Engineering & Technology · Digital Design, Simulation & Prototyping
Discretisation, finite-element and other numerical models
A computer stores finitely many numbers, while a displacement, temperature or flow field can vary continuously across a region. Numerical models make that problem manageable by representing it with finite pieces or unknowns. Understanding what those pieces approximate helps us ask whether a detailed image deserves its apparent precision.
Review sketches and models, measurement evidence and design decisions. The numerical lesson introduces its own midpoint calculation; no prior finite-element mathematics is assumed. The thermal expression is supplied to compare assumptions, not to prescribe an experiment or equipment setting.
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Explain what discretisation approximates and use a small refinement example without confusing convergence with physical validity.
A mesh is a representation of a mathematical problem
In finite-element analysis, a region is divided into elements connected through nodes. Chosen interpolation functions describe field variation within elements using a finite set of unknowns. Equations, material relationships, loads and boundary conditions connect those unknowns. “Finite” describes this approximation, not a claim that a physical object consists of little pieces joined by actual pins.
Other numerical methods organise approximations differently. Finite differences approximate derivatives through nearby values; finite volumes express balances over control volumes. The appropriate method depends on the problem. The shared beginner question is which mathematical quantities have been approximated, at what resolution, and with what checks.
Approximate one changing force using midpoint values
To see discretisation without solving field equations, use the original force law F(x)=(1 N/m²)x² for 0≤x≤2 m. Work is the area under its force-versus-position curve. Replace each short interval by a rectangle whose height is the force at the interval midpoint. This is numerical integration, an analogy for finite approximation rather than an FEM computation.
With two 1 m intervals, the midpoints are 0.5 and 1.5 m, so the forces are 0.25 and 2.25 N. Their areas sum to(0.25+2.25)×1=2.5 J. One interval gives 2 J; four intervals give 2.625 J. The supplied exact mathematical integral is 8/3 J≈2.6667 J. Refinement reduces error in this particular smooth example.
A stable answer may still represent the wrong physics
The midpoint errors are approximately−0.6667,−0.1667 and−0.0417 J for one, two and four intervals. The sign means underestimation relative to the stated mathematical reference. These values check numerical approximation of the assumed force law. They do not establish that any real mechanism follows that law.
For a field model, compare quantities relevant to the decision as the discretisation and solution tolerances change. A visually smoother image is not a numerical error estimate. Some local peaks near idealised sharp features or concentrated loads may not settle in the same way as an overall displacement. Revisit the mathematical idealisation and the output being claimed. More elements cannot repair an incorrect load, material model or boundary condition.
Reconstruct the four-interval estimate
Each interval is 0.5 m wide. At x=0.25, 0.75, 1.25 and 1.75 m, the forces are 0.0625, 0.5625, 1.5625 and 3.0625 N. Their sum is 5.25 N; multiply by 0.5 m to obtain 2.625 J.
The difference from the two-interval result is 0.125 J. That difference is useful refinement evidence but is not itself an automatically guaranteed error bound. Here an independently supplied exact result lets us calculate the actual mathematical error. A general problem needs a justified error-estimation method.
Try the reasoning
Is the rectangle example a finite-element simulation?
Show the answer and reasoning
No. It is midpoint numerical integration used to make finite approximation visible and checkable.
What has the four-interval result been checked against?
Show the answer and reasoning
The exact integral of a supplied mathematical force law, not an observation of a physical device.
Why can an extremely fine mesh still give an unsuitable answer?
Show the answer and reasoning
It can solve an unsuitable mathematical model very accurately. Incorrect boundaries, inputs or physical assumptions remain incorrect after refinement.
What to carry forward
Refinement checks the approximation of a stated mathematical model; the model’s relationship to reality needs separate evidence.
References
Teaching resources and research supporting this lesson’s explanations and examples.