How do LiveSimulators numerical physics engines work?

LiveSimulators runs client-side Float64 numerical integrators using 4th-Order Runge-Kutta (RK4) and symplectic algorithms. Differential equations and boundary conditions are solved live at 60 FPS directly in the browser with zero server latency.

Are LiveSimulators models aligned with international engineering standards?

Yes. Simulations reference established industry standards including IEEE Std 1459/3002, ASME MFC-3M, ISO 5167, IEC 60076/60381, AISC 360, and SI-CODATA fundamental physical constants.

Can professors and universities use LiveSimulators for classroom lectures and syllabi?

Yes. LiveSimulators is an open-access educational platform. Educators can freely demonstrate live simulations, project workbenches during lectures, and integrate simulation links into syllabus coursework.

Who created LiveSimulators?

LiveSimulators was founded and engineered by Anil Sharma (email: 0808miracle@gmail.com, LinkedIn: https://www.linkedin.com/in/toanilsharma/) to transform mathematical pedagogy into visual physical intuition.

LiveSimulators - Interactive Engineering Simulations

Engineering Disciplines

Electrical & Electronic Systems (EE-200)

Analog circuits, switching power topologies, RF transmission lines, 3-phase grid power, and semiconductor dynamics.

Core Formulation: v(t) = L\frac{di}{dt} + Ri(t) + \frac{1}{C}\int i(t)dt

Active Simulators: 18 | Subfields: Power Electronics, RF & Transmission Lines, Analog Active Filters, 3-Phase AC Grids, Semiconductor Physics

Control Systems & Robotics (CS-300)

Closed-loop feedback stability, PID gain tuning, root locus analysis, state-space representations, and inverted pendulum dynamics.

Core Formulation: u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt}

Active Simulators: 12 | Subfields: PID Controllers, Bode & Nyquist Plots, State-Space Feedback, Inverted Pendulum, Robotic Kinematics

Mechanical & Thermal Dynamics (ME-400)

Damped harmonic oscillators, multi-degree-of-freedom vibrations, thermodynamic Carnot cycles, and heat transfer convection.

Core Formulation: m\ddot{x} + c\dot{x} + kx = F_0 \cos(\omega t)

Active Simulators: 14 | Subfields: Vibration & Resonance, Carnot & Rankine Cycles, Heat Exchanger Conduction, Gear Train Dynamics, Fluid Bernouilli Flow

Chemical & Process Engineering (CH-250)

Continuous Stirred Tank Reactors (CSTR), McCabe-Thiele distillation columns, TEMA shell-and-tube heat exchangers, and packed gas absorbers.

Core Formulation: \frac{dC_A}{dt} = \frac{F}{V}(C_{A0} - C_A) - k_0 e^{-E_a/RT} C_A^2

Active Simulators: 12 | Subfields: CSTR Chemical Kinetics, McCabe-Thiele Distillation, TEMA Shell & Tube Exchangers, Packed Gas Absorbers, Reaction Thermodynamics

Civil & Structural Mechanics (CE-320)

Truss axial stress distribution, Euler-Bernoulli beam deflection under uniform distributed loads, and seismic response spectra.

Core Formulation: EI \frac{d^4 w}{dx^4} = q(x)

Active Simulators: 11 | Subfields: Beam Bending & Shear, Warren & Pratt Trusses, Buckling Instability, Seismic Base Isolation, Soil Consolidation

Quantum & Semiconductor Physics (PH-500)

P-N junction energy band bending, SiC MOSFET switching dynamics, IGBT multi-layer thermal Foster networks, and BSIM4 inversion channels.

Core Formulation: i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r},t)

Active Simulators: 11 | Subfields: P-N Diode Band Diagram, SiC MOSFET Double-Pulse, IGBT Thermal Foster Model, MOSFET Inversion Channel, Quantum Well Tunneling

Complete Interactive Simulation Library

Explore our comprehensive collection of first-principles client-side solvers:

