Stiff arc length constraint in nonlinear FEA

Summary.

Arc length constraints enable iterative solution procedures in nonlinear finite element analysis to converge even at critical points. The arc length constraint replaces the conventional m×m stiffness matrix with an augmented (m+1)×(m+1) stiffness matrix. Its use is referred to as arc length control, in contrast to load control which furnishes the conventional stiffness matrix. In the current article, an apparently new arc length constraint is introduced. It identifies arc length parameters maximizing the stiffness (absolute value of the determinant) of the augmented matrix. The parameters, viewed as a vector, must be perpendicular to the rows of the stiffness matrix, likewise considered vectors. The augmented stiffness matrix is nonsymmetric and lacks the small bandwidth of the conventional stiffness matrix. However, using a block triangularization, it is demonstrated that a solution may be attained by standard finite element operations, namely triangularization of a banded nonsingular portion of the stiffness matrix followed by forward and backward substitutions involving banded lower and upper triangular matrices. The proposed constraint is expected to permit convergence under longer arc lengths than currently implemented methods. A simple example is given illustrating the application of the constraint.