About the Use of the HdHr Algorithm Group in Integrating the Movement Equation with Nonlinear Terms

Algorithm 1

Steps for the HdHr algorithm in non-linear problems.

 1. Start

   Read 𝑥₀,𝑥^•₀,Δ𝑡,𝑡_𝑓,𝜉_lim

   Calculate ^•𝑥^•₀=[𝑓_𝑒𝑥𝑡(𝑥₀,𝑥^•₀,𝑡₀)−𝑘𝑥₀](1/𝑚)

 2. Advance in movement

              𝑡𝑛=^𝑓,Δ𝑡𝑖=0,𝑛

 2.1. Estimate of the (restorative) and external forces:

 2.1.1. Estimate of displacement in the 𝑖+1 instant

         𝑥_{𝑖+1,𝑒𝑠𝑡}=𝑥_𝑖+Δ𝑡𝑥^•_𝑖+Δ𝑡²2^••𝑥_𝑖,_•𝑥_{𝑖+1,𝑒𝑠𝑡}=𝑥^•_𝑖+Δ𝑡𝑥^••_𝑖.

 2.1.2. Estimate of the matrixes for the internal forces

           𝑘_𝑒𝑠𝑡=𝑘(𝑥_{𝑖+1,𝑒𝑠𝑡})

 2.1.3. Estimate of the external forces

   Calculate 𝑓_𝑒𝑥𝑡(𝑥_{𝑖+1,𝑒𝑥𝑡},𝑥^•_{𝑖+1,𝑒𝑠𝑡},𝑡_𝑖+1)

 2.2. Evaluation of approximate displacements and velocities

      {𝑥_{𝑖+1,𝑎𝑝}}=[𝑀_𝑒𝑠𝑡]⁻¹([𝑁]{𝑥_𝑖}+[𝐿]{𝑓_{𝑖+1,𝑒𝑠𝑡}}+[𝐿𝑖]{𝑓_𝑖})

 2.3. Evaluation of approximate acceleration

  {𝑥^••_{𝑖+1,𝑎𝑝}}=[𝑓_𝑒𝑥𝑡(𝑥_{𝑖+1,𝑎𝑝},𝑥^•_{𝑖+1,𝑎𝑝},𝑡_𝑖+1)−𝑘_𝑒𝑠𝑡𝑥_{𝑖+1,𝑎𝑝}]∗(1/𝑚)

 2.4. Updating of the internal force vector

           𝑘_𝑒𝑠𝑡=𝑘(𝑥_{𝑖+1,𝑎𝑝})

 2.5. Evaluation of the remainder

  𝑓_𝑒𝑥𝑡(𝑥_{𝑖+1,𝑎𝑝},𝑥^•_{𝑖+1,𝑎𝑝},𝑡_𝑖+1)−𝑘_𝑒𝑠𝑡𝑥_{𝑖+1,𝑎𝑝}−𝑚𝑥^••_{𝑖+1,𝑎𝑝}=𝜀(𝑥_{𝑖+1,𝑎𝑝},𝑥^•_{𝑖+1,𝑎𝑝})

 2.6. Remainder test

            if𝜀(𝑥_{𝑖+1,𝑎𝑝},𝑥^•_{𝑖+1,𝑎𝑝})>𝜀_lim:

             returns to 2.2.

            if𝜀(𝑥_{𝑖+1,𝑎𝑝},𝑥^•_{𝑖+1,𝑎𝑝})≤𝜀_lim:

            𝑥_𝑖+1⟵𝑥_{•𝑖+1,𝑎𝑝}𝑥_𝑖+1⟵𝑥^•_{𝑖+1,𝑎𝑝••}𝑥_𝑖+1⟵𝑥^••_{𝑖+1,𝑎𝑝}

  if𝑖<𝑛

             returns to 2.1.

  if𝑖=𝑛

End