Iterative Solution Methods for Nonlinear Studies

Nonlinear Static Studies

In nonlinear static analysis, the basic set of equations to be solved at any “time” step, t+Δt, is:

^t+Δt{R} - ^t+Δt{F} = 0,

where:

^t+Δt{R} = Vector of externally applied nodal loads

^t+Δt{F} = Vector of internally generated nodal forces.

Since the internal nodal forces ^t+Δt{F} depend on nodal displacements at time t+Δt, ^t+Δt{U}, an iterative method must be used. The following equations represent the basic outline of an iterative scheme to solve the equilibrium equations at a certain time step, t+Δt,

{ΔR}^(i-1) = ^t+Δt{R} - ^t+Δt{F}^(i-1)

^t+Δt[K]⁽ⁱ⁾ {ΔU}⁽ⁱ⁾ = {ΔR}^(i-1)

^t+Δt{U}⁽ⁱ⁾ = ^t+Δt{U}^(i-1) + {ΔU}⁽ⁱ⁾

^t+Δt{U}⁽⁰⁾ = ^t{U}; ^t+Δt{F}⁽⁰⁾ = ^t{F}

where:

^t+Δt{R} = Vector of externally applied nodal loads

^t+Δt{F}^(i-1) = Vector of internally generated nodal forces at iteration (i)

{ΔR}^(i-1) = The out-of-balance load vector at iteration (i)

{ΔU}⁽ⁱ⁾ = Vector of incremental nodal displacements at iteration (i)

^t+Δt{U}⁽ⁱ⁾ = Vector of total displacements at iteration (i)

^t+Δt[K]⁽ⁱ⁾ = The Jacobian (tangent stiffness) matrix at iteration (i).

There are different schemes to perform the above iterations. A brief description of two methods of the Newton type are presented below: