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#+TITLE: The pugs user manual
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* Introduction
~pugs~[fn:pugs-def] is a general purpose solver collection built to
approximate solutions of partial differential equations. It is mainly
(but not only) designed to deal with finite-volume methods.
~pugs~ is a parallel software that uses ~MPI~[fn:MPI-def] and multi-threading
for parallelism. Multi-threading is achieved through an encapsulation
of some [[https:github.com/kokkos/kokkos][Kokkos]] mechanisms.
The philosophy of ~pugs~ is to provide "simple" numerical tools that are
assembled together through a high-level language (a DSL[fn:DSL-def]
close to the mathematics) to build more complex solvers. This approach
is inspired by the success of [[http://freefem.org][FreeFEM]], which use a similar approach.
Before detailing the leading concepts and choices that we have made to
develop ~pugs~, we give a simple example.
** An example
For instance, the following code builds a uniform Cartesian grid of
$]-1,1[^2$ made of $20\times20$ cells and transforms it by displacing the
mesh nodes according to a user-defined vector field $T: \mathbb{R}^2
\to \mathbb{R}^2$.
#+NAME: introduction-example
#+BEGIN_SRC pugs :exports both :results output
import mesh;
import math;
import writer;
let pi:R, pi = acos(-1);
let theta:R^2 -> R, x -> 0.5*pi*(x[0]*x[0]-1)*(x[1]*x[1]-1);
let M:R^2 -> R^2x2, x -> [[cos(theta(x)), -sin(theta(x))],
[sin(theta(x)), cos(theta(x))]];
let T: R^2 -> R^2, x -> x + M(x)*x;
let m:mesh, m = cartesianMesh([-1,-1], [1,1], (20,20));
m = transform(m, T);
write_mesh(gnuplot_writer("transformed"), m);
#+END_SRC
The example is quite easy to read.
- First, some *modules* are loaded: the ~mesh~ module, which contains some
mesh manipulation functions. The ~math~ module provides a set of
classical mathematical functions ($\sin$, $\cos$, ...). The ~writer~
module is used to save meshes or discrete functions to files using
various formats.
- The second block of data defines variables of different types
- ~pi~ is a real value that is initialized by an approximation of $\pi$.
- ~theta~ is the real value function $\theta$ defined by
\begin{equation*}
\theta: \mathbb{R}^2 \to \mathbb{R},\quad\mathbf{x} \mapsto
\frac{\pi}{2} (x_0^2-1)(x_1^2-1)
\end{equation*}
- ~M~ is the $\mathbb{R}^{2\times2}$ matrix field $M$ defined by
\begin{equation*}
M: \mathbb{R}^2 \to \mathbb{R}^{2\times2},\quad\mathbf{x} \mapsto
\begin{pmatrix}
\cos(\theta(\mathbf{x})) & -\sin(\theta(\mathbf{x})) \\
\sin(\theta(\mathbf{x})) & \cos(\theta(\mathbf{x}))
\end{pmatrix}
\end{equation*}
- ~T~ is the vector field $T$ defined by
\begin{equation*}
T: \mathbb{R}^2 \to \mathbb{R}^2, \quad\mathbf{x} \mapsto (I+M(\mathbf{x}))\mathbf{x}
\end{equation*}
- Finally ~m~ is defined as the uniform Cartesian mesh grid of
$]-1,1[^2$. The last argument: ~(20,20)~ sets the number of cells in
each direction: $20$.
- The third block is the application of the transformation $T$ to the
mesh. Observe that if the resulting mesh is stored in the same
variable ~m~, the old one was not modified in the process. It is
important to already have in mind that the ~pugs~ language *does not
allow* the modifications of values of variables of *non-basic*
types. This is discussed in the section [[high-level-types]].
- Finally, the last block consists in saving the obtained mesh in a
~gnuplot~ file. The result is shown on Figure [[fig:intro-example]].
#+NAME: intro-transform-mesh-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/intro-transform-mesh.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set terminal png truecolor enhanced
set size square
plot '<(sed "" $PUGS_SOURCE_DIR/doc/transformed.gnu)' w l
#+END_SRC
#+CAPTION: Obtained transformed mesh
#+NAME: fig:intro-example
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: intro-transform-mesh-img
Even if this first example is very simple, some key aspects can
already be discussed.
- There is no predefined constant in ~pugs~. Here a value is provided
for ~pi~.
- There are two kinds of variable in ~pugs~: variables of basic types
and variable of high-level types. This two kinds of variable behave
almost the same but one must know their differences to understand
better the underlying mechanisms and choices that we made. See
section [[basic-types]] and [[high-level-types]] for details.
- Also, there are two types of functions: *user-defined* functions and
*builtin* functions. In this example, ~theta~, ~M~ and ~T~ are user-defined
functions. All other functions (~cos~, ~cartesianMesh~,...) are
builtin functions and are generally defined when importing a
module. These functions behave similarly, one should refer to
[[functions]] for details.
** Concepts and design
As it was already stated, ~pugs~ can be viewed as a collection of
numerical methods or utilities that can be assembled together, using a
user friendly language, to build simulation scenarios.
Utilities are tools that are often used in numerical
simulations. Examples of such utilities are mesh manipulations,
definition of initial conditions, post-processing, error
calculations,...
*** A C++ toolbox driven by a user friendly language
Numerical simulation packages are software of a particular
kind. Generally, in order to run a calculation, one has to define a
set of data and parameters. This can simply be definition of a
discretization parameter such as the mesh size. One can also specify
boundary conditions, equations of state, source terms for a specific
model. Choosing a numerical method or even more, setting the model
itself, is common in large codes.
In ~pugs~, all these "parameters" are set through a
DSL[fn:DSL-def]. Thus, when ~pugs~ is launched, it actually executes a
provided script. A ~C++~ function is associated to each instruction of
the script. The ~C++~ components of ~pugs~ are completely unaware one of
the others. ~pugs~ interpreter is responsible of data flow between the
components: it manages the data transfer between those ~C++~ components
and ensures that the workflow is properly defined.
**** Why?
In this section we motivate the choice of a language and not of a more
standard approach.
***** Data files are evil
There are lots of reasons not to use data files. By data file, we
refer to a set of options that describe physical models, numerical
methods or their settings.
- Data files are not flexible. This implies in then one hand that
application scenarios must be known somehow precisely to reflect
possible option combinations and in the other hand even defining a
specific initial data may require the creation of a new option and
the associated code (in ~C++~ for instance). \\
Usually, the last point is addressed by adding a local interpreter
to evaluate user functions.
- Data files may contain irrelevant information. Actually, it is quite
common to allow to define options that are valid but irrelevant to a
given scenario. This policy can be changed but it is generally not
an easy task and requires more work from the user (which can be a
good thing).
- Generally data files become rapidly obsolete. An option was not the
right one, or its type changed to allow other contexts... This puts
pressure on the user.
- Even worst, options meaning can depend on other
options. Unfortunately, this happens commonly. For instance, a
global option can change implicitly the treatment associated to
another one. This is dangerous since writing or reading the data
file requires an important and up to date knowledge of the code's
internals.
- Another major difficulty when dealing with data files is to check
the compatibility of provided options.
***** Embedded "data files" are not a solution
Using directly the general purpose language (~C~, ~C++~, ~Fortran~,...) used
to write the code can be tempting. It has the advantage that no
particular treatment is necessary to build a parser (to read data
files or a script), but it presents several drawbacks.
- The first one is probably that it allows to much freedom. While
defining the model and numerical options, the user has generally
access to the whole code and can change almost everything, even
things that should not be changed.
- Again, one can easily have access to irrelevant options and it
requires a great knowledge of the code to find important ones.
- With that regard, defining a simulation properly can be a difficult
task. For instance, in the early developments of ~pugs~ (when it was
just a raw ~C++~ code) it was tricky to change boundary conditions for
coupled physics.
- Another difficulty is related to the fact that code's internal API
is likely to change permanently in a research code. Thus valid
constructions or setting may become rapidly obsolete. In other
words keeping up to date embedded "data file" might be difficult.
- Finally it requires recompilation of pieces of code (which can be
large in some cases) even if one is just changing a simple
parameter.
***** Benefits of a DSL
Actually, an honest analysis cannot conclude that a DSL is the
solution to all problems. However, it offers some advantages.
- First, it allows a fine control on what the user can or cannot
perform. In some sense, it offers a chosen level of flexibility.
- It allows to structure the code in the sense that new developments
have to be designed not only focusing on the result but also on the
way it should be used (its interactions with the scripting language).
- In the same idea, it provides a framework that drives the desired
principle of "do simple things and do them well".
- There are no hidden dependencies between numerical options: the DSL
code is easier to read (than data files) and is less likely to
become obsolete (this is not that true in early developments since
the language itself and some concepts are still likely to change).
- The simulation scenario is *defined* by the script, it is the
responsibility of the user to check its meaning (not to the charge
of the code).
***** ~pugs~ language purpose
~pugs~ language is used to assemble ~C++~ tools which are well
tested. These tools should ideally be small pieces of ~C++~ code that
do one single thing and do it well.
Another purpose of the language is to allow to perform high-level
calculations. In other words, the language defines a data flow and
checks that each ~C++~ piece of code is used correctly. Since each piece
of code acts as a pure function (arguments are unchanged by calls),
the calling context is quite easy to check.
Finally it aims at simplifying the definition of new methods since
common utilities are available directly in scripts.
***** The framework: divide and conquer
~pugs~ is a research oriented software, thus generally the user is also
a developer. If this paragraph is indeed more dedicated to the
developer, by reading it, the user will have a better understanding of
the development choices and the underlying policy that the code
follows.
Actually the development framework imposed by the DSL tends to guide
writing of new methods.
- In the process of writing a *new numerical methods*, one must create
*new functions in the language*. Moreover, if a new method is close to
an existing one, it is generally *better* to use completely new
underlying ~C++~ code than to patch existing methods. Starting from a
*copy* of the existing code ~C++~ is *encouraged* for developments. This
may sound weird since classical development guidelines encourage
inheritance or early redesign. Actually, this policy is the result
of the following discussion.
- Using this approach, one *separates* clearly the *design* of a
numerical method and the *design* of the code.
- From the computer science point of view, early design for new
numerical methods is generally wrong: usually one cannot
anticipate precisely enough eventual problems or method
corrections.
- It is much more difficult to introduce bugs in existing methods,
since previously validated methods are unchanged!
- For the same reason, existing methods performances are naturally
unchanged by new developments.
- Also, when comparing methods, it is better to compare to the
original existing code.
- If the new method is abandoned or simply not kept, the existing
code is not polluted.
- Finally when a method is validated and ready to integrate the
mainline sources of the code, it is easier to see differences with
existing ones and *at this time* one can redesign the ~C++~ code
checking that results are unchanged and performances not
deteriorated. At this time, it is likely that the numerical method
design is finished, thus (re)designing the source code makes more
sense.
- Another consequence is that utilities are not be developed again and
again.
- This implies an /a priori/ safer code: utilities are well tested and
validated.
- It saves development time obviously.
- The code of numerical methods is not polluted by environment
instructions (data initialization, error calculation,
post-processing,...)
- The counterpart is somehow classical. In the one hand, the
knowledge of existing utilities is required, this document tries
to address a part of it. In the other hand, if the developer
requires a new utility, a good practice is to discuss with the
other ones to check if it could benefit to them. Then one can
determine if it should integrate rapidly or not the main
development branch.
***** Why not python or any other scripting language?
As it was already pointed out above, general purpose languages offer
to much freedom: it is not easy to protect data. For instance in the
~pugs~ DSL, non basic variables are constant (see paragraph
[[high-level-types]]). It is important since it prevents the user from
modifying data in inconsistent ways. Also, one must keep in mind that
constraining the expressiveness is actually a strength. As said
before, one can warranty coherence of the data, perform calculations
without paying attention to the parallelism aspects,... Observe that
it is not a limitation: if the DSL's field of application needs to be
extended, it is always possible. But these extensions should never
break the rule that a DSL must not become a general purpose language.
Providing a language close to the application reduces the gap between
the application and its expression (code): Domain Specific Languages
are made for that.
Finally, python is ugly.
*** A high-level language
Following the previous discussion, the reader should now understand
the motivations that drove the design choices that conduct to build
~pugs~ as a ~C++~ toolbox driven by a user friendly language.
#+begin_verse
Keep it simple and relevant!
#+end_verse
In the development process of an application, the easy step is
generally the implementation itself. It is even more true when the
developer feels that changes to the code are natural and that the
modifications themselves look easy. Obviously, any experienced
programmer knows that writing the code is only the first step of a
much longer process which requires rigorous tests and validations (the
enrichment of a non-regression database). Generally one also desires
to ensure that the development is used within the correct bounds which
requires to implement data verification. In a perfect world, an
up-to-date documentation of the functionality and its domain of
validity should be provided.
This is even more true when defining a language (or a DSL). Enriching
a language syntax (or grammar) is not something that must be done to
answer a specific need. It must not be done /because it is possible to
do it/!
#+begin_verse
When designing a language, the difficulty is not to offer new functionalities,\\
it is generally to decide not to offer them.\\
--- Bjarne Stroustrup, C++ conference 2021.
#+end_verse
If the grammar of ~pugs~ is still likely to be extended, it should *never*
integrate low-level instructions. Low-level instructions give too much
freedom and thus are a source of errors. Several things are already
done to forbid this kind of evolution. The constness of high-level
data is a good illustration. For instance, meshes or discrete
functions *cannot* be modified. This is not only a security to protect
the user from doing "dangerous" manipulations, but it also permits to
define high-level optimizations.
- Since meshes are constant objects, one can for instance compute
geometric data on demand. These data are kept into memory as long as
the mesh lives.
- Forbidding the modification of values of a discrete function ensures
that parallel communication instructions should never appear in a
~pugs~ script.
Another benefit of not providing low-level instructions is that the
scripts are more easy to write and read, and it is more difficult to
write errors.
* Language
** Variables
In order to simplify the presentation, before going further, we
introduce the syntax that allows to print data to the terminal. It
follows ~C++~ streams construction for convenience. For instance
#+NAME: cout-preamble-example
#+BEGIN_SRC pugs :exports both :results output
cout << "2+3 = " << 2+3 << "\n";
#+END_SRC
produces the following output
#+results: cout-preamble-example
The code is quite obvious for ~C++~ users, note that ~"\n"~ is the
linefeed string (there is no character type in ~pugs~, just strings).
Actually, ~cout~ is itself a variable, we will come to this later.
~pugs~ is a strongly typed language. It means that a variable *cannot*
change of type in its lifetime.
*** Declaration and affectation syntax
**** Declaration of simple variables
To declare a variable ~v~ of a given type ~V~, one writes
#+BEGIN_SRC pugs :exports source
let v:V;
#+END_SRC
This instruction is read as
#+begin_verse
Let $v\in V$.
#+end_verse
Actually,
- ~let~ is the declaration keyword,
- ~v~ is the variable name,
- ~:~ is a separation token (it can be read as "/in/" in this context),
- ~V~ is the identifier of the type, and
- ~;~ marks the end of the instruction.
For instance to declare a real variable ~x~, one writes
#+NAME: declare-R
#+BEGIN_SRC pugs :exports both :results none
let x:R;
#+END_SRC
The instructions that follow the declaration of a variable can use it
while defined in the same scope, but it cannot be used before. Also,
after its declaration, one cannot declare another variable with the
same name in the same scope (see [[blocks-and-life-time]]).
#+NAME: redeclare-variable
#+BEGIN_SRC pugs-error :exports both :results output
let x:R; // first declaration
let x:R; // second declaration
#+END_SRC
produces the following error
#+results: redeclare-variable
**** Affectation of simple variables
To affect the value of an expression ~expression~ to variable ~v~ one
simply uses the ~=~ operator. Thus assuming that a variable ~v~ has
already been /declared/, one writes simply
#+BEGIN_SRC pugs :exports source
v = expression;
#+END_SRC
There is not to much to comment, reading is quite natural
- ~v~ is the variable name,
- ~=~ is the affectation operator,
- ~expression~ is some code that provides a value of the same type as ~v~
(it can be another variable, an arithmetic expression, the result of
a function,...), and
- ~;~ marks the end of the instruction.
For instance, one can write
#+NAME: simple-affectations-example
#+BEGIN_SRC pugs :exports both :results output
import math; // to load the sin function
let a:N;
let b:N;
let x:R;
a=3;
b=2+a;
x=sin(b)+a;
cout << "a = " << a << " b = " << b << " x = " << x << "\n";
#+END_SRC
In this example, we import the ~math~ module which provides the ~sin~
function and we use the ~N~ data type of natural integers
($\mathbb{N}\equiv\mathbb{Z}_{\ge0}$).
Running the example gives the following result.
#+results: simple-affectations-example
#+BEGIN_warning
Actually, *separating* the *declaration* from the *initialization* of a
variable is quite *dangerous*. This is prone to errors and can lead to
the use of undefined values. During the compilation of scripts, ~pugs~
detects uninitialized values. For instance,
#+NAME: uninitialized-variable
#+BEGIN_SRC pugs-error :exports both :results output
let x:R;
let y:R;
y=x+1;
#+END_SRC
produces the following compilation error
#+results: uninitialized-variable
For more complex constructions, it can be very difficult to detect it
at /compile time/, this is why it is *encouraged* to use variable
definition (see [[definition-simple-variables]]).
Observe nonetheless that ~pugs~ checks at /run time/ that used variables
are correctly defined. If not, a /runtime/ error is produced.
