (c) 利用泰勒Taylor級數分析來計算某一給定方法的局部精度。
   7. (a) 確定特徵值穩定性,和
       (b) 確定多步法或多階段法用於一個線性常微分方程組時的穩定性邊界。
   8. 就一個具體問題建議一個適當的常微分方程積分方法。
   9. 用多步法和多階段法來解決一個工程應用中具有代表性的常微分方程組問題。


偏微分方程的有限差分和有限體積法

 1. (a) 定義一個問題的物理影響域,
     (b) 定義和決定一個離散問題的數值域,和
     (c) 解釋CFL條件並確定其產生的時間步長約束。
 2.  利用泰勒Taylor級數分析解釋偏微分方程的有限差分估計局部截斷誤差。
 3. 解釋中心格式和單邊格式(上風格式)的離散方式的區別。
 4. 描述二維對流在非結構網格上的戈多諾夫Godunov有限體積離散。
 5. 用馮諾依曼Von Neumann分析或半離散分析(線化理論),完成偏微分方程有限差分估計的特徵值穩定性分析。
 6. 使用有限差分或有限體積離散法來解決工程應用中具有代表性的偏微分方程(或者是一系列偏微分方程)。


偏微分方程的有限元法

 1. (a) 描述加權餘量法如何用來計算偏微分方程的近似解,
     (b) 描述加權餘量法、排列法和最小二乘法在求偏微分方程近似解時的差別,和
     (c) 描述什麼是Galerkin加權餘量法。
 2. (a) 描述選擇有限元法使用的近似解法(如:試驗函數或插值),和
     (b) 舉例說明近似解題的基礎,尤其是包含至少可用於線性和二次解題的一個節點。
 3. (a) 描述如何用參考單元來做積分,
     (b) 描述如何推導出高斯Gaussian求積法,以及
     (c) 描述高斯求積法如何用於參考單元中作近似積分。
 4. 解釋 Dirichlet邊界條件和Neumann邊界條件如何用到有限元法離散後的拉普拉斯Laplace方程。
 5. (a) 描述有限元法離散如何做到離散方程組;對於線性問題,這如何導致剛度矩陣;以及
     (b) 描述剛度矩陣的條目(列和行)以及線性問題右邊向量的意思。


機率法

注意: 要求所有學生必須從先修課中全面理解:機率論、隨機變量、機率密度函數(PDF)累積分布函數(CDF)、平均值(預期)、方差、標準差、百分比、均勻分佈、正態分佈和x2分佈。

1. 描述蒙特卡洛法Monte Carlo如何從多變量,均勻分佈中採樣。
2. 描述蒙特卡洛法如何把均勻分佈採樣修改為一般分佈。
3. (a) 描述什麼是無偏差估計;
    (b) 敘述平均值、方差和機率的無偏差估計;以及
    (c) 敘述這些無偏差估計的分佈。
4. (a) 定義標準誤差;
    (b) 給出平均值、方差和機率的標準差;
    (c) 給出平均值、方差和機率的標準差的置信區間;以及
    (d) 闡明蒙特卡洛法在隨機數輸入時的收斂與用以上誤差估計的採樣數目輸入時的關係。
5. (a) 描述單獨輸入和多元輸入的分層採樣,
    (b) 描述拉丁超立方體採樣法(LHS),以及
    (c) 描述LHS根據採樣數目平均值標準誤差的收斂性在近線性輸出的優點。
6. (a) 描述響應面法(RSM);
    (b) 描述經過泰勒Taylor級數、用最小二乘法設計實驗和用最小二乘法隨機採樣的響應面的架構;以及
    (c) 描述R2度量空間,及其用於測量響應面質量方面的用處和潛在問題。

作業問題

作業會在定期講座結束時給出,並要求在下一課開始時上交。這些作業需要花1-2小時完成。每次作業會以下面的尺度標準給分:

3: 足以說明對概念有出色理解的完整解答。

2: 足以說明對概念有足夠理解的解答,但有一些次要錯誤。

1: 足以說明對概念有一定理解的完整或近乎完整的解答,但犯了重大錯誤。

0: 非常不完整的解答或者一點都沒有解答。

注意:每次作業給分只是整數。在學期結束時,作業得分最高的2/3將計算平均分以確定作業總分等級。大致上是:A: 2.5-3; B: 2-2.5; C: 1.5-2; D: 1-1.5; F: 0-1.

