Non-Linear FEA and Solution Methods

Many a times we assume neither the stiffness of the structure nor the loads or boundary conditions change in a structural system on progressive application of loads. It means we are assuming system to behave linearly with external loads. This is a fairly good assumption to make sometimes and it produces accurate enough results to make a quick design decision. Simple rule to make a linearity assumption is that system should have small displacements, rotations and infinitesimal strains.

Non-linearity in a structural system

A load-displacement behavior which also includes instability in the system could be depicted by following figure,

Not a typical load-displacement curve, but shows some important features necessary to set up the context of non-linearity and instability in a structural system.

Typically a structure when loaded progressively from reference state exhibits a linear behavior for small load increment on primary equilibrium path. At primary equilibrium path for small loads system is linear but as the loads increase system starts getting into non-linear zone.

In systems with large deformations, system might follow zero stiffness point which is referred as "limiting point". A common phenomena observed with column buckling under high loads. System behavior after limiting point is the "secondary path" with negative stiffness. Secondary part reflects the conversion of potential energy stored into the system into kinetic energy. A sudden change in deformation state happens, a buckled column under compressive load for example. Secondary path could be followed by another stable zone of positive stiffness and structure gets the stability again.

Under heavy loads a structure might cross the normally assumed linear elastic zone to plastic zone, crossing of the yield point, a non-linear material behavior. Post yield point material stiffness reduces which leads to large plastic deformation. Rubbers and other elastomers exhibit a nonlinear elasticity. Characterized by large deflections and reversal to original shape on loads elimination.

Equations of a linear FE model

A finite element model is collection of simultaneous algebraic equations representing a reference state of structure on equilibrium path. Simultaneous equations can be solved by matrix methods used for solution of simultaneous algebraic equations, for a given state, if structure stiffness and applied loads & boundary conditions don't change with displacements, a solution method for linear FE model.

Solution method for non-linear FE equations

When expected a non-linear system behavior, equations are solved for progressive small loads and for each small increment of load equilibrium is said to be achieved if a predefined criteria of convergence is achieved. In each small load increment stiffness is approximated with the displacement values obtained by previous load increment to achieve a new equilibrium state.

Direct iteration method

A non-linear system can be assumed to have a two components in stiffness term. One being a constant term and another a non-linear, value of which depends on the displacement vector. Let's consider a simple example, an axially loaded bar having stiffness k with two components showing a non-linear load-displacement curve (green in color in figure below). The stiffness of the bar is reducing with the displacement, the dropping k curve. It simply means the non-linear part of the stiffness is negative (kn < 0).

To solve this system iteratively for a given load Pa we need to have some initial estimate of the stiffness k. With the initial estimate ko, displacement vector u1 can be calculated. With this displacement value we can estimate kn and hence k = k0 + kn1 (which is nothing but tangent at u1 on load-displacement curve).

If the process of calculating displacements with new estimate of stiffness is repeated, as shown in figure below, we would approach the solution point. This method is known as Direct iteration method. For the solution to converge residual force should converge.

Newton-Raphson method

Another method of solving non-linear is Newton-Raphson method. In N-R method instead of calculating new displacement value in each iteration, displacement increments are calculated in each iteration based on tangent stiffness, as shown in figure below. Displacement increments are then added to previously calculated displacement to get new estimate of displacement vector.

Convergence criteria

For convergence to occur residual forces and displacement increments should converge. Convergence is achieved when conv, based on residual forces or incremental displacements, falls below a certain defined value, normally a very small value.

There are other alternative forms of solving non-linear finite element equations, which uses initial stiffness in each iteration and calculates effective load to arrive at displacement values. All methods can be used separately or in combination to solve FE equations. This article is intended to give a short introduction to non-linearity in Structural systems and Solution techniques for non-linear FE equations. I will try to write detailed separate article on different aspects of Non-Linear FEA.