#+END_warning
**** Definition of simple variables<<definition-simple-variables>>
The best way to define variables is the following. To define a
variable ~v~ of a given type ~V~, from an expression one writes
#+BEGIN_SRC pugs :exports source
let v:V, v = expression;
#+END_SRC
- ~let~ is the declaration keyword,
- ~v~ is the variable name,
- ~:~ is a separation token (can be read as "/in/" in this context),
- ~V~ is the identifier of the type,
- ~,~ is the separator that indicates the beginning of the affectation,
- ~expression~ is some code that provides a value of type ~V~ (it can be
another variable, an expression, the result of a function,...), and
- ~;~ marks the end of the instruction (here the definition of ~v~).
A practical example is
#+NAME: simple-definition-example
#+BEGIN_SRC pugs :exports both :results output
import math; // to load the sin function
let a:N, a=3;
let b:N, b=2+a;
let x:R, x=sin(b)+a;
cout << "a = " << a << " b = " << b << " x = " << x << "\n";
#+END_SRC
which produces the result
#+results: simple-definition-example
*** Blocks and lifetime of variables<<blocks-and-life-time>>
In pugs scripts, variables have a precise lifetime. They are defined
within scopes. The main scope is implicitly defined and sub-scopes are
enclosed between curly bracket pairs: ~{~ and ~}~. Following ~C++~, a
variable exists (can be used) as soon as it has been declared and
until the end of the scope (where it has been declared). At this point
we give a few examples.
**** A variable cannot be used before its declaration
#+NAME: undeclare-variable
#+BEGIN_SRC pugs-error :exports both :results output
n = 3;
let n:N;
#+END_SRC
#+results: undeclare-variable
**** A variable cannot be used after its definition scope
#+NAME: out-of-scope-variable-use
#+BEGIN_SRC pugs-error :exports both :results output
{
let n:N, n = 3;
n = n+2;
}
cout << n << "\n";
#+END_SRC
#+results: out-of-scope-variable-use
**** Variable name can be reused in an enclosed scope
#+NAME: nested-scope-variable-example
#+BEGIN_SRC pugs :exports both :results output
let n:N, n = 0; // global variable
{
cout << "global scope n = " << n << "\n";
let n:N, n = 1; // scope level 1 variable
{
cout << "scope level 1 n = " << n << "\n";
let n:N, n = 2; // scope level 2 variable
cout << "scope level 2 n = " << n << "\n";
}
{
cout << "scope level 1 n = " << n << "\n";
let n:N, n = 4; // scope level 2.2 variable
cout << "scope level 2.2 n = " << n << "\n";
}
cout << "scope level 1 n = " << n << "\n";
}
cout << "global scope n = " << n << "\n";
#+END_SRC
#+results: nested-scope-variable-example
This example is self explanatory. Obviously such constructions are
generally *bad ideas*. This kind of constructions can appear in loops
where the variables defined in blocks follow the same lifetime rules.
*** Basic types<<basic-types>>
Basic types in ~pugs~ are boolean (~B~), natural integers (~N~), integers (~Z~),
real (~R~), small vectors (~R^1~, ~R^2~ and ~R^3~), small matrices (~R^1x1~, ~R^2x2~
and ~R^3x3~) and strings (~string~).
#+BEGIN_note
Observe that while mathematically, obviously $\mathbb{R} = \mathbb{R}^1
= \mathbb{R}^{1\times1}$, the data types ~R~, ~R^1~ and ~R^1x1~ are different
in ~pugs~ and are *not implicitly* convertible from one to the other!
This may sound strange but there are few reasons for that.
- First, these are the reflect of internal ~pugs~ ~C++~-data types that
are used to write algorithms. In its core design pugs aim at writing
numerical methods generically with regard to the dimension. One of
the ingredients to achieve this purpose is to use dimension $1$
vectors and matrices when some algorithms reduce to dimension $1$
instead of ~double~ values. To avoid ambiguity that may arise in some
situations (this can lead to very tricky code), we decided to forbid
automatic conversions of these types with ~double~. When designing the
language, we adopted the same rule to avoid ambiguity.
- A second reason is connected to the first one. Since ~pugs~ aims at
providing numerical methods for problems in dimension $1$, $2$ or
$3$, this allows to distinguish the nature of the underlying objects.
- It is natural to consider that the coordinates of the vertices
defining a mesh in dimension $d$ are elements of $\mathbb{R}^d$,
- or that a velocity or a displacement are also defined as
$\mathbb{R}^d$ values.
Thus using ~R^1~ in dimension $1$ for this kind data precise their
nature in some sense .
#+END_note
**** Expression types
The language associates statically some types to several special
expressions.
- Special boolean (type ~B~) expressions. Two *keywords* allow to define
boolean values: ~true~ and ~false~.
- Integers (type ~Z~) are defined as a contiguous list of digits.
- For instance, the code ~0123~ is interpreted as the integer $123$.
- However the sequence ~123 456~ is *not interpreted* as $123456$ but as
the two integers $123$ and $456$.
- Real values (type ~R~) use the same syntax as in ~C++~. For instance,
the following expressions are accepted to define the number $1.23$.
#+BEGIN_SRC pugs :exports both
1.23;
0.123E1;
.123e+1;
123e-2;
12.3E-1;
#+END_SRC
- Small vectors values (types ~R^1~, ~R^2~ and ~R^3~) are written through
square brackets. Each component of vector values must be a scalar
expression.
#+BEGIN_SRC pugs :exports both
[1]; // R^1 value
[1,2]; // R^2 value
[1,2,3]; // R^3 value
#+END_SRC
- Small matrices values (types ~R^1x1~, ~R^2x2~ and ~R^3x3~) are written by
lines through square brackets. Each line is enclosed between square
brackets.
#+BEGIN_SRC pugs :exports both
[[1]]; // R^1x1 value
[[1,2],[3,4]]; // R^2x2 value
[[1,2,3],[4,5,6],[7,8,9]]; // R^3x3 value
#+END_SRC
- ~string~ values are defined as the set of characters enclosed between
two double quotes ( ~"~ ). The string /Hello world!/ would be simply
written as ~"Hello world!"~. Strings support the following escape
sequences (similarly to ~C++~):
#+LATEX: \definecolor{contiYellow}{RGB}{255,165,0}
#+LATEX: \rowcolors[]{2}{contiYellow!5}{contiYellow!20}
| escape | meaning |
|--------+-----------------|
| ~\'~ | single quote |
| ~\"~ | double quote |
| ~\?~ | question mark |
| ~\\~ | backslash |
| ~\a~ | audible bell |
| ~\b~ | backspace |
| ~\f~ | new page |
| ~\n~ | new line |
| ~\r~ | carriage return |
| ~\t~ | horizontal tab |
| ~\v~ | vertical tab |
These special characters are not interpreted by ~pugs~ itself but
interpreted by the system when an output is created. They are just
allowed in the definition of a ~string~.
**** Variables of basic types are stored by value
This means that each variable of basic type allocates its own memory
to store its data. This is the natural behavior for variables. It is
illustrated in the following example
#+NAME: basic-type-value-storage
#+BEGIN_SRC pugs :exports both :results output
let a:N, a = 3;
let b:N, b = a;
a = 1;
cout << "a = " << a << " b = " << b << "\n";
#+END_SRC
which produces the expected result
#+results: basic-type-value-storage
When defining ~b~, the *value* contained in ~a~ is copied to set the value
of ~b~. Thus changing ~a~'s value does not impact the variable ~b~.
**** Variables of basic types are mutable
In ~pugs~ the only variables that are mutable (their value can be
*modified*) are of basic types. Executing the following code
#+NAME: basic-type-mutable-value
#+BEGIN_SRC pugs :exports both :results output
let a:N, a = 2;
a += 3;
cout << "a = " << a << "\n";
#+END_SRC
gives
#+results: basic-type-mutable-value
which is not a surprise. However, the use of the ~+=~ operator results
in the modification of the stored value. There is no copy.
Actually, this is not really important from the user point of
view. One just have to keep in mind that, as it will be depicted
after, high-level variables *are not mutable*: their values can be
*replaced* by a new ones but *cannot be modified*.
*** Implicit type conversions<<implicit-conversion>>
In order to avoid ambiguities, in ~pugs~, there is *no implicit*
conversion in general.
#+BEGIN_note
Actually, there are only three situations for which implicit type
conversion can occur: when the value is
- given as a parameter of a function,
- used as the returned value of a function, or
- used to define a tuple
This will be illustrated in section [[functions]] and [[tuples]]
#+END_note
This means that all affectations, unary operators and binary operators
are defined explicitly for supported types.
Here is a table of implicit type conversions *when allowed*.
| expected type | convertible types |
|---------------+------------------------------------------------|
| ~N~ | ~B~, ~Z~ |
| ~Z~ | ~B~, ~N~ |
| ~R~ | ~B~, ~N~, ~Z~ |
| ~R^1~ | ~B~, ~N~, ~Z~, ~R~ |
| ~R^2~ | ~0~ (special value) |
| ~R^3~ | ~0~ (special value) |
| ~R^1x1~ | ~B~, ~N~, ~Z~, ~R~ |
| ~R^2x2~ | ~0~ (special value) |
| ~R^3x3~ | ~0~ (special value) |
| ~string~ | ~B~, ~N~, ~Z~, ~R~, ~R^1~, ~R^2~, ~R^3~, ~R^1x1~, ~R^2x2~, ~R^3x3~ |
*** Operators
**** Affectation operators
In the ~pugs~ language, the affectation operators are the following.
| operator | description |
|----------+----------------------|
| ~=~ | affectation operator |
|----------+----------------------|
| ~+=~ | increment operator |
| ~-=~ | decrement operator |
| ~*=~ | multiply operator |
| ~/=~ | divide operator |
#+BEGIN_note
It is important to note that in ~pugs~ language, affectation operators
have *no* return value. This is a notable difference with ~C~ or
~C++~. Again, this is done to avoid common mistakes that can be
difficult to address. For instance, the following ~C++~ code is valid
but does not produce the expected result.
#+BEGIN_SRC C++ :exports source
bool b = true;
// do things ...
if (b=false) {
// do other things ...
}
#+END_SRC
Obviously the mistake is that the test should have been ~(b==false)~,
since otherwise, the conditional block is never executed (in ~C++~, the
result of an affectation is the value affected to the variable, which
is always false in that case.
This cannot happen with ~pugs~. A similar example
#+NAME: no-affectation-result
#+BEGIN_SRC pugs-error :exports both :results output
let b:B, b=true;
// do things
if (b=false) {
// do other things
}
#+END_SRC
produces the following error
#+results: no-affectation-result
Actually, affectations are /expressions/ in ~C++~. In ~pugs~, affectations
are /instructions/.
#+END_note
***** List of defined operator ~=~ for basic types.
As already mentioned, operator ~=~ is defined for *all* types in ~pugs~ if
the right hand side expression has the *same* type as the left hand side
variable. This is true for basic types as well as for high-level
types.
We now give the complete list of supported ~=~ affectations. The lists
are sorted by type of left hand side variable.
- ~B~: boolean left hand side variable. One is only allowed to affect boolean
values.
| ~B =~ allowed expression type |
|-----------------------------|
| ~B~ |
- ~N~: natural integer ($\mathbb{N}$ or $\mathbb{Z}_{\ge0}$) left hand side variable.
| ~N =~ allowed expression type |
|-----------------------------|
| ~B~ |
| ~N~ |
| ~Z~ (for convenience) |
- ~Z~: integer ($\mathbb{Z}$) left hand side variable.
| ~Z =~ allowed expression type |
|-----------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
- ~R~: real ($\mathbb{R}$) left hand side variable.
| ~R =~ allowed expression type |
|-----------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^1~: vector of dimension 1 ($\mathbb{R}^1$) left hand side variable.
| ~R^1 =~ allowed expression type |
|-------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
| ~R^1~ |
- ~R^2~: vector of dimension 2 ($\mathbb{R}^2$) left hand side variable.
| ~R^2 =~ allowed expression type |
|---------------------------------------------|
| ~R^2~ |
| ~0~ (special value) |
| list of 2 scalar (~B~, ~N~, ~Z~ or ~R~) expressions |
An example of initialization using an $\mathbb{R}^2$ value or the special value ~0~ is
#+NAME: affectation-to-R2-from-list
#+BEGIN_SRC pugs :exports both :results output
let u:R^2, u = [-3, 2.5];
let z:R^2, z = 0;
cout << "u = " << u << "\n";
cout << "z = " << z << "\n";
#+END_SRC
which produces
#+RESULTS: affectation-to-R2-from-list
Observe that ~0~ is a special value and *treated as a keyword*. There is no conversion from
integer values. For instance:
#+NAME: R2-invalid-integer-affectation
#+BEGIN_SRC pugs-error :exports both :results output
let z:R^2, z = 1-1;
#+END_SRC
produces the compilation error
#+results: R2-invalid-integer-affectation
- ~R^3~: vector of dimension 3 ($\mathbb{R}^3$) left hand side variable.
| ~R^3 =~ allowed expression type |
|---------------------------------------------|
| ~R^3~ |
| ~0~ (special value) |
| list of 3 scalar (~B~, ~N~, ~Z~ or ~R~) expressions |
An example of initialization is
#+NAME: affectation-to-R3-from-list
#+BEGIN_SRC pugs :exports both :results output
let u:R^3, u = [-3, 2.5, 1E-2];
let z:R^3, z = 0;
cout << "u = " << u << "\n";
cout << "z = " << z << "\n";
#+END_SRC
the output is
#+RESULTS: affectation-to-R3-from-list
- ~R^1x1~: matrix of dimensions $1\times1$ ($\mathbb{R}^{1\times1}$) left hand side variable.
| ~R^1x1 =~ allowed expression type |
|---------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
| ~R^1x1~ |
- ~R^2x2~: matrix of dimension $2\times2$ ($\mathbb{R}^{2\times2}$) left hand side variable.
| ~R^2x2 =~ allowed expression type |
|---------------------------------------------|
| ~R^2x2~ |
| ~0~ (special value) |
| list of 4 scalar (~B~, ~N~, ~Z~ or ~R~) expressions |
An example of initialization using an $\mathbb{R}^{2\times2}$ value or
the special value ~0~ is
#+NAME: affectation-to-R2x2-from-list
#+BEGIN_SRC pugs :exports both :results output
let u:R^2x2, u = [[-3, 2.5],
[ 4, 1.2]];
let z:R^2x2, z = 0;
cout << "u = " << u << "\n";
cout << "z = " << z << "\n";
#+END_SRC
which produces
#+RESULTS: affectation-to-R2x2-from-list
- ~R^3x3~: matrix of dimension $3\times3$ ($\mathbb{R}^{3\times3}$) left hand side variable.
| ~R^3x3 =~ allowed expression type |
|---------------------------------------------|
| ~R^3x3~ |
| ~0~ (special value) |
| list of 9 scalar (~B~, ~N~, ~Z~ or ~R~) expressions |
An example of initialization is
#+NAME: affectation-to-R3x3-from-list
#+BEGIN_SRC pugs :exports both :results output
let u:R^3x3, u = [[ -3, 2.5, 1E-2],
[ 2, 1.7, -2],
[1.2, 4, 2.3]];
let z:R^3x3, z = 0;
cout << "u = " << u << "\n";
cout << "z = " << z << "\n";
#+END_SRC
the output is
#+RESULTS: affectation-to-R3x3-from-list
- ~string~ left hand side variable. Expressions of any basic type can be
used as the right hand side.
| ~string =~ allowed expression type |
|----------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
| ~R^1~ |
| ~R^2~ |
| ~R^3~ |
| ~R^1x1~ |
| ~R^2x2~ |
| ~R^3x3~ |
| ~string~ |
***** List of defined operator ~+=~ for basic types.
- ~B~: the ~+=~ operator is not defined for left hand side boolean
variables.
- ~N~: natural integer ($\mathbb{N}$ or $\mathbb{Z}_{\ge0}$) left hand side variable.
| ~N +=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ (for convenience) |
- ~Z~: integer ($\mathbb{Z}$) left hand side variable.
| ~Z +=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
- ~R~: real ($\mathbb{R}$) left hand side variable.
| ~R +=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^1~: vector of dimension 1 ($\mathbb{R}^1$) left hand side variable.
| ~R^1 +=~ allowed expression type |
|--------------------------------|
| ~R^1~ |
- ~R^2~: vector of dimension 2 ($\mathbb{R}^2$) left hand side variable.
| ~R^2 +=~ allowed expression type |
|--------------------------------|
| ~R^2~ |
- ~R^3~: vector of dimension 3 ($\mathbb{R}^3$) left hand side variable.
| ~R^3 +=~ allowed expression type |
|--------------------------------|
| ~R^3~ |
- ~R^1x1~: matrix of dimensions $1\times1$ ($\mathbb{R}^{1\times1}$) left hand side variable.
| ~R^1x1 +=~ allowed expression type |
|----------------------------------|
| ~R^1x1~ |
- ~R^2x2~: matrix of dimension $2\times2$ ($\mathbb{R}^{2\times2}$) left hand side variable.
| ~R^2x2 +=~ allowed expression type |
|----------------------------------|
| ~R^2x2~ |
- ~R^3x3~: matrix of dimension $3\times3$ ($\mathbb{R}^{3\times3}$) left hand side variable.
| ~R^3x3 +=~ allowed expression type |
|----------------------------------|
| ~R^3x3~ |
- ~string~ left hand side variable. Expressions of any basic type can be
used as the right hand side.
| ~string +=~ allowed expression type |
|-----------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
| ~R^1~ |
| ~R^2~ |
| ~R^3~ |
| ~R^1x1~ |
| ~R^2x2~ |
| ~R^3x3~ |
| ~string~ |
***** List of defined operator ~-=~ for basic types.
- ~B~: the ~-=~ operator is not defined for left hand side boolean variables.
