Week - 2 - Explicit and Implicit Analysis

Aim

F(u) = u3+9u2+4u use this equation and solve using both Explicit and Implicit Methods

Objectives

problem is solved by both an explicit and implicit analysis. In both cases an incremental load control scheme is used.

Explicit method

Its a spatial discretization or we can say infinite to finite. Usually we say time discretization. This method has to solve the physical problem

that is provide.In explicit method we doesnt need the iterations ,its completely stable. It needs low memory and need only smaller time steps. In explicit, when use smaller or fine mesh ie, stable time is going up or computational time goes up.

Implicit method

in implicit method, find asolution by dealing an equation involvng both the current state of the system and the later one. In implicit we cannot apply the load in one step. need more time steps to complete. If we apply in one step it will lead to convergance.So we divide into smaller steps like pseudo steps.

Explicit and implicit analysis

Explicit method

F(u)=u^3+9U2+4U

K(u)= df/du=1+u^2

Apply 3 equal forces, ΔF=1 to load the tension bar.

ΔF=kΔu

Step 1

U0=0,ku=1+u^2 ΔF=1,Δu1=ΔF/k(u0)

KU0=1+0^2=1 =1/1=1

Step 2

Ku1=1+1^2=2 ΔF=1,ΔU2=ΔF/KU1=1/2

U2=U1+ΔU2=1+1/2=1.5

Step 3

KU2=1+U^2

KU2=1+1.5^2=3.25 ΔF=1,ΔU3=ΔF/KU2=1/3.25

U3=U2+ΔU3=1.5+1/3.25=1.81

F(external)=ΔF+ΔF+ΔF=3

F(internal)=U^3+9U2+4U=1.81^3+9*1.5+4*1=22.32

Implicit Analysis

R(u)=F int-F ext

Rj+1=Rj(uj)+dr(Uj)/du*Δuj+1+.......=Rj+K(uj)Δuj+1

Rj+1=0,ΔUj+1

Step1

U0=0, K(U0)=1, ΔF=1,ΔU1=ΔF/KU0, U1=U0+ΔU1=1