How Does FEA Work? A Step-by-Step Guide to Finite Element Analysis

You create a CAD model, assign a material, apply some loads and boundary conditions, click Solve, and suddenly your screen is covered with colourful stress contours.

But what actually happens between clicking Solve and seeing those results?

How does FEA work?

The answer is surprisingly elegant. Although commercial FEA packages such as Abaqus,

ANSYS and other solvers can look very different on the surface, underneath they are solving the same fundamental mathematical problem.

In this article, we will follow that process from beginning to end, using a single finite element to see what happens inside an FEA solver.

What Is FEA?

FEA stands for Finite Element Analysis.

It is a numerical method used to predict how physical structures behave under conditions such as forces, pressures, temperatures and other loads.

The basic idea is simple:

Real object → Mesh → Equations → Solve → Displacement → Strain → Stress

A real structure is continuous. In principle, every point inside the structure could move by a different amount when a load is applied.

Solving the exact behaviour of every point directly is usually impractical.

Instead, FEA divides the structure into a large number of smaller pieces called finite elements.

These elements are connected at specific points called nodes.

The computer then uses the behaviour of these elements to approximate the behaviour of the entire structure.

This is the fundamental idea behind the finite element method.

Step 1: Divide the Structure into Finite Elements

The first major step in FEA is discretisation.

Instead of treating the entire structure as one continuous object, the geometry is divided into many smaller elements.

Depending on the problem, these might be:

• Triangular or quadrilateral elements in 2D

• Tetrahedral or hexahedral elements in 3D

• Beam elements

• Shell elements

• Other specialised element types

  
  

Each element has nodes, and the solver calculates unknown quantities such as displacement at these nodes.

For example, the interactive example below follows a single 3-node triangular element.

The three nodes are:

Node 1: (0, 0) mm

Node 2: (5, 0) mm

Node 3: (0, 2) mm

The triangle is a Constant Strain Triangle (CST), meaning that the strain calculated from this particular element is constant throughout the element.

Interactive: How FEA Works Inside One Element

The simulation follows the mathematical chain inside an FEA calculation. Click through each step to see how nodal displacement eventually becomes strain and then stress.

Step 2: Build the Element Stiffness Matrix

Once the structure has been divided into elements, the solver needs to determine how each element responds to displacement.

This is represented by the element stiffness matrix.

The fundamental equation for an individual element is:

Fᵉ = kᵉuᵉ

Here:

• Fᵉ = element nodal force vector

• kᵉ = element stiffness matrix

• uᵉ = element nodal displacement vector

  
  

You can think of the stiffness matrix as describing how difficult it is to deform that particular element.

A stiffer element requires more force to produce the same displacement.

Importantly, the element stiffness depends on the element's geometry and material properties, rather than the applied load.

For the triangular element in our example, the stiffness matrix is obtained from:

kᵉ = ∫ BᵀDB dV

The exact calculation depends on the element formulation, material model and dimensional assumptions.

An FEA solver performs this calculation automatically for every element in the mesh.

Step 3: Assemble the Global System

A real engineering model does not contain just one element.

It may contain hundreds, thousands or even millions of elements.

Each element has its own stiffness matrix, but the elements are connected to one another through their shared nodes.

The solver therefore assembles all of the individual element stiffness matrices into one large global stiffness matrix.

The resulting system can be written in the familiar form:

KU = F

where:

• K = global stiffness matrix

• U = global displacement vector

• F = global force vector

  
  

This is one of the most important equations in finite element analysis.

The solver's job is essentially to determine the unknown values in U.

But there is a problem.

Without additional information, the structure could simply move freely through space.

This is where boundary conditions become essential.

Step 4: Apply Boundary Conditions

An FEA model needs to know how the structure is constrained.

These constraints are called boundary conditions.

For example, a fixed support might prevent movement in certain directions.

In our triangular example, Node 1 is fixed:

u₁ = 0

v₁ = 0

This means that the node cannot move in either the x or y direction.

The boundary conditions modify the global system so that the solver knows which degrees of freedom are fixed and which are unknown.

The remaining unknown displacements can then be calculated.

This is an important point to remember:

FEA does not simply calculate stress from the loads you entered. It first solves for the structure's displacement.

Step 5: Calculate the Displacement Field

The solver calculates displacement at the nodes.

But what happens between the nodes?

This is where shape functions are used.

For our triangular element, the displacement at any point inside the element can be written as:

u(x,y) = N uᵉ

The matrix N contains the element's shape functions.

The shape functions interpolate the displacement between the nodes.

In other words, the solver does not need to calculate the displacement independently at every point inside the element.

Instead, it calculates the nodal displacements and uses the shape functions to determine what happens between them.

For our triangular element, the displacement at a point P is effectively a weighted combination of the displacements at Nodes 1, 2 and 3.

This is one of the key ideas behind the finite element method.

A finite number of unknowns can be used to approximate a continuous displacement field.

Step 6: Calculate Strain

Once the displacement field is known, the solver can calculate strain.

Strain describes how much the material has deformed relative to its original dimensions.

The relationship is:

ε = Buᵉ

where:

• ε = strain

• B = strain-displacement matrix

• uᵉ = nodal displacement vector

• 
  

The important idea is that strain is obtained from displacement.

Mathematically, strain is related to the spatial derivatives of displacement.

Because the displacement field is already defined using the shape functions, the solver can differentiate that field to obtain strain.

