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232
Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods
, 2004
"... We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the JacobiDavidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new li ..."
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Cited by 46 (4 self)
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We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the JacobiDavidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear eigenvalue problems arise and some numerical results.
Quasicontinuum models of interfacial structure and deformation.
 Phys. Rev. Lett.
, 1998
"... Microscopic models of the interaction between grain boundaries (GBs) and both dislocations and cracks are of importance in understanding the role of microstructure in altering the mechanical properties of a material. A recently developed mixed atomistic and continuum method is reformulated to allow ..."
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Cited by 40 (3 self)
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Microscopic models of the interaction between grain boundaries (GBs) and both dislocations and cracks are of importance in understanding the role of microstructure in altering the mechanical properties of a material. A recently developed mixed atomistic and continuum method is reformulated to allow for the examination of the interactions between GBs, dislocations, and cracks. These calculations elucidate plausible microscopic mechanisms for these defect interactions and allow for the quantitative evaluation of critical parameters such as the force needed to induce GB migration. [S00319007(97)05134X] PACS numbers: 68.45.Nj, 81.40.Lm With the continuing development of more accurate, less expensive models for atomistic interactions and expansion of computational resources, there is growing interest in the modeling of materials from fundamental principles rather than phenomenological approaches. An outstanding problem in this regard is the role of microstructure in determining material properties. The influence of microstructure (e.g., grain size and shape) on the mechanical properties of materials is clearly revealed, for example, in the yield strength and the fracture toughness In this Letter, we present a reformulation of one such method for treating multiple scales and demonstrate its application to two examples: The interaction of lattice dislocations with grain boundaries (GBs) and the interaction of cracks with GBs. The quasicontinuum method [2], a mixed atomisticcontinuum formulation, is based on a finite element discretization of a continuum mechanics variational principle. The finite element method serves as the numerical engine for determining the energy minimizing displacement fields, while atomistic analysis is used to determine the energy of a given configuration. This is in contrast to standard finite element approaches, where the constitutive input is made via phenomenological models. The method is successful in capturing the structure and energetics of dislocations. In this paper we consider a reformulation of the method that allows for the treatment of interfaces, and show how it allows for the simultaneous treatment of dislocations, material interfaces, and cracks. The key new idea in the formulation is that rather than considering an atomistic scheme for providing constitutive input to a continuum model, which requires the definition of an energy density near the grain boundary, we begin with the recognition that from the microscopic perspective the body may be regarded as a collection of N atoms. The total potential energy of such a collection is given by where r i is the position of the atom i, f i is the external force on that atom, and E i is its energy as would be computed from an atomistic model such as the embedded atom method (EAM) [3] used here. One of the primary objectives in the formulation of the method is to eliminate the redundant atomistic degrees of freedom associated with the regions of the body far from extended defects and hence subject to displacement fields which are slowly varying on the atomic scale. To achieve the requisite degree of freedom reduction, we select M representative atoms from the N atoms ͑M ø N͒, chosen to best represent the energetics of the body, the positions r a ͑a 1, . . . , M͒ of which serve as the reduced set of degrees of freedom. The body is now divided into disjoint cells such that each cell contains exactly one representative atom. The key energetic approximation is that the energy of all of the atoms in a given cell is the same as that of the cell's representative atom. The positions of the atoms that are not treated explicitly are obtained by interpolating the nodal values of the displacements using a finite element mesh which is constructed with the representative atoms as the nodal points. (One possible implementation of this strategy in two dimensions is to use the Voronoi polygons [4] surround the representative atoms as the cells and the geometric dual of the Voronoi tiling, the Delaunay triangulation 742 00319007͞98͞80(4)͞742(4)$15.00
A visual model for blast waves and fracture
 In Proc. of Graph. Interface
, 1999
"... Key words: Fracture, blast waves, explosions, physicallybased modelling, animation. 1 Introduction Few phenomena are at once so short lived, so powerfuland so awe inspiring as explosions. They are a source of wonder, delight, and destruction. Used widely in manyindustries, the special effects indust ..."
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Cited by 37 (0 self)
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Key words: Fracture, blast waves, explosions, physicallybased modelling, animation. 1 Introduction Few phenomena are at once so short lived, so powerfuland so awe inspiring as explosions. They are a source of wonder, delight, and destruction. Used widely in manyindustries, the special effects industry among them, real explosions are costly, dangerous, difficult to control, andimpossible to undo. That makes them excellent candidates for visual and physical simulation, but the timescales at which they operate, and the sheer number of physical interactions involved makes this a difficult problem. The benefits are significant, since having a consistent mathematical formulation of a phenomenon can givereproducible usercontrolled simulations. Our work is primarily concerned with deriving a mathematical modelthat gives rise to convincing visual depictions. It is unlikely our work will be of significant relevance to the accurate physical simulation of explosion mechanics.
Kinking of a Crack Out of an Interface,”
 ASME J. Appl. Mech.,
, 1989
"... Kinking of a plane strain crack out of the interface between two dissimilar isotropic ..."
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Cited by 36 (0 self)
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Kinking of a plane strain crack out of the interface between two dissimilar isotropic
size and distribution
 of Synechococcus in the North Atlantic and Pacific Oceans, Limnology and Oceanography
, 1990
"... diagenesis and mechanics to quantify fracture aperture ..."