- ~N~: natural integer ($\mathbb{N}$ or $\mathbb{Z}_{\ge0}$) left hand side variable.
| ~N -=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ (for convenience) |
- ~Z~: integer ($\mathbb{Z}$) left hand side variable.
| ~Z -=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
- ~R~: real ($\mathbb{R}$) left hand side variable.
| ~R -=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^1~: vector of dimension 1 ($\mathbb{R}^1$) left hand side variable.
| ~R^1 -=~ allowed expression type |
|--------------------------------|
| ~R^1~ |
- ~R^2~: vector of dimension 2 ($\mathbb{R}^2$) left hand side variable.
| ~R^2 -=~ allowed expression type |
|--------------------------------|
| ~R^2~ |
- ~R^3~: vector of dimension 3 ($\mathbb{R}^3$) left hand side variable.
| ~R^3 -=~ allowed expression type |
|--------------------------------|
| ~R^3~ |
- ~R^1x1~: matrix of dimensions $1\times1$ ($\mathbb{R}^{1\times1}$) left hand side variable.
| ~R^1x1 -=~ allowed expression type |
|----------------------------------|
| ~R^1x1~ |
- ~R^2x2~: matrix of dimension $2\times2$ ($\mathbb{R}^{2\times2}$) left hand side variable.
| ~R^2x2 -=~ allowed expression type |
|----------------------------------|
| ~R^2x2~ |
- ~R^3x3~: matrix of dimension $3\times3$ ($\mathbb{R}^{3\times3}$) left hand side variable.
| ~R^3x3 -=~ allowed expression type |
|----------------------------------|
| ~R^3x3~ |
- ~string~: the ~-=~ operator is not defined for left hand side string variables.
***** List of defined operator ~*=~ for basic types.
- ~B~: the ~*=~ operator is not defined for left hand side boolean variables.
- ~N~: natural integer ($\mathbb{N}$ or $\mathbb{Z}_{\ge0}$) left hand side variable.
| ~N *=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ (for convenience) |
- ~Z~: integer ($\mathbb{Z}$) left hand side variable.
| ~Z *=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
- ~R~: real ($\mathbb{R}$) left hand side variable.
| ~R *=~ allowed expression type |
|------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^1~: vector of dimension 1 ($\mathbb{R}^1$) left hand side variable.
| ~R^1 *=~ allowed expression type |
|--------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^2~: vector of dimension 2 ($\mathbb{R}^2$) left hand side variable.
| ~R^2 *=~ allowed expression type |
|--------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^3~: vector of dimension 3 ($\mathbb{R}^3$) left hand side variable.
| ~R^3 *=~ allowed expression type |
|--------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^1x1~: matrix of dimensions $1\times1$ ($\mathbb{R}^{1\times1}$) left hand side variable.
| ~R^1x1 *=~ allowed expression type |
|----------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^2x2~: matrix of dimension $2\times2$ ($\mathbb{R}^{2\times2}$) left hand side variable.
| ~R^2x2 *=~ allowed expression type |
|----------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^3x3~: matrix of dimension $3\times3$ ($\mathbb{R}^{3\times3}$) left hand side variable.
| ~R^3x3 *=~ allowed expression type |
|----------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
#+BEGIN_note
Observe that for these small matrix types ($\mathbb{R}^{d\times d}$) the
construction ~A *= B;~ where ~B~ is a matrix of the same type as ~A~ is not
allowed. The main reason for that is that for $d>1$ this operation has
no interests since it requires a temporary. One will see bellow that
it is possible to write ~A = A*B;~ if needed.
#+END_note
- ~string~: the ~*=~ operator is not defined for left hand side string variables.
***** List of defined operator ~/=~ for basic types.
- ~B~: the ~/=~ operator is not defined for left hand side boolean variables.
- ~N~: natural integer ($\mathbb{N}$ or $\mathbb{Z}_{\ge0}$) left hand side variable.
| ~N /=~ allowed expression type |
|------------------------------|
| ~N~ |
| ~Z~ (for convenience) |
- ~Z~: integer ($\mathbb{Z}$) left hand side variable.
| ~Z /=~ allowed expression type |
|------------------------------|
| ~N~ |
| ~Z~ |
- ~R~: real ($\mathbb{R}$) left hand side variable.
| ~R /=~ allowed expression type |
|------------------------------|
| ~N~ |
| ~Z~ |
| ~R~ |
- ~R^d~: the ~/=~ operator is not defined for left hand side vector (of
dimension $d\in\{1,2,3\}$) variables.
- ~R^dxd~: the ~/=~ operator is not defined for left hand side matrix (of
dimension $d\times d$ with $d\in\{1,2,3\}$) variables.
**** Unary operators
The ~pugs~ language allows the following tokens as unary operators
| operator | description |
|----------+----------------------|
| ~not~ | not operator |
| ~+~ | plus unary operator |
| ~-~ | minus unary operator |
|----------+----------------------|
| ~++~ | increment operator |
| ~--~ | decrement operator |
|----------+----------------------|
| ~[]~ | access operator |
The ~not~, ~+~ and ~-~ operators apply to the *expression* on their right. ~++~
and ~--~ operators apply only to a *variable* that can be positioned
before (pre increment/decrement) or after the token (post
increment/decrement). These operators are also inspired from their ~C++~
counterparts for commodity.
The ~+~ unary operator is a convenient operator that is *elided* when
parsing the script.
For basic types, when the operators ~not~, ~+~, ~-~, ~++~ or ~--~ are defined,
they return a value of the same type as the argument (except for the
operator ~-~ if the argument is a ~B~ or a ~N~, then the result is a
~Z~). These operators can be defined for high-level types.
- The ~not~ operator is only defined for boolean values (~B~).
- The ~-~ unary operator is defined for numeric basic types: ~B~,
~N~, ~Z~, ~R~, ~R^1~, ~R^2~, ~R^3~, ~R^1x1~, ~R^2x2~ and ~R^3x3~. It is not defined
for ~string~ variables.
- Pre and post increment operators, ~--~ and ~++~, are defined for integer
types: ~N~ and ~Z~. They are not defined for ~B~, ~R~, ~R^1~, ~R^2~, ~R^3~, ~R^1x1~,
~R^2x2~, ~R^3x3~ and ~string~ variables.
Note that the pre increment/decrement operators behave slightly
differently than their ~C++~ counterparts since they are not allowed to
be chained. In ~C++~, the following code is allowed
#+BEGIN_SRC C++ :exports source
int i = 0;
int j = ++ ++i;
#+END_SRC
In ~pugs~, it is forbidden:
#+NAME: double-pre-incr-result
#+BEGIN_SRC pugs-error :exports both :results output
let i:N, i=0;
let j:N, j = ++ ++i;
#+END_SRC
produces the compilation error
#+results: double-pre-incr-result
Again, this is done to simplify the syntax and to avoid weird
constructions.
- Access operators are only defined for small vectors ~R^d~ and small
matrices ~R^dxd~. To avoid use of uninitialized variables (or
partially uninitialized variables), these are ~read-only~ access
operators. Their syntax is the following.
#+NAME: Rd-Rdxd-access-operator
#+BEGIN_SRC pugs :exports both :results output
let x:R^2, x = [1,2];
let A:R^3x3, A = [[1,2,3],[4,5,6],[7,8,9]];
cout << "x[0] = " << x[0] << "\nx[1] = " << x[1] << "\n";
cout << "A[0,0] = " << A[0,0] << "\nA[2,1] = " << A[2,1] << "\n";
#+END_SRC
This code produces
#+results: Rd-Rdxd-access-operator
**** Binary operators
Syntax for binary operators follows again a classical structure: if
~exp1~ and ~exp2~ denotes two expressions and if ~op~ denotes a binary
operator, one simply writes ~exp1 op exp2~.
Here is the list of binary operators
| keyword | operator |
|---------+-----------------------|
| ~and~ | logic and |
| ~or~ | logic or |
| ~xor~ | logic exclusive or |
|---------+-----------------------|
| ~==~ | equality |
| ~!=~ | non-equality |
| ~<~ | lower than |
| ~<=~ | lower or equal than |
| ~>~ | greater than |
| ~>=~ | greater or equal than |
|---------+-----------------------|
| ~<<~ | shift left |
| ~>>~ | shift right |
|---------+-----------------------|
| ~+~ | sum |
| ~-~ | difference |
| ~*~ | product |
| ~/~ | division |
Binary operators can be defined for high-level types. For basic types,
they follow a few rules.
- Logical operators ~and~, ~or~ and ~xor~ are defined for boolean operands
(type is ~B~) only. The result of the expression is a boolean.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\left|
\begin{array}{rl}
\mathtt{and}:&\quad \mathbb{B} \times \mathbb{B} \to \mathbb{B}\\
\mathtt{or}:& \quad\mathbb{B} \times \mathbb{B} \to \mathbb{B}\\
\mathtt{xor}:& \quad \mathbb{B} \times \mathbb{B} \to \mathbb{B}
\end{array}
\right.
\end{equation*}
#+end_src
- Comparison operators ~==~, ~!=~, ~<~, ~<=~, ~>~ and ~>=~ are defined for all
basic scalar type and return a boolean value.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S}_1, \mathbb{S}_2 \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R}\},
\quad
\left|
\begin{array}{rl}
\mathtt{==}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{B}\\
\mathtt{!=}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{B}\\
\mathtt{<}: & \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{B}\\
\mathtt{<=}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{B}\\
\mathtt{>}: & \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{B}\\
\mathtt{>=}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{B}
\end{array}
\right.
\end{equation*}
#+end_src
When comparing a boolean value (type ~B~) with another scalar value
type (~N~, ~Z~ or ~R~), the value ~true~ is interpreted as $1$ and the value
~false~ as $0$.
\\
For vector and matrix basic types, the only allowed operators are ~==~
and ~!=~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall d \in \{1,2,3\},\quad
\left|
\begin{array}{rl}
\mathtt{==}:& \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{B}\\
\mathtt{==}:& \mathbb{R}^{d \times d} \times \mathbb{R}^{d \times d} \to \mathbb{B}\\
\mathtt{!=}:& \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{B}\\
\mathtt{!=}:& \mathbb{R}^{d \times d} \times \mathbb{R}^{d \times d} \to \mathbb{B}
\end{array}
\right.
\end{equation*}
#+end_src
\\
This is also the case for ~string~ values: only allowed operators are
~==~ and ~!=~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\left|
\begin{array}{rl}
\mathtt{==}:& \mathtt{string} \times \mathtt{string} \to \mathbb{B}\\
\mathtt{!=}:& \mathtt{string} \times \mathtt{string} \to \mathbb{B}
\end{array}
\right.
\end{equation*}
#+end_src
- Shift operators ~<<~ and ~>>~ are not used to define binary operators
between two basic types.
- Arithmetic operators (~+~, ~-~, ~*~ and ~/~) are defined for a set of
combinations of basic types. We classify them by their returned
types.
- Their is no arithmetic operation that returns a boolean ~B~.
- Operators that return a natural integer ~N~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S}_1, \mathbb{S}_2 \in \{\mathbb{B},\mathbb{N}\},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{N}\\
\mathtt{*}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{N}
\end{array}
\right.
\end{equation*}
#+end_src
Boolean values (type ~B~) are not allowed as the right operand of a
division.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S} \in \{\mathbb{B},\mathbb{N}\},
\mathtt{/}: \mathbb{S} \times \mathbb{N} \to \mathbb{N}
\end{equation*}
#+end_src
Observe that ~-~ is *not* in the list.
- Operators that return an integer ~Z~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S}_1, \mathbb{S}_2 \in \{\mathbb{B},\mathbb{N},\mathbb{Z}\},
\mbox{ such that }\mathbb{S}_1 = \mathbb{Z}\mbox{ or }\mathbb{S}_2 = \mathbb{Z},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{Z}\\
\mathtt{-}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{Z}\\
\mathtt{*}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{Z}\\
\mathtt{/}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{Z}
\end{array}
\right.
\end{equation*}
#+end_src
The divide operator does not allow boolean values as a right
operand.
#+begin_src latex :results drawer :exports results
\begin{align*}
\forall \mathbb{S}_1 \in \{\mathbb{B},\mathbb{N},\mathbb{Z}\}&\mbox{ and } \mathbb{S}_2 \in \{\mathbb{N},\mathbb{Z}\},
\mbox{ such that }\mathbb{S}_1 = \mathbb{Z}\mbox{ or }\mathbb{S}_2 = \mathbb{Z},\\
&\mathtt{/}: \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{Z}.
\end{align*}
#+end_src
Finally the following operator is also defined
#+begin_src latex :results drawer :exports results
\begin{equation*}
\mathtt{-}: \mathbb{N} \times \mathbb{N} \to \mathbb{Z}
\end{equation*}
#+end_src
The difference of two natural integers returns a /signed/ integer.
- Operators that return a real ~R~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S}_1, \mathbb{S}_2 \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R}\},
\mbox{ such that }\mathbb{S}_1 = \mathbb{R}\mbox{ or }\mathbb{S}_2 = \mathbb{R},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{R}\\
\mathtt{-}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{R}\\
\mathtt{*}:& \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{R}
\end{array}
\right.
\end{equation*}
#+end_src
The divide operator does not allow boolean values as a right
operand.
#+begin_src latex :results drawer :exports results
\begin{align*}
\forall \mathbb{S}_1 \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R}\}&\mbox{ and } \mathbb{S}_2 \in \{\mathbb{N},\mathbb{Z},\mathbb{R}\},
\mbox{ such that }\mathbb{S}_1 = \mathbb{R}\mbox{ or }\mathbb{S}_2 = \mathbb{R},\\
&\mathtt{/}: \mathbb{S}_1 \times \mathbb{S}_2 \to \mathbb{Z}.
\end{align*}
#+end_src
- Operators that return a small vector ~R^d~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall d \in\{1,2,3\},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d\\
\mathtt{-}:& \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d\\
\mathtt{*}:& \mathbb{B} \times \mathbb{R}^d \to \mathbb{R}^d\\
\mathtt{*}:& \mathbb{N} \times \mathbb{R}^d \to \mathbb{R}^d\\
\mathtt{*}:& \mathbb{Z} \times \mathbb{R}^d \to \mathbb{R}^d\\
\mathtt{*}:& \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}^d\\
\mathtt{*}:& \mathbb{R}^{d\times d} \times \mathbb{R}^d \to \mathbb{R}^d
\end{array}
\right.
\end{equation*}
#+end_src
- Operators that return a small matrix ~R^dxd~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall d \in\{1,2,3\},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{R}^{d\times d} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}\\
\mathtt{-}:& \mathbb{R}^{d\times d} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}\\
\mathtt{*}:& \mathbb{B} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}\\
\mathtt{*}:& \mathbb{N} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}\\
\mathtt{*}:& \mathbb{Z} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}\\
\mathtt{*}:& \mathbb{R} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}\\
\mathtt{*}:& \mathbb{R}^{d\times d} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d}
\end{array}
\right.
\end{equation*}
#+end_src
- Operators that return a ~string~.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S} \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R}, \mathbb{R}^1, \mathbb{R}^2, \mathbb{R}^3, \mathbb{R}^{1\times1}, \mathbb{R}^{2\times2}, \mathbb{R}^{3\times3}\},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{S} \times \mathtt{string} \to \mathtt{string}\\
\mathtt{+}:& \mathtt{string} \times \mathbb{S} \to \mathtt{string}\\
\mathtt{+}:& \mathtt{string} \times \mathtt{string} \to \mathtt{string}
\end{array}
\right.
\end{equation*}
#+end_src
**** Operator precedence and associativity
To avoid confusions, the operators precedence in ~pugs~ language follows
the same rules as in ~C++~.
This is summarized in the following table, where ~a~ and ~b~ denotes two
expressions.
| Precedence | Operator |
|------------+----------|
| 1 | ~a++~ |
| | ~a--~ |
| | ~a[]~ |
|------------+----------|
| 2 | ~++a~ |
| | ~--a~ |
| | ~not a~ |
|------------+----------|
| 3 | ~a*b~ |
| | ~a/b~ |
|------------+----------|
| 4 | ~a+b~ |
| | ~a-b~ |
|------------+----------|
| 5 | ~a<<b~ |
| | ~a>>b~ |
|------------+----------|
| 6 | ~a<b~ |
| | ~a<=b~ |
| | ~a>b~ |
| | ~a>=b~ |
|------------+----------|
| 7 | ~a==b~ |
| | ~a!=b~ |
|------------+----------|
| 8 | ~a=b~ |
| | ~a+=b~ |
| | ~a-=b~ |
| | ~a*=b~ |
| | ~a/=b~ |
As already said, we forbid some constructions to try to avoid spurious
constructions. By construction associativity of many operators makes
no sense in ~pugs~ language (affectations and increment/decrement
operators for instance).
For all other operators, the associativity rule is *left to right*. Thus
the following code
#+BEGIN_SRC pugs :exports source
- 1 + 3 - 4 + 2;
#+END_SRC
is equivalent to
#+BEGIN_SRC pugs :exports source
(((- 1) + 3) - 4) + 2;
#+END_SRC
Obviously ~pugs~ allows the use of parenthesis to write
expressions. It is enough to give a simple example.
