Multi-domain simulation
Why it matters
Coupled subsystems can produce plausible yet incorrect outputs when their interfaces disagree.
Learning objectives
- Compute first thermal step for ηP=80 W, C=400 J/°C, Δt=10 s at T=Ta.
- For T=22 °C and h=2 W/°C, Ta=20 °C, compute loss and next thermal step.
- Verify the calculation and define independent validation evidence.
Prerequisites
Course 9.2
Concept, definitions and key terms
A coupled simulation exchanges state across domains at a declared interface. For a thermal-electrical system, electrical power P produces heat Q̇=ηP; temperature changes with C dT/dt=ηP−h(T−Ta). Define P in W, C in J/°C, h in W/°C, T in °C, time in s. One-way coupling fixes P; feedback lets resistance or a controller change P as T changes. Identify boundary, sign convention, exchange variables, different time scales and interface assumptions before selecting a numerical step. Model equations are not the physical system.
Notation, state, units and assumptions
Thermal state Tₖ₊₁=Tₖ+Δt[ηP−h(Tₖ−Ta)]/C. Interface heat ηP in W=J/s. Controller state u∈{0,1}; optional P=uPmax. Check that Δt·W/(J/°C) is °C.
Simulation method and procedure
Electrical/thermal device: T₀=20 °C, Ta=20 °C, η=.8, P=100 W, C=400 J/°C, h=2 W/°C, Δt=10 s. Q̇=80 W and loss initially 0 W; Euler yields T₁=22 °C. Next loss=4 W and T₂=23.9 °C. Verify energy: 10·80/400=2 °C and 10·76/400=1.9 °C. Validate against an independent temperature sensor; constant η and h are assumptions.
Define model and units → calculate a baseline → vary or estimate inputs → record results → verify arithmetic → validate against untouched evidence → state limits.
Worked simulation examples
Worked example 1
System, variables, parameters, units, assumptions and calculation: Electrical/thermal device: T₀=20 °C, Ta=20 °C, η=.8, P=100 W, C=400 J/°C, h=2 W/°C, Δt=10 s. Q̇=80 W and loss initially 0 W; Euler yields T₁=22 °C. Next loss=4 W and T₂=23.9 °C. Verify energy: 10·80/400=2 °C and 10·76/400=1.9 °C. Validate against an independent temperature sensor; constant η and h are assumptions.
Verification and expected result: ΔT=10·80/400=2 °C; new T=22 °C.
Validation and limitations: Failure: feeding 100 kW into a component expecting W, or reversing heat-loss sign. A numerically stable trajectory can still be physically impossible. Correct interface unit conversion, energy balance and feedback timing, then compare held-out observations.
Worked example 2
System, variables, parameters, units, assumptions and calculation: Motor/control coupling: a controller turns a 50 W motor on for 2 s and off for 1 s; electrical input over 3 s is 100 J. With a 2 kg load, neglecting losses, energy would permit v=√(2·100/2)=10 m/s from rest. Real motor efficiency and controller timing reduce it. Pass on/off state and torque or power across the interface; verify joules on both sides and validate motion with independent measurements.
Verification and expected result: Loss 4 W; ΔT=10·(80−4)/400=1.9 °C; T=23.9 °C.
Validation and limitations: Failure: feeding 100 kW into a component expecting W, or reversing heat-loss sign. A numerically stable trajectory can still be physically impossible. Correct interface unit conversion, energy balance and feedback timing, then compare held-out observations.
Common simulation error / counterexample
Failure: feeding 100 kW into a component expecting W, or reversing heat-loss sign. A numerically stable trajectory can still be physically impossible. Correct interface unit conversion, energy balance and feedback timing, then compare held-out observations.
Guided practice
Compute first thermal step for ηP=80 W, C=400 J/°C, Δt=10 s at T=Ta.
Solution / evidence
ΔT=10·80/400=2 °C; new T=22 °C.
For T=22 °C and h=2 W/°C, Ta=20 °C, compute loss and next thermal step.
Solution / evidence
Loss 4 W; ΔT=10·(80−4)/400=1.9 °C; T=23.9 °C.
A/B/C/D practice
Define domain, interface variable and units for coupled thermal/electrical model.
Solution / evidence
Electrical P [W] enters thermal as ηP [W]; thermal state T [°C].
Choose a controller on/off state and thermal state; define feedback direction.
Solution / evidence
u∈{0,1} controls P; T influences threshold decision; record synchronization.
Diagnose a thermal component receiving kW but treating value as W.
Solution / evidence
Convert kW×1000 to W; test energy conservation and magnitude.
Design a motor-control coupled simulation with separate time scales and an independent validation observation.
Solution / evidence
State velocity and controller timing separately; check energy transfer and compare held-out motion trace.
Mastery checks and corrective feedback
Mastery: 4/4 checks plus a correct method, units, implementation verification and one independent validation boundary. A reviewer determines mastery.
Related laboratory and next class
Parameter estimation