## 12 August 2014

### SubDomain

As said in my previous post, a missing feature in fem-fenics was the marking of subdomains. Indeed, I proposed an example that needed a file generated with a run of the corresponding Python code, which is not, honestly, the best approach. In order to address this issue, these days I have implemented a new class, subdomain, which can be used to mark mesh entities. In the following I will describe how to use this new functionality. Here is the code:

ufl start Subdomains
ufl fe = FiniteElement "(""CG"", triangle, 2)"
ufl u = TrialFunction (fe)
ufl v = TestFunction (fe)
ufl
ufl a0 = Coefficient (fe)
ufl a1 = Coefficient (fe)
ufl g_L = Coefficient (fe)
ufl g_R = Coefficient (fe)
ufl f = Coefficient (fe)
ufl
ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1)
ufl end

# Create mesh and define function space
x = y = linspace (0, 1, 65);
[msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1));

V = FunctionSpace ("Subdomains", msh);

# Define boundary conditions
bc1 = DirichletBC (V, @(x, y) 5.0, facets, 2);
bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4);

# Define problem coefficients
a0 = Constant ("a0", 1.0);
a1 = Constant ("a1", 0.01);
g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2));
g_R = Constant ("g_R", 1.0);
f = Constant ("f", 1.0);

# Define subdomains - Here are the edits #
domains = MeshFunction ("dx", msh, 2, 0);
obstacle = SubDomain (@(x,y) (y >= 0.5) && (y <= 0.7) && ...
(x >= 0.2) && (x <= 1.0), false);
domains = mark (obstacle, domains, 1);

# Define variational form
a = BilinearForm ("Subdomains", V, V, a0, a1, domains);
L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains);

# Assemble system
[A, b] = assemble_system (a, L, bc1, bc2);
sol = A \ b;
u = Function ("u", V, sol);

# Save solution in VTK format
save (u, "subdomains");

# Plot solution
[X, Y] = meshgrid (x, y);
U = u (X, Y);
surf (X, Y, U);

As you can see, it is basically the same as in the previous post, except the line used to import the meshfunction. I wrote in the corresponding comment where the edits are to be found. Now the workflow comprises these steps: first of all, a meshfunction needs to be created, then a subdomain, in the end we should mark cells.

The call to MeshFunction is something new, since it is now possible to instantiate a meshfunction given a mesh, the required topological dimension and the value to initialise it with. Moreover, the optional label "dx" means that it can be used in calls to BilinearForm or LinearForm to supply markers for subsets of the integration domains. In the example, this instruction returns a meshfunction of dimension 2, which means it holds values associated with each triangle in the mesh, initialised to be 0 in every entry.

The subsequent instruction, instead, defines a subdomain, passing as arguments a function handle and a logical flag. The former will be the override of the dolfin::SubDomain::inside method, so it must return true for entities contained in the subset and false otherwise. In facts it checks whether the coordinates are inside the 2-interval defining the obstacle. The latter, instead, can be used to ask for a boundary subdomain, when set to true.

At last, mark is called to set the entries corresponding to cells inside the subdomain to 1, so that the returned meshfunction now represents the obstacle: after these lines, the variable named domains assumes value 1 on cells inside the obstacle region and 0 outside. Thus, it is now possible to solve a problem whose formulation entails subdomains entirely using fem-fenics.

## 09 August 2014

### New features of meshfunction

As you may recall from my last post, for DirichletBC to work in parallel runs I had to implement a new class, meshfunction. However it was still quite unfinished, with no way for the user to create one, except extracting it from a mesh produced by the msh package, no description to display, no way to save it. These days I have been tackling this issue: while at it I wondered what one could do with meshfunction and found out that they can come in handy when you are dealing with obstacles.