#+NAME: parenthesis-precedence
#+BEGIN_SRC pugs :exports both :results output
cout << " 2 + 3 * 4 = " << 2 + 3 * 4 << "\n";
cout << "(2 + 3) * 4 = " << (2 + 3) * 4 << "\n";
#+END_SRC
the output is
#+RESULTS: parenthesis-precedence
*** High-level types<<high-level-types>>
Aside from the basic types described in the previous section, ~pugs~
also deals with "high-level" types. This term is more to understand as
"non-basic types". The ~pugs~ language is not object oriented to keep it
simple.
To fix ideas, let us give examples of a few high-level data
types. These can be meshes (the ~mesh~ type), output streams (the
~ostream~ type), boundary descriptors (~boundary~), quadrature formula
(~quadrature~), discrete functions (~Vh~),...
One can already see that the complexity of these types may vary a lot.
The main difference between these types and basic types is that,
high-level types are not available in directly the language but they
are loaded on demand (using the ~import~ keyword in the preamble of the
script).
The second difference is that data of these types are *constant*. More
precisely, the content of high-level variables can be replaced by a
new one *but* it cannot be modified. For this reason, the following
operators can never be applied to variables of this kind
| forbidden operators | description |
|---------------------+-----------------------------|
| ~++~ | pre/post increment operator |
| ~--~ | pre/post decrement operator |
| ~+=~ | assignment by sum |
| ~-=~ | assignment by difference |
| ~*=~ | assignment by product |
| ~/=~ | assignment by quotient |
We conclude by stating that if access operator ~[]~ can eventually be
overloaded for high-level types, it should be done with care. It is
not recommended.
Finally, the last difference lies in the fact that high-level types
use shallow copies and not value copies as it is the case for basic
types. This is transparent to the user and provides the intuitive
(similar) behavior since data of high-level variables are constant. To
illustrate this, let us consider the following example.
#+BEGIN_SRC pugs :exports both
import mesh;
let m1:mesh, m1 = cartesianMesh(0, [1,1], (10,10));
let m2:mesh, m2 = m1;
let m3:mesh, m3 = cartesianMesh(0, [1,1], (10,10));
m3 = m1;
#+END_SRC
In this example, we are dealing with 3 ~mesh~ variables.
- First, ~m1~ is defined as a uniform cartesian mesh in dimension 2 of
$]0,1[^2$ made of $10\times10$ identical square cells. Let us call this
resident mesh $\mathcal{M}_a$ for clarity.
- Next, the variable ~m2~ is defined as a copy of the variable ~m1~. It is
a copy of the variable, but its content is *the same*. The variable ~m2~
is also referring to the mesh $\mathcal{M}_a$: there is no
duplication of the mesh in memory.
- Then, the variable ~m3~ is defined to refer to a *new* mesh (called
$\mathcal{M}_b$. It is /similar/ to $\mathcal{M}_a$, but it is *not* the
same one! When ~m3~ is defined two meshes $\mathcal{M}_a$ and
$\mathcal{M}_b$ (and two distinct connectivities) are resident in
memory.
- Finally, the last instruction (~m3 = m1;~) sets ~m3~ to also refer
$\mathcal{M}_a$. Since no other variable refers to $\mathcal{M}_b$,
this mesh is destroyed (memory is freed). At the end of the program,
all the variables ~m1~, ~m2~ and ~m3~ are referring to $\mathcal{M}_a$
which is the only mesh that resides in memory.
*** Compound types
The ~pugs~ language allows to deal with compound types. The idea is to
define a list of variables as a member of a product space (each
variable belongs to one of them, each variable has a simple type:
basic or high-level).
**** Compound declaration
Let us provide an example to fix ideas.
#+NAME: compound-declaration
#+BEGIN_SRC pugs :exports both :results output
let (A,x,n): R^2x2*R^3*N;
A = [[1,2],[3,4]];
x = [2,4,6];
n = 2;
cout << "A = " << A << "\nx = " << x << "\nn = " << n << "\n";
#+END_SRC
the output is
#+RESULTS: compound-declaration
This is completely equivalent to declaring the variables one after the
other.
**** Compound definition
One can also use the following definition instruction
#+NAME: compound-definition
#+BEGIN_SRC pugs :exports both :results output
let (A,x,n): R^2x2*R^3*N, (A,x,n) = ([[1,2],[3,4]], [2,4,6], 2);
cout << "A = " << A << "\nx = " << x << "\nn = " << n << "\n";
#+END_SRC
which gives the same result
#+RESULTS: compound-definition
A potential mistake with this construction, is that variables
must appear in the same order in the declaration part and in the
affectation part of a definition. For instance, the following code is
invalid.
#+NAME: invalid-compound-definition
#+BEGIN_SRC pugs-error :exports both :results output
let (x,y):R*R, (y,x) = (0,1);
#+END_SRC
It produces the following error
#+results: invalid-compound-definition
which is easy to fix.
Another potential mistake is that all variables of the list are marked
as defined at the same time. Thus one cannot use construction like
this:
#+NAME: undeclared-compound-definition
#+BEGIN_SRC pugs-error :exports both :results output
let (x,y):R*R, (x,y) = (0,2+x);
#+END_SRC
It produces the following error
#+results: undeclared-compound-definition
While the variable ~x~ is defined *before* ~y~, this kind of construction is
forbidden. From a technical point of view, this behavior would be easy
to change (allow to use the fresh value of ~x~ in the definition of ~y~),
but this make the code unclear and this is not the purpose of compound
types.
#+BEGIN_note
Observe that there is no implicit conversion when dealing with
compound types. The ~=~ operators are used sequentially to set the
different variables.
#+END_note
**** Compound affectation
The last way to use compound types is illustrated by the following
example.
#+NAME: compound-affectation
#+BEGIN_SRC pugs :exports both :results output
let A: R^2x2;
let x: R^3;
let n: N;
(A,x,n) = ([[1,2],[3,4]], [2,4,6], 2);
cout << "A = " << A << "\nx = " << x << "\nn = " << n << "\n";
#+END_SRC
It produces again
#+results: compound-affectation
#+BEGIN_note
Observe that the only operator allowed for this kind of construction
is the operator ~=~.
#+END_note
**** Use of compound types
Actually using compound types the way it is presented in this
paragraph is not recommend. The purpose of compounds types and
compound affectations is related to functions. As one will see in
section [[functions]], functions can return compound values, thus compound
affectations (or definitions) are needed to get returned values in
that case.
*** Tuple types<<tuples>>
The last special type construction is the ability to deal with tuples
in the ~pugs~ language. The tuples we are describing here are lists of
data of a *unique* and *simple* type (one cannot use compound types to
define tuples). Tuples in the ~pugs~ language have *no relationship* with
~C++~'s ~std::tuple~ these are two completely different notions.
The list of values given to the tuple must be *implicitly* convertible
to the type of tuple elements (see the conversion table in section
[[implicit-conversion]]). There is no ambiguity, the implicit conversions
follow the rules of operator ~=~.
**** Tuple declaration and affectation
For instance, one may write
#+NAME: tuple-declaration-affectation
#+BEGIN_SRC pugs :exports both :results output
let x:(R);
x = (1,2,3.4,2);
cout << x << "\n";
#+END_SRC
Executing this code, one gets
#+results: tuple-declaration-affectation
**** Tuple definition
The definition syntax is also possible.
#+NAME: tuple-definition
#+BEGIN_SRC pugs :exports both :results output
let x:(R^2), x = ([1,2],[3.4,2], 0, [2,3]);
cout << x << "\n";
#+END_SRC
This code gives
#+results: tuple-definition
Observe that the special value ~0~ is used there.
**** Tuple purpose
Tuples variables are just lists of data of the same type. In the ~pugs~
language, one cannot access to a specific value of the list nor alter
one of them. This is not something that should ever change. Tuples are
not arrays! The ~pugs~ language is not meant to allow low-level
instructions.
The use case of tuples is to provide lists of data to the ~C++~
underlying methods. A classical example is to provide a set of
boundary conditions to a method.
** Statements
The ~pugs~ language supports classical statements to control the data
flow. For simplicity, these statements syntax follow their ~C++~
counterpart. The only statement that is not implemented in ~pugs~ is the
~switch...case~. This may change but in the one hand, up to now it was
never necessary (up to now, we did not encountered the need to chain
~if...else~ statements), and on the other hand, keeping the language as
simple as possible remains the policy in ~pugs~ development.
*** ~if...else~ statement
The simplest statement is the conditional statement ~if...else~. If a
condition is satisfied, the ~truestatement~ (a single instruction or
block of instructions) is executed. In the other case, if the /optional/
~else~ keyword is used, the ~falsestatement~ (a single instruction or
block of instructions) is executed.
#+BEGIN_SRC pugs :exports code
if (condition) truestatement
if (condition) truestatement else falsestatement
#+END_SRC
The condition itself *must be* a boolean value (of type ~B~).
#+NAME: if-instruction
#+BEGIN_SRC pugs :exports both :results output
if (2>1) cout << "yes: 2 > 1\n";
if (not (2>1)) cout << "hmm... 2 <= 1 ?\n";
#+END_SRC
generates the following expected output. The second output is
hopefully not executed.
#+results: if-instruction
It is probably better to use the ~else~ keyword to obtain the same
result.
An example of the use of the ~else~ keyword follows.
#+NAME: if-else-instruction
#+BEGIN_SRC pugs :exports both :results output
if (2<=1)
cout << "hmm... 2 <= 1 ?\n";
else
cout << "ok: 2 > 1\n";
#+END_SRC
#+results: if-else-instruction
However, it is recommended even in the case of single instructions in
the conditional statements to use instruction blocks:
#+NAME: if-else-block-of-one
#+BEGIN_SRC pugs :exports both :results output
import math;
let sin100_positive:B;
if (sin(100) > 0){
sin100_positive = true;
} else {
sin100_positive = false;
}
cout << "sin(100)>0: " << sin100_positive << "\n";
#+END_SRC
#+results: if-else-block-of-one
We give a final illustration
#+NAME: if-block
#+BEGIN_SRC pugs :exports both :results output
let m:R, m = 2.5;
let M:R, M = 2;
let has_swapped:B, has_swapped = false;
if (m>M){
let tmp:R, tmp = m;
m = M;
M = tmp;
has_swapped = true;
}
cout << "min = " << m << " max = " << M << " has_swapped = " << has_swapped << "\n";
#+END_SRC
#+results: if-block
*** ~for~ loops
~pugs~ allows to write ~for~ loops. It follows the ~C++~ syntax
#+BEGIN_SRC pugs :exports code
for (declarationinstruction ; condition ; postinstruction) statement
#+END_SRC
Following ~C++~, ~declarationinstruction~, ~condition~ and ~postinstruction~
are optional. The ~condition~ argument, if it is present, *must* be a
boolean value (type ~B~). If it is absent, the default value ~true~ is
used.
- The ~declarationinstruction~ is execute only *once* /before/ the beginning
of the loop. The lifetime of the variable declared here is defined
by the ~for~ instruction itself.
- The ~condition~ is evaluate /before/ each loop.
- The ~postinstruction~ is executed /after/ each loop.
- The ~statement~ is either a single instruction or a block of
instructions. The ~statement~ is executed if the ~condition~ has the
value ~true~.
For instance, one can write
#+NAME: for-block
#+BEGIN_SRC pugs :exports both :results output
let n:N, n = 10;
let sum:N, sum = 0;
for (let i:N, i=1; i<=n; ++i) {
sum += i;
}
cout << "sum = " << sum << "\n";
#+END_SRC
which gives as expected
#+results: for-block
The lifetime of the declared variable (in the ~declarationinstruction~
statement) is illustrated by the following example
#+NAME: for-scope-error
#+BEGIN_SRC pugs-error :exports both :results output
for (let i:N, i=0; i<2; ++i) {
cout << "i = " << i << "\n";
}
cout << "i = " << i << "\n";
#+END_SRC
Running this example produces the following error
#+results: for-scope-error
To fix the previous code, one can write
#+NAME: for-no-decl
#+BEGIN_SRC pugs :exports both :results output
let i:N, i=0;
for (; i<2; ++i) {
cout << "i = " << i << "\n";
}
cout << "i = " << i << "\n";
#+END_SRC
One gets
#+results: for-no-decl
*** ~do...while~ loops
The second kind of loops that is supported is the ~do...while~
construction which executes /at least/ one time the enclosed ~statement~.
#+BEGIN_SRC pugs :exports code
do statement while (condition);
#+END_SRC
The ~statement~ is either a single instruction or a block of
instructions. The ~condition~ is an expression of boolean value (type
~B~).
#+NAME: do-while-block
#+BEGIN_SRC pugs :exports both :results output
let sum:N, sum = 0;
let i:N, i = 0;
do {
sum += i;
++i;
} while (i<=10);
cout << "sum = " << sum << "\n";
#+END_SRC
It gives
#+results: do-while-block
*** ~while~ loops
The last kind of loops that is allowed in ~pugs~ language is the ~while~
loop.
#+BEGIN_SRC pugs :exports code
while (condition) statement
#+END_SRC
The ~statement~ is either a single instruction or a block of
instructions. The ~condition~ is an expression of boolean value (type
~B~).
This time is the ~condition~ is not satisfied (~false~ when reaching the
~while~ instruction), the ~statment~ is never executed.
An example of the ~while~ loop is the following.
#+NAME: while-block
#+BEGIN_SRC pugs :exports both :results output
let sum:N, sum = 0;
let i:N, i = 1;
while (i<=10) {
sum += i;
++i;
}
cout << "sum = " << sum << "\n";
#+END_SRC
The result is
#+results: while-block
*** ~break~ and ~continue~ keywords.
These *keywords* are used to control loops behavior from enclosed loop
statements. They follow their ~C++~ counterparts.
- The ~continue~ keyword is used to skip the instructions that follow in
the loop statement and continue the loop.
- The ~break~ keyword leaves the current loop.
An example of use of the ~continue~ keyword is
#+NAME: nested-continue
#+BEGIN_SRC pugs :exports both :results output
for (let i:N, i = 0; i < 5; ++i) {
cout << i << ": ";
for (let j:N, j=0; j < 5; ++j) {
if (j<i) continue;
cout << j << " ";
}
cout << "\n";
}
#+END_SRC
The result is
#+results: nested-continue
Replacing the ~continue~ keyword by a ~break~ (and changing the test)
#+NAME: nested-break
#+BEGIN_SRC pugs :exports both :results output
for (let i:N, i = 0; i < 5; ++i) {
cout << i << ": ";
for (let j:N, j=0; j < 5; ++j) {
if (j>i) break;
cout << j << " ";
}
cout << "\n";
}
#+END_SRC
changes the output to
#+results: nested-break
Obviously the behavior is the same using ~do...while~ or ~while~ loops.
** Functions<<functions>>
The ~pugs~ language allows the manipulation and definition of
functions. These are *mathematical functions* and are not functions as
functions in ~C++~: functions in the ~pugs~ language are *not subroutines*.
To be more precise, a function $f$ follows the following structure
\begin{align*}
\mbox{let }f:& X_1\times \cdots \times X_n \to Y_1\times \cdots \times Y_m,\\
& (x_1,\ldots,x_n)\mapsto (y_1,\ldots,y_m)=f(x_1,\ldots,x_n),
\end{align*}
where $n,m\in\mathbb{N}_{\ge1}$, and where ${(X_i)}_{1\le i\le n}$ and ${(Y_i)}_{1\le
i\le m}$ are /simple/ types. Actually $X_1\times \cdots \times X_n$ and
$Y_1\times \cdots \times Y_m$ are /compound/ types.
Thus assuming that the function $f$ is defined in the language as ~f~,
one can use the following syntax to evaluate it. This is a
pseudo-code, real examples will follow. Assuming that ~(x1...,xn)~ has
been properly defined in ~X1*...*Xn~ one can write
#+BEGIN_SRC pugs :exports code
let (y1,...,ym):Y1*...*Ym, (y1,...,ym) = f(x1,...,xn);
#+END_SRC
or if ~(y1,...,ym)~ has already been declared in ~Y1*...*Ym~
#+BEGIN_SRC pugs :exports code
(y1,...,ym) = f(x1,...,xn);
#+END_SRC
The ~pugs~ language handles two kinds of functions. User-defined
functions (see [[user-defined-functions]]) and builtin functions (see
[[builtin-functions]]). They behave essentially the same and are both
constant. In this context, it means that one cannot just declare or
modify a function.
*** Pure functions
In the ~pugs~ language, functions are *pure functions* in the sense that
arguments given to a function are *never* modified by the function. They
act as operators.
#+BEGIN_note
Actually these functions are not strictly /pure functions/ in the
computer science sense. The reason is that they can eventually have
side effects. As an example, it is possible to modify the random seed
used by the code. In that case, the modified value is not a variable
of the language itself but the internal random seed.
#+END_note
*** Implicit type conversion for parameters and returned values
As already said, in general, there is no implicit type conversion in
the ~pugs~ language. The only exceptions are when initializing tuples
(see [[tuples]]), passing arguments to functions and dealing with returned
values. See the conversion table in section [[implicit-conversion]].
*** User-defined functions<<user-defined-functions>>
To define user functions, the syntax mimics mathematics. The
definition of the function itself is a *single* expression. This means
that one cannot use ~if...else~, loops and there is not such keyword as
~return~ in the ~C++~.
