v0.9.1 |

Hello world

In this tutorial, basic ideas about MoFEM::Simple interface and programming with user data operators (UDO) are presented. This article aims at providing a big picture of the sequence of how MoFEM works in a simple and typical problem. Before proceeding to the next parts, readers are strongly encouraged to look at the Basic design to have fundamental ideas of MoFEM and its ecosystem.

MoFEM is designed for quick development of finite element application for multi-physics problems. We designed MoFEM to enable decomposition of the complex problem into small sub-problems, wherein each individual sub-problem can be independently tested and debugged or used in different context. That is achieved by pipelines of user data operators (UDO), see Figure 1. In this figure, there are operators in green, finite element instances which are presented in blue, and red shows finite element objects. Typical operators including OpRow, OpRowCol and OpVolume are mainly used to construct the force vector (the right hand side) and the stiffness matrix and to assemble the system, respectively. Meanwhile, the finite element instances include *domain_fe*, *boundary_fe*, and *skeleton_fe*. Operators can be set to volume or face elements on one of the finite element instances.

In MoFEM, there is a clear separation of declaration, definition and implementation of a field, finite element and problem. Such approach allows to use the same implementation in different problems, e.g. use the same implementation of an elastic finite element in a mechanical problem and thermo-mechanical problem without the need to introduce changes into a code and maximising code reusability. On the other hand for the same problem declaration and definition, one can test various formulations and implementations. In this example problem declaration and definition is managed by MoFEM::Simple interface, in which we focus attention only on field declaration and implementation of finite elements, in particular, UDO.

/**

* \file hello_world.cpp

* \ingroup mofem_simple_interface

* \example hello_world.cpp

*

* Prints basic information about users data operator.

* See more details in \ref hello_world_tut1

*

*/

/* This file is part of MoFEM.

* MoFEM is free software: you can redistribute it and/or modify it under

* the terms of the GNU Lesser General Public License as published by the

* Free Software Foundation, either version 3 of the License, or (at your

* option) any later version.

*

* MoFEM is distributed in the hope that it will be useful, but WITHOUT

* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or

* FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public

* License for more details.

*

* You should have received a copy of the GNU Lesser General Public

* License along with MoFEM. If not, see <http://www.gnu.org/licenses/>. */

#include <BasicFiniteElements.hpp>

using namespace MoFEM;

static map<EntityType, std::string> type_name;

OpRow(const std::string &field_name)

: ForcesAndSourcesCore::UserDataOperator(field_name, field_name, OPROW) {}

DataForcesAndSourcesCore::EntData &data) {

if (type == MBVERTEX) {

// get number of evaluated element in the loop

std::cout << std::endl

<< "**** " << getNinTheLoop() << " **** " << std::endl;

std::cout << "**** Operators **** " << std::endl;

}

std::cout << "Hello Operator OpRow:"

<< " field name " << rowFieldName << " side " << side << " type "

<< data.getIndices().size() << std::endl;

}

};

const bool symm)

: ForcesAndSourcesCore::UserDataOperator(row_field, col_field, OPROWCOL,

symm) {}

EntityType col_type,

DataForcesAndSourcesCore::EntData &row_data,

DataForcesAndSourcesCore::EntData &col_data) {

std::cout << "Hello Operator OpRowCol:"

<< " row field name " << rowFieldName << " row side " << row_side

<< row_data.getIndices().size() << " : "

<< " col field name " << colFieldName << " col side " << col_side

<< col_data.getIndices().size() << std::endl;

}

};

OpVolume(const std::string &field_name)

: VolumeElementForcesAndSourcesCore::UserDataOperator(field_name,

field_name, OPROW) {

}

DataForcesAndSourcesCore::EntData &data) {

if (type == MBVERTEX) {

std::cout << "Hello Operator OpVolume:"

<< " volume " << getVolume() << endl;

}

}

};

OpFace(const std::string &field_name)

: FaceElementForcesAndSourcesCore::UserDataOperator(field_name, OPROW) {}

DataForcesAndSourcesCore::EntData &data) {

if (type == MBVERTEX) {

std::cout << "Hello Operator OpFace:"

<< " normal " << getNormal() << endl;

}

}

};

boost::shared_ptr<VolumeElementForcesAndSourcesCoreOnSide> &feSidePtr;

const std::string &field_name,

boost::shared_ptr<VolumeElementForcesAndSourcesCoreOnSide> &fe_side_ptr)