At this link you can find a detailed explanation of the problem. It is a Poisson equation with variable diffusion coefficient on the unit square. Precisely, on [0.2, 1]x[0.5, 0.7] its value is 0.01, otherwise it is 1. The mentioned subset is the obstacle to diffusion, so we study its effect applying u = 0 on the y = 0 edge and u = 5 on y = 1. Here is the fem-fenics code:

``` pkg load fem-fenics msh ufl start Subdomains ufl fe = FiniteElement "(""CG"", triangle, 2)" ufl u = TrialFunction (fe) ufl v = TestFunction (fe) ufl ufl a0 = Coefficient (fe) ufl a1 = Coefficient (fe) ufl g_L = Coefficient (fe) ufl g_R = Coefficient (fe) ufl f = Coefficient (fe) ufl ufl a = "inner(a0*grad (u), grad (v))*dx(0) + inner(a1*grad (u), grad (v))*dx(1)" ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1) ufl end # Create mesh and define function space x = y = linspace (0, 1, 65); [msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1)); V = FunctionSpace ("Subdomains", msh); # Define boundary conditions bc1 = DirichletBC (V, @(x, y) 5.0, facets, 2); bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4); # Define problem coefficients a0 = Constant ("a0", 1.0); a1 = Constant ("a1", 0.01); g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2)); g_R = Constant ("g_R", 1.0); f = Constant ("f", 1.0); # Define subdomains domains = MeshFunction ("dx", msh, "cells.xdmf"); # Define variational form a = BilinearForm ("Subdomains", V, V, a0, a1, domains); L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains); # Assemble system [A, b] = assemble_system (a, L, bc1, bc2); sol = A \ b; u = Function ("u", V, sol); # Save solution in VTK format save (u, "subdomains"); # Plot solution [X, Y] = meshgrid (x, y); U = u (X, Y); surf (X, Y, U); ```

In the beginning there is the now familiar ufl block. As you might have noticed, subscripted measures appear in the definition of the bilinear form a and of the linear functional L. This is UFL notation for the integration on specific subsets of the computational domain. For instance, dx(1) is an integral over the subdomain marked with label 1, while ds(3) is an integral over the exterior edges marked with label 3. A third possibility, even if not used in this example, is to use dS for integrals on interior facets, which could be of use for interior penalty methods. Going back to the example, you can see that markers are used to enforce non-homogeneous Neumann conditions on the side edges and to assign the proper coefficient on the two subdomains.

After defining the problem in UFL language, there are instructions to define the mesh, the function space, the essential boundary conditions and all the coefficients involved. All such lines come from fem-fenics before this summer or have been described in my previous posts, so I will not cover them in detail. The same applies for the assembly, solve and all the output in the end of the script. The only note is that the very last lines will error out in parallel runs: point-wise evaluations in DOLFIN can be performed only on local cells, but with meshgrid we are providing to every process the whole domain.
 The computed solution
In between there are my latest efforts. At first, the brand new MeshFunction. With this, providing a mesh and a file name you can import a dolfin::MeshFunction. In this case it was saved in the XDMF format, here you can find the files needed to execute the script. DOLFIN uses this format for parallel input/output. It comprises a .h5 file storing data and a .xdmf with metadata useful to read the other one. The optional first argument is a string identifying the role of the returned meshfunction in the variational problem. In this case, with "dx" it will be searched for markers of the integrals on cells. All the previously mentioned measures are available, and "ds" is automatically attached to the meshfunction returned by Mesh. In the example this behaviour is exploited for the measure on edges.

Afterwards, the mesh functions are passed as arguments to BilinearForm and LinearForm, so that the markers are available to assemble the system. In addition to the usual parameters, such as the name of the imported UFL problem, the function spaces and the coefficients, it is now possible to provide mesh functions properly labeled and they will be used.

Currently fem-fenics allows for easily marking subdomains and exterior edges copying markers from the PDEtool representation returned by the functions of the msh package, which makes it quite tricky to properly identify the obstacle in the example. The approach used in the python interface to DOLFIN entails subclassing dolfin::Subdomain with the proper implementation of the inside method, then use an object of the derived class to mark a dolfin::MeshFunction. This could be an interesting feature to implement in the future also in fem-fenics.