The syntax of the definition of functions is
#+BEGIN_SRC pugs :exports code
let f1 : X -> Y, x -> e;
let f2 : X -> Y1*...*Ym, x -> (e1,...,em);
let f3 : X1*...*Xn -> Y, (x1,...,xn) -> e;
let f4 : X1*...*Xn -> Y1*...*Ym, (x1,...,xn) -> (e1,...,em);
#+END_SRC
~X~, ~Y~, ~X1~, ..., ~Xn~ and ~Y1~, ..., ~Ym~ are /simple/ data types (not
compound). The type of ~x~ is ~X~ and for lists, each argument ~xi~ belongs
to ~Xi~. The function themselves are defined by the expressions ~e~ (or
~e1~, ..., ~em~) which types are set by ~Y~ (or ~Y1~, ..., ~Ym~).
Let us give a few examples.
#+NAME: R-to-R-function
#+BEGIN_SRC pugs :exports both :results output
let f: R -> R, x -> -x;
cout << "f(2.3) = " << f(2.3)
<< "\nf(1) = " << f(1)
<< "\nf(true) = " << f(true)
<< "\n";
#+END_SRC
One observes the implicit conversion of the values passed by argument.
#+results: R-to-R-function
The following example illustrates the implicit conversion to the
returned type
#+NAME: R-to-R1-function
#+BEGIN_SRC pugs :exports both :results output
let f: R -> R^1, x -> 2*x;
cout << "f(3.2) = " << f(3.2) << "\n";
#+END_SRC
#+results: R-to-R1-function
Using compound types as input and output, one can write
#+NAME: R22-R-string-to-R-string-function
#+BEGIN_SRC pugs :exports both :results output
let f : R^2x2*R*string -> R*string*R^3x3,
(A,x,s) -> (x*A[0,0]*A[1,1]-A[1,0],
A+","+x+","+s,
[[A[0,0], A[0,1], 0],
[A[1,0], A[1,1], 0],
[ 0, 0, x]]);
let x : R, x=0;
let s : string, s= "";
let A : R^3x3, A = 0;
(x,s,A) = f([[1,2],[3,4]], 2.3, "foo");
cout << "x = " << x
<< "\ns = " << s
<< "\nA = " << A << "\n";
let (y,t,A2):R*string*R^3x3, (y,t,A2) = f([[3,1],[4,2]], -5.2, "bar");
cout << "y = " << y
<< "\nt = " << t
<< "\nA2 = " << A2 << "\n";
#+END_SRC
This meaningless example produces the following result.
#+results: R22-R-string-to-R-string-function
**** Lifetime of functions arguments
The arguments used to define a function are *local* variables that
exists only during the evaluation of the function.
Let us consider the following example
#+NAME: lifetime-of-function-args
#+BEGIN_SRC pugs :exports both :results output
let a:R, a = 1.4;
let plus: R -> R, a -> (a>0)*a;
cout << "plus(1) = " << plus(1) << "\n";
cout << "plus(-3.2) = " << plus(-3.2) << "\n";
cout << "a = " << a << "\n";
#+END_SRC
This gives the expected result: the value of the variable ~a~ is
unchanged.
#+results: lifetime-of-function-args
**** Non-arguments variables in function expressions
Here we discuss rapidly of using variables (which are not arguments)
in function expressions.
#+NAME: non-arg-variables-in-functions
#+BEGIN_SRC pugs :exports both :results output
let a:R, a = 1.4;
let plus: R -> R, x -> (x>0)*a;
cout << "a = " << a << "\n";
cout << "plus(2.2) = " << plus(2.2) << "\n";
cout << "plus(-3.2) = " << plus(-3.2) << "\n";
a = 2;
cout << "a = " << a << "\n";
cout << "plus(2.2) = " << plus(2.2) << "\n";
cout << "plus(-3.2) = " << plus(-3.2) << "\n";
#+END_SRC
Running the example, one gets
#+results: non-arg-variables-in-functions
While the function itself is a constant object, one sees that since
the value of ~a~ is changed, the value function is implicitly
modified. /This is a dangerous feature and should be avoid!/
Since functions themselves are variables one can use functions in
function expressions.
#+NAME: functions-in-functions
#+BEGIN_SRC pugs :exports both :results output
let plus: R -> R, x -> (x>0)*x;
let minus: R -> R, x -> -(x<0)*x;
let pm : R -> R*R, x -> (plus(x), minus(x));
let toR2: R*R -> R^2, (x,y) -> [x,y];
cout << "pm(2) = " << toR2(pm(2)) << " pm(-3) = " << toR2(pm(-3)) << "\n";
#+END_SRC
One observes the utility function ~toR2~ that is used to perform the
output since ~cout~ does not handle compound types output. One gets
#+results: functions-in-functions
**** Lifetime of user-defined functions
Since functions are somehow variables, the lifetime of functions
follows the similar rules.
Let us give an example
#+NAME: functions-lifetime
#+BEGIN_SRC pugs :exports both :results output
let f: R -> R, x -> 2.3*x+2;
{
let f: R -> R, x -> 1.25*x+3;
cout << "block 1: f(2) = " << f(2) << "\n";
{
let f: R -> R, x -> 3.25*x-0.3;
cout << "block 2: f(2) = " << f(2) << "\n";
}
}
for (let i:N, i = 1; i<=2; ++i) {
let f: R -> R, x -> i*x+2;
cout << "for(i=" << i << "): f(2) = " << f(2) << "\n";
}
cout << "global: f(2) = " << f(2) << "\n";
#+END_SRC
Running this example produces
#+results: functions-lifetime
**** Recursion
Since functions cannot be simply declared but must be defined, double
recursion is not possible. Thus, there is no equivalent construction
in ~pugs~ of the following ~C++~ code
#+BEGIN_SRC C++ :exports source
int f(int i);
int g(int i)
{
if (i < 0)
return 0;
else
return f(i-1);
}
int f(int i) { return g(i); }
#+END_SRC
Moreover simple recursion is forbidden for similar reasons, since
functions are made of a single expression (one cannot use statements),
there would be no way to end the recursion. Thus, the code
#+NAME: no-recursion
#+BEGIN_SRC pugs-error :exports both :results output
let f: N->N, n -> f(2*n);
#+END_SRC
produces the following compilation time error
#+results: no-recursion
*** Builtin functions<<builtin-functions>>
In ~pugs~ language, builtin functions are ~C++~ pieces of code that can be
called in scripts. There usage is very similar to user-defined
functions. They differ from user-defined functions in three points.
- Builtin functions can have no parameter or no returned value.
- Builtin functions are polymorphic. More precisely, this means that
the signature of a builtin function is also defined by its expected
arguments types.
- Builtin functions can take user-defined functions as parameters.
- user-defined functions cannot take functions as parameters
- builtin functions cannot take builtin functions as parameters
(actually, this is not a limitation since it is trivial to embed a
builtin function into a user-defined one).
Here is a simple example of builtin function embedding in a user
function.
#+NAME: builtin-function-embedding
#+BEGIN_SRC pugs :exports both :results output
import math;
let cosinus: R -> R, x -> cos(x);
cout << "cosinus(2) = " << cosinus(2) << "\n";
#+END_SRC
Running this example produces
#+results: builtin-function-embedding
Builtin functions are defined when modules are imported, see [[modules]].
** Modules<<modules>>
Modules are sets of *builtin functions* and *high-level types* that are
not provided by default by ~pugs~. To load a module, one has to use an
~import~ instruction in the preamble of the script.
For instance to load the ~math~ module which contains builtin
mathematical functions, one writes in the preamble of the script
#+BEGIN_SRC pugs :exports source
import math;
#+END_SRC
#+BEGIN_warning
A work in progress
- At the time of writing this documentation, one should note that
module inter-dependencies is still not implemented.
- Also, (and especially with regard to the ~scheme~ module), module
contents are likely to change and to be reorganized.
- Finally it is almost sure that modules will be equipped with a
/namespace/-like functionality to avoid conflicts. This actually
should make the scripts cleaner since, the naming of functions would
be more natural.
#+END_warning
One can access to the list of available modules inside the language.
#+NAME: get-available-modules
#+BEGIN_SRC pugs :exports both :results output
cout << getAvailableModules() << "\n";
#+END_SRC
The output lists all available modules
#+RESULTS: get-available-modules
Let us comment a bit this output. One notices that there are two kind
of modules. Modules that are automatically imported (tagged with a ~*~)
and the other ones.
In this section we will not describe exhaustively the whole module
contents but will give the basic information that should allow the
user to find his way. To do so, it is important to examine carefully
the content of the ~core~ module, since it contains some helper
functions.
*** The ~core~ module
As already said, the ~core~ module cannot be imported manually since it
is imported automatically.
**** ~core~ provided types
***** ~ostream~
The ~core~ module provides an important type: ~ostream~. This type has
already been encountered in this documentation. This is the type of
some specific predefined objects: ~cout~, ~cerr~ and ~clog~. These objects
are very similar to their ~C++~ counterparts, respectively ~std::cout~,
~std::cerr~ and ~std::clog~.
Objects of type ~ostream~ can be used as the left operand for the binary
operator ~<<~ for basic types and their associated tuples.
| ~ostream <<~ allowed expression type |
|------------------------------------|
| ~B~ |
| ~N~ |
| ~Z~ |
| ~R~ |
| ~R^1~ |
| ~R^2~ |
| ~R^3~ |
| ~R^1x1~ |
| ~R^2x2~ |
| ~R^3x3~ |
| ~string~ |
As one should have noticed, when the left operand of the binary
operator ~<<~ is an ~ostream~, the result of the operation is also an
~ostream~, which allows to chain output.
One can overload the ~ostream <<~ construction for high-level types.
Other variables of the type ~ostream~ can be created (in order to write
to files for instance) as one should see below.
**** ~core~ provided functions
***** ~getAvailableModules: void -> string~
This function that is used in the preamble of this section is a
function that returns a ~string~ that contains the list of modules that
are available in the current version of pugs.
***** ~getModuleInfo: string -> string~
This is a *very* helpful function. It lists *all* the available *builtin
functions* of a module and *all* the provided *types*. It is important to
remark that the name of the module is given as a ~string~ and not as is.
#+BEGIN_note
This function provides a very simple documentation of the
modules. Hopefully in future developments it would be really useful to
obtain a complete documentation of each function to offer a precise
online documentation.
#+END_note
In the following of the documentation we will run this function on all
modules.
#+NAME: get-module-info-core
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("core") << "\n";
#+END_SRC
For the ~core~ module, it gives
#+RESULTS: get-module-info-core
This output is quite rustic, but it contains the main information: the
name of the function and its input and output sets.
#+BEGIN_warning
Observe that this function does not provide the list of operators that
are defined in the module (eventually associated to the defined
types).
#+END_warning
***** ~getPugsBuildInfo: void -> string~
Sets the building information of the executable into a ~string~.
#+NAME: get-pugs-build-info
#+BEGIN_SRC pugs :exports both :results output
cout << getPugsBuildInfo() << "\n";
#+END_SRC
It gives for instance
#+RESULTS: get-pugs-build-info
***** ~getPugsVersion: void -> string~
Sets the ~pugs~ version of the executable into a ~string~.
#+NAME: get-pugs-version
#+BEGIN_SRC pugs :exports both :results output
cout << getPugsVersion() << "\n";
#+END_SRC
The output contains also ~git~ information
#+RESULTS: get-pugs-version
***** ~resetRandomSeed: void -> void~
At the start of ~pugs~, a *common* random seed is defined for all ~MPI~
processes. The seed is written automatically written in the preamble
of the listing (in the documentation the ~--no-preamble~ option is used
this is why it is not displayed in examples).
The ~resetRandomSeed~ creates a *new shared* random and displays its value.
#+NAME: reset-random-seed
#+BEGIN_SRC pugs :exports both :results output
resetRandomSeed();
#+END_SRC
The output is
#+RESULTS: reset-random-seed
***** ~setRandomSeed: Z -> void~ <<set-random-seed>>
In order to reproduce exactly the same results of a calculation, it
can be interesting to set the random seed to some given value.
#+NAME: set-random-seed
#+BEGIN_SRC pugs :exports both :results output
setRandomSeed(123456789);
#+END_SRC
Running this example gives
#+RESULTS: set-random-seed
***** ~ofstream: string -> ostream~
This function is used to create an ~ostream~ that actually write to the
file which name is given by the ~string~ argument. One should notice
that the file is *created* at the function call. If a file with the same
name existed, it is *erased*.
#+NAME: ofstream-example
#+BEGIN_SRC pugs :exports both :results output
let fout:ostream, fout = ofstream("filename.txt");
fout << [1,2] << " is a vector of R^2\n";
#+END_SRC
Running this example produces no output
#+RESULTS: ofstream-example
But a file is created (in the execution directory), with the name
~"filename.txt"~. Its content is
#+NAME: cat-filename-txt
#+BEGIN_SRC shell :exports results :results output
cat filename.txt
#+END_SRC
#+RESULTS: cat-filename.txt
Following ~C++~, the file is closed when the variable is destroyed.
*** The ~dev_utils~ module
This module contains utilities for ~pugs~ developers.
**** ~dev_utils~ provided functions
#+NAME: get-module-info-dev-utils
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("dev_utils") << "\n";
#+END_SRC
#+RESULTS: get-module-info-dev-utils
***** ~getAST: void -> string~
Gets the abstract syntax tree (AST) that will be executed into a ~string~.
***** ~getFunctionAST: function -> string~
Gets the abstract syntax tree associated to a user function into a
~string~.
***** ~saveASTDot: string -> void~
Saves the AST of the script into the file (whose name is given as
argument) using the dot format.
*** The ~math~ module
The ~math~ module is a small utility module that provides a set of
standard mathematical functions.
#+NAME: get-module-info-math
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("math") << "\n";
#+END_SRC
For the ~math~ module, it gives
#+RESULTS: get-module-info-math
There is not much to say. One can see that this module does not
provide new types of data. All the classical mathematical functions
follow their ~C++~ counterparts. One can see that there are integer
versions (type ~Z~) of ~abs~, ~min~ and ~max~ functions.
The ~dot~ function family provides the dot product for vectors of
$\mathbb{R}^1$, $\mathbb{R}^2$ and $\mathbb{R}^3$.
#+NAME: dot-examples
#+BEGIN_SRC pugs :exports both :results output
import math;
cout << "([1],[2]) = " << dot([1],[2]) << "\n";
cout << "([1,2],[3,4]) = " << dot([1,2],[3,4]) << "\n";
cout << "([1,2,3],[4,5,6]) = " << dot([1,2,3],[4,5,6]) << "\n";
#+END_SRC
The output is
#+RESULTS: dot-examples
#+BEGIN_note
Observe that the use of a proper rounding or truncation function is
the right way to convert a real value to an integer one. Available
rounding or truncation functions are ~ceil~, ~floor~, ~round~ and ~trunc~. See
their ~C++~ man pages for details.
#+END_note
#+BEGIN_note
Let us comment the use of the ~pow~ function. Actually one can wonder
why we did not use a syntax like ~x^y~? The reason is that if
mathematically ${x^y}^z = x^{(y^z)}$, many software treat it (by mistake)
as ${(x^y)}^z$. Thus, using the ~pow~ function avoids any confusion.
#+END_note
*** The ~mesh~ module
This is an important module. It provides mesh utilities tools.
#+NAME: get-module-info-mesh
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("mesh") << "\n";
#+END_SRC
#+RESULTS: get-module-info-mesh
**** ~mesh~ provided types
***** ~boundary~
The ~boundary~ type is a boundary descriptor: it refers to a boundary of
a ~mesh~ that is either designated by an integer or by a ~string~.
A ~boundary~ can be used refer to an interface, it can designate a set
of nodes, edges or faces. The ~boundary~ (descriptor) itself is not
related to any ~mesh~, thus the nature of the ~boundary~ is precised when
it is used with a particular ~mesh~.
***** ~zone~
Following the same idea, a ~zone~ is a descriptor of set of cells. It
can be either defined by an integer or by a ~string~. Its meaning is
precised when it is associated with a ~mesh~.
***** ~mesh~
The type ~mesh~ is an *abstract* type that is used to store meshes. A
variable of that type can refer for instance unstructured meshes of
dimension 1, 2 or 3.
The following binary operator is provided.
| ~ostream <<~ allowed expression type |
|------------------------------------|
| ~mesh~ |
It enables to write mesh information to an ~ostream~, here is a simple
example.
#+NAME: ostream-mesh-example
#+BEGIN_SRC pugs :exports both :results output
import mesh;
let m:mesh, m = cartesianMesh(0,[1,1,2], (2,3,2));
cout << m << "\n";
#+END_SRC
Running this script generates the following output.
#+RESULTS: ostream-mesh-example
The information produced concerns
- the dimension of the connectivity,
- the number of cells and their references,
- the number of faces and their references,
- the number of edges and their references,
- and the number of nodes and their references.
**** ~mesh~ provided functions
***** Boundary descriptor functions
****** ~boundaryName: string -> boundary~
Sets a boundary descriptor to a boundary name
#+BEGIN_SRC pugs :exports both :results none
import mesh;
let b:boundary, b = boundaryName("boundary_name");
#+END_SRC
****** ~boundaryTag: Z -> boundary~
Creates a boundary descriptor to a boundary reference
#+BEGIN_SRC pugs :exports both :results none
import mesh;
let b:boundary, b = boundaryTag(12);
#+END_SRC
***** Zone descriptor functions
****** ~zoneName: string -> zone~
Creates a zone descriptor from a ~string~ name
#+BEGIN_SRC pugs :exports both :results none
import mesh;
let z:zone, z = zoneName("zone_name");
#+END_SRC
****** ~zoneTag: Z -> zone~
Associates a zone descriptor from zone tag
#+BEGIN_SRC pugs :exports both :results none
import mesh;
let z:zone, z = zoneTag(5);
#+END_SRC
***** ~cartesianMesh: Rˆd*Rˆd*(N) -> mesh~ (with $d\in\{1, 2, 3\}$)
Creates a cartesian mesh of dimension $d\in\{1, 2, 3\}$. The produced
cartesian grid is aligned with the axis and made of identical cells.
