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RestitutionWall

The RigidWall constraint essentially implements artificial springs to alter the trajectories of nodes. The physical implication is clear but the main issue is that the conservation of energy/momentum cannot be guaranteed.

The RestitutionWall constraint adopts a different approach to ensure the assigned restitution is satisfied. If the coefficient of restitution is set to unity, conservation of energy/momentum is guaranteed.

Syntax

The rigid wall constraints are single sided. Travelling against the outer normal direction is not allowed while the other direction is permitted.

1D

The 1D version takes the origin and the side of the wall as the inputs.

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! infinite rigid wall by penalty
restitutionwall (1) (2) (3) (4) [5]
constraint restitutionwall (1) (2) (3) (4) [5]
# (1) int, unique constraint tag
# (2) double, coordinate of origin of rigid wall
# (3) double, sign of normal direction +1 or -1
# (4) double, restitution coefficient
# [5] double, multiplier, default: 1E4

2D

The 2D version takes the origin and either the edge vector or the normal vector as the inputs.

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! infinite rigid wall by penalty
restitutionwall (1) (2...3) (4...5) (6) [7]
constraint restitutionwall (1) (2...3) (4...5) (6) [7]
# (1) int, unique constraint tag
# (2...3) double, coordinates of origin of rigid wall
# (4...5) double, vector of normal direction
# (6) double, restitution coefficient
# [7] double, multiplier, default: 1E4

! finite rigid wall by penalty
finiterestitutionwall (1) (2...3) (4...5) (6) [7]
constraint finiterestitutionwall (1) (2...3) (4...5) (6) [7]
# (1) int, unique constraint tag
# (2...3) double, coordinates of origin of rigid wall
# (4...5) double, vector of wall edge
# (6) double, restitution coefficient
# [7] double, multiplier, default: 1E4

3D

The 3D version takes the origin and the normal vector as the inputs. Alternatively, two edges can be specified to define a finite wall.

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! infinite rigid wall by penalty
restitutionwall (1) (2...4) (5...7) (8) [9]
constraint restitutionwall (1) (2...4) (5...7) (8) [9]
# (1) int, unique constraint tag
# (2...4) double, coordinates of origin of rigid wall
# (5...7) double, vector of normal direction
# (8) double, coefficient of restitution
# [9] double, multiplier, default: 1E4

! finite rigid wall by penalty
finiterestitutionwall (1) (2...4) (5...7) (8...10) (11) [12]
constraint finiterestitutionwall (1) (2...4) (5...7) (8...10) (11) [12]
# (1) int, unique constraint tag
# (2...4) double, coordinates of origin of rigid wall
# (5...7) double, vector of first edge
# (8...10) double, vector of second edge
# (11) double, coefficient of restitution
# [12] double, multiplier, default: 1E4

Assumptions

It is assumed that the collision occurs within a brief time. As a result, the change of acceleration is not reflected at either \(t_n\) or \(t_{n+1}\). The assumptions adopted are shown in the following figure.

assumptions

Example

See Bouncing of A Ball.

Another validation can be downloaded.

validation

The kinetic energy is conserved.

conservation