 

專題

目前,本學期有三個編程專題(除了常微分方程,每部份一個)。這些專題將是關於應用於航空工程中的數值算法。強烈推薦大家用Matlab®完成編程。以下列出專題的到期日。

專題

到期日

專題1

16

專題2

30

專題3

38

專題作業在到期日之前至少一星期佈置。專題到期那一周不會佈置任何作業。每專題評分按照麻省理工的等級評分細則。(請看課程評分)。

共同研究作業和專題

雖然我們鼓勵同學之間討論作業和專題,但是上交評分的作業必須能代表你自己對課程內容的理解。若通過其它途徑得到重大幫助必須註明。

口試

將會有期中口試和期末口試。期中口試在講座20和講座21之間。期末口試在期末考試周舉行。我會在二月底根據各同學的選擇來編排期中試日程。學校公佈期末考試的日程後,我會定出本課程的期末考試日程安排。每次口試根據麻省理工標準等級分數細則來評分。(請看課程)。

課程評分

學科總分由作業、專題和口試的分數組成。大致來說,各部份的權重如下所示:

評分準則

活動

明細

作業評分

總分的1/8

專題評分

每個項專題佔總分的1/8

口試評分

每次測驗佔總分的1/4


對於學科的評分,我是根據麻省理工的給分指導,以下是具體描述。

A: 異常出色的表現,展示對學科內容有深度理解、有廣泛知識的基礎以及靈活使用概念和資料。

B: 良好的表現,展示有能力利用合適的概念、對學科內容有很好的瞭解以及有能力解決與學科相關的問題。

C: 及格的表現,展示對學科內容有足夠的理解、能解決相對簡單的問題以及有能力繼續在這領域做進一步的工作。

D:可以接受的最低限度表現,展示對課程內容有點熟悉,有一些能力解決一些簡單問題,但可見有嚴重不足,如不痛下功夫不建議在這領域進行深入工作。

教科書

筆記將會發給大家。有需要時會根據具體內容建議參考書目。

 

教學時程

 

課程單元

重要日期

常微分方程Ordinary Differential Equations

1

常微分方程的數值積分:介紹Numerical Integration of Ordinary Differential Equations: An Introduction

 

2

收斂和精度Convergence and Accuracy

 

3

多步法收斂Convergence of Multi-Step Methods

 

4

(續3)

作業 1到期

5

(續4)

作業 2到期

6

常微分方程系統及其本征值的穩定性Systems of ODE's and Eigenvalue Stability

作業 3到期

7

剛度法和隱式格式法Stiffness and Implicit Methods

 

8

(續7)

作業 4到期

9

龍格-庫塔法Runge-Kutta Methods

 

有限體積/差分法 Finite Volume/Difference Methods

10

有限體積法Finite Volume Method

作業 5到期

11

(續11)

作業 6到期

12

(續12)

作業 7到期

13

有限差分法Finite Difference Method

作業 8到期

14

(續13)

 

15

(續14)

 

16

矩陣穩定性分析 Matrix Stability Analysis

專題 1到期

17

(續16)

 

18

傅立葉穩定性分析 Fourier Stability Analysis

 

19

(續19)

 

20

(續20)

 
 

期中口試 Midterm Oral Exam

 

有限元法Finite Element Methods

21

加權餘量法 Method of Weighted Residuals

 

22

(續22)

作業 9到期

23

一維擴散的有限元法 Finite Element Method for 1-D Diffusion

 

