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2D1263 : Scientific Computing
Lecture 1
2D1263 : Scientific Computing
(3)
What is object-orientation?
Lecture 1
(1)
Object Oriented Program Construction
for Scientific Computing
“Organisation of software as a collection of discrete objects that
incorporate both data structure and behaviour.”
2D1263
Some key concepts:
http://www.nada.kth.se/kurser/kth/2D1263/
• Identity – data quantized into distinguashable entities (objects).
Marco Kupiainen
marcok@nada.kth.se
http://www.nada.kth.se/marcok
• Abstraction – separate external aspects of an object from
internal (functionality from implementation)
• Inheritance – build new abstractions out of existing ones
• Polymorphism – ability to provide the same interface to objects
with different implementations.
Objects are similar to C structures but have associated functions.
NADA
There are no widely accepted definitions of OO requirements.
Marco Kupiainen
marcok@nada.kth.se
Marco Kupiainen
marcok@nada.kth.se
NADA
NADA
2D1263 : Scientific Computing
Lecture 1
2D1263 : Scientific Computing
NADA
Golub and Ortega: “Scientific computing is the collection of tools,
techniques and theories required to solve on a computer mathematical
models of problems in science and engineering.”
Or a more narrow definition: “Development and use of numerical
methods and mathematical models to solve real-world problems
efficiently on computers.”
Goals
To give students knowledge of
Interdisciplinary field requiring:
• knowledge about the underlying (physical) problem,
• ability to formulate a mathematical model,
• stable & accurate numerical schemes,
Lecture 1
• formulation of PDEs from fluid dynamics,
• their numerical approximation on
non-trivial grids using finite differences,
• object-oriented implementation in C++,
• parallel programming for distributed
memory architectures (SPMD)
Prerequisites
• a second course in numerical analysis, e.g.
Applied Numerical Methods (2D1250)
• previous programming experience
(preferably but not necessarily C)
Marco Kupiainen
marcok@nada.kth.se
Familiarity with basic C can help, see e.g.
http://www.nada.kth.se/kurser/master/intro/
lectures 2–4 (apart from input/output).
What is scientific computing?
• efficient implementation on high performance computers.
(4)
Marco Kupiainen
marcok@nada.kth.se
NADA
(2)
2D1263 : Scientific Computing
Lecture 1
(5)
2D1263 : Scientific Computing
Lecture 1
(6)
Literature
The course literature will consist of handouts,
Examination
• lecture notes written by Bj¨
orn Sj¨
ogren
Written exam (2 credits):
• brief notes on topics not in lecture notes
• first exam by end of May or beginning of
June 2002
• copies of transparencies
Additional online references can be found on
the course homepage.
• at least three times per year
Computer laborations (2 credits):
Good C++ books (≈ $50 each):
1. Algebraic grid generation (due 15/2)
2. Distributed parallel arrays (due 22/3)
• “The C++ Programming Language”,
Bjarne Stroustrup
3. A parallel PDE solver (due 17/5)
• “C++ Primer”, Lippman & Lajoie
It will not be assumed that you buy any C++
books for the course.
The laborations can be done individually or in
groups of two students.
There are many other C++ books (e.g. at the
KTH library). Ask me if you’d like a second
opinion on a book.
Marco Kupiainen
marcok@nada.kth.se
Marco Kupiainen
marcok@nada.kth.se
NADA
NADA
2D1263 : Scientific Computing
Lecture 1
(7)
2D1263 : Scientific Computing
(Preliminary) schedule
(Preliminary) schedule
Computer exercises
Lectures
Monday
21/1
13.15 – 15.00
4523
Period 3
Weeks 5, 7, 9
Friday
25/1
13.15 – 15.00
4523
Period 4
Weeks 12, 17, 19
Monday
28/1
13.15 – 15.00
4523
Friday
1/2
13.15 – 15.00
4523
Monday
4/2
13.15 – 15.00
4523
Friday
8/2
13.15 – 15.00
4523
Monday
11/2
13.15 – 15.00
4523
Friday
15/2
13.15 – 15.00
4523
Wednesday
20/2
10.15 – 12.00
4523
Wednesday
27/2
10.15 – 12.00
4523
Wednesday
13/3
10.15 – 12.00
4523
Wednesday
20/3
10.15 – 12.00
4523
Tuesday or Thursday mornings? Afternoons?
