Tutorials

Table of Contents

• Fundamentals
• Mesh creation
• Scalar-field problems
• Problems with complex variable fields
• Vector-field problems
• Mixed field problems
• Maxwell problems
• Advanced topics
• Tutorials using core interfaces or low level implementation
• Tutorials implemented as a separate module

The tutorials contain a number of basic programs that are built on top of each other. Those with the full source code has the following structure:

1. Introduction: Problem being solved, mathematical equations and derivations leading to finite element implementation
2. Implementation: Detailed explanation of the source code where tasks are separated in different functions
3. Results: How to run the program, output, visualisation, interpretation and comments, possible extensions.
4. Plain program: Full source code without extended comments

If you install MoFEM using either instructions in Installation with Docker - JupyterHub or the script provided in Installation with Spack (Scripts) the source code and corresponding binary files of the tutorials are located in the following directories:

• Source code: $HOME/mofem_install/mofem-cephas/mofem/users_modules/tutorials
• Binary files (build directory): $HOME/mofem_install/mofem-cephas/mofem/users_modules/um-build-Release-5sehreo/tutorials/

In the build directory described above, 5sehreo is the hash of the particular build, and it may be different on your machine.

Each tutorial in this page includes code name and keywords for quick reference and search within the browser.

Note

You do not need to go through all the tutorials in the order they are listed in this page before jumping into the topic in which you are interested. However, we do recommend you to have a look at the first few tutorials solving scalar-field problems to have a general idea how the finite element implementation is done in MoFEM. We also make a recommendation at the beginning of each tutorial regarding which tutorial(s) you should read (prerequisites) before continuing with the one you are most excited about.

Note

The tutorials are under development. Send us your feedback on Q&A regarding the tutorials that already done and the missing ones that you most prefer to see.

Source code    Description    Jupyter Notebook

Fundamentals

	MoFEM Interfaces	Brief introduction about MoFEM interfaces which enable users to implement the codes with different level of complexity. Keywords: Design
	How to compile a program	Brief introduction about compiling MoFEM - Developer version Keywords: Compilation
	How to add a new module and program	Tutorial on adding a new MoFEM module (you might want to skip this tutorial for now and return after reading other tutorials) Keywords: modules
	FUN-0: Hello world	Creating Simple problem, and pushing operators to pipelines Keywords: UDOs, pipelines
	FUN-1: Integration on finite element mesh	Numerical integration is essential for most of the numerical methods employed to solve partial differential equations (PDEs) Keywords: UDOs and integration
	FUN-2: Hierarchical approximation	Introduce elementary concepts of the Finite Element Method (FEM) with hierarchical shape functions and their implementation in MoFEM Keywords: Hierarchical approximation

Mesh creation

	MSH-1: Create a 2D mesh from Gmsh	This document shows how to create a MoFEM-compatible 2D input mesh from Gmsh Keywords: Gmsh, block definition, config file, read_med
	MSH-2: Create a 3D mesh from Gmsh	This document shows how to create a MoFEM-compatible 3D input mesh from Gmsh Keywords: Gmsh, block definition, config file, read_med

Scalar-field problems

	SCL-0: Least square approximaton	Solve the least square problem to approximate scalar function Keywords: Simple Interface, KSP solver, mofem_part, mbconvert, 3D extension
	SCL-1: Poisson's equation (homogeneous BC)	Solve the Poisson's equation with zero value boundary conditions Keywords: Simple Interface, KSP solver, mofem_part, mbconvert, 3D extension
	SCL-2: Poisson's equation (non-homogeneous BC)	An expansion of previous tutorial to cover non-homogeneous (non-zero value) boundary condition of Poisson's equation Keywords: Least square approximation
	SCL-3: Poisson's equation (Lagrange multiplier)	Solve the Poisson's equation using Lagrange multiplier for the non-homogeneous boundary condition Keywords: PCFIELDSPLIT block solver
	SCL-4: Nonlinear Poisson's equation	Solve nonlinear Poisson's equation using Newton iterative scheme Keywords: SNES solver
	SCL-5: Minimal surface equation	A variant of the nonlinear Poisson's equation Keywords: SNES solver
	SCL-6: Heat equation	This is the first tutorial to solve a time-dependent problem with first-order time derivative Keywords: TS solver, implicit scheme, remove dofs on skin
	SCL-7: Wave equation	Time-dependent problem with second-order time derivative Keywords: TS solver, implicit scheme, non-homegenous time dependent boundaru condition
	SCL-8: Radiation boundary conditions	Nonlinear problem at the boundary Keywords: axisymmetric problem, linearisation
	SCL-9: Heat method	Finding geodesic distance with heat method Keywords: face elements embedded in 3d space
	SCL-10: Photon diffusion	Photon diffusion equation Keywords: inital boundary conditions, parabolic equation, adding user element, postprocessing on skin, initial vector
	SCL-11: Discontinuous Galerkin for Poisson problem	Discontinuous Galerkin for Poisson problem Keywords: integartion on skeleton

