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1937-1632
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Discrete & Continuous Dynamical Systems - S
April 2016 , Volume 9 , Issue 2
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2016, 9(2): i-ii
doi: 10.3934/dcdss.201602i
+[Abstract](1666)
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Abstract:
This issue is specially dedicated to our colleague and friend Alain Cimetière.
For more information please click the “Full Text” above.
This issue is specially dedicated to our colleague and friend Alain Cimetière.
For more information please click the “Full Text” above.
2016, 9(2): 363-381
doi: 10.3934/dcdss.2016001
+[Abstract](1939)
+[PDF](402.8KB)
Abstract:
The `stick-slip' phenomenon is the unsteady relative motion of two solids in frictional contact. Tentative explanations were given in the past by enriching the friction law (for example, introducing static and dynamic friction coefficients). In this article, we outline an approach for the analysis of the `stick-slip' phenomenon within the simple framework of the coupling of linear elasticity with the Coulomb dry friction law. Simple examples, both discrete and continuous, show that the solutions of the steady sliding frictional contact problem may exhibit bifurcations (loss of uniqueness) when the friction coefficient is taken as a control parameter. It is argued that such a bifurcation could account, in some cases, for the `stick-slip' phenomenon. The situations of a single point particle, of a linear elastic bounded body with homogeneous friction coefficient and of the elastic half-space with both homogenous and piecewise constant friction coefficient are analysed and compared.
The `stick-slip' phenomenon is the unsteady relative motion of two solids in frictional contact. Tentative explanations were given in the past by enriching the friction law (for example, introducing static and dynamic friction coefficients). In this article, we outline an approach for the analysis of the `stick-slip' phenomenon within the simple framework of the coupling of linear elasticity with the Coulomb dry friction law. Simple examples, both discrete and continuous, show that the solutions of the steady sliding frictional contact problem may exhibit bifurcations (loss of uniqueness) when the friction coefficient is taken as a control parameter. It is argued that such a bifurcation could account, in some cases, for the `stick-slip' phenomenon. The situations of a single point particle, of a linear elastic bounded body with homogeneous friction coefficient and of the elastic half-space with both homogenous and piecewise constant friction coefficient are analysed and compared.
2016, 9(2): 383-392
doi: 10.3934/dcdss.2016002
+[Abstract](2363)
+[PDF](1414.6KB)
Abstract:
This paper presents some insights into the determination, using the Proper Generalized Decomposition, of multiple solutions of nonlinear parametric partial differential equations. Although the Proper Generalized Decomposition (PGD) is well suited for computing the solution of, possibly nonlinear, parametric problems that vary smoothly with a physical parameter, no work has been achieved for the case of problems that exhibit multiple solutions for some values of a parameter. For two representative cases, we show how an appropriate parametrization, combined to a nonlinear solution procedure can be devised to describe and compute the multiple solutions of a PDE.
This paper presents some insights into the determination, using the Proper Generalized Decomposition, of multiple solutions of nonlinear parametric partial differential equations. Although the Proper Generalized Decomposition (PGD) is well suited for computing the solution of, possibly nonlinear, parametric problems that vary smoothly with a physical parameter, no work has been achieved for the case of problems that exhibit multiple solutions for some values of a parameter. For two representative cases, we show how an appropriate parametrization, combined to a nonlinear solution procedure can be devised to describe and compute the multiple solutions of a PDE.
2016, 9(2): 393-408
doi: 10.3934/dcdss.2016003
+[Abstract](2326)
+[PDF](509.2KB)
Abstract:
We compare the performance of two algorithms of computing the Borel sum of a time power series. The first one uses Padé approximants in Borel space, followed by a Laplace transform. The second is based on factorial series. These algorithms are incorporated in a numerical scheme for time integration of differential equations.
We compare the performance of two algorithms of computing the Borel sum of a time power series. The first one uses Padé approximants in Borel space, followed by a Laplace transform. The second is based on factorial series. These algorithms are incorporated in a numerical scheme for time integration of differential equations.
2016, 9(2): 409-425
doi: 10.3934/dcdss.2016004
+[Abstract](1934)
+[PDF](811.6KB)
Abstract:
Based on an extension of Fenchel's inequality, the bipotential approach is a non smooth mechanics tool used to model various non associative multivalued constitutive laws of dissipative materials (friction contact, soils, cyclic plasticity of metals, damage). Generally, such constitutive laws are given by a graph $M$. We propose a simple necessary and sufficient condition for the existence of a bipotential $b$ for which $M$ is the set of couples $(x,y)$ of dual variables such that $b(x,y) = \langle x,y \rangle$, and a method to construct such a bipotential by covering $M$ with cyclically monotone graphs which are not necessarily maximal (bipotential convex cover). As application, we show how to obtain the bipotential of the law of unilateral contact with Coulomb's friction by a bipotential convex cover. Introduced to extend the classical calculus of variation, the bipotential concept is also useful to construct numerical schemes for friction contact laws. In recents works, we extended the bipotential approach to a certain class of orthotropic frictional contact with a non-associated sliding rule proposed by Michałowski and Mróz. The bipotential suggests a predictor-corrector numerical scheme.
