Research Directions in Computational Mechanics (1991) / Chapter Skim
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8 Nonlinear Equations and Bifurcation
8 Nonlinear Equations and Bifurcation
Pages 84-93
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From page 84... ...
force nonlinearities due to nonlinear stress boundary conditions, and (4) kinematic constraint nonlinearities due to nonlinear displacement boundary conditions.
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From page 85... ...
This is the case, for instance, in finite deformation plasticity where there are profound mathematical and computational difficulties in modeling phase changes, viscous effects, cracks, singularities, etc. At the other end of the spectrum, there are many novel questions relating to scientific computing, such as those concerning the effect of different types of computational resources and, especially, the development of new computer architectures on the selection, implementation, and performance of different numerical methods.
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From page 86... ...
Some challenging questions of this type arise when, as a secondary process, a multigrid technique is applied that utilizes the fact that the nonlinear system is the discretization of some differential equation. In connection with locally convergent iterative processes, it must be noted that the attraction region for a specific solution x*
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From page 87... ...
Various forms of continuation processes have been proposed that differ in choice of the predicted point, form of the augmented equations, and type of local iterative process. Thus, all the errors for local iterative processes arise also in connection with continuation methods.
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From page 88... ...
represents the equilibrium equations of a system of differential equations, there may be interest in locating Hopf bifurcation points on a solution path where periodic solutions of the dynamic system branch off from the static equilibrium. In recent years much research has been devoted- to these various tasks, but numerous questions remain.
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From page 89... ...
While the cases of simple turning points and simple bifurcation points have been treated exhaustively in the theoretical and numerical literature, there is as yet little material available regarding calculation of all the branches emanating from multiple bifurcation points. Calculation of Paths of Singular Points Singularity theory has provided some rigorous basis for the intuitive idea that a hierarchy exists among singularities.
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From page 90... ...
Another important question concerns numerical identification of the invariant subspaces and, in finite element methods, of suitable bases of such subspaces. Here it may be noted that the required symmetry properties of the basis functions may significantly enlarge the size of their supports, which in turn may have a negative effect on the sparsity of the matrices involved in the calculations.
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From page 91... ...
Since most of these equations in computational mechanics represent discretizations of certain differential equations, the question arises whether any computed critical phenomena, such as a particular bifurcation point and its branching behavior, correspond to similar phenomena of the original problem and, if so, what errors are encountered. These questions have as yet only partial answers and represent considerable research challenges.
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From page 92... ...
While there has been considerable progress in the development of specific computational methods for DAEs in general, at present there exist no broadly applicable and robust integrators suitable for automated application of multibody dynamic simulation methods in a computer-aided engineering environment. Recent developments of methods of generalized coordinate partitioning methods for local parametrization of the kinematic constraints appear to offer particular promise for the effective solution of (~.5)
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From page 93... ...
Much theoretical and numerical work remains to be done in this little explored area.
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