Problems with complex variable fields

CLX-0: Helmholtz problem

Helmholtz problem for acoustics Keywords: complex variable, linear solver, boundary markers, remove dofs on skin

Vector-field problems

	VEC-0: Linear elasticity	Linear elasticity Keywords: Hooke equation
	VEC-1: Eigen elastic	Eigen elastic Keywords: eigen values, SLEPC
	VEC-2: Nonlinear elastic	Nonlinear elastic with Hencky material Keywords: nonlinear problem, SNES
	VEC-3: Nonlinear dynamic elastic	Time-dependent nonlinear elastic (dynamics) Keywords: Alpha2 method
	VEC-4: Shallow wave equation on sphere	Shallow wave equation on mainfold (atmospheric processes) Keywords: Alpha method, initail conditions, hyperbolic equation, sphere approximation, explicit/implicit time integration, restart vector
	VEC-5: Phase field model and Navier-Stokes	Phase field model and Navier-Sokes equations Keywords: phase field, Navier-Stokes, hyberbolic equation, inital boundary condition
	VEC-6: Discontinuous Galerkin for Kirchoff-Love Plate	Solving forth order eliptic equation describing plate bending Keywords: discontinuous Galerkin method, plate theroy

Mixed field problems

MIX-0: Mixed formulation of Poisson equation

A mixed problem Keywords: H-div space, homogenous boundary conditions by DOFs removal

	MIX-1: Retrive phase solving transport-of-intensity equation.	A mixed problem for transport intensity equation Keywords: Hetergenous paramterers and source function
	MIX-2: Mixed formulation for incompressible elasticity	A mixed problem for incompressible elasticity Keywords: incompressibility

Maxwell problems

	MAX-0: Magnetostatics	Magnetostatics Keywords:
	MAX-1: Lorenz force	Lorenz force Keywords:

Advanced topics

	ADV-0: Plastic problem	Plasticity problem Keywords: Nonlinear problem, linearisation, L2 space, boundary markers, displacement control, force control
	ADV-1: Contact problem	Contact problem Keywords: H-curl/H-div space, mix formulations
	ADV-2: Thermo-elastic example	Solve thermo-elasto-plastic probelm Keywords: Keywords: multi-physics
	ADV-3: DG Upwind for advection problem or level set	Upwind Discontinuous Galerkin for Level-Set Keywords: Keywords: upwind dg, level set
	ADV-5: Seepage in elsatic body example	Seepgae problem in deformable porous media Keywords: Keywords: multi-physiscs

Tutorials using core interfaces or low level implementation

	COR-0: Mixed formulation and integration on skeleton (h-adaptivity)	Mix formulation for transport problem Keywords: h-adaptivity and error estimator
	COR-1: Time dependent nonlinear mix formulation (unsaturated flow)	Mix formulation for capillary flow Keywords: capillary flow, unsaturated soil, nonlinear mix-formulation
	COR-2: Solving the Poisson equation	Poison equation again with different implementations Keywords: Poisson problem
	COR-3: Implementing operators for the Poisson equation	Poison equation implementation of differentiation operators Keywords: Poisson problem
	COR-4: Using fieldsplit solver and DM sub problem.	Poison equation implementation of differentiation operators Keywords: Using subproblems and nested matrices
	COR-5: A nonlinear Poisson equation	Poison nonlinear equation again Keywords: SNES solver and linearisation
	COR-6: Solid elasticity	Older implementation of elastic problem Keywords: Operators and low level implementation
	COR-7: Mixed formulation for incompressible elasticity	Older implementation of mixed elastic problem Keywords: pressure field, incompressibility
	COR-8: Implementation of spring element	Implementation of boundary element with springs Keywords: face element, spring, user data operators
	COR-9: Reaction-diffusion equation	Reaction-diffusion Fisher's equation Keywords: time solver, monitor, nonlinearity
	COR-10: Navier-Stokes equation	Navier-Stokes equations Keywords: viscous fluid flow, mixed formulation, drag force computation
	COR-11: Mixed formulation of nonlinear elasticity	Mixed formulation for large strain elasticity Keywords: H-div space, Schur complement, ADOL-C (automatic differentiation)

Tutorials implemented as a separate module

	MOD-0: Soap film spanned on wire	Soap film as an external module Keywords: Module implementations
	VEC-7: Automatic Differentiation for Plasticity	Automatic differentiation for elato-plasticity Keywords: ADOL-C, plasticity, nonlinear problem, dg projection, user module