591 research outputs found
Generalized Newton's Method based on Graphical Derivatives
This paper concerns developing a numerical method of the Newton type to solve
systems of nonlinear equations described by nonsmooth continuous functions. We
propose and justify a new generalized Newton algorithm based on graphical
derivatives, which have never been used to derive a Newton-type method for
solving nonsmooth equations. Based on advanced techniques of variational
analysis and generalized differentiation, we establish the well-posedness of
the algorithm, its local superlinear convergence, and its global convergence of
the Kantorovich type. Our convergence results hold with no semismoothness
assumption, which is illustrated by examples. The algorithm and main results
obtained in the paper are compared with well-recognized semismooth and
-differentiable versions of Newton's method for nonsmooth Lipschitzian
equations
An improved return-mapping scheme for nonsmooth yield surfaces: PART I - the Haigh-Westergaard coordinates
The paper is devoted to the numerical solution of elastoplastic constitutive
initial value problems. An improved form of the implicit return-mapping scheme
for nonsmooth yield surfaces is proposed that systematically builds on a
subdifferential formulation of the flow rule. The main advantage of this
approach is that the treatment of singular points, such as apices or edges at
which the flow direction is multivalued involves only a uniquely defined set of
non-linear equations, similarly to smooth yield surfaces. This paper (PART I)
is focused on isotropic models containing: yield surfaces with one or two
apices (singular points) laying on the hydrostatic axis; plastic
pseudo-potentials that are independent of the Lode angle; nonlinear
isotropic hardening (optionally). It is shown that for some models the improved
integration scheme also enables to a priori decide about a type of the return
and investigate existence, uniqueness and semismoothness of discretized
constitutive operators in implicit form. Further, the semismooth Newton method
is introduced to solve incremental boundary-value problems. The paper also
contains numerical examples related to slope stability with available Matlab
implementation.Comment: 25 pages, 10 figure
A New Exceptional Family of Elements and Solvability of General Order Complementarity Problems
By using the concept of exceptional family, we propose
a sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in Németh, 2010 (Theorem 3.1). Then
we study some sufficient conditions for the nonexistence of exceptional family for
GOCP in Hilbert space. Moreover, we prove that without exceptional family is
a sufficient and necessary condition for the solvability of pseudomonotone general
order complementarity problems
Analysis of friction coupling and the Painlevé paradox in multibody systems
Multibody models are useful to describe the macroscopic motion of the elements of physical systems. Modeling contact in such systems can be challenging, especially if friction at the contact interface is taken into account. Furthermore, the dynamics equations of multibody systems with contacts and Coulomb friction may become ill-posed due to friction coupling, as in the Painlevé paradox, where a solution for system dynamics may not be found. Here, the dynamics problem is considered following a general approach so that friction phenomena, such as dynamic jamming, can be analyzed. The effect of the contact forces on the velocity field of the system is the cornerstone of the proposed formulation, which is used to analyze friction coupling in multibody systems with a single contact. In addition, we introduce a new representation of the so-called generalized friction cone, a quadratic form defined in the contact velocity space. The geometry of this cone can be used to determine the critical cases where the solvability of the system dynamic equations can be compromised. Moreover, it allows for assessing friction coupling at the contact interface, and obtaining the values of the friction coefficient that can make the dynamics formulation inconsistent. Finally, the classical Painlevé example of a single rod and the multibody model of an articulated arm are used to illustrate how the proposed cone can detect the cases where the dynamic equations have no solution, or multiple solutions.Postprint (author's final draft
Convergence of time - stepping schemes for passıie and extended linear complementarity systems
Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established
Convergence of Time-Stepping Schemes for Passive and Extended Linear Complementarity Systems
Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established.</p
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