DyMESH Collision Model¶
This chapter describes the technical model implemented by DyMESH: the mesh representation, the contact (Location) and restoration algorithm, the node force–deflection law, the three-dimensional stiffness, the vertex-displacement (crush-depth) treatment, restitution, and the coupling to the vehicle equations of motion. The description follows the DyMESH Collision Simulation deck, updated to match the current version of HVE.
1. Mesh representation¶
Each body is represented by the same triangular surface mesh used for its display: an array of vertices and an array of triangular polygons.
- Each polygon carries its three vertex indices, an earth-fixed outward normal, an earth-fixed center, and its current and original area.
- Each vertex carries undamaged and current (damaged) coordinates in both the earth-fixed and vehicle-fixed frames, its current deformation vector, its accumulated maximum deformation, the force acting on it, and a per-vertex stiffness set and saturation crush.
During a run, one body is designated the Primary (P) and the other the Secondary (S). Primary surfaces (polygons) are tested against Secondary nodes (vertices): the algorithm finds, for each secondary node, the primary surface it has penetrated. The roles can be swapped every time step, or multiple times per time step, so that both bodies accumulate damage.
Because the search over all polygons would be expensive, DyMESH restricts each test to a search box around each secondary vertex, and to precomputed lists of neighboring surfaces. The neighbor lists are only recomputed every few time steps, for efficiency. The box size is either computed automatically from the vehicle extents or set by the user (Search Option: Automatic or Set Box Size).
2. Contact algorithm¶
DyMESH uses a contact algorithm similar to that used in explicit, dynamic finite-element simulation codes. The contact algorithm has two main functions:
- Location — determining mesh inter-penetration (which secondary nodes are inside which primary surfaces).
- Restoration — altering the positions of the nodes of the mesh to enforce the no-penetration condition.
The sequence over a contact is:
- The bodies approach one another.
- Location phase: the meshes inter-penetrate; the region of overlap is identified.
- Restoration phase: the meshes are adjusted (nodes pushed back to the contact surface).
- Forces are applied due to the deformations; these forces produce accelerations that separate the bodies.
Figure: three-panel contact-algorithm example — (1) bodies approach, (2) Location phase overlap box, (3) Restoration phase adjustment.
Figure: force application and separation — deformation forces at overlapping nodes drive the bodies apart.
Locating the primary surface for a secondary node¶
A secondary node is matched to the primary surface it has penetrated. A candidate contact point is the intersection of a search ray from the secondary node with the primary triangle; an inside-polygon test, using the cross-product method, confirms the intersection lies within the triangle. The direction of the search ray can be chosen among several methods — the local average normal, the velocity direction, the polygon normal, or a specified direction (a vertex-normal method also exists but is disabled) — and the local search may stop after the first, best-of-two, or best-of-all candidates. These are exposed on the Advanced tab of the DyMESH Options dialog.
Restoration (pushback)¶
Once a secondary node is found interior to a primary surface, its penetration depth is the distance from the node to the primary surface along the restoration direction. The node is restored (pushed back) toward the surface; the amount of that pushback, accumulated over time, is the node's permanent deformation. To prevent numerical instabilities, the pushback can be reduced by a limited amount, and is constrained not to restore across implausible distances.
(updated: the pushback reduction is only active when DyMESH Version 4 is selected. Under Version 3 the reduction is not applied.)
3. Node force–deflection law¶
A central design point is that any force-deflection relationship can be used; the force–deflection behavior is prescribed a priori per surface. The standard \(A\) and \(B\) crush coefficients prescribe the three-dimensional force-deflection relationship. DyMESH Version 3 extends the force-deflection relationship to a third-order polynomial by adding \(C\) and \(D\) coefficients together with a saturation deflection \(\delta_\mathrm{sat}\).
The force on a single node is computed as follows. Let \(\delta\) be the node's current total deflection (penetration depth), \(A_v\) be the area associated with the node, and \(\delta_0\) be a small null-band offset (0.5 in) below which only the constant term acts. For increasing penetration (\(\delta > \delta_\mathrm{max}\)) the node force per unit area follows a cubic:
where \(A\) (constant, lb/in²), \(B\) (linear, lb/in³), \(C\) (quadratic, lb/in⁴), and \(D\) (cubic, lb/in⁵) are the four per-node stiffnesses. Below the null band the force is simply \(F = A\,A_v\). Once the deflection reaches the node's saturation crush \(\delta_\mathrm{sat}\), the force is held at its saturation value and does not increase further:
Figure: Force vs. Deflection curve rising then flattening at the saturation deflection \(\delta_\mathrm{sat}\).
The stiffnesses are assigned per body surface — front, right, back, left, top and bottom — so different faces of a vehicle can have different crush behavior.
4. Three-dimensional (per-area) force-deflection¶
The published crush coefficients (\(A\), \(B\)) are two-dimensional — force per unit crush width. DyMESH needs force per unit crush area. The 3-D stiffness coefficients are obtained by dividing the standard (2-D) coefficients by the conversion height \(H\) of the crush region that was used to derive the 2-D coefficients:
Here \(A\) has units of force per width (lb/in) and \(\tilde{A}\) has units of force per area (lb/in²); likewise \(B\) (lb/in²) becomes \(\tilde{B}\) (lb/in³). The default conversion height is 30 in, matching the Crush Height For Stiffness dialog default.