24

(續24)

作業 10到期

25

(續25)

作業 11到期

26

二維擴散的有限元法 Finite Element Method for 2-D Diffusion

 

27

(續27)

 

28

(續28)

 

29

高階有限元法 Higher-order Finite Element Method

 

30

(續30)

專題 2到期

機率模擬技術 Probabilistic Simulation Techniques

31

蒙特卡洛法的介紹 Introduction to Monte Carlo Method

 

32

(續32)

作業 12到期

33

對蒙特卡洛法的誤差估計 Error Estimates for Monte Carlo Method

 

34

(續33)

作業 13到期

35

(續34)

 

36

拉丁超立方抽樣法 Latin Hypercube Sampling

 

37

響應面法 Response Surface Methods

 

38

自舉法 Bootstrapping

專題 3到期

39

綜述 Wrap Up

 
 

期末口試 Final Oral Exam

 
       

 

課堂講稿

MATLAB® software運行課程的.m文件。

課程單元

課堂講稿

常微分方程Ordinary Differential Equations

1

常微分方程的數值積分:介紹

Numerical Integration of Ordinary Differential Equations: An Introduction

課堂講稿(英語PDF
drop_fe.m (M)
drop_mp.m (M)
drop_rhs.m (M)

2

收斂和精度

Convergence and Accuracy

課堂講稿(英語PDF
ga_fe.m (M)
ga_mp.m (M)

3

多步法的收斂

Convergence of Multi-Step Methods

課堂講稿(英語PDF
ga_ma2.m (M)

4

常微分方程系統及其本征值的穩定性

Systems of ODE's and Eigenvalue Stability

課堂講稿(英語PDF
fe_stab.m (M)
nonpen_fe.m (M)
nonpen_mp.m (M)

5

剛度法和隱式格式法

Stiffness and Implicit Methods

課堂講稿(英語PDF
dif1d.m (M)
dif1d_fun.m (M)
dif1d_main.m (M)
eig_dif1d.m (M)
mstepstab.m (M)
stiff_err.m (M)
stiff_forced.m (M)

6

龍格-庫塔法

Runge-Kutta Methods

課堂講稿(英語PDF
rkstab.m (M)

有限體積/差分法 Finite Volume/Difference Methods

7

守恆定律和有線體積法

Conservation Laws and Finite Volume Methods

課堂講稿(英語PDF
convect1d.m (M)
convect2d.m (M)

8

對流-擴散的有限差分法

Finite Difference Methods for Convection-Diffusion

課堂講稿(英語PDF
convect1d_ftcs.m (M)

9

矩陣穩定性分析

Matrix Stability Analysis

課堂講稿(英語PDF
condif1d_ftcs_eig.m (M)
convect1d_ftcs_eig.m (M)

 

期中考試 Midterm Exam

 

10

傅立葉穩定性分析

Fourier Stability Analysis

課堂講稿(英語PDF

有限元法 Finite Element Methods

11

加權餘量法

Method of Weighted Residuals

課堂講稿(英語PDF
MWR_dif1d.m (M)

12

一維擴散的有限元法

Finite Element Method for 1-D Diffusion

課堂講稿(英語PDF
fem_dif1d.m (M)
fem_dif1d_gq.m (M)

13

二維擴散的有限元法

Finite Element Method for 2-D Diffusion

課堂講稿(英語PDF

14

高階有限元法

Higher-order FEM

課堂講稿(英語PDF
fem1D_hier.m (M)
fem1D_quad.m (M)

機率方法 Probabilistic Methods

15

蒙特卡洛法的介紹

Introduction to Monte Carlo Method

課堂講稿(英語PDF
blade1D.m (M)
bladedet.m (M)
bladeLtbc.m (M)
bladeuni.m (M)

16

蒙特卡洛法的誤差估計

Error Estimates for Monte Carlo Method

課堂講稿(英語PDF


作業

MATLAB® software運行課程的.m文件。

以下是整套作業及答案,並附上適用的MATLAB®文本。

作業

補充檔案

解答

問題集1(英語PDF

MATLAB® scripts (TXT)

