16.225 - Computational Mechanics of Materials
Homework assignment # 1
1. Verify that the Euler-Lagrange equations corresponding to the HuWashizu functional are the eld equations of linear elasticity.
2. Consistency test for constitutive models with a pot
16.225 - Computational Mechanics of Materials
Homework assignment # 2
The main objective of this assignment is to add capability to your version
of sumMIT to solve linear elasticity problems. Details of the data structures
and additional aspects of the pr
16.225 - Computational Mechanics of Materials
Homework assignment # 3
1. Behavior of the 3-node simplex element in the incompressible limit.
Using the 3-node simplex element, repeat the plane-strain, plate-withhole calculations of the previous assignment
16.225 - Computational Mechanics of Materials
Homework assignment # 4
Consider a nonlinear elastic solid described by the strain energy density:
W = k[J log(J ) J ] + trCdev
2
where
Cdev = J 2/3 C
and k and are material constants. For this model material,
16.21 Techniques of structural analysis and
design
Homework assignment # 3
Warmup exercises (not for grade)
Problem 3.23 from textbook
Problem 3.24 from textbook
Problem 3.25 from textbook
(Compliments of C. Gra.) Create a solid model of
the at plate
16.21 Techniques of Structural Analysis and
Design
Unit #9 Calculus of Variations
Let u be the actual conguration of a structure or mechanical system. u
satises the displacement boundary conditions: u = u on Su . Dene:
u = u + v, where:
: scalar
v : arbi
16.21 Techniques of structural analysis and design
Homework assignment # 4
Solidworks practice problem (Compliments of C. Gra. ) Create a solid model
for each of the following objects using Solidworks (you may turn in your le
electronically for feedback p
16.21 Techniques of structural analysis and
design
Homework assignment # 2
1. Determine whether the following stress elds are possible in a structural
member free of body forces:
(a) (not for grade)
11 = 3x1 + 6x2
12 = 4x1 + 3x2
22 = 5x1 + 4x2
(1)
(2)
(3)
16.21 Techniques of structural analysis and
design
Homework assignment # 1
Only problems 2, 3 and 4 are part of the assessment (contribute to the grade).
It is recommended that the assignment be done in Mathematica in its entirety
and turned in electronic
16.21 Techniques of Structural Analysis and
Design
Unit #1
In this course we are going to focus on energy and variational methods
for structural analysis. To understand the overall approach we start by con
trasting it with the alternative vector mechanics
16.21 Techniques of Structural Analysis and
Design
Unit #3 Kinematics of deformation
Figure 1: Kinematics of deformable bodies
Deformation described by deformation mapping :
x = (x) = x + u
1
(1)
We seek to characterize the local state of deformation of t
16.21 Techniques of Structural Analysis and
Design
Unit #2 Stress and Momentum balance
Stress at a point
We are going to consider the forces exerted on a material. These can be
external or internal. External forces come in two avors: body forces (given
pe
16.21 Techniques of Structural Analysis and
Design
Unit #2 Mathematical aside: Vectors,
indicial notation and summation convention
Indicial notation
In 16.21 well work in a an euclidean threedimensional space R3 .
Free index: A subscript index ()i wil
16.21 Techniques of Structural Analysis and
Design
Unit #5 Constitutive Equations
Constitutive Equations
For elastic materials:
ij = ij () =
U0
ij
(1)
If the relation is linear:
ij = Cijkl kl , Generalized Hookes Law
(2)
In this expression: Cijkl fourth
16.21 Techniques of Structural Analysis and
Design
Unit #4 Thermodynamics Principles
First Law of Thermodynamics
d
K +U =P +H
dt
(1)
where:
K: kinetic energy
U : internal energy
P : Power of external forces
H: hear exchange per unit time
1
K=
2
V
u u
Massachusetts Institute of Technology
16.410 Principles of Automated Reasoning
and Decision Making
Problem Set #4
Background for Problems 1 and 2
In Problems 1 and 2, you will implement backtracking algorithms for solving CSPs.
The implementation will be
Massachusetts Institute of Technology
16.410 Principles of Automated Reasoning
and Decision Making
Problem Set #6
Problem 1 (50 points)
Consider the following quote, taken from (Barwise and Etchemendy, 1993):
If the unicorn is mythical, then it is immorta
16.410-13 Principles of Automated
Reasoning and Decision Making
Problem Set 3
Problem 1: Modeling for Constraint Programming (40 Points)
You are a screen writer designing the story board for an episode of a sitcom Buddies,
a Friends knocko, with the ever
Massachusetts Institute of Technology
16.410-13 Principles of Autonomy and
Decision Making
Jump Starting lpsolve in Java
Introduction
This jumpstart shows you the basics of using a Linear Program solver, lpsolve, in Java.
Please note that this jumpstart w
Massachusetts Institute of Technology
16.410-13 Principles of Autonomy and
Decision Making
Problem Set #1
Objective
The primary objective of this problem set is to begin to exercise your skill at designing
and implementing Java programs. This includes dev
Massachusetts Institute of Technology
16.410 Principles of Automated Reasoning
and Decision Making
Problem Set #5
Problem 1: Planning for Space Operations (30 points)
In this problem you will construct a plan graph for a very simple spacecraft control
pro
Massachusetts Institute of Technology
16.410-13 Principles of Autonomy and
Decision Making
Problem Set #2
Objective
You will implement in Java the following search algorithms:
1. Depth-first
2. Breadth-first.
You will then experiment with these algorithms
Massachusetts Institute of Technology
16.410-13 Principles of Autonomy and
Decision Making
Problem Set #7 Part 2
Problem 3: Search (20 points)
You are trying to find a path from vertex 0 to vertex 7 in the following directed graph
using several search alg
16.410-13 Problems Set 7
Due in class on Wed. Nov. 10th, 2010.
Problem 1: Model-based Diagnosis (20 points)
Part A
Consider a Mars rover on which the diagnosis algorithm that you have learned in the class is running.
The electrical system components of th
16.410/13 Problem Set 10
Problem 1: Markov Chains (50 points)
Part A: Markov Chain Representation
A die is rolled repeatedly. Which of the following can be represented as a Markov Chain? For those that can be,
provide the transition matrix
The largest nu
Massachusetts Institute of Technology
16.410-13 Principles of Automated Reasoning
and Decision Making
Problem Set #11
Do not Turn In
Problem 1 MDPs: Tortoise and Hare
The following question is taken from the 2004 final. We all know, as the story goes, tha
16.410/13 Problem Set 9
Problem 1: Mixed Integer Linear Programming (50 points)
Part A: 3-SAT (10 points)
In this problem we will prove that solving a Mixed Integer Linear Program is NP-hard. Consider the (in)famous 3-SAT
problem. This problem is to nd an
16.410 Principles of Autonomy and Decision-Making
Problem Set. # 8
Please remember to include the time have spent for each problem in your solutions.
Problem 1: LP Formulation (20pts)
In one of the lunar bases that you are building, you are facing a probl