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Navier-Stokes equations

Named after Claude-Louis Navier and George Gabriel Stokes, Navier-Stokes equations are a set of partial differential equations that accurately and reliably describe the motion of liquids and gases, allowing them to be recreated in simulation.

The equations work by determining rates of change, and movement in the substance when external forces are applied, rather than modelling the substance itself. This actually considerably simplifies matters, as the shape of the body of fluid or gas, or cluster of particles does not have a bearing on how it reacts. The density does, the shape does not.

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Hacked: The Mathematics of Leaf Vein Architecture
The math by which the veins of leaves are formed has been cracked. By extension, the reasons for leaf size and shape have been cracked, and equations for creating the types of leaves that would service a given plant have been laid bare. Ideal material for anyone wishing to model a realistic but not physically real, plant.

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Creating Polygonal Shapes through a Merger with Topology
By merging the traditionally distinct fields of topology and geometry, through the discovery of a new type of mathematics called persistent homology, math researchers have created a set of equations which simply describe the pattern and placement of complex fractals as part of a polygonal model - such as the unique froth on a wave, within the reach of real-time rendering


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Since 2006, Beverley McKeon, professor of aeronautics and associate director of the Graduate Aerospace Laboratories at the California Institute of Technology (Caltech) and collaborator Ati Sharma, a senior lecturer in aerodynamics and fligh...

The way fabric moves has always been a pain to recapture virtually. Creating realistic folds and drapes in anything from skirts to sails has always been computationally expensive, and impossible to produce in realtime. Now, thanks to Mah...

Simulation of mechanical systems

Mechanical systems simulation has always typically been a primitive branch of simulation. Individual moving parts have long been simulated working together, but a complete working engine in re...

Grassed waterways are placed in agricultural fields where runoff water tends to concentrate because they can substantially reduce soil erosion. Mapping techniques that help identify where erosion channels will likely form could help farmers...

Tiny optical components the heart of modern communications systems might one day increase the operational speed of computers. When designing these components, optical engineers rely on mathematical simulations to predict the performance...