(PDF)

問題集2(英語PDF

 

(PDF)

問題集3(英語PDF

MATLAB® scripts (TXT)
MATLAB® scripts (TXT)

(PDF)

問題集4(英語PDF

 

(PDF)

問題集5(英語PDF

MATLAB® scripts (TXT)

(PDF)

問題集6(英語PDF

MATLAB® scripts (TXT)

(PDF)

問題集7(英語PDF

MATLAB® scripts (TXT)
convect1d.m (M)

(PDF)

問題集8(英語PDF

MATLAB® scripts (TXT)
convect2d.m (M)

(PDF)

問題集9(英語PDF

 

(PDF)

問題集10(英語PDF

MATLAB® scripts (TXT)

(PDF)
hw10_MATLAB®sol.txt (TXT)

問題集11(英語PDF

MATLAB® scripts (TXT)

(PDF)
hw11_MATLAB®sol.txt (TXT)

問題集12(英語PDF

MATLAB® scripts (TXT)
blade1D.m (M)
bladeLtbc_tri.m (M)
trirnd.m (M)

(PDF)

問題集13(英語PDF

MATLAB® scripts (TXT)

(PDF)

 

專題

MATLAB® software運行課程的.m文件。

課程有三個編程專題,除了常微分方程,每部份一個。專題集中於航空工程應用的數值算法。

專題文件

專題

補充檔案

解答

專題1(英語PDF

CalcForces.m (M)
cyl_adaptmesh.m (M)
cyl_initmesh.m (M)
cylgeom.mat (MAT)
eulerflux.m (M)
FVM.m (M)
SetRefineList.m (M)
SetupEdgeList.m (M)
SetupMesh.m (M)
wallflux.m (M)

(PDF - 1.4 MB)

專題2(英語PDF

bladeheat.m (M)
bladeplot.m (M)
hpblade_coarse.mat (MAT)
hpblade_fine.mat (MAT)
hpblade_medium.mat (MAT)
findloc.m (M)
Thgas.m (M)

(PDF)

p2_matlabsol.txt (TXT)

專題3(英語PDF

calcblade.m (M)
DesignIntent.m (M)
hpblade_coarse.mat (MAT)
loadblade.m (M)
MCdriver.m (M)
Screen.m (M)
Thgas.m (M)
trirnd.m (M)

(PDF)

 

研習資料

 

以下材料是為問題集和期末考試做準備而提供。

例題集

問題集1(英語PDF

問題集1解答(英語PDF

問題集2(英語PDF

問題集2解答(英語PDF

 期末考試準備

準備的材料(英語PDF)

(中譯本完)

 

Mach 2 inviscid flow simulation over a cylinder, from Project #1. (Image courtesy of Professor Darmofal.)


Course Highlights

This course features a complete set of lecture notes and assignments, and also a variety of study materials.

Course Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.

Technical Requirements

Special software is required to use some of the files in this course: .m, .mat.

*Some translations represent previous versions of courses.


Syllabus

Course Objectives

Students successfully completing 16.901 should have:

  1. A conceptual understanding of computational methods commonly used for analysis and design of aerospace systems.
  2. A working knowledge of computational methods including experience implementing them for model problems drawn from aerospace engineering applications.
  3. A basic foundation in theoretical techniques to analyze the behavior of computational methods.

