The simulation models

Water finds the low ground. The simplest useful question is which places sit below a given water level, and answering it needs terrain and a level and nothing else.

1. Bathtub inundationSimple bathtub inundation modelIf the water comes up a metre and a half, which streets and which doorways are under it?

Traffic can be treated as a fluid with a density and a speed, or as individual vehicles following the one in front. Both produce queues and stop-and-go waves. They differ in what you can ask afterwards.

1. Cell transmission modelCell Transmission Model (Daganzo 1994): min(sending, receiving) cell transfers, proportional junction merge/diverge, CFL-sized cellsWhen a lane closes, where does the queue form and how far back does it reach?
2. Intelligent driver modelIntelligent Driver Model - realistic car-followingHow do individual drivers follow the car in front, and when does stop-and-go appear out of nothing?
3. Lighthill-Whitham-RichardsLighthill-Whitham-Richards first-order continuum: Godunov scheme with demand/supply interface fluxes and link-coupled boundary conditionsHow does a slowdown travel back up the road as a wave after the cause has gone?
4. Nagel-SchreckenbergNagel-Schreckenberg cellular automaton microsimulationCell by cell, with drivers braking at random. How does phantom congestion come out of nowhere?

What a place feels like on a hot day is not the air temperature. It is temperature, humidity, wind and radiation acting together on a body standing in that spot.

1. Universal Thermal Climate IndexUniversal Thermal Climate Index screeningOn a hot afternoon, what does it feel like standing at this bus stop, once you count sun, wind and humidity?

Sound leaves a source and loses energy on the way: with distance, into soft ground, into the air itself, and around whatever stands in between. Modelling it means accounting for that loss from the source to a receiver you actually care about, which is usually somebody's window.

1. ISO 9613-2 outdoor propagationISO 9613-2 outdoor sound propagationWill a new rooftop plant push night-time noise at the flats opposite past the limit in the planning condition?
2. Distance attenuationSimple distance-only attenuationRoughly how far back does a generator have to sit before it stops mattering? No barriers, no ground effect, distance only.

Something released into the air is carried by the wind and spread by turbulence. Where it comes down depends on the release height, the weather, and the buildings in the way. The three methods trade cost against how much of that geometry they can honour.

1. Computational fluid dynamics2D steady-state advection-diffusion solver (upwind finite differences, vectorized Jacobi)Given the buildings actually standing around that stack, where does the plume come down?
2. Gaussian plumeSteady-state Gaussian plume model (Pasquill-Gifford)A depot is proposed upwind of a school. Roughly what concentration reaches the playground on an ordinary day?
3. Lagrangian particleStochastic Lagrangian particle trackingSmoke from a fire, followed as particles while the wind shifts. Where does it end up?

People in a crowd steer around each other. Model each person as pulled towards where they want to go and pushed away from everyone close by, and lanes, jams and bottlenecks appear without anyone having programmed them.

1. Agent-based pedestriansORCA velocity-obstacle collision avoidance with behavioral layersIn a crowded concourse, do people get through the gates, or does a jam form and hold?
2. Helbing social force modelClassic Helbing social force modelHow does a crowd move down a corridor, and where does it bunch up?

Graph measures, not physics.

Streets, paths and connections form a graph, and much of what people mean by access is a property of that graph: how far you can get, what is near, which junctions carry the traffic of the whole network, and where the neighbourhoods actually lie.

1. Borrowed sizePopulation potential / borrowed size (van Oort/Burger)A small town sits beside a big one. How much of the big one's amenity does it effectively borrow?
2. Capability relatednessBipartite relatedness and adjacent-possible analysisThis city already does these things. Which industries are genuinely adjacent for it, and which are a reach?
3. CentralityNode/edge centrality (betweenness, closeness, etc.)Which junctions carry the most of the network's paths, so that closing one hurts most?
4. Community detectionCommunity detection (Louvain/label propagation)Which parts of the street network hang together as neighbourhoods, rather than on the map?
5. Fifteen-minute cityPOI accessibility within walking isochrones (Moreno)From this doorstep, what can you reach on foot in fifteen minutes: school, surgery, groceries?
6. Green space accessibility2SFCA park/green space accessibility (WHO benchmarks)Does this neighbourhood meet the WHO benchmark for park access, once you count everyone competing for the same parks?
7. IsochronesTravel-time isochrones / accessibility contoursHow far can you get from here in ten minutes, twenty, thirty?
8. Shortest pathSingle-objective shortest/cheapest pathWhat is the quickest way from here to there, and how does it change when a road shuts?

