LA-UR-05-5471

B. Plohr and J. Plohr

We examine the approximations made in using Hooke's law as a constitutive relation for an isotropic thermoelastic material subjected to large deformation. For a general thermoelastic material, we employ the volume-preserving part of the deformation gradient to facilitate volumetric/shear strain decompositions of the free energy, its first derivatives (the Cauchy stress and entropy), and its second derivatives (the specific heat, Gruneisen tensor, and elasticity tensor). Specializing to isotropic materials, we calculate these constitutive quantities more explicitly. For deformations with limited shear strain, but possibly large changes in volume, we show that the differential equations for the stress involve new terms in addition to the traditional Hooke's law terms. These new terms are of the same order in the shear strain as the objective derivative terms needed for frame indifference; unless the latter terms are negligible, the former cannot be neglected. We also demonstrate that accounting for the new terms requires that the deformation gradient be included as a field variable.

Los Alamos National Laboratory, Report No. LA-UR-05-5471.

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LA-UR-05-6333

B. Plohr and J. Plohr

We consider shock loading of a thermo-visco-plastic material; the strain is not restricted to be small. We show how to reduce the Rankine-Hugoniot jump conditions to a simplified form analogous to that used in fluid dynamics. Just as for fluids, the shock conditions can be separated into purely thermodynamic conditions (the Rayleigh and Hugoniot equations), a condition determining the velocity, and a condition determining the strain, which can be solved sequentially to determine the shock wave.

In Proceedings of the 14th APS Topical Conference on Shock Compression of Condensed Matter, July 31 -- August 5, 2005, Baltimore, Maryland, Part One, eds. M. Furnish, M. Elert, T. Russel, C. White, pp. 343-346, American Institute of Physics, Melville, New York, 2006.

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LA-UR-03-9128

J. Plohr and B. Plohr

We present a study of Richtmyer--Meshkov flow for elastic materials. This flow, in which a material interface is struck by a shock wave, was originally investigated for gases, where growth of perturbations of the interface is observed. Here we consider two elastic materials in frictionless contact. The governing system of equations comprises conservation laws supplemented by constitutive equations. To analyze it, we linearize the equations around a one-dimensional background solution under the assumption that the perturbation is small. The background problem defines a Riemann problem that is solved numerically; its solution contains transmitted and reflected shock waves in the longitudinal modes. The linearized Rankine--Hugoniot condition provides the interface conditions at the longitudinal and shear waves; the frictionless material interface conditions are also linearized. The resulting equations, a linear system of partial differential equations, is solved numerically using a finite difference method supplemented by front tracking. In verifying the numerical code, we reproduce growth of the interface in the gas case. For the elastic case, in contrast, we find that the material interface remains bounded: the nonzero shear stiffness stabilizes the flow. In particular, the linear theory remains valid at late time. Moreover, we identify the principal mechanism for the stability of Richtmyer-Meshkov flow for elastic materials: the vorticity deposited on the material interface during shock passage is propagated away by the shear waves, whereas for gas dynamics it stays on the interface.

J. Fluid Mech., vol. 537, pp. 55-89, 2005.

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S. Schecter, B. Plohr, and D. Marchesin

We present a numerical method, based on the Dafermos regularization, for computing a one-parameter family of Riemann solutions of a system of conservation laws. This family corresponds to varying either the left or right state of the Riemann problem. The Riemann solutions are required to have shock waves that satisfy the viscous profile criterion prescribed by the physical model. The method uses standard continuation software to solve a boundary-value problem in which the left and right states of the Riemann problem appear as parameters. Because the continuation method can proceed around limit point bifurcations, it can sucessfully compute multiple solutions of a particular Riemann problem, including ones that correspond to unstable solutions of the viscous conservation laws.

Discrete and Continuous Dynamical Systems, vol. 10, pp. 965-986, 2004.

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SUNYSB-AMS-02-14

B. Plohr, D. Marchesin, P. Bedrikovetsky, J. E. Altoé Fº, and A. J. de Souza

We consider a model for immiscible three-phase (*e.g.*, water,
oil, and gas) flow in a porous medium. We allow the relative
permeability of the gas phase to exhibit hysteresis, in that it varies
irreversibly along two extreme paths (the imbibition and drainage
curves) that bound a region foliated by reversible paths (scanning
curves). By numerically solving one-dimensional flow problems involving
simultaneous and alternating injection of water and gas into a rock
core, we demonstrate the effects of hysteresis.