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Cited by 26 (0 self)
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diagenesis and mechanics to quantify fracture aperture
Dynamic earthquake ruptures in the presence of lithostatic normal stresses: Implications for friction models and heat production
, 2001
"... Abstract We simulate dynamic ruptures on a strikeslip fault in homogeneous and layered halfspaces and on a thrust fault in a layered halfspace. With traditional friction models, sliding friction exceeds 50 % of the fault normal compressive stress, and unless the pore pressures approach the lithos ..."
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Cited by 21 (5 self)
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Abstract We simulate dynamic ruptures on a strikeslip fault in homogeneous and layered halfspaces and on a thrust fault in a layered halfspace. With traditional friction models, sliding friction exceeds 50 % of the fault normal compressive stress, and unless the pore pressures approach the lithostatic stress, the rupture characteristics depend strongly on the depth, and sliding generates large amounts of heat. Under application of reasonable stress distributions with depth, variation of the effective coefficient of friction with the square root of the shear modulus and the inverse of the depth creates distributions of stress drop and fracture energy that produce realistic rupture behavior. This ad hoc friction model results in (1) lowsliding friction at all depths and (2) fracture energy that is relatively independent of depth. Additionally, friction models with rateweakening behavior (which form pulselike ruptures) appear to generate heterogeneity in the distributions of final slip and shear stress more effectively than those without such behavior (which form cracklike ruptures). For surface rupture on a thrust fault, the simple slipweakening friction model, which lacks rateweakening behavior, accentuates the dynamic interactions between the seismic waves and the rupture and leads to excessively large ground motions on the hanging wall. Waveforms below the center of the fault (which are associated with waves radiated to teleseismic distances) indicate that source inversions of thrust events may slightly underestimate the slip at shallow depths.
Mechanics of adhesion through a fibrillar microstructure
 Integrative and Comparative Biology
"... SYNOPSIS. Many organisms have evolved a fibrillated interface for contact and adhesion as shown by several of the papers in this issue. For example, in the Gecko, this structure appears to give them the ability to adhere and separate from a variety of surfaces by relying only on weak van der Waals f ..."
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Cited by 19 (2 self)
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SYNOPSIS. Many organisms have evolved a fibrillated interface for contact and adhesion as shown by several of the papers in this issue. For example, in the Gecko, this structure appears to give them the ability to adhere and separate from a variety of surfaces by relying only on weak van der Waals forces. Despite the low intrinsic energy of separating surfaces held together by van der Waals forces, these organisms are able to achieve remarkably strong adhesion. To help understand adhesion in such a case, we consider a simple model of a fibrillar interface. For it, we examine the mechanics of contact and adhesion to a substrate. It appears that this structure allows the organism, at the same time, to achieve good, ‘universal ’ contact and adhesion. Due to buckling, a carpet of fibrils behaves like a plastic solid under compressive loading, allowing intimate contact in the presence of some roughness. As an adhesive, we conjecture that energy in the fibrils is lost upon decohesion and unloading. This mechanism can add considerably to the intrinsic work of fracture, resulting in good adhesion even with only van der Waals forces. Analysis of the mechanics of adhesion through such a fibrillar interface provides rules for the design of the microstructure for desired performance as an adhesive.
ThreeDimensional NonPlanar Crack Growth by a Coupled Extended Finite Element and Fast Marching Method
"... A numerical technique for nonplanar threedimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method and the fast marching method. In crack modeling using the extended finite element method, the framework of partition of unity is used ..."
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Cited by 13 (1 self)
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A numerical technique for nonplanar threedimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method and the fast marching method. In crack modeling using the extended finite element method, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the twodimensional asymptotic cracktip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated
Crackling dynamics in material failure as the signature of a selforganized dynamics phase transition
 Phys. Rev. Lett
"... We derive here a linear elastic stochastic description for slow crack growth in heterogeneous materials. This approach succeeds in reproducing quantitatively the intermittent crackling dynamics observed recently during the slow propagation of a crack along a weak heterogeneous plane of a transparen ..."
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Cited by 12 (5 self)
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We derive here a linear elastic stochastic description for slow crack growth in heterogeneous materials. This approach succeeds in reproducing quantitatively the intermittent crackling dynamics observed recently during the slow propagation of a crack along a weak heterogeneous plane of a transparent Plexiglas block [K. J. Måløy et al., Phys. Rev. Lett. 96, 045501 (2006)]. In this description, the quasistatic failure of heterogeneous media appears as a selforganized critical phase transition. As such, it exhibits universal and to some extent predictable scaling laws, analogous to that of other systems such as, for example, magnetization noise in ferromagnets. DOI: 10.1103/PhysRevLett.101.045501 PACS numbers: 62.20.mt, 46.50.+a, 68.35.Ct Driven by both technological needs and the challenges of unresolved questions in fundamental physics, the effect of material heterogeneities on their failure properties has been extensively studied in the recent past (see Here we will focus our study on the dynamics of cracks. In heterogeneous materials under slow external loading, this propagation displays a jerky dynamics with seemingly random sudden jumps spanning over a broad range of length scales We will demonstrate here that this crackling dynamics can be fully reproduced through a stochastic description rigorously derived from linear elastic fracture mechanics (LEFM) Theoretical description. LEFM is based on the fact thatin an elastic medium under tensile loadingthe mechanical energy G released as a fracture occurs is entirely dissipated within a small zone at the crack tip. Defining the fracture energy ÿ as the energy needed to create two crack surfaces of a unit area, under the quasistatic condition, we assume that the local crack velocity v is proportional to the excess energy locally released: