University:
Massachusetts Institute of Technology
Course:
16.225 | Computational Mechanics of Materials
Academic year:

2020
Views:

449

Pages:

9
Author:

Felipe Stafford

**147** **Connection between Newmark algorithm and multistep methods** **Ma + f(x) = fext** **①** Xn+1 = Xn + Δt Un + Δt² [(½-β) an + β an+1] **②** Un+1 = Un + Δt [(1-γ) an + γ an+1] **Write as a difference equation for "Xn":** Multiply M × ②: 0 = M (Xn+1 - Xn - Δt Un) + Δt² [(½-β) an + β an+1] **②n-1** Xn = Xn-1 + Δt Vn-1 + Δt² [(½-β) an-1 + β an] Subtract: 0 = M (Xn+1 - Xn - Δt Un + Δt Vn-1 - Xn) + Δt² [(½-β) (an+1 - an-1) + β (an+1 - an)] 0 = M (Xn+1 - 2Xn + Xn-1) + M (Un+1 - Un) + Δt² [(½-β) (an+1 - an-1) + β (an+1 - an)] Un+1 - Un = Δt [(1-γ) an + γ an+1] from ② M (Un+1 - Un) = Δt [(1-γ) M an+1 + γ M an] = Δt [(1-γ) (fnext - fint) + γ (fnext - fint)] **Replacing in the previous equation:** 0 = M (Xn+1 - 2Xn + Xn-1) - Δt [(1-γ) (fnext - fint) + γ (fnext - fint)] Δt² [(½-β) (an+1 - an-1) + β (an+1 - an)] 0 = M (Xn+1 - 2Xn + Xn-1) + (½-β-1+γ) (fnext - fint) + β (fnext - fint) 0 = M (Xn+1 - 2Xn + Xn-1) + (½-β-1+β) (fnext - fint) + (β-1) (fnext - fint)

Connection Between Newmark Algorithm and Multistep Methods

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