The two first arguments are two opposite corners of the box (or
segment in 1d) and the list of natural integers (type ~(N)~) sets the
number of *cells* in each direction. Thus size of the list of ~N~ is $d$.
For instance one can write:
#+BEGIN_SRC pugs :exports both :results none
import mesh;
let m1d:mesh, m1d = cartesianMesh([0], [1], 10);
let m2d:mesh, m2d = cartesianMesh([0,0], [1,1], (3,6));
let m3d:mesh, m3d = cartesianMesh([-1,-1,-1], [1,1,1], (3,2,4));
#+END_SRC
- The ~m1d~ variable contains a uniform grid of $]0,1[$ made of 10 cells.
- The ~m2d~ variable refers to a uniform grid of $]0,1[^2$, made of 3
cells in the $x$ direction, and of 6 cells along the $y$ axis.
- The ~m3d~ variable designates a mesh of $]-1,1[^3$ made of $3\times2\times4$
cells.
***** ~readGmsh: string -> mesh~
Reads a ~mesh~ from a file.
#+BEGIN_warning
The file must conform to the mesh format ~msh2~.
#+END_warning
#+BEGIN_SRC shell :exports results :results none
cat << EOF > hybrid-2d.geo
//+
Point(1) = {0, 0, 0, 1.0};
//+
Point(2) = {1, 0, 0, 1.0};
//+
Point(3) = {1, 1, 0, 1.0};
//+
Point(4) = {0, 1, 0, 1.0};
//+
Point(5) = {2, 0, 0, 1.0};
//+
Point(6) = {2, 1, 0, 1.0};
//+
Line(1) = {4, 1};
//+
Line(2) = {2, 3};
//+
Line(3) = {5, 6};
//+
Line(4) = {3, 4};
//+
Line(5) = {6, 3};
//+
Line(6) = {1, 2};
//+
Line(7) = {2, 5};
//+
Curve Loop(1) = {1, 6, 2, 4};
//+
Surface(1) = {1};
//+
Curve Loop(2) = {-2, 7, 3, 5};
//+
Surface(2) = {2};
//+
Characteristic Length {4, 3, 6, 5, 2, 1} = 0.3;
//+
Recombine Surface {1};
//+
Physical Curve("XMIN") = {1};
//+
Physical Curve("XMAX") = {3};
//+
Physical Curve("YMAX") = {4, 5};
//+
Physical Curve("YMIN") = {6, 7};
//+
Physical Curve("INTERFACE") = {2};
//+
Physical Surface("LEFT") = {1};
//+
Physical Surface("RIGHT") = {2};
//+
Physical Point("XMINYMIN") = {1};
//+
Physical Point("XMINYMAX") = {4};
//+
Physical Point("XMAXYMIN") = {5};
//+
Physical Point("XMAXYMAX") = {6};
#+END_SRC
#+BEGIN_SRC shell :exports results :results none
gmsh -2 hybrid-2d.geo -format msh2
#+END_SRC
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import writer;
let m:mesh, m = readGmsh("hybrid-2d.msh");
write_mesh(gnuplot_writer("hybrid-2d"),m);
#+END_SRC
The ~mesh~ is represented on Figure [[fig:gmsh-hybrid-2d]].
#+NAME: gmsh-hybrid-2d-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/gmsh-hybrid-2d.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set terminal png truecolor enhanced size 960,480
plot '<(sed "" $PUGS_SOURCE_DIR/doc/hybrid-2d.gnu)' w l
#+END_SRC
#+CAPTION: The mesh that was read from the file ~hydrid-2d.msh~ and then saved to the ~gnuplot~ format
#+NAME: fig:gmsh-hybrid-2d
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: gmsh-hybrid-2d-img
***** ~diamondDual: mesh -> mesh~
This function creates the diamond dual ~mesh~ of a primal ~mesh~.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import writer;
let primal_mesh:mesh, primal_mesh = readGmsh("hybrid-2d.msh");
let dual_mesh:mesh, dual_mesh = diamondDual(primal_mesh);
write_mesh(gnuplot_writer("diamond"), dual_mesh);
#+END_SRC
The diamond dual mesh is defined by joining the nodes of the faces to
the center of the adjacent cells of the primal mesh.
The ~mesh~ is represented on Figure [[fig:gmsh-hybrid-2d]].
#+NAME: diamond-dual-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/diamond-dual.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set terminal png truecolor enhanced size 960,480
plot '<(sed "" $PUGS_SOURCE_DIR/doc/hybrid-2d.gnu)' lt rgb "green" w l, '<(sed "" $PUGS_SOURCE_DIR/doc/diamond.gnu)' lt rgb "black" w l
#+END_SRC
#+CAPTION: The primal mesh in green and the diamond dual mesh in black
#+NAME: fig:diamond-dual
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: diamond-dual-img
#+BEGIN_note
The mesh storage mechanisms in ~pugs~ is such that the diamond dual mesh
is built only once. This means that is one writes for instance
#+BEGIN_SRC pugs :exports both :results none
import mesh;
let primal_mesh:mesh, primal_mesh = readGmsh("hybrid-2d.msh");
let dual_mesh_1:mesh, dual_mesh_1 = diamondDual(primal_mesh);
let dual_mesh_2:mesh, dual_mesh_2 = diamondDual(primal_mesh);
#+END_SRC
then the variables ~dual_mesh_1~ and ~dual_mesh_2~ will refer to the same
~mesh~ in memory.
#+END_note
#+BEGIN_warning
The diamond dual mesh construction is not yet implemented in parallel
#+END_warning
***** ~medianDual: mesh -> mesh~
This function creates the median dual ~mesh~ of a primal ~mesh~.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import writer;
let primal_mesh:mesh, primal_mesh = readGmsh("hybrid-2d.msh");
let dual_mesh:mesh, dual_mesh = medianDual(primal_mesh);
write_mesh(gnuplot_writer("median"), dual_mesh);
#+END_SRC
The median dual mesh is defined by joining the centers of the faces to
the centers of the adjacent cells of the primal mesh.
The ~mesh~ is represented on Figure [[fig:gmsh-hybrid-2d]].
#+NAME: median-dual-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/median-dual.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set terminal png truecolor enhanced size 960,480
plot '<(sed "" $PUGS_SOURCE_DIR/doc/hybrid-2d.gnu)' lt rgb "green" w l, '<(sed "" $PUGS_SOURCE_DIR/doc/median.gnu)' lt rgb "black" w l
#+END_SRC
#+CAPTION: The primal mesh in green and the median dual mesh in black
#+NAME: fig:median-dual
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: median-dual-img
#+BEGIN_note
In ~pugs~, the storage mechanisms of median dual meshes follows the same
rules as the diamond dual meshes. As long as the primary mesh lives
and as long as the median dual mesh is referred, it is kept in memory,
thus constructed only once.
#+END_note
#+BEGIN_warning
The median dual mesh construction is not yet implemented in 3d and not
available in parallel
#+END_warning
***** ~transform: mesh*function -> mesh~
This function allows to compute a new mesh as the transformation a
given mesh by displacing its nodes through a user defined function.
For a mesh of dimension $d$, the mesh must be a function $\mathbb{R}^d
\to\mathbb{R}^d$.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import math;
import writer;
let m0:mesh, m0 = cartesianMesh([0,0], [1,1], (10,10));
let pi_over_3: R, pi_over_3 = acos(-1)/3;
let t: R^2 -> R^2, x -> [(0.2+0.8*x[0])*cos(pi_over_3*x[1]), (0.2+0.8*x[0])*sin(pi_over_3*x[1])];
let m1: mesh, m1 = transform(m0, t);
write_mesh(gnuplot_writer("transformed"), m1);
#+END_SRC
#+BEGIN_note
One should keep in mind that the mesh produced the ~transform~ function
*shares* the same connectivity than the given mesh. This means that in
~pugs~ internals, there is only one connectivity object.
#+END_note
The result of the previous script is given on Figure [[fig:transformed]].
#+NAME: transformed-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/transformed.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set square
set terminal png truecolor enhanced size 640,640
plot '<(sed "" $PUGS_SOURCE_DIR/doc/transformed.gnu)' w l
#+END_SRC
#+CAPTION: The transformed unit square
#+NAME: fig:transformed
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: transformed-img
***** ~relax: mesh*mesh*R -> mesh~ <<relax-function>>
This function is a simple utility that computes a ~mesh~ as the /mean/ of
two other mesh that share the same connectivity. The coordinates of
the vertices of the relaxed mesh $\mathcal{M}_2$, are given by
\begin{equation*}
\forall r\in\mathcal{R},\quad\mathbf{x}_r^{\mathcal{M}_2} = (1-\theta) \mathbf{x}_r^{\mathcal{M}_0} + \theta \mathbf{x}_r^{\mathcal{M}_1}.
\end{equation*}
#+BEGIN_warning
The connectivity must be the *same in memory*, this means that
constructing two identical meshes with /equivalent/ connectivity is not
allowed.
#+END_warning
Thus for instance, the following code
#+NAME: relax-with-similar-connecticities
#+BEGIN_SRC pugs-error :exports both :results output
import mesh;
let m1:mesh, m1 = cartesianMesh(0, [1,1], (10,10));
let m2:mesh, m2 = cartesianMesh(0, [1,1], (10,10));
let m3:mesh, m3 = relax(m1,m2,0.3);
#+END_SRC
produces the runtime error
#+results: relax-with-similar-connecticities
A proper use is
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import math;
import writer;
let m0:mesh, m0 = cartesianMesh([0,0], [1,1], (10,10));
write_mesh(gnuplot_writer("relax_example_m0"), m0);
let t: R^2 -> R^2, x -> [x[0]+0.25*x[1]*x[1], x[1]+0.3*sin(x[0])];
let m1: mesh, m1 = transform(m0, t);
write_mesh(gnuplot_writer("relax_example_m1"), m1);
let m2: mesh, m2 = relax(m0, m1, 0.3);
write_mesh(gnuplot_writer("relax_example_m2"), m2);
#+END_SRC
In this example, the relaxation parameter is set to $\theta=0.3$. The
different meshes produced in this example are displayed on Figure
[[fig:relax]].
#+NAME: relax-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/relax.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set square
set terminal png truecolor enhanced size 640,640
plot '<(sed "" $PUGS_SOURCE_DIR/doc/relax_example_m0.gnu)' lt rgb "green" w l, '<(sed "" $PUGS_SOURCE_DIR/doc/relax_example_m1.gnu)' lt rgb "blue" w l, '<(sed "" $PUGS_SOURCE_DIR/doc/relax_example_m2.gnu)' lt rgb "black" w l
#+END_SRC
#+CAPTION: Example of meshes relaxation. The relaxed mesh $\mathcal{M}_2$ (black) and the original meshes in green ($\mathcal{M}_0$) and blue ($\mathcal{M}_1$).
#+NAME: fig:relax
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: relax-img
This function is mainly useful to reduce the displacement of nodes
when using the ~randomizeMesh~ functions (see section [[scheme]]) for
instance.
*** The ~scheme~ module<<scheme>>
This module provides a lot of numerical tools.
#+NAME: get-module-info-scheme
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("scheme") << "\n";
#+END_SRC
#+RESULTS: get-module-info-scheme
#+BEGIN_warning
This module is very large and will be split in smaller ones in the future
#+END_warning
This module provides various types and functions. It also provides a
set of operators overloading.
**** ~scheme~ provided types
***** ~Vh~
The ~Vh~ type is the /abstract/ type of variables that refer to *discrete
functions*.
****** $\mathbb{P}_0$ scalar functions
These functions are defined on a ~mesh~ and have a constant value in
each cell.
The type of values in each cells for a $\mathbb{P}_0$ function can be
$\mathbb{R}$, $\mathbb{R}^1$, $\mathbb{R}^2$, $\mathbb{R}^3$,
$\mathbb{R}^{1\times1}$, $\mathbb{R}^{2\times2}$ or $\mathbb{R}^{3\times3}$.
For simplicity in this document, we denote these types specific
$\mathbb{P}_0$ types as $\mathbb{P}_0(\mathbb{R})$,
$\mathbb{P}_0(\mathbb{R}^1)$, $\mathbb{P}_0(\mathbb{R}^2)$,
$\mathbb{P}_0(\mathbb{R}^3)$, $\mathbb{P}_0(\mathbb{R}^{1\times1})$,
$\mathbb{P}_0(\mathbb{R}^{2\times2})$ or
$\mathbb{P}_0(\mathbb{R}^{3\times3})$.
****** $\mathbb{P}_0$ vector functions
Additionally to scalar values per cell, one can define vectors of real
values per cell. The size of the vectors is the same for all
cells. This kind of variables is useful to define mass fractions for
instance.
Again for convenience, these types are denoted as
$\vec{\mathbb{P}}_0(\mathbb{R})$.
***** ~Vh_type~
The ~Vh_type~ allows to describe a type of discretization. The available
types of discretization are
- ~P0~ for $\mathbb{P}_0(\mathbb{R})$, $\mathbb{P}_0(\mathbb{R}^1)$,
$\mathbb{P}_0(\mathbb{R}^2)$, $\mathbb{P}_0(\mathbb{R}^3)$,
$\mathbb{P}_0(\mathbb{R}^{1\times1})$, $\mathbb{P}_0(\mathbb{R}^{2\times2})$
or $\mathbb{P}_0(\mathbb{R}^{3\times3})$.
- ~P0Vector~ for $\vec{\mathbb{P}}_0(\mathbb{R})$
***** ~boundary_condition~
This type is used to describe boundary conditions.
***** ~quadrature~
This type is used to describe quadrature types.
**** ~scheme~ provided functions
The ~scheme~ module provides a lot of functions, we categorize their
description.
***** Mathematical functions
****** ~Vh -> Vh~
These functions are defined for $\mathbb{P}_0(\mathbb{R})$ data and the
return value is also a $\mathbb{P}_0(\mathbb{R})$ function. The value
is simply the application of the function to the cell values.
Here is the list of the functions
- ~abs: Vh -> Vh~
- ~acos: Vh -> Vh~
- ~acosh: Vh -> Vh~
- ~asin: Vh -> Vh~
- ~asinh: Vh -> Vh~
- ~atan: Vh -> Vh~
- ~atanh: Vh -> Vh~
- ~cos: Vh -> Vh~
- ~cosh: Vh -> Vh~
- ~exp: Vh -> Vh~
- ~log: Vh -> Vh~
- ~sin: Vh -> Vh~
- ~sinh: Vh -> Vh~
- ~sqrt: Vh -> Vh~
- ~tan: Vh -> Vh~
- ~tanh: Vh -> Vh~
****** ~Vh*Vh -> Vh~
These functions are defined for $\mathbb{P}_0(\mathbb{R})$ data and the
return value is also a $\mathbb{P}_0(\mathbb{R})$ function. These
functions require that the two arguments are defined one the *same
mesh*. The result is obtained by applying the function cell by cell.
Here is the function list
- ~atan2: Vh*Vh -> Vh~
- ~max: Vh*Vh -> Vh~
- ~min: Vh*Vh -> Vh~
- ~pow: Vh*Vh -> Vh~
Let us mention another function that applies to
$\mathbb{P}_0(\mathbb{R}^1)$, $\mathbb{P}_0(\mathbb{R}^2)$,
$\mathbb{P}_0(\mathbb{R}^3)$ and to $\vec{\mathbb{P}}_0(\mathbb{R})$
vector functions.
- ~dot: Vh*Vh -> Vh~
This function requires that both arguments are defined on the same
mesh and have the same data type. The result is a
$\mathbb{P}_0(\mathbb{R})$ function.
****** ~R*Vh -> Vh~ and ~Vh*R -> Vh~
These functions are defined for $\mathbb{P}_0(\mathbb{R})$ data and the
return value is also a $\mathbb{P}_0(\mathbb{R})$ function.
The following functions can be applied using a scalar ~R~ and a ~Vh~
operand.
- ~atan2: Vh*R -> Vh~
- ~atan2: R*Vh -> Vh~
- ~max: Vh*R -> Vh~
- ~max: R*Vh -> Vh~
- ~min: Vh*R -> Vh~
- ~min: R*Vh -> Vh~
- ~pow: Vh*R -> Vh~
- ~pow: R*Vh -> Vh~
****** ~R^1*Vh -> Vh~ and ~Vh*R^1 -> Vh~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^1)$ data and the
return value is also a $\mathbb{P}_0(\mathbb{R})$ function.
The following functions
- ~dot: Rˆ1*Vh -> Vh~
- ~dot: Vh*Rˆ1 -> Vh~
****** ~R^2*Vh -> Vh~ and ~Vh*R^2 -> Vh~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^2)$ data and the
return value is also a $\mathbb{P}_0(\mathbb{R})$ function.
The following functions
- ~dot: Rˆ2*Vh -> Vh~
- ~dot: Vh*Rˆ2 -> Vh~
****** ~R^3*Vh -> Vh~ and ~Vh*R^3 -> Vh~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^3)$ data and the
return value is also a $\mathbb{P}_0(\mathbb{R})$ function.
The following functions
- ~dot: Rˆ3*Vh -> Vh~
- ~dot: Vh*Rˆ3 -> Vh~
****** ~Vh -> R~
These functions are defined for $\mathbb{P}_0(\mathbb{R})$ data and the
return value is real ($\mathbb{R}$).
