# 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:

## Iterative Solution Methods - Newton-Raphson (NR) Scheme

In this scheme, the tangential stiffness matrix is formed and decomposed at each iteration within a particular step as shown in the figure below. The NR method has a high convergence rate and its rate of convergence is quadratic. However, since the tangential stiffness is formed and decomposed at each iteration, which can be prohibitively expensive for large models, it may be advantageous to use another iterative method.

## Iterative Solution Methods - Modified Newton-Raphson (MNR) Scheme

In this scheme, the tangential stiffness matrix is formed and decomposed at the beginning of each step (or as specified when defining the properties of the study) and used throughout the iterations as shown in the figure below.