The time integration method currently used for dynamic analysis is the explicit central difference method. It was initially chosen because it is well known, easy to implement and seemed to work reasonably well.
Although implicit methods haven't yet been tested, it is likely that they would perform better for this type of problem based on the tradeoffs of explicit vs implicit and the usual selection criteria given in literature. The ADINA Theory and Modelling Guide, Volume I (December 2012), for example, states the following:
7.5 Choosing between implicit and explicit formulations
• The main criterion governing the selection of the implicit or explicit formulations is the time scale of the solution.
• The implicit method can use much larger time steps since it is unconditionally stable. However, it involves the assembly and
solution of a system of equations, and it is iterative. Therefore, the computational time per load step is relatively high. The explicit method uses much smaller time steps since it is conditionally stable, meaning that the time step for the solution has to be less than a certain critical time step, which depends on the smallest element size and the material properties. However, it involves no matrix solution and is non-iterative. Therefore, the computational time per load step is relatively low.
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• For low-speed dynamic problems, the solution time spans a period of time considerably longer than the time it takes the wave to propagate through an element. The solution in this case is dominated by the lower frequencies of the structure. This class of problems covers most structural dynamics problems, certain metal forming problems, crush analysis, earthquake response and biomedical problems. When the explicit method is used for such problems the resulting number of time steps will be excessive, unless mass-scaling is applied, or the loads are artificially applied over a shorter time frame. No such modifications are needed in the implicit method. Hence, the implicit method is the optimal choice.
• For high-speed dynamic problems, the solution time is comparable to the time required for the wave to propagate through the structure. This class of problems covers most wave propagation problems, explosives problems, and high-speed impact problems. For these problems, the number of steps required with the explicit method is not excessive. If the implicit method uses a similar time step it will be much slower and if it uses a much larger time step it will introduce other solution errors since it will not be capturing the pertinent features of the solution (but it will remain stable). Hence, the explicit method is the optimal choice.
Additionally, the following points about the bow model (including planned changes) favor implicit methods
Damping reduces the maximum stable time step of the central difference method. This is already a problem when (unrealistically) high damping values are chosen.
Longitudinal (high frequency) vs bending motion (low frequency) of the limb
High frequencies from penalty coefficients
Non-uniform sizing of limb elements (#212)
Switch to the absolute nodal coordinate formulation (#189)
Higher order/more accurate elements -> evaluation (possibly) more expensive
Consistent (non-diagonal) mass matrix, although still constant
Requirements
Adaptive time stepping
It's not yet clear if this is a strict requirement or a "nice to have". Does the optimal time step actually vary much over the course of the simulation? If so, an adaptive step size could make the simulation more efficient. Another advantage would be that it is easier to find good numerical default settings that work for most users (e.g. a sensible error bound instead of a fixed time step).
Another alternative would be to estimate the timestep once before the simulation, as it is done now.
Form of the damping terms
This is not yet clear either. Most method descriptions assume a constant damping matrix, even for otherwise nonlinear problems. A decision needs to be made how damping is handled in the future before a method can be implemented.
Fully nonlinear damping terms (possible with the current method)
Rayleigh damping with the initial tangent stiffness matrix
Rayleigh damping with the current tangent stiffness matrix
Motivation
The time integration method currently used for dynamic analysis is the explicit central difference method. It was initially chosen because it is well known, easy to implement and seemed to work reasonably well.
Although implicit methods haven't yet been tested, it is likely that they would perform better for this type of problem based on the tradeoffs of explicit vs implicit and the usual selection criteria given in literature. The ADINA Theory and Modelling Guide, Volume I (December 2012), for example, states the following:
Additionally, the following points about the bow model (including planned changes) favor implicit methods
Damping reduces the maximum stable time step of the central difference method. This is already a problem when (unrealistically) high damping values are chosen.
Large range of stiffness/frequency in the system (Wikipedia: Stiff equation)
Switch to the absolute nodal coordinate formulation (#189)
Requirements
Adaptive time stepping
It's not yet clear if this is a strict requirement or a "nice to have". Does the optimal time step actually vary much over the course of the simulation? If so, an adaptive step size could make the simulation more efficient. Another advantage would be that it is easier to find good numerical default settings that work for most users (e.g. a sensible error bound instead of a fixed time step).
Another alternative would be to estimate the timestep once before the simulation, as it is done now.
Form of the damping terms
This is not yet clear either. Most method descriptions assume a constant damping matrix, even for otherwise nonlinear problems. A decision needs to be made how damping is handled in the future before a method can be implemented.
Literature Review
K.J. Bathe: Finite Element Procedures
M.A. Chrisfield: Non-linear Finite Element Analysis of Solids and Structures, Volume 2
P. Wriggers: Nichtlineare Finite-Element-Methoden
Interesting links
https://ckadapa.wordpress.com/2017/06/23/newmark-beta-method-disadvantages/
https://ethz.ch/content/dam/ethz/special-interest/baug/ibk/structural-mechanics-dam/education/femII/presentation_05_dynamics_v3.pdf
https://people.duke.edu/~hpgavin/StructuralDynamics/NumericalIntegration.pdf
Generalized alpha method: https://www.youtube.com/watch?v=wUsTSm-DY1g
Software Review
Overview of implicit integration methods used in major FEM software packages
Generalized HHT-α
Euler backwards
Wilson-θ
Bathe
HHT-α
Krenk
Wilson-θ
Houbolt
Single Step Houboldt
Generalized Alpha