Travel demand is built in stages: how many trips a place produces, where they go, which mode they take, and which roads they end up on. The stages separate cleanly, which is why this has been the standard planning method for decades.

1. Four-step travel demandTraditional 4-step model: generation, distribution, mode choice, assignmentA housing block goes in. How many trips does it generate, where do they go, by what mode, and which roads carry them?

Infection moves through a population that is not uniform. People mix mostly within their own age group, commute between places, and recover at different rates. These models follow the whole population moving between states rather than following individuals.

1. Metapopulation SIRDeterministic susceptible-infected-recovered flow across caller-supplied population groups.An outbreak starts in one borough. How does it move between boroughs, given who commutes where?
2. Stratified SEIRDeterministic susceptible-exposed-infected-recovered flow across heterogeneous caller-defined strata.How differently does the same outbreak run through children, working adults, and people over seventy?

People move between states: waiting, being seen, finished. Following the flow between those states over time is what tells you where a backlog builds and how long it takes to clear.

1. Continuous four-stateContinuous flow across Idle, Queued, Active, and Cleared states.The same, with a queue stage before service, so you can see where the backlog actually sits.
2. Continuous three-stateContinuous flow across Idle, Active, and Cleared states.People waiting, being seen, done. How does the queue at a vaccination centre drain across a day?
3. Stochastic three-stateStochastic flow across Idle, Active, and Cleared integer-unit states; does not represent individual attributes, schedules, movement, or explicit interaction events.The same flow counted in whole people rather than fractions, run many times, so you can see how much the day varies.

Something spreads across a network of connected places: heat, money, a rumour, a habit. Each method here makes a different assumption about what makes the next place catch it, and on the same graph they give you noticeably different answers.

1. Independent cascadeIndependent Cascade model for information/influence spreadingOne shop on the street goes contactless only. If each neighbour gets one chance to persuade the next, how far does it travel?
2. Linear heat conductionHeat equation on graph LaplacianHeat is dumped into one district. How does it even out across the connected districts over the next few hours?
3. Linear thresholdLinear Threshold model for opinion/behavior adoptionPeople join a recycling scheme only once enough neighbours already have. Does it reach the whole neighbourhood, or stall halfway?

When one part of an infrastructure network fails, the load it was carrying has to go somewhere. If its neighbours cannot absorb it they fail in turn, and the failure walks across the network.

1. Load redistribution cascadeLinearized load redistribution cascadeOne substation trips and its load moves to its neighbours. Do they trip too?

Graph measures, not physics.

Take a network and start removing pieces. For a while very little happens, and then it comes apart. Where that threshold sits, and how far it moves between random loss and deliberate attack, is what these measure.

1. Random failureUniform random node removalIf parts of the network fail at random, at what point does it break into disconnected pieces?
2. Targeted removalIterative highest-betweenness node removalIf someone takes out the most important junctions one at a time, how many before the whole thing comes apart?

Materials and perishable cargo have a clock running on them. Heat speeds up chemical decay, and repeated loading accumulates fatigue. Both are cumulative, and neither shows until something fails, so the model keeps the running total rather than a snapshot.

1. Arrhenius temperature agingArrhenius kinetics for temperature-time (organs, food)A donor organ sat in a broken cold chain for forty minutes. How much viable time did that cost?
2. Miner’s rule fatigue accumulationMiner's rule for cumulative fatigueA bridge deck has taken twenty years of lorry traffic. How much fatigue life is left in it?

Construction has geometry and it has shaking. A crane sweeps a volume that must not intersect anything; piling and demolition put vibration into the buildings next door. Both get checked against published limits.

1. Combined constraintsRun both crane clearance and PPV vibration in one passRun the crane clearance check and the vibration screen for one site in a single pass.
2. Crane clearance3D geometric crane clearance checkCan this crane slew without its jib crossing the neighbouring building or the railway?
3. Peak particle velocity vibrationPPV vibration screening (DIN 4150-3)Will piling next door shake that listed building past the DIN 4150-3 limit?

Graph measures, not physics.

A city rarely stops where its boundary says it stops. Delineation infers the real extent from where people are and where they travel, rather than from an administrative line.

1. Density clusteringDensity-based cluster delineationWhere does the built-up area actually end, going by where people are rather than by the council boundary?
2. Flow-based functional urban areasFlow-based FUA delineationWhich towns commute into this city enough to count as part of it?