: FaceElementForcesAndSourcesCore::UserDataOperator(field_name, OPROW),

feSidePtr(fe_side_ptr) {}

DataForcesAndSourcesCore::EntData &data) {

if (type == MBVERTEX) {

std::cout << "Hello Operator OpSideFace" << endl;

CHKERR loopSideVolumes("dFE", *feSidePtr);

}

}

};

struct OpVolumeSide

OpVolumeSide(const std::string &field_name)

field_name, field_name, OPROW) {}

DataForcesAndSourcesCore::EntData &data) {

if (type == MBVERTEX) {

std::cout << "Hello Operator OpVolumeSide:"

<< " volume " << getVolume() << " normal " << getNormal()

<< endl;

}

}

};

// initialize petsc

try {

type_name[MBVERTEX] = "Vertex";

type_name[MBEDGE] = "Edge";

type_name[MBTRI] = "Triangle";

type_name[MBTET] = "Tetrahedra";

// Register DM Manager

DMType dm_name = "DMMOFEM";

CHKERR DMRegister_MoFEM(dm_name);

// Create MoAB database

moab::Core moab_core;

moab::Interface &moab = moab_core;

// Create MoFEM database and link it to MoAB

MoFEM::Core mofem_core(moab);

MoFEM::Interface &m_field = mofem_core;

// Simple interface

// get options from command line

// load mesh file

// add fields

// set fields order

// setup problem

// create elements instances

boost::shared_ptr<ForcesAndSourcesCore> domain_fe(

new VolumeElementForcesAndSourcesCore(m_field));

boost::shared_ptr<ForcesAndSourcesCore> boundary_fe(

new FaceElementForcesAndSourcesCore(m_field));

boost::shared_ptr<ForcesAndSourcesCore> skeleton_fe(

new FaceElementForcesAndSourcesCore(m_field));

boost::shared_ptr<VolumeElementForcesAndSourcesCoreOnSide> side_fe(

new VolumeElementForcesAndSourcesCoreOnSide(m_field));

// set operator to the volume element instance

// set operator to the face element instance

// set operator to the face element on skeleton instance

// set operator to the volume on side of the skeleton face

auto dm = simple->getDM();

// iterate domain elements and execute element instance with operator on

// mesh entities

// iterate boundary elements and execute element instance with operator on

// mesh entities

boundary_fe);

// iterate skeleton elements and execute element instance with operator on

// mesh entities

skeleton_fe);

}

// finish work cleaning memory, getting statistics, etc.

return 0;

}

While the details of different parts of the code are discussed later in this section, the main structure of the code is as follows:

- Initialization: PETSc, MoFEM, and MoAB are initialized
- Adding fields: Three groups of fields are added including fields defined in the domain, fields defined on boundary, and fields defined on the skeleton
- Setting-up: Building of fields, finite elements and problems, and all other data structures
- Allocating finite element instances: Finite element instances are created including
*domain_fe*,*boundary_fe*, and*skeleton_fe* - Accessing discrete manager: Get the discrete manager which is the interface developed for PETSc to manage complexities associated with topology (mesh) and algebra
- Iterating finite elements: Make iteration over finite elements and run sequences of UDO for each of the element. For demonstation, operator names and entity information are printed out in this part
- Implementation of UDO: Define different user data operators

We initialize PETSc internal variables and register MoFEM discrete manager in PETSc. Next MoAB instance is created and assigned to it. Similarly, MoFEM instance is created and assigned. MoAB and MoFEM interfaces are abstract classes used to access data in database.

// Initialise PETSc

PetscInitialize(&argc,&argv,(char *)0,help);

// Register DM Manager

DMType dm_name = "DMMOFEM";

CHKERR DMRegister_MoFEM(dm_name);

// Create MoAB database

moab::Core moab_core;

moab::Interface& moab = moab_core;

// Create MoFEM database and link it to MoAB

MoFEM::Core mofem_core(moab);

MoFEM::Interface& m_field = mofem_core;

Next, we get access to database by MoFEM::Simple interface

and get options from the command line and load mesh file. Default mesh file has name *mesh.h5m*. Particular name of file can be given in command line using option *-file_name* *my_file.h5m*. The visualization of the meshed object is given in Figure 2.

The indication that MoFEM database has been initialized is that the MoFEM version and git commit id are shown on the terminal as below.

MoFEM version 0.8.13 (MOAB 5.0.2 Petsc Release Version 3.9.3, Jul, 02, 2018 ) git commit id 549c206e489d605e1c9d531f5d463b99ec2adc32

It should be noted that the message can be different depending on the time of installation and update.

