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Physical simulation is the branch of computer graphics concerned with generating motion and deformation by computing the behavior of physical systems according to mathematical models of mechanics. Where traditional animation specifies motion directly—through keyframes, inverse kinematics, or procedural rules—physical simulation treats motion as the outcome of forces, constraints, and material properties acting over time. The central question of the subfield is: given a description of an object or fluid, its initial state, and the forces acting on it, what motion or shape results, and how can that result be computed accurately and efficiently enough to be useful in images?
The stakes are practical as well as scientific. Simulated cloth, hair, water, smoke, fire, rigid bodies, and deformable objects appear throughout film, television, games, and interactive applications. The field's enduring tension is between physical fidelity and computational cost. A simulation that exactly solves the governing equations of a material is often far too slow for production; a simulation that is fast enough may look wrong, behave unstably, or require extensive hand-tuning. The history of the field is largely the history of negotiating this trade-off, with different research programmes choosing different points along the spectrum.
Physical simulation in graphics is built on continuum mechanics, the branch of physics that describes materials as continuous media rather than collections of discrete particles. The central objects are fields—quantities defined at every point in space, such as density, velocity, stress, and temperature—and the central equations are partial differential equations (PDEs) expressing conservation laws. The Navier–Stokes equations for fluids, the equations of linear and nonlinear elasticity for solids, and the heat equation for temperature diffusion are the most common foundations.
Because computers cannot represent continuous fields exactly, simulation requires discretization: the continuous domain is replaced by a finite set of unknowns, and the PDEs are replaced by algebraic equations that approximate them. The choice of discretization is the first major fork in the field. Eulerian methods fix a grid in space and track how quantities at grid points change over time. Lagrangian methods track material points that move with the flow or deformation. Hybrid methods combine both, using a grid for some quantities and particles for others. Each choice carries consequences for accuracy, stability, and the types of phenomena that can be represented naturally.
The time dimension is also discretized. A simulation advances in small steps, computing the state at each new time from the previous one. The size of the time step is constrained by stability: if it is too large, numerical errors grow exponentially and the simulation "explodes" with visibly nonphysical behavior. Implicit methods, which solve for the new state using information about the future, allow larger steps at the cost of solving a system of equations at each step. Explicit methods are simpler but require smaller steps. Much of the craft of physical simulation lies in choosing discretizations and time-stepping schemes that are stable, accurate, and fast for the specific phenomenon at hand.
The earliest and most intuitive approach to simulating motion in graphics is to track material points directly. A rigid body is represented by its position, orientation, linear velocity, and angular velocity; its motion is computed by integrating Newton's laws under gravity, contact, and friction. Rigid-body simulation, well established by the 1990s, handles tumbling objects, stacked crates, and collisions. Its central difficulty is contact: determining when bodies touch, computing the contact forces that prevent interpenetration, and resolving the resulting impulses. Contact resolution remains a subtle problem, with issues of stability, friction modeling, and the handling of many simultaneous contacts.
Deformable objects extend the Lagrangian idea to materials that change shape. The simplest models treat an object as a collection of masses connected by springs. Mass–spring systems are easy to implement and fast, but their behavior depends on the chosen spring constants and topology rather than on a physically meaningful material model. The more principled approach is to discretize the equations of elasticity directly, typically using the finite element method (FEM). The object is divided into tetrahedra or hexahedra; each element's deformation is measured by a strain tensor; and the resulting internal forces are computed from a constitutive model relating strain to stress. Linear elasticity, which assumes small deformations, is simple and stable but fails for large bending or stretching. Nonlinear elasticity models, such as the Saint-Venant Kirchhoff model or the neo-Hookean model, handle large deformations more realistically but introduce additional computational cost and the risk of instability under inversion or extreme compression.
Cloth and hair are special cases of deformable bodies with distinctive geometry. Cloth is a thin sheet with high in-plane stiffness but low bending resistance; hair and fur are one-dimensional rods with complex bending and twisting behavior. Simulating these materials requires specialized models that capture their anisotropic response to deformation. The field has developed dedicated approaches—often based on discrete differential geometry, where the continuous surface is replaced by a mesh and curvature is computed from the relative positions of neighboring vertices—that are both efficient and visually convincing.
Fluid simulation in graphics was revolutionized in the late 1990s by the introduction of methods from computational fluid dynamics, adapted to the needs of visual effects. The standard Eulerian approach solves the incompressible Navier–Stokes equations on a regular grid. At each time step, the solver computes the velocity field, enforces incompressibility by projecting out the divergent component (which requires solving a Poisson equation), and advects quantities such as density, temperature, or dye through the velocity field. The semi-Lagrangian advection scheme, which traces characteristics backward through the flow, is stable even for large time steps, though it introduces numerical diffusion that smears fine detail.