For our Constant Strain Triangle, the shape functions are linear.

Their derivatives are therefore constant.

As a result, the strain is also constant throughout this particular element.

This is why the element is called a Constant Strain Triangle.

Step 7: Calculate Stress

We now have the strain.

The next step is to calculate stress.

For a linear elastic material, the relationship can be written as:

σ = Dε

where:

• σ = stress

• D = constitutive or elasticity matrix

• ε = strain

  
  

The matrix D contains the material properties required to relate strain to stress.

For a typical isotropic material, these properties include:

• Young's modulus, E

• Poisson's ratio, ν

  
  

For example, our interactive element uses:

E = 200 GPa

ν = 0.3

The constitutive relationship is essentially Hooke's law written in matrix form.

So the complete chain is now:

Nodal displacement → Strain → Stress

This is a crucial concept for understanding how FEA works.

The Four-Equation Chain

We can now summarise the entire process using four equations.

1. Equilibrium

Fᵉ = kᵉuᵉ

The element's forces are related to its stiffness and nodal displacements.

2. Interpolation

u(x,y) = Nuᵉ

The nodal displacements are used to determine displacement throughout the element.

3. Strain-displacement relationship

ε = Buᵉ

The displacement field is differentiated to obtain strain.

4. Constitutive relationship

σ = Dε

The strain is converted into stress using the material properties.

Together:

Forces → Displacement → Strain → Stress

This is the mathematical chain that sits underneath the colourful contours we see in an FEA program.

So What Does an FEA Solver Actually Do?

When you click Solve, the software is performing a much more involved process than simply applying a formula.

A simplified version of the workflow looks like this:

1. Create the mesh

The geometry is divided into finite elements.

2. Define the element properties

The solver determines the geometry, material properties and formulation for each element.

3. Calculate element stiffness matrices

Each element receives its own stiffness matrix.

4. Assemble the global stiffness matrix

The individual element matrices are combined into one large system.

5. Apply loads and boundary conditions

The solver incorporates forces, supports and other prescribed conditions.

6. Solve for nodal displacements

The global system is solved for the unknown degrees of freedom.

7. Calculate strains

Strains are obtained from the calculated displacements.

8. Calculate stresses

Stresses are obtained from the strains using the material's constitutive relationship.

9. Display the results

The solver can then present quantities such as displacement, stress and strain as contours, vectors or other visualisations.

Why Does the Mesh Matter?

If FEA can approximate a continuous structure using finite elements, an obvious question follows:

How many elements do we need?

This is where mesh quality becomes important.

A very coarse mesh may not represent the geometry or deformation accurately.

A finer mesh can provide a better approximation, but it also increases the computational cost.

The goal is therefore not simply to create the largest possible number of elements.

Instead, engineers often perform a mesh convergence study.

The idea is to refine the mesh and observe whether the quantity of interest, such as displacement or stress, is approaching a stable value.

A result that changes significantly every time the mesh is refined should not be treated as reliable.

Why Can FEA Results Be Wrong?

FEA is extremely powerful, but the software does not automatically know whether your model represents reality correctly.

The solver will faithfully solve the mathematical problem you give it.

If the model contains incorrect assumptions, the results can also be misleading.

Common sources of error include:

• Incorrect boundary conditions

• Incorrect loads

• Incorrect material properties

• An inappropriate element type

• Poor mesh quality

• An insufficiently refined mesh

• Incorrect contact definitions

• Material behaviour that is more complicated than the chosen model

• Stress singularities

• Failure to perform a convergence study

  
  

This leads to one of the most important principles in FEA:

A converged solution to the wrong model is still the wrong answer.

Why Is Stress Often More Difficult to Interpret Than Displacement?

Displacement is obtained directly from the solution of the global system.

Stress, however, is calculated further down the chain.

The process is approximately:

Solve displacement → Calculate strain → Calculate stress

This means that stress depends on the accuracy of the calculations that came before it.

It also means that local stress results can be particularly sensitive to mesh quality, geometry and modelling assumptions.

This is one reason engineers should not simply look at the maximum stress value on a contour plot and immediately conclude that the component will fail.

The location of the stress, the type of stress, the mesh, the boundary conditions and the physical situation all matter.

The Big Picture: How Does FEA Work?

So, how does FEA work?

At its core, FEA takes a continuous physical problem and converts it into a finite system of mathematical equations.

The structure is divided into elements.

Each element is described mathematically.

The individual element equations are assembled into a global system.

Boundary conditions and loads are applied.

The solver calculates the unknown nodal displacements.

From those displacements, strain can be calculated.

From strain, stress can be calculated.

In simplified form:

Physical structure

↓

Mesh

↓

Element equations

↓

Global stiffness matrix

↓

Boundary conditions + loads

↓

Nodal displacements

↓

Strain

↓

Stress

↓

Engineering results

The colourful contour plot that appears at the end is therefore only the final step of a much larger mathematical process.

Once you understand that chain, FEA becomes much less mysterious.

It is not a black box that magically predicts stress.

It is a numerical method that uses a carefully constructed mathematical model to approximate how a physical system behaves.

And that is the fundamental idea behind the Finite Element Method. Disclaimer

For the avoidance of doubt, I have absolutely no affiliation with Senpai Corner and have never met, spoken to, or otherwise encountered this mysterious person known as CJ. Any similarities are purely coincidental. 😌