We add fields to the database. In MoFEM::Simple interface, three groups of fields can be declared. Fields defined in the domain, fields defined on boundary and fields defined on the skeleton. The same field can be defined on domain, boundary and skeleton. Fields are declared by giving its name, approximation space, base and number of coefficients. See for details here MoFEM::Simple::addDomainField.

Next, we set approximation order to those fields

// set fields order

For more details see MoFEM::Simple::setFieldOrder. If needed, function MoFEM::Simple::setFieldOrder can be supplemented by the additional parameter to set order to particular field entities, enabling heterogeneous approximation base. Basics of hierarchical approximation provides more details on the different entities used in MoFEM.

When one runs the program later in Running the program, the output of this part of the code is printed on terminal as follows

add: name U BitFieldId 1 bit number 1 space H1 approximation base AINSWORTH_LEGENDRE_BASE rank 3 meshset 12682136550675316768 add: name L BitFieldId 2 bit number 2 space H1 approximation base AINSWORTH_LEGENDRE_BASE rank 3 meshset 12682136550675316769 add: name S BitFieldId 4 bit number 3 space H1 approximation base AINSWORTH_LEGENDRE_BASE rank 3 meshset 12682136550675316770

where at the end entity handle to meshset is printed. This meshset consisting all entities to which field is set. The effect of the setting of approximation order will be viable when fields are constructed during set-up stage.

The fields, finite elements and problems and all other data structures are built with the following code

Similarly, when one runs the code, the following outputs are expected on the terminal

- Three finite elements structures are added to the database including domain finite element structure, boundary finite element structure and skeleton boundary element structure. Those elements are added by default via MoFEM::Simple interface.

add finite element: dFE add finite element: bFE add finite element: sFE

- Next problem is added to the database. This is the default name for MoFEM::Simple interface

add problem: SimpleProblem

- In the following lines, degrees of freedom (DOFs) are constructed for fields U, L and S added in Adding fields. The number of DOFs on each field entity depends on the approximation order and approximation space which have also been defined in Adding fields.
Build Field U (rank 0) nb added dofs (vertices) 27 (inactive 0) nb added dofs (edges) 234 (inactive 0) nb added dofs (triangles) 270 (inactive 0) nb added dofs (tets) 36 (inactive 0) nb added dofs 567 (number of inactive dofs 0) Build Field L (rank 0) nb added dofs (vertices) 24 (inactive 0) nb added dofs (edges) 108 (inactive 0) nb added dofs (triangles) 36 (inactive 0) nb added dofs 168 (number of inactive dofs 0) Build Field S (rank 0) nb added dofs (vertices) 24 (inactive 0) nb added dofs (edges) 108 (inactive 0) nb added dofs (triangles) 36 (inactive 0) nb added dofs 168 (number of inactive dofs 0) Nb. dofs 903

- The following lines indicate domain, boundary and skeleton finite element structures have been succesfully constructed.
Build Finite Elements dFE id 00000000000000000000000000000001 name dFE f_id_row 00000000000000000000000000000001 f_id_col 00000000000000000000000000000001 BitFEId_data 00000000000000000000000000000001 Nb. FEs 12 Build Finite Elements bFE id 00000000000000000000000000000010 name bFE f_id_row 00000000000000000000000000000111 f_id_col 00000000000000000000000000000111 BitFEId_data 00000000000000000000000000000111 Nb. FEs 12 Build Finite Elements sFE id 00000000000000000000000000000100 name sFE f_id_row 00000000000000000000000000000111 f_id_col 00000000000000000000000000000111 BitFEId_data 00000000000000000000000000000111 Nb. FEs 30

- Finally, the problem data structure are constructed, in particular knowing finite element DOFs adjacencies and enumeration of DOFs, matrices and vectors can be created.
Nb. entFEAdjacencies 918 partition_problem: rank = 0 FEs row ghost dofs problem id 00000000000000000000000000000001 FiniteElement id 00000000000000000000000000000111 name SimpleProblem Nb. local dof 903 nb global row dofs 903 partition_problem: rank = 0 FEs col ghost dofs problem id 00000000000000000000000000000001 FiniteElement id 00000000000000000000000000000111 name SimpleProblem Nb. local dof 903 nb global col dofs 903 problem id 00000000000000000000000000000001 FiniteElement id 00000000000000000000000000000111 name SimpleProblem Nb. elems 54 on proc 0 partition_ghost_col_dofs: rank = 0 FEs col ghost dofs problem id 00000000000000000000000000000001 FiniteElement id 00000000000000000000000000000111 name SimpleProblem Nb. col ghost dof 0 Nb. local dof 903 partition_ghost_row_dofs: rank = 0 FEs row ghost dofs problem id 00000000000000000000000000000001 FiniteElement id 00000000000000000000000000000111 name SimpleProblem Nb. row ghost dof 0 Nb. local dof 903