In Proceedings of the Eighth European Conference on the Mathematics of Oil Recovery, September 3-6, Freiberg, Germany, 2002.

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SUNYSB-AMS-02-01

Jane Hurley Wenstrom and Bradley J. Plohr

In this paper, we study systems of two conservation laws with homogeneous quadratic flux functions. We use the viscous profile criterion for shock admissibility. This criterion leads to the occurrence of non-classical transitional shock waves, which are sensitively dependent on the form of the viscosity matrix. The goal of this paper is to lay a foundation for investigating how the structure of solutions of the Riemann problem is affected by the choice of viscosity matrix.

Working in the framework of the fundamental wave manifold, we derive a necessary and sufficient condition on the model parameters for the presence of transitional shock waves. Using this condition, we are able to identify the regions in the wave manifold that correspond to transitional shock waves. Also, we determine the boundaries in the space of model parameters that separate models with differing numbers of transitional regions.

Comput. Appl. Math., vol. 26, pp. 1-33, 2007.

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SUNYSB-AMS-00-19

B. Plohr, D. Marchesin, P. Bedrikovetsky, and P. Krause

Two-phase flow in a porous medium can be modeled, using Darcy's law,
in terms of the relative permeability functions of the two fluids
(say, oil and water). The relative permeabilities generally depend
not only on the fluid saturations but also on the direction in which
the saturations are changing. During water injection, for example,
the relative oil permeability *k _{ro}* falls gradually
until a threshold is reached, at which stage the

In this work, we describe two models of permeability hysteresis.
Common to both models is that the scanning flow regime is modeled with
a family of curves along which the flow is reversible. In the Scanning
Hysteresis Model (SHM), the scanning curves are bounded by two curves,
the drainage and imbibition curves, where the flow can only occur in
a specific direction. The SHM is a heuristic model consistent with
experiments, but it does not have a nice mathematical specification.
For instance, the algorithm for constructing solutions of Riemann
problems involves several

The Scanning Hysteresis Model with Relaxation (SHMR) augments the SHM by (a) allowing the scanning flow to extend beyond the drainage and imbibition curves and (b) treating these two curves merely as attractors of states outside the scanning region. The attraction, or relaxation, occurs on a time scale that corresponds to the redistribution of phases within the pores of the medium driven by capillary forces. By means of a formal Chapman-Enskog expansion, we show that the SHM with additional viscosity arises from the SHMR in the limit of vanishing relaxation time, provided that the diffusion associated with capillarity exceeds that induced by relaxation. Moreover, through a rigorous study of traveling waves in the SHMR, we show that the shock waves used to solve Riemann problems in the SHM are precisely those that have diffusive profiles. Thus the analysis of the SHMR justifies the SHM model. Simulations based on a simple numerical method for the simulation of flow with hysteresis confirm our analysis.

Computational Geosciences, vol. 5, pp. 225-256, 2001.

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SUNYSB-AMS-00-20

Dan Marchesin and Bradley J. Plohr

We review recent progress in the theory of mixed-type systems of conservation laws with small diffusive terms, with emphasis on results pertinent to three-phase flow. In particular, we show that this theory can be applied to increase the rate of oil recovery, during certain production periods, in a recovery method commonly employed in petroleum engineering that is based on alternate injection of water and gas (WAG). The nonclassical "transitional" shock wave generated in the flow is the key to this improvement.

In Hyperbolic Problems: Theory, Numerics, Applications, vol. II, eds. H. Freistühler and G. Warnecke, pp. 693-702, International Series of Numerical Mathematics, vol. 141, Birkhäuser Verlag, Basel, 2001.

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SUNYSB-AMS-00-21

Arthur Azevedo, Dan Marchesin, Bradley J. Plohr, and Kevin Zumbrun

We investigate long-lasting solutions of two-component systems of conservation laws. These solutions arise from quasi-Riemann problems, which are perturbations of Riemann initial-value problems for the conservation laws augmented by a small parabolic term. Our study focuses on solutions containing a sequence of waves with the same speed, each wave having a viscous profile that is a saddle-saddle connection. Particular examples arise from homoclinic cycles, 2-cycles of shock waves, and 3-cycles of shock waves. Such wave sequences collapse in the large-time limit, but can survive for long time periods if the viscosity is small. This phenomenon sheds light on the ill-posedness of the problem of finding time-asymptotic solutions and on the related occurrence of Riemann problems that have multiple solutions. Moreover, long-lasting solutions depend continuously on the initial data; they therefore provide a global picture of quasi-Riemann solutions as a continuum even when the corresponding Riemann solutions do not depend continuously on the data.

