Explicit and Implicit Analysis
Explicit and Implicit methods
1.Explicit
An Explicit FEM analysis does the incremental procedure and at the end of each increment updates the stiffness matrix based on geometry changes (if applicable) and material changes (if applicable). Then a new stiffness matrix is constructed and the next increment of load (or displacement) is applied to the system. In this type of analysis the hope is that if the increments are small enough the results will be accurate. One problem with this method is that you do need many small increments for good accuracy and it is time consuming. If the number of increments are not sufficient the solution tends to drift from the correct solution. Futhermore this type of analysis cannot solve some problems. Unless it is quite sophisticated it will not successfully do cyclic loading and will not handle problems of snap through or snap back. Perhaps most importantly, this method does not enforce equilibrium of the internal structure forces with the externally applied loads.
2.Implicit
An Implicit FEM analysis is the same as Explicit with the addition that after each increment the analysis does Newton-Raphson iterations to enforce equilibrium of the internal structure forces with the externally applied loads. The equilibirium is usually enforced to some user specified tolerance. So this is the primary difference between the two types of anlysis, Implicit uses Newton-Raphson iterations to enforce equilibrium. This type of analysis tends to be more accurate and can take somewhat bigger increment steps. Also, this type of analysis can handle problems better such as cyclic loading, snap through, and snap back so long as sophisticated control methods such as arc length control or generalized displacement control are used. One draw back of the method is that during the Newton-Raphson iterations one must update and reconstruct the stiffness matrix for each iteration. This can be computationally costly. (As a result there are other techniques that try to avoid this cost by using Modified Newton-Raphson methods.) If done correctly the Newton-Raphson iterations will have a quadratic rate of convergence which is very desireable.
Problem-
F(u) = u3+9u2+4u use this equation and solve using both Explicit and Implicit Methods ( have a tolerence of 10-2).
Problem solution
F(u) = u3+9u2+4u use this equation and solve using both Explicit and Implicit Methods ( have a tolerance of 10-2.
1.Solving with Explicit Method-
`F(u)=u^3+9u^2+4u`
Differentiating force with respect to displacement.
`k(u)=(dF)/(du)=3u^2+18u+4`
Here `DeltaF`=1 for used for every step
Stiffness-force relation used to find incremental displacement-
`DeltaF=k.Deltau`
Step-1-
`u0=0, k(u0)= 4, DeltaF=1, Delta u1= (DeltaF)/(k(u0))=0.25, u1= u0+Delta u1= 0.25`
`Fext (DeltaF)= 1`
`Fint= u^3+9u^2+4u= 1.5781` at u=0.2
Step-2-
`u1=0.25, k(u1)= 8.688, DeltaF=1, Delta u2= (DeltaF)/(k(u1))=0.115, u2= u1+Delta u2= 0.3651`
`Fext (DeltaF)= DeltaF+DeltaF=1+1=2`
`Fint= u^3+9u^2+4u= 2.708` at u= 0.3651
Step-3-
`u2=0.3651, k(u2)= 10.98, DeltaF=1, Delta u3= (DeltaF)/(k(u2))=0.0911, u3= u2+Delta u3= 0.4562`
`Fexternal (DeltaF)= DeltaF+DeltaF+DeltaF=1+1+1=3`
`Fternal= u^3+9u^2+4u= 3.7929` at u= 0.4562
Implicit method
Implicit analysis is similar to explicit analysis but we will use the Newton-Raphson iteration for each step before moving to the next step for 10^-2 places. These changes made to reduce the error in solution and make the forces close to equilibrium, By minimization of Residue R= Finternal-Fexternal.
For this Residue will expand in the form of Taylors Series with neglecting the higher-order,
`deltau=-[k(u)]^-1.R`
Step-1-
`u0=0, k(u0)= 4, DeltaF=1, Delta u1= (DeltaF)/(k(u0))=0.25, u1= u0+Delta u1= 0.25`
`deltau^(1)=-[k(u^(0))]^-1.R^(
the updated value= 0.25-0.0665= 0.1835
R1= 0.0432
`deltau^(2)=delta(u)^(1)-[k(u^
`u1^(2)= u1^(0)+deltau^(2)=0.25-0.0723=
Again Iteration with this we can get the value of minimum error= 0.00059
From this value, we can get the value of u1= 0.177
Step-2- and Step-3 can be calculated by the same procedure,