## 04 August 2014

### MPI parallelism

After quite a struggle, I have been able to obtain a working implementation of fem-fenics supporting MPI parallelism. Let's go through an example and highlight what has changed lately.

ufl start Poisson
ufl element = FiniteElement '("Lagrange", triangle, 1)'
ufl u = TrialFunction (element)
ufl v = TestFunction (element)
ufl f = Coefficient (element)
ufl g = Coefficient (element)
ufl L = f*v*dx + g*v*ds
ufl end

# Create mesh and define function space
x = y = linspace (0, 1, 33);
[mesh, facets] = Mesh (msh2m_structured_mesh (x, y, 1, 1:4));

V = FunctionSpace ('Poisson', mesh);

# Define boundary condition
bc = DirichletBC (V, @(x, y) 0.0, facets, [2;4]);

f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
g = Expression ('g', @(x,y) sin (5.0 * x));

a = BilinearForm ('Poisson', V, V);
L = LinearForm ('Poisson', V, f, g);

# Compute solution
[A, b] = assemble_system (a, L, bc);
sol = A \ b;
u = Function ('u', V, sol);

# Save solution in VTK format
save (u, 'poisson');

The basic structure has remained the same. DOLFIN boasts the capability to be run both in serial and in parallel execution without intervening on the code, so I did my best to have the same behaviour from fem-fenics. The Poisson.m m-file above can be run either as you usually would do with any other m-file, or from the command line with an invocation such as:

mpiexec -np 4 octave --eval Poisson

Now, how is this possible? In the beginning, with the ufl block, the variational problem is defined in UFL language, written to an .ufl file and compiled via FFC. Since IO is performed, ufl.m ensures that only process zero will open and write to the file. Moreover, a MPI barrier makes sure that no process will proceed before the .ufl file is imported.

As soon as the just-in-time compilation is over, there are two instructions to build the mesh, in this case on the unit square. For this, we rely on the msh package, which returns a PDE-tool-like representation of it. Mesh.oct must, then, convert it to DOLFIN internal representation and distribute it among processes. Here comes an issue: fem-fenics relies on markers present in the PDE-tool format to impose essential boundary conditions, and in serial runs dolfin::Mesh can store them, so that DirichletBC.oct needs just to know the boundary subset label. Unfortunately, this feature is not supported yet in parallel by the DOLFIN library, then Mesh.oct has been edited to return, if requested, also a meshfunction holding this information, in the example above facets. This way markers can be conveyed to DirichletBC.oct and boundary conditions can be applied on the correct edges.

Further intervention was needed for the assembly and solve phase. In assemble_system.oct both the matrix and the vector are assembled locally on each portion of the mesh and, afterwards, gathered on process zero and joined, so that the system can be solved with the backslash instruction of Octave. In order to allow output in VTK format, in Function.oct the solution is split up and properly distributed among processes, so that each one holds the portion of degrees of freedom related to its subdomain and to the neighbouring vertices. After save.oct has written the solution to poisson.pvd and its auxiliary files, it can be visualised with ParaView.

## 30 July 2014

### Support for DOLFIN 1.4.0

Lately I have not been very active on the blog since I am encountering some difficulties in the attempt to introduce MPI parallelism. Meanwhile, I have extended support to the latest version of the DOLFIN library.

Among other changes, one that strongly affects fem-fenics is the shift from the shared pointer implementation by the Boost libraries to the one included in the Standard Template Library with the new C++11 standard. This change alone calls for edits in almost all the codebase of the package, as basically all DOLFIN data structures are stored via smart pointers in the corresponding fem-fenics classes. However, currently version 1.3.0 is still present in the official repositories of the main Linux distributions, thus switching abruptly to the latest version would have prevented further releases of the package for a while.

In order to tackle the above mentioned issue, I resorted to the preprocessor capabilities, so as to infer from the DOLFIN version available on the compiling computer the right kind of pointers to use. Among other options, the preprocessor flags obtained using pkg-config define also a macro reporting the DOLFIN version. It is, then, possible to check it and choose the correct pointer implementation right before compilation. Currently in fem-fenics every occurrence of boost::shared_ptr has been replaced by a SHARED_PTR macro, which in turn is defined in a new header that takes care of setting it to the right value. There is just a catch: preprocessor conditionals cannot compare strings, but the DOLFIN_VERSION macro is indeed defined as a string. In order for this approach to work, the package Makefile, for the initial compilation, and the get_vars.m function, for the just-in-time ones, perform the actual check and define an auxiliary macro if the latest version is found on the system.

