Discretisation Concepts in Finite Element Analysis

Welcome, students 👋 In this lesson, you will learn one of the most important ideas in Finite Element Analysis (FEA): discretisation. The main idea is simple, but powerful: instead of trying to solve a complex solid mechanics problem all at once, we break the object into many small pieces and solve the problem piece by piece.

By the end of this lesson, you should be able to:

• explain what discretisation means in FEA,
• use correct terms such as nodes, elements, and mesh,
• connect discretisation to the accuracy of an FEA model,
• understand why mesh choices matter for real engineering work,
• describe how discretisation fits into the wider FEA process.

Think of it like mapping a country using tiny squares on a grid. A tiny number of large squares gives a rough picture, while many small squares give more detail. In FEA, the same idea helps engineers predict stress, strain, and deformation in parts like beams, brackets, bolts, and car frames 🚗.

What Discretisation Means

In Solid Mechanics 2, discretisation means dividing a continuous solid body into a finite number of smaller parts so that the body can be analysed numerically. A real engineering component is continuous: every point in it can, in principle, have a displacement, strain, and stress. But computers cannot solve the full continuous problem directly for every point without approximation.

So, FEA turns the solid into a collection of small subdomains called elements. These elements are connected at special points called nodes. Together, the elements and nodes make a mesh. The mesh is the discretised representation of the original object.

This process matters because the equations of elasticity are usually too complicated to solve exactly for real-life shapes, loads, and boundary conditions. Discretisation lets us turn a complicated continuum problem into many simpler algebraic equations.

For example, imagine analysing a metal bracket with a hole. The area around the hole is where stress may concentrate. If we discretise the bracket into elements, we can estimate how displacement and stress vary across the part. More elements near the hole often give a better result than using only a few large elements.

Nodes, Elements, and Degrees of Freedom

To understand discretisation properly, students, you need to know the main terms.

A node is a point in the mesh where unknown values are stored. In solid mechanics, these unknowns are usually degrees of freedom (DOF), such as displacement components. For a simple two-dimensional solid element, a node may have displacement variables like $u_x$ and $u_y$. In three dimensions, a node may also have $u_z$.

An element is a small section of the body used to approximate the behaviour of that section. Elements can be triangles, quadrilaterals, tetrahedra, hexahedra, and other shapes. The element uses mathematical shape functions to estimate how displacement changes inside it.

A degree of freedom is one independent variable needed to describe the state of the model at a node. In structural FEA, the most common DOF are translations, such as $u_x$, $u_y$, and $u_z$. Some problems may also include rotations, especially in beam or shell elements.

The key idea is that the whole solid is represented by the values at the nodes, and the element formula is used to estimate what happens between them. This is why discretisation is an approximation: the real body is continuous, but the model uses finite points and finite elements.

Why We Need Approximation

A solid mechanics problem is governed by equations of equilibrium, compatibility, and material behaviour. For linear elasticity, the relationships are based on stress, strain, and displacement. In many real cases, the geometry is irregular, the loading is complex, and the material may have complicated properties.

If a part has a curved shape, holes, fillets, or contact regions, it becomes very difficult to solve exactly by hand. Discretisation helps because each small element is easier to analyse than the whole part.

The approximation works like this:

1. Assume a displacement pattern inside each element.
2. Use the nodal values to describe that pattern.
3. Convert the continuous problem into a set of algebraic equations.
4. Solve those equations for nodal displacements.
5. Use the displacements to calculate strains and stresses.

A common way to describe the displacement inside an element is with interpolation. For example, the displacement field may be written as $u(x) = \sum_i N_i(x)u_i$, where $N_i(x)$ are shape functions and $u_i$ are nodal displacements. You do not need to memorise this formula yet, but it shows the heart of discretisation: values at nodes are used to approximate the field inside the element.

This is why the quality of the discretisation affects accuracy. If the elements are too large or poorly shaped, the approximation may miss important details, especially near stress concentrations.

Mesh Design and Its Effect on Accuracy

The mesh is the practical result of discretisation. Choosing a mesh is not just a computer step; it is an engineering decision.

A fine mesh uses many small elements. This can capture geometric detail and stress changes more accurately, especially in regions where stress changes quickly. A coarse mesh uses fewer, larger elements. It is faster to solve but may miss important local effects.

For example, consider a plate with a circular hole under tension. The stress near the hole edge is much higher than in the rest of the plate. If the mesh is too coarse around the hole, the stress concentration may be underestimated. A finer mesh around the hole usually improves the result.

