A typical vertex‑based crack‑updating simulation proceeds through the following loop:
| Step | Description | |------|-------------| | (1) Initialization | Define geometry, material properties, initial crack (set of vertices). | | (2) Solve Governing Equations | Finite‑element solution of balance equations (static or dynamic). | | (3) Post‑Processing – Crack Driving Force | Compute ( \mathcalG_i ) for each vertex using J‑integral, VCE, or cohesive traction. | | (4) Propagation Decision | Compare ( \mathcalG_i ) with ( \mathcalG_c ); mark active vertices. | | (5) Direction & Length Determination | Solve the local optimization to obtain ( \mathbfn_i, \Delta a_i ). | | (6) Vertex Update | Move active vertices using the update rule. | | (7) Mesh Adaptation | Perform local remeshing or enrich the FE space. | | (8) Convergence Check | If the crack has reached a termination condition (e.g., prescribed length, load drop, or simulation time), stop; otherwise return to (2). | vertex bd crack upd
A flowchart is shown below (textual representation): The next decade is likely to witness a
┌─────────────┐
│ Initialize │
└─────┬───────┘
▼
┌─────────────┐
│ Solve FE │
└─────┬───────┘
▼
┌─────────────┐
│ Compute G │
└─────┬───────┘
▼
┌─────────────┐
│ Check G>Gc │
└─────┬───────┘
Yes │ No
▼ ▼
Update ──► End
Vertices
└─────┬───────┘
▼
Local Remesh / Enrich
└─────┬───────┘
▼
Loop back to Solve FE
The next decade is likely to witness a convergence of three technological trends that will reshape vertex‑based crack updating: These advances will push vertex‑based crack updating from
These advances will push vertex‑based crack updating from a high‑fidelity research tool toward a predictive, operational technology in safety‑critical industries.
| Era | Key Development | Relevance to Vertex‑Based Methods | |-----|----------------|-----------------------------------| | 1970s‑80s | Cohesive Zone Models (CZM) and Linear Elastic Fracture Mechanics (LEFM) | Established the concept of tracking crack fronts via displacement or stress discontinuities. | | Early 1990s | Extended Finite Element Method (XFEM) | Introduced enrichment functions that allow cracks to cut through elements without remeshing, inspiring later vertex‑centric strategies. | | Late 1990s – early 2000s | Discrete Element and Lattice Models | Treated material as a network of interacting vertices, laying the groundwork for vertex‑based fracture formulations. | | Mid‑2000s | Vertex‑Based Crack Propagation (VBCP) | First explicit algorithms that updated the crack geometry by moving mesh vertices rather than re‑meshing whole elements. | | 2010s – present | Hybrid Phase‑Field / Vertex Approaches, GPU‑accelerated implementations | Integrated vertex updating with diffuse‑interface representations for superior scalability. |
The evolution from classical mesh‑dependent crack tracking to vertex‑centric updating reflects a broader trend: the desire to maintain mesh quality while capturing the inherently discrete nature of fracture.