## 12 July 2014

### Factories for assembly

fem-fenics provides a couple of utilities for assembling matrices and vectors, which compose the algebraic system weak formulations are reduced to when applying FEMs. Left aside all the checks needed to verify inputs, their job boils down to creating proper DOLFIN objects to store these matrices or vectors, possibly applying boundary conditions, then building an Octave-friendly representation. This last task is quite critical for the implementation of the MPI parallelisation, as the underlying DOLFIN representation of matrices and vectors is transparently distributed among processes, thus making the serial code used to pass them on to Octave useless. Lately I implemented some new classes to manage this aspect, so I will highlight my design considerations.

The translation from DOLFIN's to Octave's data structures is logically a well defined task, whilst its implementation needs to vary according to its serial or parallel execution. Furthermore, it strictly depends on the linear algebra back-end used, for each of them stores a different representation and exposes a different interface to access it. To address these difficulties, I wrote a hierarchy of factories to provide the adequate methods, based on the run-time necessities. Moreover, this way the code is easily expandable to allow for more back-ends to be used in fem-fenics (currently only uBLAS is available). There is an abstract class, to declare the interface of its derived ones, and a concrete factory implementing the uBLAS-specific methods.

Since in a future fem-fenics there will be several algebraic back-ends available for use, the hierarchy will expand. This means that the checks of the run-time configuration will eventually become more complex. Another issue comes from the need to use different types depending on information available only at run-time. Both to encapsulate those checks, avoiding code duplication, and to solve the problem of choosing the right class, I added to the hierarchy a class implementing the Pimpl idiom. With this design, the "user code" in the C++ implementation of assemble.oct and assemble_system.oct needs just to create a femfenics_factory object and use it to extract the data structures of interest, while every other hassle is dealt with behind the scenes by this class.

 UML diagram of the new hierarchy

In the diagram above you can see the already implemented classes and an example class to point out where others will collocate amongst them. femfenics_factory has private methods to check which is the right concrete class to use each time, and implements the public methods of the abstract class dispatching the call through a reference. uBLAS_factory, as other concrete classes are expected to do, holds the real code for creating Octave matrices and vectors and exposes a static method, instance, which allows for access to the singleton object of this type. femfenics_factory, in turn, obtains with it the reference needed for dispatching.

## 08 July 2014

### MPI and the problems to face

These days I have started my investigations on the actual implementation of the MPI parallelisation in fem-fenics. I found out some information that I will point out here, together with my goals for the next weeks.

First of all, apparently MPI can be used without user intervention on the serial code. This is a feature that DOLFIN boasts, but I would expect it not to pass on to fem-fenics, at least not without some effort on the implementation side. Furthermore, DOLFIN offers also a wrapper for MPI functionalities, thus probably it can be helpful in managing data transfers among threads in C++ code.

An issue that will need to be addressed is making ufl.m robust to parallel execution, since its serial implementation leads to all workers trying to open the same file, thus leading to an error that stops computation. Anyway, even if they could all open the file and write to it, this would entail that lines are copied in random order or more than once, so it must be fixed.

In the end, it seems that the partitioning procedure produces matrices that are not slices of the one assembled in serial execution. Due to this fact, I must go deep in the algorithm to find out the proper way to merge the pieces and obtain the complete matrix, which will be stored as octave_value to allow for further computation using Octave's features.