Measurable Outcomes

The subject is divided into four sections:

  • Integration of Systems of Ordinary Differential Equations (ODE's)
  • Finite Volume and Finite Difference Methods for Partial Differential Equations (PDE's)
  • Finite Element Methods for Partial Differential Equations
  • Probabilistic Simulation Techniques

For each of these sections, the measurable outcomes are described below. Specifically, a student successfully completing 16.901 will be able to:

Integration Methods for ODE's

  1. (a) Describe the Adams-Bashforth, Adams-Moulton, and Backwards Differentiation families of multi-step methods;
    (b) Describe the form of the Runge-Kutta family of multi-stage methods; and
    (c) Explain the relative computational costs of multi-step versus multi-stage methods.
  2. (a) Explain the concept of stiffness of a system of equations, and
    (b) Describe how it impacts the choice of numerical method for solving the equations.
  3. (a) Explain the differences and relative advantages between explicit and implicit methods to integrate systems of ordinary differential equations; and
    (b) For nonlinear systems of equations, explain how a Newton-Raphson can be used in the solution of an implicit method.
  4. (a) Define a convergent method;
    (b) Define a consistent method;
    (c) Explain what (zero) stability is; and
    (d) Demonstrate an understanding of the Dahlquist Equivalence Theorem by describing the relationship between a convergent method, consistency, and stability.
  5. Determine if a multi-step method is stable and consistent.
  6. (a) Define global and local order of accuracy for an ODE integration method,
    (b) Describe the relationship between global and local order of accuracy, and
    (c) Calculate the local order of accuracy for a given method using a Taylor series analysis.
  7. (a) Define eigenvalue stability, and
    (b) Determine the stability boundary for a multi-step or multi-stage method applied to a linear system of ODE's.
  8. Recommend an appropriate ODE integration method based on the features of the problem being solved.
  9. Implement multi-step and multi-stage methods to solve a representative system of ODE's from an engineering application.

Finite Difference and Finite Volume Methods for PDE's

  1. (a) Define the physical domain of dependence for a problem,
    (b) Define and determine the numerical domain of dependence for a discretization, and
    (c) Explain the CFL condition and determine the timestep constraints resulting from the CFL conditions.
  2. Determine the local truncation error for a finite difference approximation of a PDE using a Taylor series analysis.
  3. Explain the difference between a centered and a one-sided (e.g. upwind) discretization.
  4. Describe the Godunov finite volume discretization of two-dimensional convection on an unstructured mesh.
  5. Perform an eigenvalue stability analysis of a finite difference approximation of a PDE using either Von Neumann analysis or a semi-discrete (method of lines) analysis.
  6. Implement a finite difference or finite volume discretization to solve a representative PDE (or set of PDE's) from an engineering application.

Finite Element Methods for PDE's

  1. (a) Describe how the Method of Weighted Residuals (MWR) can be used to calculate an approximate solution to a PDE,
    (b) Describe the differences between MWR, the collocation method, and the least-squares method for approximating a PDE, and
    (c) Describe what a Galerkin MWR is.
  2. (a) Describe the choice of approximate solutions (i.e. the test functions or interpolants) used in the Finite Element Method, and
    (b) Give examples of a basis for the approximate solutions in particular including a nodal basis for at least linear and quadratic solutions.
  3. (a) Describe how integrals are performed using a reference element,
    (b) Explain how Gaussian quadrature rules are derived, and
    (c) Describe how Gaussian quadrature is used to approximate an integral in the reference element.
  4. Explain how Dirichlet and Neumann boundary conditions are implemented for Laplace's equation discretized by FEM.
  5. (a) Describe how the FEM discretization results in a system of discrete equations and, for linear problems, gives rises to the stiffness matrix; and
    (b) Describe the meaning of the entries (rows and columns) of the stiffness matrix and of the right-hand side vector for linear problems.

Probabilistic Methods

Note: all students are expected to have a thorough understanding of probability, random variables, PDF's, CDF's, mean (expectation), variance, standard deviation, percentiles, uniform distributions, normal distributions, and x2-distributions from the prerequisite coursework.