RLC Resonant Circuit & Damping Transient — Observe underdamped ringing, critical damping, and impedance resonance in real time.
3-Phase AC Synchronous Motor & Rotating Field — Track how three balanced 120° sinusoidal currents generate a continuous rotating stator flux vector.
Buck-Boost DC-DC Switching Regulator — Explore inductor energy storage, Continuous vs Discontinuous mode, and voltage inversion.
2nd-Order Sallen-Key Active Low-Pass Filter — Tweak component values to shape Butterworth, Chebyshev, or Bessel frequency responses.
RF Transmission Line & Reflection Coefficient — See incident and reflected TEM voltage waves collide to form standing wave envelopes.
Fourier Series Harmonic Decomposition & THD — Sum discrete sinusoidal harmonics to assemble square, triangle, and pulse waveforms.
Planar 4-Bar Grashof Kinematics & Coupler Curves — Synthesize crank-rocker vs double-crank inversions and trace continuous coupler point trajectories.
Thermodynamic Rankine Steam Power Cycle — Track steam expansion through high-pressure boilers, turbines, condensers, and feed pumps on a T-s diagram.
Damped Forced Harmonic Oscillator & Resonance Peak — Visualize mass-spring-damper resonance, phase lag φ(ω), and displacement amplification M(ω).
Involute Spur Gear Mesh Dynamics & Backlash — Track tooth contact along the line of action, verify contact ratio CR ≥ 1.4, and check sliding velocity.
Industrial Closed-Loop PID Tuning & Anti-Windup — Tune proportional, integral, and derivative gains with integrator anti-windup clamping.
Pneumatic Control Valve Flow Cv & Cavitation — Compare Linear, Equal-Percentage, and Quick-Opening trims under varying differential pressures.
4-20mA Current Loop Transmitter & Line Resistance — Simulate 2-wire smart transmitters, cable loop resistance drops, and NAMUR fault alarms.
Differential Pressure Orifice Plate Flowmeter — Calculate volumetric flow through concentric thin-plate square-edge orifices and tap dP.
Euler-Bernoulli Beam Deflection, Shear & Moment — Calculate Shear Force Diagrams (SFD), Bending Moments (BMD), and elastic deflection profiles.
Warren & Pratt Truss Bridge Analysis — Solve Method of Joints 2D nodal equilibrium under moving deck truck live loads and check Euler buckling.
Seismic Base Isolation & Dynamic Multi-Story Response — Compare conventional fixed-base vs elastomeric Lead-Rubber Bearing (LRB) building lateral drift.
Mohr’s Circle of Stress & Soil Shear Slip Failure — Transform 2D normal and shear stresses to principal axes (σ₁, σ₂) and evaluate Coulomb failure envelopes.
Non-Isothermal CSTR & Thermal Runaway Dynamics — Simulate continuous reaction kinetics, cooling jacket heat transfer, and thermal runaway thresholds.
Binary Distillation Column & McCabe-Thiele Stages — Step through McCabe-Thiele theoretical equilibrium trays, adjust reflux ratio, and evaluate minimum reflux R_min.
Shell-and-Tube Heat Exchanger & ε-NTU Rating — Evaluate countercurrent thermal gradients, logarithmic mean temperature difference (LMTD), and ε-NTU effectiveness.
Packed Gas Absorption Tower & Flooding Hydraulics — Simulate countercurrent solvent scrubbing, calculate packing height Z via HTU-NTU, and detect Sherwood flooding limits.
P-N Junction Diode Band Bending & Carrier Transport — Visualize energy band bending (Ec, Ev, Ef), space-charge depletion width, and carrier drift-diffusion.
Silicon Carbide (SiC) MOSFET Double-Pulse Switching & Slew Rates — Capture ultra-fast dv/dt (up to 100 V/ns), Miller plateau dynamics, inductive ringing, and switching energy losses E_on / E_off.
IGBT Module Multi-Layer Thermal Foster Network & SOA — Model multi-layer conduction from Silicon Die through DBC ceramic and baseplate to heatsink, tracking T_j ripple.
MOSFET Inversion Channel, Band Bending & 2D Electron Gas — Solve surface potential band bending, visualize 2D electron sheet accumulation, and observe channel pinch-off.

Platform Architecture & Numerical Integrity

Every simulation executes directly inside the user's browser using 4th-Order Runge-Kutta (RK4) integration algorithms. Discrete parameter adjustments trigger continuous differential solutions evaluated at 60 frames per second without network lag.

            STANDARDS: IEEE Std 1459, ASME MFC-3M, ISO 5167, IEC 60076, AISC 360, SI-CODATA 2022
          