Improved numerical dissipation for time integration algorithms in structural dynamics
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...Twist also provides a standard implementation of a finite difference time-stepping algorithm that is commonly used in the computational mechanics community: the Hilber–Hughes–Taylor (HHT) method (Hilber et al., 1977)....
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...A fixed time step of ∆t = 1.25 × 10−8 was chosen, which is slightly less than 0.9∆tcrit, where the critical time step, ∆tcrit, is computed as ∆tcrit = Ωcrit ωmax (92) with ωmax the maximum natural frequency of the momentum equation determined from the undamped eigenproblem and (considering only the undamped case) HHT-α (see Miranda, Ferencz, and Hughes (1989)) Ωcrit = √ 2(γ + 2α(γ − β) γ + 2α(γ − β) (93) Explicit generalized-α (see Hulbert and Chung (1996)) Ωcrit = √ 12(1 + ρb)3(2− ρb) 10 + 15ρb − ρ2b + ρ3b − ρ4b ....
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...The HHT-α method, parameterized by α, provides a second-order accurate family of algorithms for linear second-order equations if α ∈ [− 13 , 0] and αm = 1, (82) αf = 1 + α, (83) β = (1− α)2 4 , (84) γ = 1− 2α 2 (85) (see Miranda, Ferencz, and Hughes (1989))....
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...The simulations were performed using the staggered integration scheme described in Section 3.3.2 with the momentum equation being solved explicitly using the HHT-α method with α = −0.3....
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...For the fully explicit case (M∗ = αmM̃) we use either the HHT-α of Hilber, Hughes, and Tayler (1977) or the explicit generalized-α method of Hulbert and Chung (1996)....
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...The simulations were performed using the staggered integration scheme described in Section 3.3.2 with the momentum equation being solved explicitly using the HHT-α method with α = −0.1....
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"Improved numerical dissipation for ..." refers background or methods in this paper
...Elaboration on these points and further properties of the Newmark family of algorithms may be found in [3L To analyze systems such as (1), or equivalently (3a), it is convenient to invoke the property of orthogonality of....
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...Mu + Ku = F (1) where M is the mass matrix, K is the....
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...for (1) consists of finding a function u = u(t), where t£[0,T], T > 0, satisfying (1) and the initial conditions: u(O)_ = (2) u<o> = where d and v are the given vectors of initial data....
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...We are interested in obtaining approximate solutions of (1) by one-step difference methods....
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...Employing the obvious notations, the singledegree-of-freedom analogs of (1) and (3a) - (3c) are: Mi....
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"Improved numerical dissipation for ..." refers background in this paper
...for (1) consists of finding a function u = u(t), where t£[0,T], T > 0, satisfying (1) and the initial conditions: u(O)_ = (2) u<o> = where d and v are the given vectors of initial data....
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