Week - 2 - Explicit and Implicit Analysis
EXPLICIT METHOD:
The explicit method was originally developed, and is primarily used, to solve dynamic problems involving deformable bodies. The finite element equations in the explicit approach are reformulated as being dynamic and in this form they can be solved directly to determine the solution at the end of the increment, without iteration. Accelerations and velocities at a particular point in time are assumed to be constant during a time increment and are used to solve for the next point in time.
In explicit method, There are no convergence criteria to check and no iterations required, instead of verifying the “global equilibrium” solver assumes that the equilibrium “simply exists”. This means that the solver goes straight into calculation of local finite element variables. Due to this each increment calculation is fast. But this comes at a cost of small time steps required to solve the problem. This time step is guided by Courant's equation which depends on on material and speed of sound in the material (the time taken by a sound wave to travel across an element).
The explicit method should be used when the strain rates/velocity is over 10 units/second or 10 m/s respectively. These events can be best exampled by extreme scenarios such as an automotive crash, ballistic event, or even meteor impact. In these cases, the material models do not only need to account for the variation of stress with strain but also the strain rate. On this scale, the strain rates play a particularly important contribution.
IMPLICIT METHOD:
In the implicit approach a solution to the set of finite element equations involves iteration until a convergence criterion is satisfied for each increment. The word ‘implicit’ refers to the method by which the state of a finite element model is updated from time t to t + Δt. A fully implicit procedure means that the state at t + Δt is determined based on information at time t + Δt, while the explicit method solves for t + Δt based on information at time t.
In implicit analysis, in each time increment the “global equilibrium” in the model is established. This means that each increment has to converge (this is happening in iterations). After global equilibrium is ok, solver calculates all the local finite element variables (stresses etc.) for this increment. Since global equilibrium is verified at each time increments, those increments can be large. Each time increment computes slowly, since iterations are needed to get to the global equilibrium. The accuracy of the solution is dictated by the convergence criterion where the updated value for variable under study must be less than a tolerance value.
In static problems, an implicit solver requires the formation of a global stiffness matrix [K]. The equilibrium is maintained when [K]−1 is multiplied to both sides of the static equillibrium. After that, nodal displacements can be easily calculated. One disadvantage of carrying out this operation is that large sizes of storage space and in-core memory are needed to find inverse of K, [K]−1. Additionally, nonlinearity causes [K] to become a function of displacement. As such, updating or factorizing the global stiffness [K] before inverting it to [K]−1 for computing the nodal displacements needed before the next time step can be proceed is computationally intensive.
These solutions are unconditionally stable and facilitate larger time steps. Despite this advantage, the implicit methods can be extremely time-consuming when solving dynamic and nonlinear problems.
Solution for the given problem is in an attached excel sheet
CONCLUSION
Explicit and implicit analysis is done for the given equation, implicit method is more accurate than the explicit but the time consumption is more in implicit method.