Mesh quality also matters. Good elements should not be too distorted. Very stretched, twisted, or badly shaped elements can reduce accuracy even if the mesh is fine. Engineers try to keep elements well-shaped because poor element quality can create numerical problems.

A useful idea is mesh refinement. This means making the mesh finer in important regions. Engineers often refine the mesh near:

• holes,
• sharp corners,
• load application points,
• supports and fixed boundaries,
• contact surfaces,
• cracks or notches.

This is because these regions often have rapid changes in stress or strain. In contrast, less critical regions may use a coarser mesh to save time.

Boundary Conditions and Discretised Models

Discretisation is closely linked to boundary conditions. A model is not complete unless the supports, loads, and constraints are defined properly.

When a solid is discretised, the boundary conditions are usually applied at nodes or over element surfaces. For example:

• a fixed end may have $u_x = 0$, $u_y = 0$, and $u_z = 0$,
• a loaded surface may carry a force or pressure,
• symmetry conditions may reduce the model size.

These conditions tell the system how the part interacts with its environment. Without them, the equations may not have a unique solution.

Imagine a cantilever bracket bolted to a wall. The wall is represented by fixed boundary conditions at the mounting region. The load may be applied at the free end. The discretised model then calculates how the bracket bends and where the highest stress occurs.

Good discretisation makes it easier to apply boundary conditions accurately. However, if loads are applied to only one node instead of over an area, the stress may become unrealistically high at that point. Engineers often use distributed loads or carefully selected constraint regions to avoid unrealistic results.

Practical Example: A Tension Bar with a Hole

Let’s use a simple example to connect everything together.

Suppose students is analysing a flat steel bar pulled in tension, and the bar has a hole in the middle. The hole weakens the bar and creates a stress concentration around its edge.

To discretise the bar, the engineer would:

1. Draw the geometry of the bar.
2. Divide it into many elements.
3. Place more elements around the hole.
4. Assign material properties such as Young’s modulus and Poisson’s ratio.
5. Apply the tensile load at one end and the support or displacement condition at the other.
6. Solve for nodal displacements.
7. Calculate stress and strain in each element.

If the mesh near the hole is too coarse, the model may smooth out the sharp increase in stress. If the mesh is refined, the stress pattern near the hole becomes more realistic.

This example shows a key engineering rule: the mesh should be dense where the physics changes quickly. That is one of the most important lessons in discretisation.

How Discretisation Fits Into the Full FEA Process

Discretisation is one stage in the larger FEA workflow. The full process usually includes:

• defining the problem,
• creating geometry,
• discretising the body into a mesh,
• assigning materials,
• applying loads and boundary conditions,
• solving the equations,
• interpreting the results.

So, discretisation sits between geometry creation and solving. It converts the physical object into a mathematical model that a computer can handle.

If discretisation is poor, the final answer may be misleading even if the solver is powerful. That is why engineers do not trust a result just because the software produced a colourful stress plot 🌈. They check whether the mesh is suitable, whether boundary conditions make sense, and whether the results are physically reasonable.

One important idea is convergence. As the mesh becomes finer, the results should approach a stable value. If the stress or displacement changes a lot when the mesh is refined, the model may not yet be accurate enough. This is why engineers often perform a mesh convergence study.

Conclusion

Discretisation is the process of dividing a continuous solid into finite elements connected by nodes so that the behaviour of the solid can be approximated numerically. It is the foundation of FEA because it makes complex solid mechanics problems solvable by computer.

A good discretisation balances accuracy, computational cost, and element quality. It also works together with boundary conditions, material properties, and loading to produce reliable results. In Solid Mechanics 2, understanding discretisation helps you move from theory to real engineering analysis.

Study Notes

• Discretisation means breaking a continuous body into a finite number of small elements.
• The mesh is the collection of elements and nodes used in the FEA model.
• Nodes store unknown values such as displacements.
• Elements use interpolation to estimate behaviour between nodes.
• Degrees of freedom are independent variables, often displacement components like $u_x$, $u_y$, and $u_z$.
• Fine meshes usually give better accuracy, especially near stress concentrations.
• Poorly shaped elements can reduce accuracy even if the mesh is fine.
• Mesh refinement is often needed near holes, corners, supports, loads, and contact regions.
• Boundary conditions must be applied correctly for the model to have a valid solution.
• FEA results should be checked for convergence as the mesh is refined.
• Discretisation is a central step in the full FEA process because it turns a real solid into a solvable mathematical model.