Based on an extension of Fenchel's inequality, the bipotential approach is a non smooth mechanics tool used to model various non associative multivalued constitutive laws of dissipative materials (friction contact, soils, cyclic plasticity of metals, damage). Generally, such constitutive laws are given by a graph $M$. We propose a simple necessary and sufficient condition for the existence of a bipotential $b$ for which $M$ is the set of couples $(x,y)$ of dual variables such that $b(x,y) = \langle x,y \rangle$, and a method to construct such a bipotential by covering $M$ with cyclically monotone graphs which are not necessarily maximal (bipotential convex cover). As application, we show how to obtain the bipotential of the law of unilateral contact with Coulomb's friction by a bipotential convex cover. Introduced to extend the classical calculus of variation, the bipotential concept is also useful to construct numerical schemes for friction contact laws. In recents works, we extended the bipotential approach to a certain class of orthotropic frictional contact with a non-associated sliding rule proposed by Michałowski and Mróz. The bipotential suggests a predictor-corrector numerical scheme.
2016, 9(2): 427-444
doi: 10.3934/dcdss.2016005
+[Abstract](1985)
+[PDF](5209.4KB)
Abstract:
This paper concerns a theoretical study on the possibility of using Love waves for non destructive testing. A mathematical model is presented and analyzed. Several numerical tests are given in order to show the mechanical behaviour of this model.
This paper concerns a theoretical study on the possibility of using Love waves for non destructive testing. A mathematical model is presented and analyzed. Several numerical tests are given in order to show the mechanical behaviour of this model.
2016, 9(2): 445-455
doi: 10.3934/dcdss.2016006
+[Abstract](2225)
+[PDF](379.8KB)
Abstract:
In this manuscript, we present optimal sensitivity results of eigenvalues and eigenspaces with respect to self-adjoint compact operators. We show that while eigenvalues depend in a Lipschitzian way in compact operators, the eigenspaces are only locally Lipschitz. Our results generalize to arbitrary dimension eigenspaces the results obtained in [19] for one-dimensional eigenspaces sensitivity and thus simplify the celebrate results by Davis and Kahan [6] developed for general Hermitian operator perturbations. Moreover, Proper Orthogonal Decomposition bases sensitivity is carried out in the case of time-interval perturbations, spatial perturbations (Gappy-POD) or parameter perturbations.
In this manuscript, we present optimal sensitivity results of eigenvalues and eigenspaces with respect to self-adjoint compact operators. We show that while eigenvalues depend in a Lipschitzian way in compact operators, the eigenspaces are only locally Lipschitz. Our results generalize to arbitrary dimension eigenspaces the results obtained in [19] for one-dimensional eigenspaces sensitivity and thus simplify the celebrate results by Davis and Kahan [6] developed for general Hermitian operator perturbations. Moreover, Proper Orthogonal Decomposition bases sensitivity is carried out in the case of time-interval perturbations, spatial perturbations (Gappy-POD) or parameter perturbations.
2016, 9(2): 457-474
doi: 10.3934/dcdss.2016007
+[Abstract](1945)
+[PDF](500.1KB)
Abstract:
We are interested by a nonlinear single lithology diffusion model adapted from ideas originally developed by the Institut Français du Pétrole (IFP). The geological stratigraphic modeling has to describe transports of sediments, erosion and sedimentation processes by taking into account a limited weathering condition; the method by which the history of a sedimentary basin is revealed relies on knowledge of both initial and final data and can be generalized to multiple lithology. For this purpose, we introduce a relaxation time related to a delayed response for establishing equilibrium states; this approach introduces regularizing effects according to the ideas of G.I. Barenblatt - S. Sobolev and J.-L. Lions - O. A. Oleinik. New well-posedness results are presented.
We are interested by a nonlinear single lithology diffusion model adapted from ideas originally developed by the Institut Français du Pétrole (IFP). The geological stratigraphic modeling has to describe transports of sediments, erosion and sedimentation processes by taking into account a limited weathering condition; the method by which the history of a sedimentary basin is revealed relies on knowledge of both initial and final data and can be generalized to multiple lithology. For this purpose, we introduce a relaxation time related to a delayed response for establishing equilibrium states; this approach introduces regularizing effects according to the ideas of G.I. Barenblatt - S. Sobolev and J.-L. Lions - O. A. Oleinik. New well-posedness results are presented.