## 25 June 2014

### Mid term accomplishments

I will try to give a comprehensive feel of what I achieved in this first part of the Google Summer of Code, since it is time for the mid term evaluation. Let's start with an example: as usual, it is the Poisson equation, but today, as a twist, we consider a fully Neumann problem. In order for such a problem to be well posed there is the need of an additional constraint, otherwise the solution would not be unique, so in the Octave code there is the Lagrange multiplier cHere you can find more details and the C++ and Python code, I will just write down the differential problem for convenience:

- Δu = f in Ω
u ⋅ n = g on ∂Ω

Here is the Octave code that solves the above mentioned problem:

ufl start NeumannPoisson
ufl CG = FiniteElement '("CG", triangle, 1)'
ufl R = FiniteElement '("R", triangle, 0)'
ufl W = CG * R
ufl
ufl "(u, c)" = TrialFunctions (W)
ufl "(v, d)" = TestFunctions (W)
ufl f = Coefficient (CG)
ufl g = Coefficient (CG)
ufl
ufl a = "(inner (grad (u), grad (v)) + c*v + u*d)*dx"
ufl L = f*v*dx + g*v*ds
ufl end

# Create mesh and function space
x = y = linspace (0, 1, 33);
mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));

W = FunctionSpace ("NeumannPoisson", mesh);

# Define variational problem
f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
g = Expression ('g', @(x,y) - sin (5.0 * x));

a = BilinearForm ("NeumannPoisson", W, W);
L = LinearForm ("NeumannPoisson", W, f, g);

# Compute solution
[A, b] = assemble_system (a, L);
sol = A \ b;
solution = Function ('solution', W, sol);

u = Function ('u', solution, 1);

# Plot solution
[X, Y] = meshgrid (x, y);
U = u (X, Y);
surf (X, Y, U);

At the very beginning you can see a block with every line starting with ufl. That is what you would have to put in a separate UFL file before this summer. In a sense it is not plain UFL, but there are extra quotes and apices. They are needed because, using the current version of Octave, those brackets with commas inside would otherwise be interpreted as function calls. After this blocks closes with the ufl end line, the resulting UFL file is compiled to obtain a FunctionSpace, a BilinearForm and a LinearForm. These are oct-files that fem-fenics will use later on to define the corresponding variables in Octave. A robust implementation of ufl.m, the function that provides this binding to the UFL language, is one of the results of the first term.

In the end of the snippet you can see that the solution u is evaluated in its domain exactly as you expect to do with a regular function taking two arguments and returning one value. This is due to the new subsref method of the function class, which is used to represent the elements of a function space. Aside from surface plots, this feature can be of interest to generalise methods that rely on analytical solutions to differential problems, or to apply basically any algorithm to such functions. Here is the plot you will obtain with this script:

I wrote in an earlier post of the interpolate function: with this you can get the representation of a Function or Expression on a given FunctionSpace. It is useful, for instance, to compare your numerical solution with an exact one you happen to know. Or, in the example above, you might want to view what is the forcing term like:

f_cg = interpolate ("f_cg", f, u);
F = f_cg (X, Y);
surf (X, Y, F);

There is one last achievement to highlight for the mid term evaluation: currently both the initial compilation of the package and all the ones performed just-in-time when importing UFL instructions proceed smoothly without user intervention. To this end, now the build system relies on pkg-config to get at once all the flags needed for proper compilation and linking, since some dependencies of dolfin, the FEniCS interface, are not to be found in standard directories. In order to exploit the extracted information also for the subsequent run time builds, the autoconf substitution is performed also in the get_vars.m auxiliary function, which in turn provides it to generate_makefile.m. An implementation detail that proved quite tricky is how to pass all the preprocessor flags to mkoctfile: only a subset of the options of g++ are hard-coded in it, so I needed to resort to a workaround. Indeed, CPPFLAGS are always passed as environment variables and not as command line flags, so that mkoctfile will just copy and deliver them to the real compiler.

To further enhance the build system, I implemented other internal functions that hash the UFL file that was compiled and, later, check it to understand if it changed between the previous and the freshly requested build. In the example above, you will find in your working directory four new files after a run: the three already mentioned oct-files and a text file storing the md5 sum of the UFL that has been imported. Until one of these files gets somehow deleted or the problem in the ufl block changes, you will not need to take on a time consuming compilation any more.