Mat. Contemp., vol. 18, pp. 1-29, 2000.

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SUNYSB-AMS-00-24

John Walter, Dahai Yu, Bradley J. Plohr, John Grove, and James Glimm

An attractive approach for simulation of large deformation solid
dynamics is to combine Eulerian finite differencing with material
interface tracking. The Eulerian computational mesh is not subject to
mesh distortion, and tracking eliminates spurious numerical diffusion
at interfaces and the need for mixed-material computational cells.
We have developed such an approach within the framework of the front
tracking method, as implemented in the

Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, Report No. SUNYSB-AMS-00-24.

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SUNYSB-AMS-99-15

A. V. Azevedo, D. Marchesin, B. Plohr, and K. Zumbrun

Standard models for immiscible three-phase flow in porous media exhibit unusual behavior associated with loss of strict hyperbolicity. Anomalies were at one time thought to be confined to the region of nonhyperbolicity, where the purely convective form of the model is ill-posed. However, recent abstract results have revealed that diffusion terms, which are usually neglected, can have a significant effect. The delicate interplay between convection and diffusion determines a larger region of diffusive linear instability. For artificial and numerical diffusion, these two regions usually coincide, but in general they do not.

Accordingly, in this paper, we investigate models of immiscible three-phase flow that account for the physical diffusive effects caused by capillary pressure differences among the phases. Our results indicate that, indeed, the locus of instability is enlarged by the effects of capillarity, which therefore entails complicated behavior even in the region of strict hyperbolicity. More precisely, we demonstrate the following results.

(1) For general immiscible three-phase flow models, if there is stability near the boundary of the saturation triangle, then there exists a Dumortier-Roussarie-Sotomayor (DRS) bifurcation point within the region of strict hyperbolicity. Such a point lies on the boundary of the diffusive linear instability region. Moreover, as we have shown in previous works, existence of a DRS point (satisfying certain nondegeracy conditions) implies nonuniqueness of Riemann solutions, with corresponding nontrivial asymptotic dynamics at the diffusive level and ill-posedness for the purely convective form of the equations.

(2) Models employing the interpolation formula of Stone (1970) to define the relative permeabilities can be linearly unstable near a corner of the saturation triangle. We illustrate this instability with an example in which the two-phase permeabilities are quadratic.

Results (1) and (2) are obtained as consequences of more general theory concerning Majda-Pego stability and existence of DRS points, developed for any two-component system and applied to three-phase flow. These results establish the need for properly modelling capillary diffusion terms, for they have a significant influence on the well-posedness of the initial-value problem. They also suggest that generic immiscible three-phase flow models, such as those employing Stone permeabilities, are inadequate for describing three-phase flow.

Zeit. angew. Math. Phys., vol. 53, pp. 713-746, 2002.

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SUNYSB-AMS-99-03

Dan Marchesin and Bradley J. Plohr

In immiscible three-phase flow, the lead oil bank can split into two, a Buckley-Leverett shock wave followed by a new type of shock wave. Such a nonclassical "transitional" shock wave is common in three-phase flow. Its sensitivity to diffusion implies that capillary pressure must be modeled correctly in order to calculate the flow. In particular, transitional waves arise in WAG flow. They can be calculated by semi-analytic methods, which are helpful in the design of effective WAG recovery strategies.

SPE J., vol. 6, pp. 209-219, 2001.

Also in Proceedings of the SPE Annual Technical Conference and Exhibition, pp. 205--216, Society of Petroleum Engineers, Richardson, Texas, 1999.

