Probabilistic and Fuzzy Arithmetic Approaches for the Treatment of Uncertainties in the Installation of Torpedo Piles

Algorithm 1

Solution procedure in the time domain.

(a) Initial calculations

       1. Calculate the torpedo mass M.

       2. Initialize 𝑑₀,𝑣₀,𝑓₀, and 𝑎₀, where 𝑎₀=𝑓₀/𝑀.

       3. Select the appropriate algorithmic parameter 𝛽 and time step Δ𝑡,

      Δ𝑡≤Δ𝑡_𝑐 and calculate the integration constants,

      𝛽₁=Δ𝑡²(12−𝛽),𝛽₂=Δ𝑡²𝛽,𝛾₁1=−2Δ𝑡,𝛾₂=32Δ𝑡.

(b) For each time step (𝑛=0,1,…,𝑁−1).

       1. Calculate the forces,

      𝑓_𝑛=𝑊_𝑠−𝐹_𝐷−𝐹_𝑇−𝐹_𝑆.

       2. Calculate the acceleration at time 𝑡_𝑛+1=𝑡_𝑛+Δ𝑡,

      𝑎_𝑛+1=𝑓_𝑛/𝑀.

       3. Calculate the displacement at time 𝑡_𝑛+1=𝑡_𝑛+Δ𝑡,

      𝑑_𝑛+1=𝑑_𝑛+Δ𝑡𝑣_𝑛+𝛽₁𝑎_𝑛+𝛽₂𝑎_𝑛+1.

       4. Calculate the velocity at time 𝑡_𝑛+1=𝑡_𝑛+Δ𝑡,

      𝑣_𝑛+1=𝑣_𝑛+𝛾₁𝑎_𝑛+𝛾₂𝑎_𝑛+1.

       5. 𝑛←𝑛+1, go to (1) of step (b).