Explicit Dynamic Finite Element Method for Failure with Smooth Fracture Energy Dissipations

Table 1

Flowchart for numerical computation procedures.

Steps	Numerical computation procedures
1	Apply initial conditions including initial boundary conditions and prestress
2	Compute initial forces: 𝐟0=𝐟^ext0−𝐟0^int
3	Compute acceleration: 𝐚𝑛=𝐌𝑛^−1𝐟𝑛
4	Time update: 𝑡𝑛+1=𝑡𝑛+Δ𝑡𝑛+(1/2)
5	First partial update for nodal velocities: 𝐯𝑛+(1/2)=𝐯𝑛+(𝑡𝑛+(1/2)−𝑡𝑛)𝐚𝑛
6	Apply velocity boundary conditions at 𝑡𝑛+(1/2)
7	Update nodal displacements: 𝐝𝑛+1=𝐝𝑛+Δ𝑡𝑛+(1/2)×𝐯𝑛+(1/2)
8	Compute strain and stress
9	Check fracture criterion and insert phantom nodes
10	Compute forces at time: 𝐟𝑛+1=𝐟^ext𝑛+1−𝐟^int𝑛+1+𝐟^coh𝑛+1+𝐟^correct𝑛+1
11	Compute acceleration: 𝐚𝑛+1=𝐌^−1𝑛+1𝐟𝑛+1
12	Second partial update for nodal velocities: 𝐯𝑛+1=𝐯𝑛+(1/2)+(𝑡𝑛+1−𝑡𝑛+(1/2))𝐚𝑛+1
13	Update counter n
14	Output: if simulation is not completed, go to Step 4