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Los Alamos National Laboratory Report Number LA-UR-99-1796

John Walter, James Glimm, John Grove, Hyun-Cheol Hwang, Xiao Lin, Bradley J. Plohr, David Sharp, and Dahai Yu

An attractive approach for simulation of solid dynamics is to
combine an Eulerian finite difference method with material interface
tracking. The fixed Eulerian computational mesh is not subject to
mesh distortion, and the tracking eliminates spurious numerical
diffusion at interfaces and the need for mixed-material
computational cells. We have developed such an approach within the
framework of the front tracking method, as implemented in the
*FronTier* code. Our two-dimensional solid dynamics code is based on
a fully conservative formulation of the governing equations for
large-strain deformation, a hyperelastic equation of state that
allows for large volumetric change, and a rate-dependent plasticity
model for high strain rates; it features conservative finite
differencing, a Riemann solver that accounts for the nonlinearity of
longitudinal waves, and an implicit method for integrating the
plastic source term. This paper presents an overview of the
*FronTier Solid* code and some preliminary applications to high-velocity
impact and shock-accelerated interface problems.

In Proceedings of the 15th US Army Symposium on Solid Mechanics, eds. K. Iyer and S. Chou, pp. 343-366, Battelle Press, 1999. Published on CD-ROM (ISBN 1-57477-083-7).

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Pontificia Universidade Catolica do Rio de Janeiro Preprint

E. Costa Monteiro, E. Andrade Lima, C. Hall Barbosa, P. H. Ornelas, E. G. Cavalcante, S. F. Santos, D. Marchesin, B. Plohr, B. Gundelach, A. Marchesin, M. Brio, P. Costa Ribeiro

The arrhythmia known as atrial flutter is usually associated with a reentrant excitation of atrium cells. In such situation, instead of the normal linear activation originating at the right atrium sinus node and propagating toward the ventricles, the electrical activity begins to continuously rotate within the atrium. The detection of the anatomical path of such reentrant excitation is of special interest for its therapy through catheter ablation [1], which is the treatment of choice for drug resistant or drug intolerant atrial flutter. This technique consists of application of radiofrequency energy in specific areas, in order to interrupt the reentry path. The curative success of the selective catheter ablation technique is presently fully acknowledged, but one of the main concerns is the optimal method for localization of the reentry path, which vary from patient to patient. The localization is traditionally performed by intracardiac electrophysiology techniques, requiring a long exposition to X-rays by both the patient and the procedure team. Thus, the possibility of non-invasive localization of the reentrant electrical path in the tissue during atrial flutter in humans is of great clinical interest. In order to locate the reentry path, an inverse problem has to be solved, that is, given the magnetic field generated by the heart tissue, determine the associated configuration of currents. Inverse problem methods specially tailored to take into account the rotating pattern characteristics of the flutter arrhythmia will make optimal usage of the available experimental data.

In the present work, a SQUID system was used to detect the magnetic field associated with atrial flutter, which has been experimentally induced in isolated rabbit hearts. Through the analysis of the magnetic field spatial behavior, the propagation pattern of activation is identified, and the location of the center of the reentrant circuit can be assessed.

In Recent Advances in Biomagnetism: Proceedings of the 11th International Conference on Biomagnetism, eds. T. Yoshimoto, M. Kotani, S. Kuriki, H. Karibe, and N. Nakasoto, Tohuku University Press, Sendai, Japan, 1999.

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SUNYSB-AMS-97-28

A. V. Azevedo, D. Marchesin, B. Plohr, and K. Zumbrun

We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic field, nonclassical waves involve two fields in an essential way. This feature is reflected in the viscous profile differential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurcation occurring in the classical case. We carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points. One consequence of our analysis is that weak transitional waves arise in pairs, with profiles forming a 2-cycle configuration previously shown to lead to nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws, rather than with change of type in the associated hyperbolic system.

Commun. Math. Phys., vol. 202, pp. 267-290, 1999.

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SUNYSB-AMS-97-13

Bradley J. Plohr and David H. Sharp

When subjected to rapid acceleration, a metal plate that is not perfectly flat displays a type of Rayleigh-Taylor instability, which is affected by shear strength. We investigate the initial stage of this instability assuming that the deviation from flatness is small and the pressure producing the acceleration is moderate. Under these assumptions, the plate can be modeled as elastic and incompressible, and the linearized form of the governing are valid. We derive a linear initial/boundary-value problem that models the flow and obtain analytical formulae for the solutions. Our solutions exhibit vorticity inside the plate, an important feature caused by shear strength that was omitted in previous solutions. The theoretical relationship between the acceleration and the critical perturbation wave length, beyond which the flow is unstable, agrees quantitatively with results of numerical simulations.