Once the problem is defined and set up, we create finite element instances. We are using generic implementation of finite elements which is tailored to the dimension of finite element entities. Note that in the following code we use MoFEM::VolumeElementForcesAndSourcesCore for domain elements and MoFEM::FaceElementForcesAndSourcesCore for boundary elements and skeleton elements. This is because we use 3d mesh in which boundary and skeleton are 2d entities emerged in 3d space.

boost::shared_ptr<ForcesAndSourcesCore> domain_fe(new VolumeElementForcesAndSourcesCore(m_field));

boost::shared_ptr<ForcesAndSourcesCore> boundary_fe(new FaceElementForcesAndSourcesCore(m_field));

boost::shared_ptr<ForcesAndSourcesCore> skeleton_fe(new FaceElementForcesAndSourcesCore(m_field));

Once finite element objects are created, we add operators to them as shown in Figure 1. Starting from *domain_fe*,

next *boundary_fe*,

and finally *skeleton_fe*

To be more explicit, as an example, we can write for *domain_fe* as follows

boost::ptr_vector<UserDataOperator>& sequence_of_operators = domain_fe->getOpPtrVector();

Note that *boost::ptr_vector* (see link) is container of pointers. Once vector is destroyed, the memory pointed by pointers is released.

We have to discuss integration over the skeleton. This functionality is used for a class of Petrov-Discontinuous Galerkin methods, or when Nitsche method is applied. Those methods involve of integration of traces of approximation functions on faces. That enforces the evaluation of derivatives of base functions on finite elements adjacent to the face. In MoFEM it is realised in a way that the generic element instance is iterated over adjacent entities to faces. We start with allocation memory for "side" finite element as before

boost::shared_ptr<VolumeElementForcesAndSourcesCoreOnSide> side_fe(new VolumeElementForcesAndSourcesCoreOnSide(m_field));

Note that this time finite element class is of type MoFEM::VolumeElementForcesAndSourcesCoreOnSide, since this element has integration points which are associated with face adjacent face. Next, to that element we add user data operators, in this particular case only one

Note that pointer to finite element instance is passed in the constructor of user data operator for skeleton element, what is discussed in detail in next section. Finally, we add operator to skeleton face element itself

Following line gives access to the discrete manager (DM). DM is the interface developed for PETSc to manage complexities associated with topology (mesh) and algebra and several implementation of this interface are available in native PETSc library. Here we are using the particular implementation developed for MoFEM. To get interface, the following lines of the code are needed

When DM interface is no longer needed, it needs to be destroyed by user in the standard way well-known from PETSc interface

CHKERR DMDestroy(&dm);

- Note
- In MoFEM implementation, in comparison to other finite element codes, we do not wrap PETSc library functions but extend PETSc interface with features native to MoFEM. You can use DM interface in the same way how you use native PETSc interface, see Solving the Poisson equation it is shown how to set up KSP solver and get the solution. That philosophy applies to MoAB functions as well.

Finally, we get to the point when we can put our machine in motion, we iterate over finite elements and run sequences of user data operator for each of them. We start with *domain_fe*

as results we get

**** 0 **** **** Operators **** Hello Operator OpRow: field name U side 0 type Vertex nb dofs on entity 12 ... Hello Operator OpRow: field name U side 0 type Tetrahedra nb dofs on entity 3 Hello Operator OpRowCol: row field name U row side 0 row type Vertex nb dofs on row entity12 : col field name U col side 0 col type Vertex nb dofs on col entity12 ... Hello Operator OpRowCol: row field name U row side 0 row type Tetrahedra nb dofs on row entity3 : col field name U col side 0 col type Tetrahedra nb dofs on col entity3 Hello Operator OpVolume: volume 0.0834851 **** 2 **** **** Operators **** ... **** 11 **** **** Operators **** ...

where dots are added for the abbreviation of output. Note that operators are called in the order we pushed them to finite element operators vector. Since we have twelve volume (Tetrahedra) elements, iteration ends on eleven as in MoFEM we always start counting from zero.