2016, 9(2): 475-500
doi: 10.3934/dcdss.2016008
+[Abstract](2363)
+[PDF](1678.5KB)
Abstract:
Separated representations allow impressive computational CPU time savings when applied in different fields of computational mechanics. They have been extensively used for solving models defined in multidimensional spaces coming from (i) its proper physics, (ii) model parameters that were introduced as extra-coordinates and (iii) 3D models when the solution can be separated as a finite sum of functional products involving lower dimensional spaces. The last route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist and were considered in our former works. In the present work, we are analyzing two alternative routes. The first one consists of immersing the real non-separable domain into a fully separable hexahedral domain. The second procedure consists in applying a geometrical transformation able to transform the real domain into a hexahedra in which the model is solved by using a fully separated representation of the unknown field.
Separated representations allow impressive computational CPU time savings when applied in different fields of computational mechanics. They have been extensively used for solving models defined in multidimensional spaces coming from (i) its proper physics, (ii) model parameters that were introduced as extra-coordinates and (iii) 3D models when the solution can be separated as a finite sum of functional products involving lower dimensional spaces. The last route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist and were considered in our former works. In the present work, we are analyzing two alternative routes. The first one consists of immersing the real non-separable domain into a fully separable hexahedral domain. The second procedure consists in applying a geometrical transformation able to transform the real domain into a hexahedra in which the model is solved by using a fully separated representation of the unknown field.
2016, 9(2): 501-527
doi: 10.3934/dcdss.2016009
+[Abstract](1772)
+[PDF](2330.6KB)
Abstract:
After previous works related to the equilibrium states, this paper goes deeper into the study of the effect of coupling between smooth and non-smooth non-linearities on the qualitative behavior of low dimensional dynamical systems. The non-smooth non-linearity is due to non-regularized unilateral contact and Coulomb friction while the smooth one is due to large strains of a simple mass spring system, which lead to a nonlinear restoring force. The main qualitative differences with the case of a linear restoring force are due to the shape of the set of equilibrium states.
After previous works related to the equilibrium states, this paper goes deeper into the study of the effect of coupling between smooth and non-smooth non-linearities on the qualitative behavior of low dimensional dynamical systems. The non-smooth non-linearity is due to non-regularized unilateral contact and Coulomb friction while the smooth one is due to large strains of a simple mass spring system, which lead to a nonlinear restoring force. The main qualitative differences with the case of a linear restoring force are due to the shape of the set of equilibrium states.
2016, 9(2): 529-536
doi: 10.3934/dcdss.2016010
+[Abstract](2032)
+[PDF](360.8KB)
Abstract:
This paper gives an overview of our results obtained from 2009 until 2014 about paradoxical stability properties of non conservative systems which lead to the concept of Kinematical Structural Stability (Ki.s.s.). Due to Fischer-Courant results, this ki.s.s. is universal for conservative systems whereas new interesting situations may arise for non conservative ones. A remarkable algebraic property of the symmetric part of linear operators may generalize this result for divergence stability but leading only to a conditional ki.s.s. By duality, the concept of geometric degree of nonconservativity is highlighting. Paradigmatic examples of Ziegler systems illustrate the general results and their effectiveness.
This paper gives an overview of our results obtained from 2009 until 2014 about paradoxical stability properties of non conservative systems which lead to the concept of Kinematical Structural Stability (Ki.s.s.). Due to Fischer-Courant results, this ki.s.s. is universal for conservative systems whereas new interesting situations may arise for non conservative ones. A remarkable algebraic property of the symmetric part of linear operators may generalize this result for divergence stability but leading only to a conditional ki.s.s. By duality, the concept of geometric degree of nonconservativity is highlighting. Paradigmatic examples of Ziegler systems illustrate the general results and their effectiveness.
2016, 9(2): 537-556
doi: 10.3934/dcdss.2016011
+[Abstract](2294)
+[PDF](461.4KB)
Abstract:
The paper concerns with the existence, uniqueness, regularity and the approximation of solutions to the nonlinear phase-field (Allen-Cahn) equation, endowed with non-homogeneous dynamic boundary conditions (depending both on time and space variables). It extends the already studied types of boundary conditions, which makes the problem to be more able to describe many important phenomena of two-phase systems, in particular, the interactions with the walls in confined systems. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual numerical algorithm is formulated in the end.
