*We customize the course outline and content to your specific needs and relevant use cases.
Day 1 – From Physical Problem to Finite Element Model
Introduction to the Finite Element Method
- What does an FEM program do internally?
- From a continuous physical problem to a discrete numerical model
- Discretisation and meshes
- Nodes and elements
- Degrees of freedom
- Physical and material parameters
- Loads, source terms and boundary conditions
- Local and global quantities
Mathematical Foundations
- Review of relevant linear algebra
- Vectors and matrices
- Linear systems of equations
- Local and global systems
- The general FEM system
K u = f
- Interpretation of the system matrix, solution vector and right-hand side
- Different physical interpretations of the same numerical structure
Depending on the application, K may represent, for example, stiffness, conductivity or another discretised physical operator.
Shape Functions and Element Approximation
- Why interpolation is required in FEM
- Introduction to shape functions
- Interpolation within an element
- Element coordinates
- Local and global coordinates
- Derivatives of shape functions
- Connection between element interpolation and physical quantities
Element-Level Calculations
- Element geometry
- Local degrees of freedom
- Element matrices
- Element vectors
- Physical parameters at element level
- Introduction to numerical integration
- Integration points and weights
- Coordinate transformations where required
Software Representation of the FEM Model
Development of suitable Python components such as:
- Node
- Element
- Material or PhysicalProperties
- BoundaryCondition
- Load or Source
- Mesh
- Model
Software design topics include:
- separating physical data from numerical algorithms,
- reusable element implementations,
- clear responsibilities between model, element and solver components,
- keeping model definition independent from the numerical solver.
Practical Exercise
Participants implement the first components of a small FEM application, including:
- creation of nodes and elements,
- definition of physical parameters,
- degree-of-freedom assignment,
- element-level calculations,
- basic automated tests.
Day 2 – Assembly and Solution of the Global System
Degree-of-Freedom Management
- Local and global degrees of freedom
- Degree-of-freedom numbering
- Mapping element quantities to global indices
- Handling constrained and unconstrained degrees of freedom
- Strategies for larger models
Global Assembly
Step-by-step implementation of the finite element system:
- calculate element matrices and vectors,
- determine the associated global degrees of freedom,
- transform element quantities where necessary,
- assemble the global system matrix,
- assemble the global right-hand-side vector.
Boundary Conditions and External Influences
- Essential and natural boundary conditions
- Prescribed values
- Loads and source terms
- Different implementation strategies
- Modification of the equation system
- Validation of boundary conditions
Solving the Global System
- Solution of K u = f
- Dense linear solvers with NumPy
- Introduction to sparse matrices with SciPy
- Why FEM matrices are typically sparse
- Common sparse matrix formats
- Sparse matrix assembly
- Sparse linear solvers
- Numerical accuracy
- Conditioning and numerical stability
Result Evaluation
Depending on the selected application:
- nodal solution values,
- gradients,
- fluxes,
- deformations,
- strains,
- stresses,
- internal forces,
- other derived engineering quantities.
Visualisation
- Visualising the mesh
- Displaying nodal solution values
- Plotting scalar fields
- Plotting deformations
- Visualising element-based results
- Creating engineering result diagrams
Practical Exercise
Participants complete a small FEM solver that can:
- define a numerical model,
- assemble a global system,
- apply boundary conditions,
- solve the equation system,
- calculate selected result quantities,
- and visualise the solution.
Day 3 – Verification, Generalisation and Maintainable FEM Software
Verification and Testing
- Why numerical software must be verified
- Testing element-level calculations
- Symmetry checks
- Consistency checks
- Equilibrium or conservation checks
- Comparison with analytical reference solutions
- Numerical tolerances
- Automated tests with pytest
- Regression tests
- Plausibility checks
Mesh Refinement and Convergence
- Influence of discretisation on results
- Mesh refinement
- Convergence concepts
- Comparing numerical solutions at different mesh resolutions
- Distinguishing modelling errors from implementation errors
Generalising the FEM Implementation
- Separating element formulation from assembly
- Supporting different element types
- Supporting different physical problems
- Generic treatment of degrees of freedom
- Extensible material and parameter models
- Reusable solver components
Structuring a Maintainable FEM Application
A possible software architecture:
Model → Elements → Assembly → Solver → Results → Visualisation
Possible project structure:
model/
Mesh, nodes, physical parameters and boundary conditions
elements/
Finite element formulations
assembly/
Global system assembly and degree-of-freedom management
solver/
Boundary conditions and equation solvers
results/
Result evaluation and derived quantities
visualization/
Model and result visualisation
tests/
Automated verification
Topics include:
- separation of responsibilities,
- clear interfaces between components,
- reusable numerical algorithms,
- maintainability and extensibility,
- avoiding tightly coupled implementations.
Adaptation to Different Engineering Disciplines
The mathematical and software concepts of the course remain generic. The practical exercises and physical interpretation can be adapted to the participants’ engineering background.
Possible application variants include:
- Structural engineering: truss, beam or simple elasticity problems
- Mechanical engineering: deformation and stress analysis
- Heat transfer: stationary heat conduction
- General field problems: Poisson-type equations
- Other engineering disciplines: comparable discretised field problems according to participant requirements
For structural engineers, for example, the exercises can use a 2D truss, beam or simple continuum model. For mechanical or thermal engineers, the same FEM software architecture can instead be demonstrated using elasticity or heat-conduction examples.
Final Practical Project
Participants consolidate the course content into a small, structured FEM application that can:
- define a discretised physical model,
- create nodes and elements,
- assign physical parameters,
- define boundary conditions and external influences,
- calculate element contributions,
- assemble the global system,
- solve the resulting equations,
- evaluate derived quantities,
- automatically verify selected calculations,
- visualise the numerical model and results,
- and be extended with additional element types or physical models.
Outlook
The course concludes with an overview of further development paths, including:
- higher-dimensional problems,
- additional element formulations,
- higher-order elements,
- more advanced numerical integration,
- nonlinear problems,
- eigenvalue problems,
- transient and dynamic simulations,
- complex material models,
- mesh generation,
- adaptive refinement,
- performance optimisation,
- parallelisation,
- graphical user interfaces,
- established FEM frameworks such as FEniCSx.