  1. Describe how Monte Carlo sampling from multivariable, uniform distributions works.
  2. Describe how to modify Monte Carlo sampling from uniform distributions to general distributions.
  3. (a) Describe what an unbiased estimator is;
    (b) State unbiased estimators for mean, variance, and probability; and
    (c) State the distributions of these unbiased estimators.
  4. (a) Define standard error;
    (b) Give standard errors for mean, variance and probability;
    (c) Place confidence intervals for estimates of the mean, variance, and probability; and
    (d) Demonstrate the dependence of Monte Carlo convergence on the number of random inputs and the number of samples using the above error estimates.
  5. (a) Describe stratified sampling for single input and multiple inputs,
    (b) Describe Latin Hypercube Sampling (LHS), and
    (c) Describe the benefits of LHS for nearly linear outputs in terms of the standard error convergence of the mean with the number of samples.
  6. (a) Describe the Response Surface Method (RSM);
    (b) Describe the construction of a response surface through Taylor series, Design of Experiments with the least-square regression, and random sampling with least-squares regression; and
    (c) Describe the R2 -metric, its use in measuring the quality of a response surface, and its potential problems.

Homework Problems

A homework problem will be given at the end of most regular lectures and will be due at the beginning of the next class. These homework problems are intended to take 1-2 hours to complete. The individual homework sets will be graded on the following scale:

3: A complete solution demonstrating an excellent understanding of the concepts.

2: A complete solution demonstrating an adequate understanding of the concepts, though some minor mistakes may have been made.

1: A complete or nearly-complete solution demonstrating some understanding of the concepts, though major mistakes may have been made.

0: A largely incomplete solution or no solution at all.

Note: the individual homework grades will only be integer values. At the end of the semester, the highest 2/3's of the grades received in the homeworks will be averaged to determine an overall homework letter grade. Roughly, the following ranges will be used. A: 2.5-3; B: 2-2.5; C: 1.5-2; D: 1-1.5; F: 0-1.

Projects

Currently, three programming projects are planned for this semester (one for each section of the course except the ODE section). The projects will focus on applying numerical algorithms to aerospace applications. The programming is highly recommended to be done in Matlab®. The expected due dates for the projects are as follows.


PROJECTS DUE DATES
Project 1 Lecture 16
Project 2 Lecture 30
Project 3 Lecture 38

The project assignments will be distributed at least one week prior to the due dates. No homeworks will be given during the week the projects are due. Each project will be assigned a letter grade based on the standard MIT letter grade descriptions (see Course Grade).

Homework and Project Collaboration

While discussion of the homework and projects is encouraged among students, the work submitted for grading must represent your understanding of the subject matter. Significant help from other sources should be noted.

Oral Exams

There will be a mid-term and final oral exam. The mid-term oral exam will be held between Lecture 20 and Lecture 21. The final oral exam will be held during Final Exam Week. I will schedule the mid-term oral exam by the end of February based on preferences from each student. I will schedule the final oral exam once the final exam schedule for the institute has been published. Each oral exam will be assigned a letter grade based on the standard MIT letter grade descriptions (see Course Grade).

Course Grade

The subject total grade will be based on the letter grades from the homework, projects, and oral exams. Roughly, the weighting of the individual letters grade is as follows:


ACTIVITIES BREAKDOWN
Homework Letter Grades 1/8 of the Subject Total Grade
Project Letter Grades Each Project is 1/8 of the Total Grade
Oral Exam Letter Grades Each Exam is 1/4 of the Total Grade

For the subject letter grade, I adhere to the MIT grading guidelines which give the following description of the letter grades:

A: Exceptionally good performance demonstrating a superior understanding of the subject matter, a foundation of extensive knowledge, and a skillful use of concepts and/or materials.

B: Good performance demonstrating capacity to use the appropriate concepts, a good understanding of the subject matter, and an ability to handle the problems and materials encountered in the subject.

C: Adequate performance demonstrating an adequate understanding of the subject matter, an ability to handle relatively simple problems, and adequate preparation for moving on to more advanced work in the field.

D: Minimally acceptable performance demonstrating at least partial familiarity with the subject matter and some capacity to deal with relatively simple problems, but also demonstrating deficiencies serious enough to make it inadvisable to proceed further in the field without additional work.