To iterate over boundary element, analogically we do,

and we get similar output to the one shown before, with one difference being that the last operator does not print volume of the element but that is normal since entity of boundary finite element in this particular case is the triangle.

The same procedure is applied to iterate over skeleton finite elements entities

We have thirty skeleton elements and output looks as follows

**** 0 **** **** Operators **** Hello Operator OpRow: field name S side 0 type Vertex nb dofs on entity 9 Hello Operator OpRow: field name S side 0 type Edge nb dofs on entity 6 Hello Operator OpRow: field name S side 1 type Edge nb dofs on entity 6 Hello Operator OpRow: field name S side 2 type Edge nb dofs on entity 6 Hello Operator OpRow: field name S side 0 type Triangle nb dofs on entity 3 Hello Operator OpSideFace Hello Operator OpVolumeSide: volume 0.0782402 normal [3](0,0,1) ... **** 13 **** **** Operators **** Hello Operator OpRow: field name S side 0 type Vertex nb dofs on entity 9 Hello Operator OpRow: field name S side 0 type Edge nb dofs on entity 6 Hello Operator OpRow: field name S side 1 type Edge nb dofs on entity 0 Hello Operator OpRow: field name S side 2 type Edge nb dofs on entity 0 Hello Operator OpRow: field name S side 0 type Triangle nb dofs on entity 0 Hello Operator OpSideFace Hello Operator OpVolumeSide: volume 0.0834851 normal [3](0,0.530559,-0.50091) Hello Operator OpVolumeSide: volume 0.0884264 normal [3](0,0.530559,-0.50091) ...

Note that first operator is *OpRow*, the second operator is *OpSideFace*, this operator prints its name and runs integration over adjacent to given face elements, which is *side_fe*. Once this element is run for each adjacent finite element entity, user data operators are run on it, i.e. *OpVolumeSide* which prints volume of the adjacent entity and normal of the face. Note that first element has only one run of *OpVolumeSide*, since skeleton finite element "0" is on the body boundary, while skeleton finite element "13" is in body volume and it has two volume neighbours.

Now we focus attention on the implementation of user data operators including *OpRow*, *OpRowCol*, *OpVolume*, etc. which have been used previously. The first operator has the structure

This user data operator class is derived from MoFEM::ForcesAndSourcesCore::UserDataOperator which can be used with any type of entity. It is the type of OPROW, which indicates that it only iterates lower dimension entities on the element. On each lower entity overload method is called

which as arguments take entity side number (local entity number on the finite element), entity type (e.g. MBVERTEX, MBEDGE, MBTET) and reference to structure MoFEM::DataForcesAndSourcesCore::EntData, which keeps information on DOFs, approximation on given entity. This type of entity is usually used to integrate the right-hand side vector.

Another type of user data operator is implemented here

const std::string row_field,

const std::string col_field,

const bool symm

):

}

int row_side,int col_side,

EntityType row_type,EntityType col_type,

DataForcesAndSourcesCore::EntData &row_data,

DataForcesAndSourcesCore::EntData &col_data

);

};

This user data operator is of type OPROWCOL, which takes an additional parameter in constructor, i.e. *symm*, which is used to set symmetry of operator. Operator of this type iterates over all unique pairs of entities. If a symmetric operator pair is set of two elements (i.e. entities), thus order of entities is not important. If an operator is not symmetric, then pairs are the sequence of two elements and all variations of entities pairs are considered. This type of operator is used to integrate matrices. Note that this time function is overloaded, which takes as argument data for rows and columns, respectively.

Performing calculations on entity of specific dimension additional data like volume, normal need to be attained, for such case derived user data operator class can be used, e.g.

and for case of operator working on adjacent to face volume entity

where from members of this class information about face normal and adjacent entity volume can be accessed.

In order to run the program, one should first go to the directory where the problem is located, compile the code and then run the executable file. This can be done as follows

cd mofem_install/um/build/basic_finite_elements/hello_world

make -j2

./hello_world

- Exercise 1: Add
*OpRow*and*OpRowCol*operator to*side_fe* - Exercise 2: Print indices in
*OpRow*operator. - Exercise 3: Print base functions and derivative of base functions on
*OpRow*. - Exercise 4: Change space of field "U", "L" and "S", how that changes a number of DOFs on entities?
- Exercise 5: Change order of field "U", "L" and "S", how that changes a number of DOFs on entities?
- Exercise 6: Modify code and calculate a volume of the body. See simple.cpp for example.

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