Civil Engineering Reference
In-Depth Information
9.6 Nonlinear Analysis Algorithm
An incremental procedure is usually used to study nonlinear finite element problems. For
nonlinear problems, however, the incremental procedure would lead to a build-up of error.
An iterative procedure using a certain solution algorithm should be employed to correct
the build-up errors. Therefore, the combination of the incremental approach and iterative
procedure are used as the basis for most of the nonlinear finite element analyses in our work.
In the analysis procedure, the integrator determines the next predictive step during the analysis
procedure, and specifies the tangent matrix and residual vector at any iteration. In this topic, the
commonly used integrators for static and dynamic analyses such as load control, displacement
control, Newmark's method, and Wilson's method are introduced and the analysis procedure
of each integrator is described. The algorithm determines the sequence of steps taken to
solve the nonlinear equations during the iterative procedures. Hence, different algorithms are
also described.
9.6.1 Load Control Iteration Scheme
To study the problem of a structure under monotonic loading, an incremental procedure
can be used. The structural tangent stiffness matrix is related to the increments of loads
to increments of displacements, which incorporates a tangential material constitutive matrix
relating the increments of stresses to the increments of strains. Under load control, the total
load is divided into small load increments. Each load step is applied in turn and iterations
are performed until convergence is achieved at the structural level. Then the next load step
is processed.
In solving nonlinear equations, the commonly used solution algorithm is the full
Newton-Raphson method. In each iteration, the stiffness matrix is iteratively refined until
the convergence criterion is achieved. The stiffness matrix is computed from the last it-
erative solution during the iterative procedure until convergence is achieved. A modified
Newton-Raphson procedure is also used in the solution algorithms. Different from the full
Newton-Raphson method, the stiffness matrix from the last converged equilibrium was used
during the iterative procedure until convergence is achieved. The Newton-Raphson method
uses the initial stiffness matrix throughout the iterative procedure.
When compared with theNewton-Raphson method and the modified Newton-Raphson
method, the full Newton-Raphson method converges more rapidly, and the process will
converge in fewer iterations and give smaller residual force at each iteration. However, it
requires that the tangent stiffness matrix be evaluated at each iteration, which can be significant
for large structures. In contrast to the full Newton-Raphson method, the initial stiffness matrix
in the Newton-Raphson method is calculated at the beginning of the load step and the stiffness
matrix remains the same throughout the procedure. A large number of iterations are required
to achieve convergence. The modified Newton-Raphson method shows the balance between
the computation and iteration numbers. Many algorithms, such as the KrylovNewton method
(Carlson and Miller, 1998), have been developed by improving the Newton-type methods
with the acceleration technology. The KrylovNewton method is a modified Newton-Raphson
method with 'Krylov subspace' acceleration, which greatly decreases the number of iterations
in the solution.