Zeit. angew. Math. Phys., vol. 49, pp. 786-804, 1998.

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SUNYSB-AMS-97-07

Stephen Schecter, Bradley J. Plohr, and Dan Marchesin

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data.

In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.

J. Dynamics and Differential Equations, vol. 13, pp. 523-588, 2001.

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SUNYSB-AMS-96-14

Bradley J. Plohr

We present a short introduction to continuum models for the plastic flow of metals. Our emphasis is on the physical principles underlying these models, the nature and validity of approximations involved, and the mathematical structure of the flow equations. Using the framework developed, we derive a simple, but realistic, model describing one-dimensional plastic flow.

Mat. Contemp., vol. 11, pp. 95-120, 1996.

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SUNYSB-AMS-95-07

Arthur Azevedo, Dan Marchesin, Bradley J. Plohr, and Kevin Zumbrun

We investigate a general mechanism, utilizing nonclassical shock waves,
for nonuniqueness of solutions of Riemann initial-value problems for
systems of two conservation laws. This nonuniqueness occurs whenever
there exists a pair of viscous shock waves forming a 2-cycle, *i.e.*, two
states *U_1* and *U_2* such that a traveling wave leads from
*U_1* to *U_2* and another leads from *U_2* to *U_1*.
We prove that a 2-cycle
gives rise to an open region of Riemann data for which there exist
multiple solutions of the Riemann problem, and we determine all
solutions within a certain class. We also present results from
numerical experiments that illustrate how these solutions arise in the
time-asymptotic limit of solutions of the conservation laws, as
augmented by viscosity terms.

Zeit. angew. Math. Phys., vol. 47, pp. 977-998, 1996.

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SUNYSB-AMS-95-04

James Glimm, Bradley J. Plohr, and David H. Sharp

We develop a model for the dynamics of a fully developed shear band that allows effective computation across several length scales. From a macroscopic point of view, a shear band is a discontinuity in tangential velocity that supports a shear stress. Numerical simulation of the full system of governing equations reveals that the internal structure of the band consists of a quasistatic core surrounded by a thermal layer. We show that the shear band can be modeled as a composite structure whose evolution is governed by an integral equation, coupled to the external flow through jump conditions. We establish the accuracy of the model equations by numerical experiments.

Mech. Materials, vol. 24, pp. 31-41, 1996.

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SUNYSB-AMS-94-21

Feng Wang, James Glimm, and Bradley J. Plohr

The purpose of this note is to propose an improvement to the widely-used Steinberg-Lund model for rate-dependent plasticity. The improved model is rate-dependent above, as well as below, the Peierls stress. As a result of this change, one material constant in the Steinberg-Lund model acquires a new value, and we thereby obtain a revised estimate for the dislocation density in tantalum.

J. Mech. Phys. Solids., vol. 43, pp. 1497-1503, 1995.

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SUNYSB-AMS-94-10

Stephen Schecter, Dan Marchesin, and Bradley J. Plohr

We study the structure of solutions of Riemann problems for systems of two conservation laws. Such a solution comprises a sequence of elementary waves, viz., rarefaction and shock waves of various types; shock waves are required to have viscous profiles. We construct a Riemann solution by solving a system of equations characterizing its component waves. A Riemann solution is ``structurally stable'' if the number and types of its component waves are preserved when the initial data and the flux function are perturbed.

Under the assumption that rarefaction waves and shock states lie in the strictly hyperbolic region, we characterize Riemann solutions for which the defining equations have maximal rank and we prove that such solutions are structurally stable. Structurally stable Riemann solutions cannot contain overcompressive shock waves, but they can contain transitional shock waves, including doubly sonic transitional shock waves that have not been observed before.

J. Differential Equations, vol. 126, pp. 303-354, 1996.

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SUNYSB-AMS-94-05

Dan Marchesin, Bradley J. Plohr, and Stephen Schecter

We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and bifurcation theory of dynamical systems, including Melnikov's method, to find all nearby shock waves that are admissible according to the viscous profile criterion. We use these results to construct a unique solution of the Riemann problem for each left and right state and parameter value in a neighborhood of the unperturbed shock wave solution; together with previous numerical work, this construction completes the solution of the three-phase flow model. In the bifurcation analysis, the unperturbed shock wave acts as an organizing center for the waves appearing in Riemann solutions.

