Explicit and Implicit Analysis
AIM: To solve using both Explicit and Implicit Methods for the Given Equation F(u) = `u^3+9u^2+4u`. have a tolerence of `10^-2`
EXPLICT AND IMPLICT ANALYSIS
There are two types of approach here IMPLICT & EXPLICT, this used to solve the Finite element analysis problems. Implicit and Explicit analysis differ in the approach to time incrementation.In Implicit analysis each time increment has to converge for [eg.,Static structural, Harmonic , Modal analysis etc..,] but you can set pretty long time increments. Explicit on the other hand doesn’t have to converge each increment, but for the solution to be accurate time increments must be super small. for [eg.,crash, blast, Imact].
EXPLICT:
In explicit dynamic analysis, nodal accelerations are solved directly (not iteratively) as the inverse of the diagonal mass matrix times the net nodal force vector where net nodal force includes contributions from exterior sources (body forces, applied pressure, contact, etc.), element stress, damping, bulk viscosity, and hourglass control. Once accelerations are known at time n, velocities are calculated at time n+1/2, and displacements at time n+1. From displacements comes strain. From strain comes stress. And the cycle is repeated. In explicit analysis, no iteration is required as the nodal accelerations are solved directly.
In nonlinear problems—those with large deformations and/or nonlinear material response—the highest frequency of the model will continually change, which consequently changes the stability limit. Explicit has two strategies for time incrementation control,fully automatic time incrementation (where the code accounts for changes in the stability limit) and fixed time incrementation.
LS DYNA is one of the software which used to solve the EXPLICT analysis problem for CRASh,impact,Penetration etc..,
ABAQUS,PAMCRASH also used for explict analysic problems.
IMPLICT:
In implict analysis, the displacement is not a function of time (i.e.X=constant), hence the velocity and accelerations which are time derivations of displacement turn out to be Zero, mass and Damping factorscan be negleted, the impilict method based on Newark's method, Newton Raphson Method etc., In order to solve an FEM problem using Implict Method, Inversion of Stifness Matrix is Required. Static analysis is done using an implicit solver in LS-DYNA. Dynamic analysis can be done via the explicit solver or the implicit solver.
In nonlinear implicit analysis, solution of each step requires a series of trial solutions (iterations) to establish equilibrium within a certain tolerance,Implicit analysis requires a numerical solver to invert the stiffness matrix once or even several times over the course of a load/time step. This matrix inversion is an expensive operation, especially for large models.
Implict methods are mainly used in softwares like ANSYS,ABACUS,NASTRAN etc..,
GIVEN EQUATION:
`F(u)`=` u^3+9u^2+4u` -------- (1)
use this equation and solve using both Explicit and Implicit Methods, have a tolerence of `10^-2`.
STIFNESS of the bar
K(u)`=`df/du`=``3u^2+18u+4` --------(2)
EXPLICT ANALYSIS:
STEP 1:
TAKE U=0.0 --------(2)
`K(U)=3(0)^3+18(0)^2+4(0) =4`