The grid-based approach excels at capturing the large-scale behavior of smoke, fire, and water. Smoke is typically modeled as a density field advected by the velocity field, with buoyancy forces driving the characteristic rising and rolling motion. Fire adds combustion chemistry, usually simplified to a reaction that converts fuel into heat and products. Water is more challenging because it has a free surface—a boundary between the water and the air—that must be tracked as it moves and deforms. The level set method represents this surface implicitly as the zero level of a scalar function, allowing topological changes such as splashing and merging to be handled naturally.
The principal limitation of pure Eulerian methods is numerical diffusion: the grid smears sharp features, such as the fine spray of a breaking wave or the detailed vorticity of a turbulent plume. The field has responded with techniques to counteract this smearing. Vorticity confinement adds a force that reinjects small-scale rotational motion that the grid has dissipated. Higher-order advection schemes reduce diffusion at the cost of complexity. And hybrid methods, discussed below, bring back some of the sharpness of Lagrangian tracking.
The most influential development in fluid simulation over the past two decades has been the combination of Lagrangian and Eulerian ideas. The smoothed particle hydrodynamics (SPH) method is fully Lagrangian: the fluid is represented by particles that carry mass, momentum, and other quantities, and forces are computed by smoothing contributions from neighboring particles. SPH handles free surfaces and complex boundaries naturally, but it is computationally expensive and can suffer from density errors and instability.
The fluid-implicit-particle (FLIP) and particle-in-cell (PIC) methods take a different tack. They use particles to track the fluid's state but perform the pressure solve on a grid. At each step, particle velocities are transferred to the grid, the incompressibility constraint is enforced, and the corrected velocities are transferred back to the particles. PIC is stable but overly diffusive; FLIP reduces diffusion by transferring only the change in velocity, but introduces noise. Modern variants blend the two to balance stability and detail. These methods have become the workhorse for water simulation in production, capable of producing the fine spray, foam, and splashing that audiences expect.
The material point method (MPM) generalizes the particle–grid idea to solids and to materials that can transition between solid and fluid behavior. MPM represents the material as particles that carry deformation state, while forces are computed on a background grid. Because the grid handles the computation of gradients and forces, MPM avoids the mesh entanglement problems that plague FEM for large deformations. It has become a standard tool for simulating snow, sand, mud, and other materials that flow like fluids but can also pile up and support stress like solids.
A recurring theme in physical simulation is the tension between physical fidelity and artistic control. A simulation that faithfully follows the laws of mechanics may produce motion that is physically correct but aesthetically wrong—too chaotic, too calm, or simply not what the director envisioned. The field has therefore developed a range of techniques for steering simulations toward desired outcomes.
Constraint-based methods allow the artist to specify targets that the simulation must satisfy. A character's hand may be constrained to follow a keyframed path while the rest of the cloth responds physically. Forcing methods add artificial forces that push the simulation toward a target state without fully constraining it. Optimization-based methods treat the simulation as an inverse problem: given a desired outcome, find the initial conditions, forces, or material parameters that produce it. These methods are computationally expensive but offer the most direct control. The practical reality in production is that simulations are rarely run once; they are iterated, with parameters adjusted and constraints added until the result is acceptable. The field's success is measured not only by the fidelity of its models but by the ease with which artists can shape them.
The current state of physical simulation in graphics reflects a mature field with a diverse toolkit. No single approach dominates; rather, practitioners choose among methods based on the material, the required detail, the available compute budget, and the need for control. Rigid-body solvers handle large numbers of interacting objects in real time for games. FEM-based deformable solvers produce high-quality cloth and soft bodies for film. Grid-based and hybrid fluid solvers generate water, smoke, and fire at scales from a dripping faucet to an ocean. MPM has become the default for granular and multiphase materials.
Two broader trends shape the present landscape. The first is the increasing importance of machine learning. Neural networks have been trained to accelerate simulations, to replace components of solvers (such as the pressure solve), and to generate plausible motion directly from latent representations. These methods are promising but not yet a replacement for traditional simulation; they are best understood as accelerants or approximations that trade some fidelity for speed. The second trend is the convergence of simulation and rendering. As real-time ray tracing and path tracing become more common, the demand for simulations that can run interactively—or that can be precomputed and replayed with varying conditions—has grown. The field continues to develop methods that balance the competing demands of accuracy, speed, stability, and control, with the same fundamental tension that has defined it since its inception: the equations of mechanics are unforgiving, but the images they produce must serve the needs of storytelling and interaction.