By Eugene le Roux, FAIRAC, and Eamonn Ryan
From closed-form equations to iterative equilibrium – this is part one of a two-part series.
For centuries, engineering and applied science have relied on classical analytical formulae to describe the physical world. yogiermansyah22 | Freepik.com
For centuries, engineering and applied science have relied on classical analytical formula to describe the physical world. In disciplines such as strength of materials, heat transfer and fluid mechanics, we learn elegant equations – Hooke’s law, Fourier’s heat conduction equation, Bernoulli’s principle – that allow us to compute stresses, temperatures, and velocities directly. These formulae are powerful because they are fast, transparent and grounded in physical intuition.
However, they come with an important limitation: they only work well under idealised conditions.
Classical engineering formulae typically assume simple geometries such as beams, plates, pipes or cylinders. They also rely on strong simplifications – materials are often assumed to be homogeneous, isotropic and linear; boundary conditions are steady; and interactions between physical effects are weak or ignored. In carefully controlled situations, these assumptions are acceptable. In the real world, they are often violated.
Modern engineering problems rarely involve neat shapes or uniform materials. Components may have complex geometries, layered or anisotropic materials, non-linear behaviour, or strong coupling between thermal, mechanical, electrical and fluid processes. In such cases, closed-form theory either becomes extremely difficult or does not exist at all.
This is where computational dynamics enters the picture.
Rather than seeking a single formula that describes the entire system, computational dynamics works from the opposite direction: it builds global behaviour from the smallest elements. This philosophy underpins several major numerical methods used across science and engineering, including the Finite Element Method (FEM), the Finite Volume Method (FVM) and the Finite Difference Method (FDM). While their mathematical details differ, their conceptual workflow is remarkably similar.
The process begins with discretisation. A continuous system – such as a solid body, a fluid domain or a thermal field – is divided into many small elements or volumes. Within each element, the governing physical laws are written in local form: stress–strain relationships, conservation of energy, or momentum balance. These local equations are then connected through interface conditions, ensuring compatibility and continuity between neighbouring elements.
Once assembled, the result is a large system of coupled equations that represents the entire problem. Unlike classical formulae, this system is rarely solved in one step. Instead, computational dynamics relies on iterative solution techniques – such as Gauss–Seidel, conjugate gradient methods, multigrid schemes or relaxation techniques.
In relaxation-based approaches, the system evolves progressively toward equilibrium. At each iteration, local imbalances are reduced until the residual error becomes negligibly small. The final solution is not guessed, it emerges as the equilibrium state of a high-dimensional system.
In part two, we will explore why this equilibrium-seeking mindset is so powerful, how it extends beyond traditional engineering, and where its limitations lie.