The following functions
- ~min: Vh -> R~\\
returns the minimal value of all the cell values
- ~max: Vh -> R~\\
returns the maximal value of all the cell values
- ~integral_of_R: Vh -> R~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R})$ discrete
function $f$. If $f_j \in\mathbb{R}$ denotes the value of $f$ in the
cell $j$, the return value is $$\sum_{j\in\mathcal{J}}V_j f_j,$$ where $V_j$
denotes the volume of the cell $j$.
- ~sum_of_R: Vh -> R~\\
computes the sum of the cell values of a $\mathbb{P}_0(\mathbb{R})$
discrete function $f$. If $f_j \in\mathbb{R}$ denotes the value of
$f$ in the cell $j$, the return value is $$\sum_{j\in\mathcal{J}} f_j.$$
****** ~Vh -> R^1~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^1)$ data and
the return value is a $\mathbb{R}^1$ vector.
- ~integral_of_R1: Vh -> R^1~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R}^1)$ discrete
function $\mathbf{u}$. If $\mathbf{u}_j \in\mathbb{R}^1$ denotes the
value of $\mathbf{u}$ in the cell $j$, the return value is
$$\sum_{j\in\mathcal{J}}V_j \mathbf{u}_j,$$ where $V_j$ denotes the volume of
the cell $j$.
- ~sum_of_R1: Vh -> R^1~\\
computes the sum of the cell values of a $\mathbb{P}_0(\mathbb{R}^1)$
discrete function $\mathbf{u}$. If $\mathbf{u}_j \in\mathbb{R}^1$
denotes the value of $\mathbf{u}$ in the cell $j$, the return value
is $$\sum_{j\in\mathcal{J}} \mathbf{u}_j.$$
****** ~Vh -> R^2~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^2)$ data and
the return value is a $\mathbb{R}^2$ vector.
- ~integral_of_R2: Vh -> R^2~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R}^2)$ discrete
function $\mathbf{u}$. If $\mathbf{u}_j \in\mathbb{R}^2$ denotes the
value of $\mathbf{u}$ in the cell $j$, the return value is
$$\sum_{j\in\mathcal{J}}V_j \mathbf{u}_j,$$ where $V_j$ denotes the volume of
the cell $j$.
- ~sum_of_R2: Vh -> R^2~\\
computes the sum of the cell values of a $\mathbb{P}_0(\mathbb{R}^2)$
discrete function $\mathbf{u}$. If $\mathbf{u}_j \in\mathbb{R}^2$
denotes the value of $\mathbf{u}$ in the cell $j$, the return value
is $$\sum_{j\in\mathcal{J}} \mathbf{u}_j.$$
****** ~Vh -> R^3~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^3)$ data and
the return value is a $\mathbb{R}^3$ vector.
- ~integral_of_R3: Vh -> R^3~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R}^3)$ discrete
function $\mathbf{u}$. If $\mathbf{u}_j \in\mathbb{R}^3$ denotes the
value of $\mathbf{u}$ in the cell $j$, the return value is
$$\sum_{j\in\mathcal{J}}V_j \mathbf{u}_j,$$ where $V_j$ denotes the volume of
the cell $j$.
- ~sum_of_R3: Vh -> R^3~\\
computes the sum of the cell values of a $\mathbb{P}_0(\mathbb{R}^3)$
discrete function $\mathbf{u}$. If $\mathbf{u}_j \in\mathbb{R}^3$
denotes the value of $\mathbf{u}$ in the cell $j$, the return value
is $$\sum_{j\in\mathcal{J}} \mathbf{u}_j.$$
****** ~Vh -> R^1x1~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^{1\times1})$ data
and the return value is an $\mathbb{R}^{1\times1}$ matrix.
- ~integral_of_R1x1: Vh -> R^1x1~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R}^{1\times1})$
discrete function $A$. If $A_j \in\mathbb{R}^{1\times1}$ denotes the
value of $A$ in the cell $j$, the return value is
$$\sum_{j\in\mathcal{J}}V_j A_j,$$ where $V_j$ denotes the volume of the cell
$j$.
- ~sum_of_R1x1: Vh -> R^1x1~\\
computes the sum of the cell values of a
$\mathbb{P}_0(\mathbb{R}^{1\times1})$ discrete function $A$. If $A_j
\in\mathbb{R}^{1\times1}$ denotes the value of $A$ in the cell $j$, the
return value is $$\sum_{j\in\mathcal{J}} A_j.$$
****** ~Vh -> R^2x2~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^{2\times2})$ data
and the return value is an $\mathbb{R}^{2\times2}$ matrix.
- ~integral_of_R2x2: Vh -> R^2x2~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R}^{2\times2})$
discrete function $A$. If $A_j \in\mathbb{R}^{2\times2}$ denotes the
value of $A$ in the cell $j$, the return value is
$$\sum_{j\in\mathcal{J}}V_j A_j,$$ where $V_j$ denotes the volume of the cell
$j$.
- ~sum_of_R2x2: Vh -> R^2x2~\\
computes the sum of the cell values of a
$\mathbb{P}_0(\mathbb{R}^{2\times2})$ discrete function $A$. If $A_j
\in\mathbb{R}^{2\times2}$ denotes the value of $A$ in the cell $j$, the
return value is $$\sum_{j\in\mathcal{J}} A_j.$$
****** ~Vh -> R^3x3~
These functions are defined for $\mathbb{P}_0(\mathbb{R}^{3\times3})$ data
and the return value is an $\mathbb{R}^{3\times3}$ matrix.
- ~integral_of_R3x3: Vh -> R^3x3~\\
computes the integral of a $\mathbb{P}_0(\mathbb{R}^{3\times3})$
discrete function $A$. If $A_j \in\mathbb{R}^{3\times3}$ denotes the
value of $A$ in the cell $j$, the return value is
$$\sum_{j\in\mathcal{J}}V_j A_j,$$ where $V_j$ denotes the volume of the cell
$j$.
- ~sum_of_R3x3: Vh -> R^3x3~\\
computes the sum of the cell values of a
$\mathbb{P}_0(\mathbb{R}^{3\times3})$ discrete function $A$. If $A_j
\in\mathbb{R}^{3\times3}$ denotes the value of $A$ in the cell $j$, the
return value is $$\sum_{j\in\mathcal{J}} A_j.$$
***** Interpolation and integration functions
These functions are ways to define discrete functions from analytic
data.
****** ~interpolate: mesh*Vh_type*(function) -> Vh~
This functions takes a ~mesh~, a type of discrete function (~Vh_type~) and
a list of user functions as arguments. It returns a $\mathbb{P}_0$
function defined at the mesh.
All the user functions of the list must be defined on $\mathbb{R}^d$
for a mesh of dimension $d$.
There are several situations according to the ~Vh_type~.
******* ~P0~
In that case the list of user functions *must* reduce to a *single* user
function.
The codomain (or image space) of the user function defines the type of
the returned discrete function.
Let us give an example.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
let m:mesh, m = cartesianMesh([0,0], [1,1], (10,10));
let u:R^2 -> R^3, x -> [x[0], 2*x[1], x[1]-x[0]];
let uh:Vh, uh = interpolate(m, P0(), u);
#+END_SRC
In this exampel the discrete function ~uh~ is a
$\mathbb{P}_0(\mathbb{R}^3)$ function defined on a 2d ~mesh~. The ~P0()~
function returns the type of interpolation ($\mathbb{P}_0$).
#+BEGIN_note
In the case of $\mathbb{P}_0$ interpolation, the function is evaluated
at the mass center of the mesh cells $$\forall j\in\mathcal{J},\quad
\mathbf{x}_j=\frac{1}{V_j}\int_j \mathbf{x}.$$
#+END_note
******* ~P0Vector~
In that case the codomain of each user function in the list must be a
real function (values in $\mathbb{R}$). The instruction will define a
$\vec{\mathbb{P}}_0(\mathbb{R})$.
A first example is
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
let m:mesh, m = cartesianMesh([0,0], [1,1], (10,10));
let f:R^2 -> R, x -> 2*x[0]-x[1];
let fh:Vh, fh = interpolate(m, P0Vector(), f);
#+END_SRC
The obtained ~fh~ is a vector $\vec{\mathbb{P}}_0(\mathbb{R})$ where each
vector (in each cell) is of dimension 1.
The number of scalar user functions sets the size of the discrete
vector function.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import math;
let m:mesh, m = cartesianMesh([0,0], [1,1], (10,10));
let f0:R^2 -> R, x -> 2*x[0]-x[1];
let f1:R^2 -> R, x -> x[0]*x[1]-2;
let f2:R^2 -> R, x -> 2*dot(x,x);
let fh:Vh, fh = interpolate(m, P0Vector(), (f0,f1,f2));
#+END_SRC
Here we defined a $\vec{\mathbb{P}}_0(\mathbb{R})$ discrete function of
dimension 3.
****** ~interpolate: mesh*(zone)*Vh_type*(function) -> Vh~
This function works exactly the same as the previous function. The
additional parameter, the ~zone~ lists is used to define the cells where
the user function (or the user function list) is interpolate. For
cells that are not in the ~zone~ list, the discrete function is set to
the value $0$.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
let m:mesh, m = readGmsh("hybrid-2d.msh");
let f:R^2 -> R, x -> 2*x[0]-x[1];
let fh:Vh, fh = interpolate(m, (zoneName("LEFT"), zoneName("RIGHT")), P0(), f);
let fh_l:Vh, fh_l = interpolate(m, zoneName("LEFT"), P0(), f);
#+END_SRC
In this example, we define two discrete functions. ~fh~ is defined as
the interpolation of the function ~f~ in *all* cells of the mesh since the
mesh is partitioned into two zones labeled ~LEFT~ and ~RIGHT~. The
discrete function ~fh_l~ has exactly the same values in the ~LEFT~ region,
but is $0$ in the cells that belong to ~RIGHT~.
For completeness, we give an example in the case of ~P0Vector~.
The number of scalar user functions sets the size of the discrete
vector function.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import math;
let m:mesh, m = readGmsh("hybrid-2d.msh");
let f0:R^2 -> R, x -> 2*x[0]-x[1];
let f1:R^2 -> R, x -> x[0]*x[1]-2;
let f2:R^2 -> R, x -> 2*dot(x,x);
let fh:Vh, fh = interpolate(m, zoneName("RIGHT"), P0Vector(), (f0,f1,f2));
#+END_SRC
Here we defined a $\vec{\mathbb{P}}_0(\mathbb{R})$ discrete function of
dimension 3.
****** ~integrate: mesh*quadrature*function -> Vh~ <<integrate-classic>>
This function integrates the given user function in each cell of a
~mesh~ using a prescribed ~quadrature~. The result is of type as
$\mathbb{P}_0(\mathbb{R})$, $\mathbb{P}_0(\mathbb{R}^1)$,
$\mathbb{P}_0(\mathbb{R}^2)$, $\mathbb{P}_0(\mathbb{R}^3)$,
$\mathbb{P}_0(\mathbb{R}^{1\times1})$, $\mathbb{P}_0(\mathbb{R}^{2\times2})$
or $\mathbb{P}_0(\mathbb{R}^{3\times3})$, according to the user function
codomain.
Let us consider the following example
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import math;
let m:mesh, m = readGmsh("hybrid-2d.msh");
let u:R^2 -> R^2, x -> [cos(x[0]), sin(x[1])];
let U:Vh, U = integrate(m, Gauss(5), u);
#+END_SRC
Here, for each cell $j$, the value of the discrete function
$\mathbf{F}_j$ is computed using a Gauss quadrature formula that is
exact for polynomials of degree $5$, $\mathbf{F}_j \approx\int_j
\mathbf{u}$. More details about quadrature formula will be given
below.
Thus if one wants to project $\mathbf{u}$ on
$\mathbb{P}_0(\mathbb{R}^2)$ to the sixth order, one can modify the
previous script by writing
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import math;
let m:mesh, m = readGmsh("hybrid-2d.msh");
let u:R^2 -> R^2, x -> [cos(x[0]), sin(x[1])];
let uh:Vh, uh = (1/cell_volume(m)) * integrate(m, Gauss(5), u);
#+END_SRC
The function ~cell_volume: mesh -> Vh~ creates a
$\mathbb{P}_0(\mathbb{R})$ function whose values are the volume of the
cells.
****** ~integrate: mesh*quadrature*Vh_type*(function) -> Vh~ <<integrate-P1-vector>>
This function behaves the same, the user function list size defines
the dimension of the vector value of the produced
$\vec{\mathbb{P}}_0(\mathbb{R})$ discrete function. Actually the
~Vh_type~ parameter is there to allow the construction of
$\vec{\mathbb{P}}_0(\mathbb{R})$ of dimension 1 (since passing a single
function as a list of user function, the previous function
[[integrate-classic]], would be used).
****** ~integrate: mesh*(zone)*quadrature*function -> Vh~
This function is an enhancement of the function defined in
[[integrate-classic]]. It allow to specify a ~zone~ list which defines the
set of cells where the integration is operated.
#+BEGIN_SRC shell :exports results :results none
cat << EOF > zones-1d.geo
//+
Point(1) = {-1, 0, 0, 0.01};
//+
Point(2) = {-0.3, 0, 0, 0.01};
//+
Point(3) = {0.3, 0, 0, 0.01};
//+
Point(4) = {1, 0, 0, 0.01};
//+
Line(1) = {1, 2};
//+
Line(2) = {2, 3};
//+
Line(3) = {3, 4};
//+
Physical Point("XMIN") = {1};
//+
Physical Point("XMAX") = {4};
//+
Physical Point("INTERFACE1") = {2};
//+
Physical Point("INTERFACE2") = {3};
//+
Physical Curve("LEFT") = {1};
//+
Physical Curve("MIDDLE") = {2};
//+
Physical Curve("RIGHT") = {3};
#+END_SRC
#+BEGIN_SRC shell :exports results :results none
gmsh -1 zones-1d.geo -format msh2
#+END_SRC
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import math;
import writer;
let pi:R, pi = acos(-1);
let m:mesh, m = readGmsh("zones-1d.msh");
let f:R^1 -> R, x -> sin(2*pi*x[0]);
let fh:Vh,
fh = (1/cell_volume(m))
,* integrate(m, (zoneName("LEFT"), zoneName("RIGHT")), Gauss(5), f);
write(gnuplot_1d_writer("zone_integrate"), name_output(fh, "f"));
#+END_SRC
In this example, the ~mesh~ provided in the file ~zones-1d.msh~ is a 1d
~mesh~ of $]-1,1[$ made of $200$ cells that is partitioned into 3
connected subdomains. The zones corresponding to these 3 subdomains
are named ~LEFT~ for $]-1,-0.3[$, ~MIDDLE~ for $]-0.3, 0.3[$ and ~RIGHT~ for
$]0.3,1[$. The result is displayed on Figure [[fig:zone-integrate-1d]]. In
the ~MIDDLE~ region, cell values are set to 0.
#+NAME: zone-integrate-1d-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/zone-integrate-1d.png")
reset
set grid
set border
unset key
set xtics
set ytics
set terminal png truecolor enhanced size 960,480
plot '<(sed "" $PUGS_SOURCE_DIR/doc/zone_integrate.gnu)' lw 2 w l
#+END_SRC
#+CAPTION: $L^1$ projection of order 6 of the function $\sin(2\pi x)$ on the zones ~LEFT~ and ~RIGHT~. The values in the zone ~MIDDLE~ are set to $0$.
#+NAME: fig:zone-integrate-1d
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: zone-integrate-1d-img
****** ~integrate: mesh*(zone)*quadrature*Vh_type*(function) -> Vh~
This function behaves essentially as the function described in
paragraph [[integrate-P1-vector]], it also adds the possibility to choose
sets of cells where to integrate the list of user functions.
***** Random mesh generators
For numerical it is often useful to create meshes with random vertices
positions. This is the aim of the functions that are described in this
section. These function share some properties.
- The generate mesh is always suitable for calculations in the sense
that cells volumes are warrantied to be positive.
- Generated cells can be non-convex.
- One has to specify boundary conditions to drive the mesh
displacement on boundaries.
- The obtained mesh does not depend on parallelism: it is exactly the
same whichever is the number of ~MPI~ processes. It only depends on
the random seed (see paragraph [[set-random-seed]] how to set the random
seed to obtain the exact same mesh through different runs).
****** ~randomizeMesh: mesh*(boundary_condition) -> mesh~
This function creates a random mesh by displacing the nodes of a given
~mesh~ and a list of ~boundary_condition~.
The supported boundary conditions are the following:
- ~fixed~: the vertices of the ~boundary~ cannot be displaced
- ~axis~: the vertices are allowed to be displaced in the direction of
the ~boundary~
- ~symmetry~: the vertices are displaced in the plane formed by the
~boundary~
One should refer to the section [[boundary-condition-descriptor]] for a
description of the boundary condition descriptors.
#+BEGIN_note
Let us precise these boundary conditions behavior
- In dimension 1, ~fixed~, ~axis~ and ~symmetry~ boundary conditions have
the same effect.
- In dimension 2, ~axis~ and ~symmetry~ behave the same. Thus, boundaries
supporting this kind of boundary conditions *must* form *straight*
lines.
- In dimension 3, boundaries describing ~axis~ conditions *must* be
*straight* lines, and boundaries describing ~symmetry~ conditions *must*
be *planar*.
If a boundary does not satisfy geometrical requirements, ~pugs~ produces
a runtime error.