You will probably have to work more than this
to complete the laborations.
Outside of scheduled classes any questions
should be directed to the teaching assistants
• Kyoung-Sook Moon (moon@nada.kth.se)
• Sandy Sefi (sandy@nada.kth.se)
They can be reached by email, and also answer
questions in person (subject to office hours).
Marco Kupiainen
marcok@nada.kth.se
NADA
Lecture 1
Marco Kupiainen
marcok@nada.kth.se
NADA
(8)
Lecture 1
(9)
• Basic C/C++ constructs
Marco Kupiainen
marcok@nada.kth.se
Marco Kupiainen
marcok@nada.kth.se
NADA
• Overview of computational grids
This course focuses mainly on structured boundary-fitted grids.
2D1263 : Scientific Computing
Today
– Structured
4. Implement a solver in e.g. C++
– Boundary-fitted
3. Find a tool for solving the discretized
PDE, or
– Unstructured
2. Approximate the PDE on the grid by e.g.
finite differences or finite elements
– Cartesian
1. Discretize the domain (generate a grid)
• Boundary-fitted
we need to
• Structured grids
• initial & boundary conditions
• Cartesian
Different types of grids
• a partial differential equation
• Unstructured grids
• a geometry
Boundary representation
Given
Structure
Numerical approximation
Subdivision of domain Ω into small cells or a finite set of points
intended for approximating PDEs by algebraic equations.
Lecture 1
(10)
2D1263 : Scientific Computing
NADA
Marco Kupiainen
marcok@nada.kth.se
Lecture 1
(12)
Unstructured grids
x = [(1/2 + ξ1 ) cos(πξ2 ), (1/2 + ξ1 ) sin(πξ2 )]
+ Generality
– Handles complex geometries
– “Straightforward” generation &
refinement
– Inefficiency
– Indirect addressing ⇒ inefficient cache
usage & many dereferences
– Parallelization difficult
double x[n],y[n]; // Node coordinates
int tri[m,3]; // Nodes in triangle
Coordinates for first node in triangle i:
x[tri[i,0]], y[tri[i,0]].
NADA
ξ¯ = [“radius” , “angle”]
1
0.9
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0.5
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0.3
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0.1
0
0
0.1
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1
2D1263 : Scientific Computing
Structured grids are indexed along coordinate directions:
−1.5
0
0.5
⇒
1
1.5
−1
A structured grid
−0.5
0
0.5
1
1.5
Lecture 1
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2D1263 : Scientific Computing
Marco Kupiainen
marcok@nada.kth.se
NADA
2D1263 : Scientific Computing
Lecture 1
(13)
2D1263 : Scientific Computing
Structured grids
Lecture 1
(14)
Cartesian grids
¯
Mapping from unit square to geometry, x(ξ).
(only cells intersecting domain shown)
+ Efficiency
Rectangular grid with objects “carved out”.
– Memory usage (regular connectivity)
+ works for any geometry (simple)
– Numerical methods (structured
matrices)
– accuracy in boundary conditions
+ Simple to use
– numerical stability
– How to do local refinement?
It is possible but cumbersome to get accurate
boundary conditions (immersed boundary
method ).
¯
– Difficulties with complex geometries (x(ξ))
⇒ may have to divide domain into smaller
blocks before grid generation.
Can be useful for “inexact” domains, e.g. a
lake shore.
Block subdivision may be good for parallel
computations (c.f. domain decomposition)
Marco Kupiainen
marcok@nada.kth.se
NADA
Marco Kupiainen
marcok@nada.kth.se
NADA