Textbooks

Notes will be distributed. Reference texts will be recommended for specific topics as needed.

Calendar

LEC # TOPICS KEY DATES
Ordinary Differential Equations
1 Numerical Integration of Ordinary Differential Equations: An Introduction  
2 Convergence and Accuracy  
3 Convergence of Multi-Step Methods  
4 Convergence of Multi-Step Methods (cont.) Homework 1 due
5 Convergence of Multi-Step Methods (cont.) Homework 2 due
6 Systems of ODE's and Eigenvalue Stability Homework 3 due
7 Stiffness and Implicit Methods  
8 Stiffness and Implicit Methods (cont.) Homework 4 due
9 Runge-Kutta Methods  
Finite Volume/Difference Methods
10 Finite Volume Method Homework 5 due
11 Finite Volume Method (cont.) Homework 6 due
12 Finite Volume Method (cont.) Homework 7 due
13 Finite Difference Method Homework 8 due
14 Finite Difference Method (cont.)  
15 Finite Difference Method (cont.)  
16 Matrix Stability Analysis Project 1 due
17 Matrix Stability Analysis (cont.)  
18 Fourier Stability Analysis  
19 Fourier Stability Analysis (cont.)  
20 Fourier Stability Analysis (cont.)  
  Midterm Oral Exam  
Finite Element Methods
21 Method of Weighted Residuals  
22 Method of Weighted Residuals (cont.) Homework 9 due
23 Finite Element Method for 1-D Diffusion  
24 Finite Element Method for 1-D Diffusion (cont.) Homework 10 due
25 Finite Element Method for 1-D Diffusion (cont.) Homework 11 due
26 Finite Element Method for 2-D Diffusion  
27 Finite Element Method for 2-D Diffusion (cont.)  
28 Finite Element Method for 2-D Diffusion (cont.)  
29 Higher-order Finite Element Method  
30 Higher-order Finite Element Method (cont.) Project 2 due
Probabilistic Simulation Techniques
31 Introduction to Monte Carlo Method  
32 Introduction to Monte Carlo Method (cont.) Homework 12 due
33 Error Estimates for Monte Carlo Method  
34 Error Estimates for Monte Carlo Method (cont.) Homework 13 due
35 Error Estimates for Monte Carlo Method (cont.)  
36 Latin Hypercube Sampling  
37 Response Surface Methods  
38 Bootstrapping Project 3 due
39 Wrap Up  
  Final Oral Exam  

Lecture Notes

Special software is required to use some of the files in this section: .m, .mat.

Lecture notes are listed by week in the table below. A complete set of lecture notes is also available and is included above the table.

Complete Lecture Notes - "Computational Methods in Aerospace Engineering" (PDF - 2.5 MB)

LEC # TOPICS LECTURE NOTES
Ordinary Differential Equations
1 Numerical Integration of Ordinary Differential Equations: An Introduction (PDF)

drop_fe.m (M)

drop_mp.m (M)

drop_rhs.m (M)
2 Convergence and Accuracy (PDF)

ga_fe.m (M)

ga_mp.m (M)
3 Convergence of Multi-Step Methods (PDF)

ga_ma2.m (M)
4 Systems of ODE's and Eigenvalue Stability (PDF)

fe_stab.m (M)

nonpen_fe.m (M)

nonpen_mp.m (M)
5 Stiffness and Implicit Methods (PDF)

dif1d.m (M)

dif1d_fun.m (M)

dif1d_main.m (M)

eig_dif1d.m (M)

mstepstab.m (M)

stiff_err.m (M)

stiff_forced.m (M)
6 Runge-Kutta Methods (PDF)

rkstab.m (M)
Finite Volume/Difference Methods
7 Conservation Laws and Finite Volume Methods (PDF)

convect1d.m (M)

convect2d.m (M)
8 Finite Difference Methods for Convection-Diffusion (PDF)

convect1d_ftcs.m (M)
9 Matrix Stability Analysis (PDF)

condif1d_ftcs_eig.m (M)

convect1d_ftcs_eig.m (M)
  Midterm Exam  
10 Fourier Stability Analysis (PDF)
Finite Element Methods
11 Method of Weighted Residuals (PDF)