now,
`Δu1=` Δf/k(u0) `=1/4 =0.25`
`U1=U0+Δu1=0.25`
STEP 2:
TAKE U=0.25 --------(2)
`K(U)=3(0.25)^3+18(0.25)^2+4(0.25) =0.8675`
now,
`Δu2=1/0.8675=0.11510`
`U2=U1+ΔU2=0.1150+0.25=0.3651`
STEP 3:
TAKE U=0.3651 --------(2)
`K(U)=3(0.3651)^3+18(0.3651)^2+4(0.3651) =10.9716`
`Δu3=1/10.9716=0.9176`
`U3=U2+ΔU2=0.1150+0.25=0.4561`
FINALLY, Determine the analysis in equlibrium
`F(ext)=ΔF+ΔF+ΔF=3`
F(int)=`(1)=(0.46)^3+(9*0.46)^2+4*0.46 =3.841`
F(int)`≠`F(int)
It is the evident from the results and inertial forces are not in Equlibrium
SPREADSHEET CALCULATION WITH GRAPH:
FIG.1 SPREAD SHEET CALCULATIONS
Following table summary of Explict analysis
| STEP | ΔFi | ΔUi | Ui | F(ext)i | F(int)i | F(int)i-F(ext)i=R |
| 1 | 1 | 0.25 | 0.25 | 1 | 1.578 | 0.578 |
| 2 | 1 | 0.1151 | 0.3651 | 2 | 2.708 | 0.708 |
| 3 | 1 | 0.09114 | 0.46 | 3 | 3.841 | 0.841 |
TABLE.1 SUMMARY FOR EXPLICT ANALYSIS
EXPLICT ANALYSIS (FORCE VS DISPLACEMENT) GRAPH:
FIG.2 GRAPH FOR EXPLICT ANALYSIS
The above plot shows the force vs displacement for explict analysis
IMPLICT ANALYSIS:
`F(U)= U^3+9U^2+4U` --------(1)
DIFFERENTIATE THE EQUATION
K=df/du`=3u^3=18^u+4` --------(2)
STEP:1
TAKE U=0.0 --------(2)
`K(U)=3(0)^3+18(0)^2+4(0) =4`
now,
`Δu1= Δf/k(u0) =1/4 =0.25`
`U1=U0+Δu1=0.25`
Checking Residual: R
F(int)`=1=(0.25)^3+9(0.25)^2+4(0.25) =1.578`
`R0=1.578-1=0.578 > 10^-2`
so, Newton Raphson Iterations are necessary
U1=U1(0)
`we know that, ∫u1=-[k(u1)(o)]^-1*Ro=[3*0.25^2+18*0.25+4]`
UPDATED `U1(1)=u1(0)+∫u(1)=0.25-0.0665=0.1835`
Checking Residual: R1
`F(ext)=1`,F(int)`=(0.1835)^3+(9X0.1835)^2+(4*0.1835)=1.0432`
`R1=1.0432-1=0.0432>10^-2` another-Raphson iteration needed
clc.the correction to U1=U1(1)
`∫u(2)=∫u(1)-[k(u1(1)]^-1*R1=0.0665-(8.6875)^-1*0.0432=-0.0714`
`updated,U1(2)=u1(1)+∫u(2)=0.25-0.0714=0.1786`
checking Residual again R2
`F(ext)=1`, F(int)`=(1)=(0.1786)^3+(9*0.1786^2)+4*0.1786)=1.0071`
`R2=1.0071-1=0.0071<10^-2`
Therefore no Iterations needed, final U1=0.1786
SPREADSHEET CALCULATION WITH GRAPH:
FIG.3 SPREADSHEET CALCULATIONS IMPLICT ANALUSIS
Step 2&3:
similiarly as step 1, we get `u2=0.2966 , u3=0.3911`
| STEP | ΔFi | ΔUi | Ui | F(ext)i | F(int)i | F(int)i-F(ext)i=R |
| 1 | 1 | 0.25 | 0.1786 | 1 | 1.0071 | 0.0071 |
| 2 | 1 | 0.1374 | 0.2966 | 2 | 2.0042 | 0.0042 |
| 3 | 1 | 0.1041 | 0.3911 | 3 | 3.0008 | 0.0008 |
TABLE 2. SUMMARY FOR IMPLICT ANALYSIS
IMPLICT ANALYSIS (FORCE VS DISPLACEMENT) GRAPH:
FIG.4 GRAPH FOR EXPLICT ANALYSIS
The above plot shows the force vs displacement for implict analysis
EXACT VALUES :
Take F(u)`=u^3+9u^2+4U=1`
we get using calculator U=0.1776
Take F(u)`=u^3+9u^2+4U=2`
we get using calculator U=0.2962
Take F(u)`=u^3+9u^2+4U=3`
we get using calculator U=0.3910
COMPARISON GRAPH EXPLICT VS IMPLICT:
FIG.5 DIFFERENCE BETWEEN EXPLICT AND IMPLICT
FIG.6 DIFFERENCE BETWEEN EXPLICT AND IMPLICT PLOT
THE IMPLICT AND EXACT ARE IN SAME LINE, WHILE EXPLICT IS AWAY FROM THE LINE
CONCLUSION:
The implict and Explict numerical Analysis is carried out from Given Equation.
From the Graphs we defines the Implict are more accurate but Time Consuming depends on element size, Whereas explict Deviates Slightly From Exact Results and needs More Iterations To Obtain Good Results