Solver

Solver#

Time integration solvers for physics simulation

This module implements semi-implicit (symplectic Euler) time integration for tetrahedral FEM simulation. The integration scheme updates velocities using forces at the current position, then updates positions using the new velocities:

\[ \begin{align}\begin{aligned}\mathbf{v}^{n+1} = \mathbf{v}^n + \Delta t \, \mathbf{M}^{-1} \mathbf{f}(\mathbf{x}^n)\\\mathbf{x}^{n+1} = \mathbf{x}^n + \Delta t \, \mathbf{v}^{n+1}\end{aligned}\end{align} \]

This is a first-order symplectic method that provides better energy conservation than explicit Euler integration.

class diffsim.solver.SemiImplicitSolver(dt=0.01, gravity=-9.8, damping=0.99, substeps=4, enable_self_collision=False, collision_method='simplified')[source]#

Bases: object

Semi-implicit (symplectic Euler) solver for dynamic FEM simulation

This solver implements a first-order symplectic integration scheme. Velocities are updated using forces at the current position, then positions are updated using the new velocities. This scheme is symplectic (preserves phase space volume) and provides better energy conservation than explicit methods.

The integration follows:

\[ \begin{align}\begin{aligned}\mathbf{v}^{n+1} &= \mathbf{v}^n + h \, \mathbf{M}^{-1} (\mathbf{f}_{\text{elastic}}(\mathbf{x}^n) + \mathbf{f}_{\text{gravity}} + \mathbf{f}_{\text{contact}})\\\mathbf{x}^{n+1} &= \mathbf{x}^n + h \, \mathbf{v}^{n+1}\end{aligned}\end{align} \]

where \(h = \Delta t / \text{substeps}\) is the substep size.

dt#

Time step size in seconds

Type:

float

gravity_value#

Gravity acceleration in m/s² (negative for downward)

Type:

float

damping#

Velocity damping coefficient (0-1, typically ~0.99)

Type:

float

substeps#

Number of substeps per timestep for stability

Type:

int

enable_self_collision#

Whether to compute self-collision forces

Type:

bool

collision_method#

Collision detection method (‘simplified’ or ‘ipc’)

Type:

str

__init__(dt=0.01, gravity=-9.8, damping=0.99, substeps=4, enable_self_collision=False, collision_method='simplified')[source]#

Initialize solver

Parameters:

  • dt – time step size

  • gravity – gravity acceleration (m/s^2)

  • damping – velocity damping factor

  • substeps – number of substeps per timestep

  • enable_self_collision – enable self-collision detection

  • collision_method – ‘simplified’ (fast) or ‘ipc’ (accurate but slower)

step(mesh, material, positions, velocities, masses, fixed_vertices=None)[source]#

Perform one semi-implicit time step with stability controls

Parameters:

  • mesh – TetrahedralMesh object

  • material – Material model (e.g., StableNeoHookean)

  • positions – \((N, 3)\) current positions

  • velocities – \((N, 3)\) current velocities

  • masses – \((N,)\) vertex masses

  • fixed_vertices – list of fixed vertex indices

Returns:

\((N, 3)\) updated positions new_velocities: \((N, 3)\) updated velocities

Return type:

new_positions

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