Figure: two force-deflection plots — Force per Width (lb/in) vs. Permanent Crush with intercept \(A\) and slope \(B\); and Force per Area (lb/in²) vs. Permanent Crush with \(\tilde{A}\), \(\tilde{B}\) and unloading slope \(K_u\).
3-D stiffness dialog¶
The 3-D stiffness dialog lets the user enter \(A\), \(B\), \(C\) and \(D\) coefficients and a Saturation Crush, and to "tune" the coefficients using the Conversion Height to arrive at accurate 3-D stiffnesses. The dialog fields are: \(K_v\) (lb/in³, unloading), Constant \(A\) (lb/in²), Linear \(B\) (lb/in³), Quadratic \(C\) (lb/in⁴), Cubic \(D\) (lb/in⁵), Saturation Crush (in), Maximum Crush (in), and Conversion Height (in, default 30). The user can now define their own third-order curve fit.
Figure: "Vehicle Exterior Stiffness" dialog showing the Force-per-area vs. Crush plot (A,B,C,D curve vs. \(K_v\) line) and the coefficient sliders.
Note on occupant response¶
The selection of stiffness coefficients is important for simulations involving occupant response: two acceleration pulses (e.g. a tall triangular pulse and a lower square pulse) can share the same velocity change \(\Delta v\) yet represent very different loading severities for the occupant. Choosing physically representative stiffness/saturation values therefore matters beyond matching the gross \(\Delta v\).
Figure: Acceleration vs. Time — a triangular pulse and a rectangular pulse with equal area (equal \(\Delta v\)).
5. Vertex displacement (crush depth) — sharing crush between bodies¶
The 1/18 "Crush Depth" update refined how the total penetration is partitioned between the two vehicles so that the forces on the two bodies are equal and opposite (Newton's third law) even when their stiffnesses differ.
Given the primary vehicle stiffness \(A_\mathrm{primary}, B_\mathrm{primary}\), the secondary vehicle stiffness \(A_\mathrm{secondary}, B_\mathrm{secondary}\), and the total secondary vertex displacement \(\delta_\mathrm{total}\) (the total penetration — always computed using the secondary vehicle), the force each vehicle would carry at the full penetration is:
To equalize the forces, define the sharing fraction
and assign the secondary vehicle its share of the crush:
There is no need to compute \(\delta_\mathrm{primary}\) separately — the primary's share is the remainder, \(\delta_\mathrm{total}(1-\theta)\). The softer body receives the larger share of crush, and the contact force is consistent for both bodies.
(updated: the primary triangle's effective stiffness used at a contact is the average of the \(A\) and \(B\) stiffnesses of its three corner vertices, consistent with this force-equalization approach.)
6. Restitution (unloading)¶
After peak penetration the node unloads along an unloading branch. Unloading is governed by a restitution coefficient \(e\): a node only carries force while its current deflection exceeds \((1-e)\,\delta_\mathrm{max}\), i.e. force goes to zero once the node has relaxed by the restitution fraction of its maximum deflection:
On the rebound branch the force is evaluated at the rebound deformation \(\delta_r = \delta - (1-e)\,\delta_\mathrm{max}\) using the same cubic law as Eq. 1 (with \(\delta_r\) in place of \(\delta-\delta_0\)), while the stored maximum deflection \(\delta_\mathrm{max}\) is not updated. The unloading state of each node is tracked through a small sequence of states — cannot unload, beginning to unload, able to unload, now unloading, and finished unloading — governed by deflection thresholds and by a limit on how much the unloading can grow in a single time step. A separately stored unloading slope \(K_u\) defines the steeper unloading line (see the \(K_u\) line in the force-per-area plot).
DyMESH Version 2 "greatly improved" the restitution model together with the accelerations and the crush-depth (general damage profile) simulation.
7. Friction¶
Tangential (frictional) forces at the contact are computed from a material friction coefficient \(\mu\) — the Inter-vehicle Friction value for vehicle-vehicle interactions, and the wheel material's own friction value for wheels — with a friction null band that suppresses spurious tangential force at very low relative sliding velocity.
8. Coupling to the equations of motion¶
For each interacting pair, DyMESH sums the node forces and moments into a resultant force and moment on each body's center of gravity: the earth-fixed node forces are accumulated into a vehicle-fixed summed force and a summed moment about the CG. These collision forces and moments are then added to the vehicle's other external forces (tires, drag, gravity, suspension) and the six-degree-of-freedom equations of motion are integrated by the host model (SIMON) over the collision time step. Because DyMESH is called every collision time step and the meshes are advanced with the vehicle motion, the collision force develops continuously through the contact rather than as a single impulse.
The vehicle orientation used to transform between the vehicle-fixed and earth-fixed frames is the standard Euler-angle direction-cosine matrix \(A\), built from roll \(\phi\), pitch \(\theta\), and yaw \(\psi\).
Source: DyMESH Collision Simulation (2026 HVE Forum). Updated to match the current version of HVE.
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