#+END_note
Let us consider a simple example
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import writer;
setRandomSeed(123456789); // not required
let m:mesh, m = cartesianMesh([-1,-1],[1,1], (20,20));
let bc_list:(boundary_condition),
bc_list = ((fixed(boundaryName("XMINYMIN"))),
(fixed(boundaryName("XMINYMAX"))),
(fixed(boundaryName("XMAXYMIN"))),
(fixed(boundaryName("XMAXYMAX"))),
(symmetry(boundaryName("XMIN"))),
(symmetry(boundaryName("XMAX"))),
(symmetry(boundaryName("YMIN"))),
(symmetry(boundaryName("YMAX"))));
m = randomizeMesh(m, bc_list);
write_mesh(gnuplot_writer("random-mesh"), m);
#+END_SRC
Running this script one gets the ~mesh~ displayed on Figure
[[fig:random-mesh]]. To reduce the vertices displacement, one can use the
~relax~ function, see section [[relax-function]].
#+NAME: random-mesh-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/random-mesh.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set square
set terminal png truecolor enhanced size 640,640
plot '<(sed "" $PUGS_SOURCE_DIR/doc/random-mesh.gnu)' w l
#+END_SRC
#+CAPTION: The obtained random mesh
#+NAME: fig:random-mesh
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: random-mesh-img
****** ~randomizeMesh: mesh*(boundary_condition)*function -> mesh~
This function is a variation of the previous one. It allows
additionally to provide a characteristic function that designates the
vertices that can be displaced.
The characteristic function *must* be a function of $\mathbb{R}^d
\to\mathbb{B}$ for a ~mesh~ of dimension $d$.
Here is a modification of the previous example, where the random
displacement is allowed for $x<2y$.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
import writer;
import math;
setRandomSeed(123456789); // not required
let m:mesh, m = cartesianMesh([-1,-1],[1,1], (20,20));
let bc_list:(boundary_condition),
bc_list = ((fixed(boundaryName("XMINYMIN"))),
(fixed(boundaryName("XMINYMAX"))),
(fixed(boundaryName("XMAXYMIN"))),
(fixed(boundaryName("XMAXYMAX"))),
(symmetry(boundaryName("XMIN"))),
(symmetry(boundaryName("XMAX"))),
(symmetry(boundaryName("YMIN"))),
(symmetry(boundaryName("YMAX"))));
let chi:R^2 -> B, x -> x[0]<2*x[1];
m = randomizeMesh(m, bc_list, chi);
write_mesh(gnuplot_writer("random-mesh-chi"), m);
#+END_SRC
Running this script one gets the ~mesh~ displayed on Figure
[[fig:random-mesh-chi]].
#+NAME: random-mesh-chi-img
#+BEGIN_SRC gnuplot :exports results :file (substitute-in-file-name "${PUGS_SOURCE_DIR}/doc/random-mesh-chi.png")
reset
unset grid
unset border
unset key
unset xtics
unset ytics
set square
set terminal png truecolor enhanced size 640,640
plot '<(sed "" $PUGS_SOURCE_DIR/doc/random-mesh-chi.gnu)' w l
#+END_SRC
#+CAPTION: Random mesh with characteristic function. The original mesh is unchanged for $x \ge 2 y$.
#+NAME: fig:random-mesh-chi
#+ATTR_LATEX: :width 0.38\textwidth
#+ATTR_HTML: :width 300px;
#+RESULTS: random-mesh-chi-img
#+BEGIN_note
Since we set the random seed to the same value in both cases (with or
without using the characteristic function $\chi$), node displacements are
the same $\forall r\in\mathcal{R}$ such that $\chi(\mathbf{x}_r)$, see Figures
[[fig:random-mesh]] and [[fig:random-mesh-chi]].
This allows for instance to build a random mesh step-by-step.
#+END_note
***** Boundary condition descriptors <<boundary-condition-descriptor>>
In this paragraph, we describe the set of boundary condition
descriptors that are provided by the ~scheme~ module.
#+BEGIN_note
These functions provide *descriptors*, these are not related to a
particular implementation. The way they are used in different
functions dependents of the context or the numerical method itself.
#+END_note
#+BEGIN_note
There a three kind of boundaries are supported by ~pugs~, boundaries
made
- of sets of nodes,
- of sets of edges, or
- of sets of faces.
In dimension 1, nodes, edges and faces denotes the same entities. In
dimension 2, edges and faces refer the same entities.
#+END_note
#+BEGIN_note
~pugs~ integrates some automatic mechanisms to translate boundary types
in other ones.
For instance, if an algorithm or a method requires a set of nodes to
set some numerical treatment, it can be deduced from a set of faces.
Obviously, these conversions can be meaningless, for instance, if one
expects a *line* in 3d, cannot be defined by a set of faces. ~pugs~ will
forbid this kind of conversion at runtime.
#+END_note
#+BEGIN_note
When a method expects a *straight* line or a *planar* surface, ~pugs~ checks
that the given boundary is actually *straight* or *planar*.
#+END_note
#+BEGIN_note
~pugs~ checks that boundaries do not contain /inner/ items.
#+END_note
We regroup the boundary condition descriptors functions according to
their arguments
****** ~boundary -> boundary_condition~
- ~axis: boundary -> boundary_condition~\\
The boundary denotes a *straight* line of the mesh
- ~fixed: boundary -> boundary_condition~
- ~symmetry: boundary -> boundary_condition~\\
The boundary denotes a *planar* surface of the mesh
****** ~boundary*function -> boundary_condition~
The provided user function type depends on the numerical method of
function that utilizes the boundary condition.
- ~pressure: boundary*function -> boundary_condition~
- ~velocity: boundary*function -> boundary_condition~
***** Quadrature formulas descriptors
~pugs~ provides a quite complete set of quadrature formulas that can be
chosen inside the script.
#+BEGIN_warning
While ~pugs~ is written to deal with general polygonal and polyhedral
meshes, quadrature formulas are not yet implemented on the general
elements. Supported elements are triangles, quadrangles, tetrahedra
pyramids, prisms and hexahedra.
#+END_warning
****** ~Gauss: N -> quadrature~
This function returns the quadrature descriptor associated to Gauss
formulas for the given ~N~.
In the following table, we summarize the *maximal degree* quadrature
that are available in ~pugs~ for various elements.
| element type | max. degree |
|------------------------+-------------|
| segment | 23 |
|------------------------+-------------|
| triangle | 20 |
| square | 21 |
|------------------------+-------------|
| tetrahedron | 20 |
| pyramid (square basis) | 20 |
| prism (triangle basis) | 20 |
| cube | 21 |
****** ~GaussLegendre: N -> quadrature~
Gets the Gauss-Legendre quadrature descriptor exact for polynomials of
degree given in argument.
The maximum allowed degree is 23.
For dimension 2 or 3 elements, Gauss-Legendre formulas are defined by
tensorization. Conform transformations are used to map the cube
$]-1,1[^d$ to supported elements.
****** ~GaussLobatto: N -> quadrature~
Gets the Gauss-Lobatto quadrature descriptor exact for polynomials of
degree given in argument.
The maximum allowed degree is "only" 13.
For dimension 2 or 3 elements, Gauss-Lobatto formulas are defined by
tensorization. Conform transformations are used to map cube $]-1,1[^d$
to supported elements.
***** ~lagrangian: mesh*Vh -> Vh~
This function is a special function whose purpose is to transport
lagrangian quantities from one mesh to the other. Obviously, this
function make lots of sense in the case of lagrangian calculations.
This is a zero cost function, since discrete functions are *constants*
in ~pugs~, it consists in associating the data of the given discrete
function to the provided ~mesh~. In other words, the underlying array of
values is shared by the two discrete functions, which are associated
to different meshes.
A good example of the use of this kind of function is mass fractions.
***** Numerical methods
We describe rapidly two functions that embed numerical methods. These
are in some sense /models/ that are used to test evolution of ~pugs~
itself and can be used as examples.
#+BEGIN_warning
These functions will become obsolete (soon?), since another interface
to numerical methods is in preparation.
#+END_warning
The functions are
- ~eucclhyd_solver: Vh*Vh*Vh*Vh*Vh*(boundary_condition)*R -> mesh*Vh*Vh*Vh~
- ~glace_solver: Vh*Vh*Vh*Vh*Vh*(boundary_condition)*R -> mesh*Vh*Vh*Vh~
Both functions share the same interface. The arguments are provided in
this order:
- the mass density $\rho$ of type $\mathbb{P}_0(\mathbb{R})$,
- the velocity $\mathbf{u}$ of type $\mathbb{P}_0(\mathbb{R}^d)$ in
dimension $d$,
- the total energy density $E$ of type $\mathbb{P}_0(\mathbb{R})$,
- the sound speed $c$ of type $\mathbb{P}_0(\mathbb{R})$,
- the pressure $p$ of type $\mathbb{P}_0(\mathbb{R})$,
- a list of boundary conditions,
- and a time step of type ~R~.
Observe that ~pugs~ checks the types of the parameters and that all
discrete functions are defined on the same mesh.
The functions return a compound type made of
- the new ~mesh~ $\mathcal{M}$,
- the new mass density ~\rho~ of type $\mathbb{P}_0(\mathbb{R})$ defined
on $\mathcal{M}$,
- the new velocity $\mathbf{u}$ of type $\mathbb{P}_0(\mathbb{R}^d)$ in
dimension $d$, defined on the mesh $\mathcal{M}$,
- and the new total energy density $E$ of type
$\mathbb{P}_0(\mathbb{R})$, defined on the mesh $\mathcal{M}$.
The time step can be calculated through the ~acoustic_dt: Vh -> R~
function.
**** Operators overloading for ~Vh~ <<Vh-operators>>
The ~scheme~ module provides overload of binary operators. Since ~Vh~ is
an abstract type, some operators may not be defined (or allowed) for
concrete types.
***** Unary operators
The only supported unary operators for ~Vh~ are given in the following
table
| operator | description |
|----------+----------------------|
| ~+~ | plus unary operator |
| ~-~ | minus unary operator |
We recall that the unary ~+~ operator is a convenience operator that has
no effect.
***** Binary operators
The supported binary operators for ~vh~ data types are arithmetic
operators.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{V}_h \times \mathbb{V}_h \to \mathbb{V}_h\\
\mathtt{-}:& \mathbb{V}_h \times \mathbb{V}_h \to \mathbb{V}_h\\
\mathtt{*}:& \mathbb{V}_h \times \mathbb{V}_h \to \mathbb{V}_h\\
\mathtt{/}:& \mathbb{V}_h \times \mathbb{V}_h \to \mathbb{V}_h
\end{array}
\right.
\end{equation*}
#+end_src
Observe that in the case of $\vec{\mathbb{P}}_0(\mathbb{R})$, the only
available operators are ~+~ and ~-~.
#+BEGIN_note
In the case of $\mathbb{P}_0$ functions, an operator is defined as soon
as it is defined for the value type.
#+END_note
For instance, one can multiply a $\mathbb{P}_0(\mathbb{R}^{2\times2})$
discrete function by a $\mathbb{P}_0(\mathbb{R}^2)$. The result is then
a $\mathbb{P}_0(\mathbb{R}^2)$ discrete function.
#+BEGIN_SRC pugs :exports both :results none
import mesh;
import scheme;
let m:mesh, m = cartesianMesh(0,[1,1],(10,10));
let A:R^2 -> R^2x2, x -> [[2*x[0], x[1]],[-x[0], 3*x[1]]];
let u:R^2 -> R^2, x -> 2*[x[0], x[1]*x[0]];
let Ah:Vh, Ah = interpolate(m, P0(), A);
let uh:Vh, uh = interpolate(m, P0(), u);
let Auh:Vh, Auh = Ah*uh;
#+END_SRC
Another illustration is: trying to add $\mathbb{P}_0(\mathbb{R})$ and
$\mathbb{P}_0(\mathbb{R}^1)$
#+NAME: invalid-Vh-sum-type
#+BEGIN_SRC pugs-error :exports both :results output
import mesh;
import scheme;
let m:mesh, m = cartesianMesh([0],[1],10);
let f:R^1 -> R, x -> 2*x[0];
let f1:R^1 -> R^1, x -> 2*x;
let fh:Vh, fh = interpolate(m, P0(), f);
let f1h:Vh, f1h = interpolate(m, P0(), f1);
fh+f1h;
#+END_SRC
produces the following error
#+results: invalid-Vh-sum-type
****** Additional ~+~ and ~-~ operators
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S}, \mathbb{S}_2 \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R},\mathbb{R}^1,\mathbb{R}^2,\mathbb{R}^3,\mathbb{R}^{1\times1},\mathbb{R}^{2\times2},\mathbb{R}^{3\times3}\},
\quad
\left|
\begin{array}{rl}
\mathtt{+}:& \mathbb{S} \times \mathbb{V}_h \to \mathbb{V}_h\\
\mathtt{-}:& \mathbb{S} \times \mathbb{V}_h \to \mathbb{V}_h\\
\mathtt{+}:& \mathbb{V}_h \times \mathbb{S} \to \mathbb{V}_h\\
\mathtt{-}:& \mathbb{V}_h \times \mathbb{S} \to \mathbb{V}_h
\end{array}
\right.
\end{equation*}
#+end_src
Let us consider the following example
#+NAME: substract-mean-value-to-Vh
#+BEGIN_SRC pugs :exports both :results output
import mesh;
import scheme;
import math;
let m:mesh, m = cartesianMesh(0,[0.3,1.1],(5,15));
let mesh_volume:R, mesh_volume = sum_of_R(cell_volume(m));
let u:R^2 -> R, x -> 2*dot(x,x);
let uh:Vh, uh = interpolate(m, P0(), u);
let uh0:Vh, uh0 = uh - (integral_of_R(uh) / mesh_volume);
cout << "integral(uh) = " << integral_of_R(uh) << "\n";
cout << "integral(uh0) = " << integral_of_R(uh0) << "\n";
#+END_SRC
Here we substract the mean value of a discrete function.
#+results: substract-mean-value-to-Vh
****** Additional ~*~ operators
The following constructions are allowed for ~*~ operator.
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S} \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R},\mathbb{R}^{1\times1},\mathbb{R}^{2\times2},\mathbb{R}^{3\times3}\},
\quad
\mathtt{*}: \mathbb{S} \times \mathbb{V}_h \to \mathbb{V}_h.
\end{equation*}
#+end_src
Obviously, if $\mathbb{S}=\mathbb{R}^{d\times d}$, for $d\in\{1,2,3\}$,
the right operand must be have a compatible type, for instance
$\mathbb{P}_0(\mathbb{R}^{d\times d})$ or $\mathbb{P}_0(\mathbb{R}^d)$.
Additionally these operators are defined
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S} \in \{\mathbb{B},\mathbb{N},\mathbb{Z},\mathbb{R},\mathbb{R}^1,\mathbb{R}^2,\mathbb{R}^3,\mathbb{R}^{1\times1},\mathbb{R}^{2\times2},\mathbb{R}^{3\times3}\},
\quad
\mathtt{*}: \mathbb{V}_h \times \mathbb{S}\to \mathbb{V}_h.
\end{equation*}
#+end_src
Following logic, if for instance the right operand is an ~R^2~
expression, the left operand ~Vh~ must be of type
$\mathbb{P}_0(\mathbb{R})$ or $\mathbb{P}_0(\mathbb{R}^{2\times 2})$.
****** Additional ~/~ operators
The following operators are defined
#+begin_src latex :results drawer :exports results
\begin{equation*}
\forall \mathbb{S} \in \{\mathbb{N},\mathbb{Z},\mathbb{R}\},
\quad
\mathtt{/}: \mathbb{S} \times \mathbb{V}_h \to \mathbb{V}_h.
\end{equation*}
#+end_src
*** The ~linear_solver~ module
This module provides the following functions
#+NAME: get-module-info-linear-solver
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("linear_solver") << "\n";
#+END_SRC
#+RESULTS: get-module-info-linear-solver
**** ~linear_solver~ provided functions
The set of provided functions is used to define the global behavior of
~pugs~ linear system solver.
***** Utility functions
An important function is
****** ~getLSAvailable: void -> string~
This shows the available options or libraries that are available. It
depends on the compilation options of the code.
#+NAME: get-ls-available-example
#+BEGIN_SRC pugs :exports both :results output
import linear_solver;
cout << getLSAvailable() << "\n";
#+END_SRC
#+results: get-ls-available-example
****** ~getLSOptions: void -> string~
This function show the current tuning of the global linear solver
#+NAME: get-ls-options-example
#+BEGIN_SRC pugs :exports both :results output
import linear_solver;
cout << getLSOptions() << "\n";
#+END_SRC
#+results: get-ls-options-example
***** Tuning functions
****** ~setLSEpsilon: R -> void~
Sets the relative epsilon criterion for iterative methods.
****** ~setLSLibrary: string -> void~
Selects the library to use.
****** ~setLSMaxIter: N -> void~
Sets the maximum number of iterations.
****** ~setLSMethod: string -> void~
Sets the method to solve linear systems.
****** ~setLSPrecond: string -> void~
Sets the preconditioner type.
****** ~setLSVerbosity: B -> void~
Sets verbosity mode to ~true~ or ~false~ for linear solvers.
*** The ~writer~ module
This module provides functionalities to numerical results to files for
post processing.
It provides the following functions and types.
#+NAME: get-module-info-writer
#+BEGIN_SRC pugs :exports both :results output
cout << getModuleInfo("writer") << "\n";
#+END_SRC
#+RESULTS: get-module-info-writer
[fn:pugs-def] ~pugs~: Parallel Unstructured Grid Solvers
[fn:MPI-def] ~MPI~: Message Passing Interface
[fn:DSL-def] ~DSL~: Domain Specific Language~