The paper concerns with the existence, uniqueness, regularity and the approximation of solutions to the nonlinear phase-field (Allen-Cahn) equation, endowed with non-homogeneous dynamic boundary conditions (depending both on time and space variables). It extends the already studied types of boundary conditions, which makes the problem to be more able to describe many important phenomena of two-phase systems, in particular, the interactions with the walls in confined systems. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual numerical algorithm is formulated in the end.
2016, 9(2): 557-584
doi: 10.3934/dcdss.2016012
+[Abstract](2735)
+[PDF](1544.2KB)
Abstract:
The aim of the present work is to study the nucleation and propagation of cohesive cracks in two-dimensional elastic structures. The crack evolution is governed by Dugdale's cohesive force model. Specifically, we investigate the stabilizing effect of the stress field non-uniformity by introducing a length $l$ which characterizes the stress gradient in a neighborhood of the point where the crack nucleates. We distinguish two stages in the crack evolution: the first one where the entire crack is submitted to cohesive forces, followed by a second one where a non-cohesive part appears. Assuming that the material characteristic length $d_c$ associated with Dugdale's model is small in comparison with the dimension $L$ of the body, we develop a two-scale approach and, using the methods of complex analysis, obtain the entire crack evolution with the loading in closed form. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal crack length jump as soon as the non-cohesive crack part appears. We also discuss the influence of the problem parameters and study the sensitivity to imperfections.
The aim of the present work is to study the nucleation and propagation of cohesive cracks in two-dimensional elastic structures. The crack evolution is governed by Dugdale's cohesive force model. Specifically, we investigate the stabilizing effect of the stress field non-uniformity by introducing a length $l$ which characterizes the stress gradient in a neighborhood of the point where the crack nucleates. We distinguish two stages in the crack evolution: the first one where the entire crack is submitted to cohesive forces, followed by a second one where a non-cohesive part appears. Assuming that the material characteristic length $d_c$ associated with Dugdale's model is small in comparison with the dimension $L$ of the body, we develop a two-scale approach and, using the methods of complex analysis, obtain the entire crack evolution with the loading in closed form. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal crack length jump as soon as the non-cohesive crack part appears. We also discuss the influence of the problem parameters and study the sensitivity to imperfections.
2016, 9(2): 585-597
doi: 10.3934/dcdss.2016013
+[Abstract](3248)
+[PDF](931.5KB)
Abstract:
The paper is concerned by multi-scale methods to describe instability pattern formation, especially the method of Fourier series with variable coefficients. In this respect, various numerical tools are available. For instance in the case of membrane models, shell finite element codes can predict the details of the wrinkles, but with difficulties due to the large number of unknowns and the existence of many solutions. Macroscopic models are also available, but they account only for the effect of wrinkling on membrane behavior. A Fourier-related method has been introduced in order to modelize the main features of the wrinkles, but by using partial differential equations only at a macroscopic level. Within this method, the solution is sought in the form of few terms of Fourier series whose coefficients vary more slowly than the oscillations. The recent progresses about this Fourier-related method are reviewed and discussed.
The paper is concerned by multi-scale methods to describe instability pattern formation, especially the method of Fourier series with variable coefficients. In this respect, various numerical tools are available. For instance in the case of membrane models, shell finite element codes can predict the details of the wrinkles, but with difficulties due to the large number of unknowns and the existence of many solutions. Macroscopic models are also available, but they account only for the effect of wrinkling on membrane behavior. A Fourier-related method has been introduced in order to modelize the main features of the wrinkles, but by using partial differential equations only at a macroscopic level. Within this method, the solution is sought in the form of few terms of Fourier series whose coefficients vary more slowly than the oscillations. The recent progresses about this Fourier-related method are reviewed and discussed.
2016, 9(2): 599-611
doi: 10.3934/dcdss.2016014
+[Abstract](2144)
+[PDF](305.4KB)
Abstract:
After a general formulation of the evolution of an elastoplastic body using duality based on the constitutive behaviour, some classes of inverse problems (estimation of the internal state, determination of an unknown history, ...) for such materials are investigated. A general formulation based on optimal control is proposed, the control variables are related to the internal state. In each class of inverse problem, the solution is obtained by introducing a adjoin state and a suitable cost function.
After a general formulation of the evolution of an elastoplastic body using duality based on the constitutive behaviour, some classes of inverse problems (estimation of the internal state, determination of an unknown history, ...) for such materials are investigated. A general formulation based on optimal control is proposed, the control variables are related to the internal state. In each class of inverse problem, the solution is obtained by introducing a adjoin state and a suitable cost function.
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