MWR_dif1d.m (M)
12 Finite Element Method for 1-D Diffusion (PDF)

fem_dif1d.m (M)

fem_dif1d_gq.m (M)
13 Finite Element Method for 2-D Diffusion (PDF)
14 Higher-order FEM (PDF)

fem1D_hier.m (M)

fem1D_quad.m (M)
Probabilistic Methods
15 Introduction to Monte Carlo Method (PDF)

blade1D.m (M)

bladedet.m (M)

bladeLtbc.m (M)

bladeuni.m (M)
16 Error Estimates for Monte Carlo Method (PDF)

Assignments

Special software is required to use some of the files in this section: .m.

A full set of assignments and solutions are posted below. Where applicable, text files of MATLAB® scripts have been included.


ASSIGNMENTS SUPPORTING FILES SOLUTIONS
Problem Set 1 (PDF) MATLAB® scripts (TXT) (PDF)
Problem Set 2 (PDF)   (PDF)
Problem Set 3 (PDF) MATLAB® scripts (TXT)

MATLAB® scripts (TXT)
(PDF)
Problem Set 4 (PDF)   (PDF)
Problem Set 5 (PDF) MATLAB® scripts (TXT) (PDF)
Problem Set 6 (PDF) MATLAB® scripts (TXT) (PDF)
Problem Set 7 (PDF) MATLAB® scripts (TXT)

convect1d.m (M)
(PDF)
Problem Set 8 (PDF) MATLAB® scripts (TXT)

convect2d.m (M)
(PDF)
Problem Set 9 (PDF)   (PDF)
Problem Set 10 (PDF) MATLAB® scripts (TXT) (PDF)

hw10_MATLAB®sol.txt (TXT)
Problem Set 11 (PDF) MATLAB® scripts (TXT) (PDF)

hw11_MATLAB®sol.txt (TXT)
Problem Set 12 (PDF) MATLAB® scripts (TXT)

blade1D.m (M)

bladeLtbc_tri.m (M)

trirnd.m (M)
(PDF)
Problem Set 13 (PDF) MATLAB® scripts (TXT) (PDF)

Projects

Special software is required to use some of the files in this section: .m, .mat.

There are three programming projects for the class, one for each section of the course except the ODE section. They focus on applying numerical algorithms to aerospace applications.


Projects Supporting Files Solutions
Project 1 (PDF) CalcForces.m (M)

cyl_adaptmesh.m (M)

cyl_initmesh.m (M)

cylgeom.mat (MAT)

eulerflux.m (M)

FVM.m (M)

SetRefineList.m (M)

SetupEdgeList.m (M)

SetupMesh.m (M)

wallflux.m (M)
(PDF - 1.4 MB)
Project 2 (PDF) bladeheat.m (M)

bladeplot.m (M)

hpblade_coarse.mat (MAT)

hpblade_fine.mat (MAT)

hpblade_medium.mat (MAT)

findloc.m (M)

Thgas.m (M)
(PDF)

p2_matlabsol.txt (TXT)
Project 3 (PDF) calcblade.m (M)

DesignIntent.m (M)

hpblade_coarse.mat (MAT)

loadblade.m (M)

MCdriver.m (M)

Screen.m (M)

Thgas.m (M)

trirnd.m (M)
(PDF)

Study Materials

The following materials are provided for preparation for the problem sets and final exam.

Sample Problems Sets

Problem Set 1 (PDF)

Solution Set 1 (PDF)

Problem Set 2 (PDF)

Solution Set 2 (PDF)

Final Exam Preparation

Preparation Materials (PDF)




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