SIAM J. Appl. Math., vol. 57, pp. 1189-1215, 1997.

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SUNYSB-AMS-93-16

Jane M. Hurley and Bradley J. Plohr

This paper concerns 2×2 systems of conservation laws with quadratic fluxes corresponding to Case II in the classification of such problems. Aided by a computer program, we have constructed the solution that satisfies the viscous profile criterion for shock admissibility. Our solution differs from that obtained using the Lax admissibility criterion, even though solutions exist and are unique for both criteria. With the viscous profile criterion, some nonlocal Lax shock waves are inadmissible; in their place, transitional waves appear in the wave patterns.

Mat. Contemp., vol. 8, pp. 203-224, 1995.

SUNYSB-AMS-93-14

James Glimm, Bradley J. Plohr, and David Sharp

We explain several ideas which may, either singly or in combination, help achieve high resolution in simulations of large-deformation plasticity. Because of the large deformations, we work in the Eulerian picture. The governing equations are written in a fully conservative form, which are correct for discontinuous as well as continuous solutions. Models of shear bands are discussed. These models describe the internal dynamics of a developed shear band in terms of time-asymptotic states; in other words, the smooth internal structure is replaced by a jump discontinuity, and the shear band evolution is determined by jump relations. This analysis is useful for high resolution numerical methods, including both shock capturing and shock tracking schemes, as well as for the understanding and validation of computations, independently of the underlying method. Preliminary computations, which illustrate the feasibility of these ideas, are presented.

Appl. Mech. Rev., vol. 46, pp. 519-526, 1993.

SUNYSB-AMS-93-04

Feng Wang, James G. Glimm, John W. Grove, Bradley J. Plohr, and David H. Sharp

We present a numerical method for computing elasto-plastic flows in metals. The method uses a conservative Eulerian formulation of elasto-plasticity together with a higher-order Godunov finite difference method combined with tracking of material boundaries. The Eulerian approach avoids the problem of mesh distortion caused by a Lagrangian remap, and can be easily extended to the computation of flows in multiple space dimensions using operator splitting. The method is validated by a comparison of computations with experiments on one-dimensional high velocity plate impact. We obtain excellent agreement between our computations and experiment.

Impact Comput. Sci. Engrg., vol. 5, pp. 285-308, 1993.

SUNYSB-AMS-92-13

Sunčica Čanić (Suncica Canic) and Bradley J. Plohr

In this work we present a new approach to the study of the stability of
admissible shock wave solutions for systems of conservation laws that
change type. The systems we treat have quadratic flux functions. We
employ the fundamental wave manifold *W* as a global framework to
characterize shock waves that comply with the viscosity admissibility
criterion. Points of *W* parametrize dynamical systems associated
with shock wave solutions.

The region of *W* comprising admissible shock waves is bounded by the
loci of structurally unstable dynamical systems. Explicit formulae are
presented for the loci associated with saddle-node, Hopf, and
Bogdanov-Takens bifurcation, and with straight-line heteroclinic
connections. Using Melnikov's integral analysis, we calculate the
tangent to the homoclinic part of the admissibility boundary at
Bogdanov-Takens points of *W*. Furthermore, using numerical methods,
we explore the heteroclinic loci corresponding to curved connecting
orbits and the complete homoclinic locus.

We find the region of admissible waves for a generic, two-dimensional slice of the fundamental wave manifold, and compare it with the set of shock points that comply with the Lax admissibility criterion, thereby elucidating how this criterion differs from viscous profile admissibility.

J. Differential Equations, vol. 118, pp. 293-335, 1995.

Los Alamos National Laboratory Report Number LA-UR-92-5

Bradley J. Plohr and David Sharp

In this paper we propose a fully conservative form for the continuum equations governing rate-dependent and rate-independent plastic flow in metals. The conservation laws are valid for discontinuous as well as smooth solutions. In the rate-dependent case, the evolution equations are in divergence form, with the plastic strain being passively convected and augmented by source terms. In the rate-independent case, the conservation laws involve a Lagrange multiplier that is determined by a set of constraints; we show that Riemann problems for this system admit scale-invariant solutions.

Adv. Appl. Math., vol. 13, pp. 462-493, 1992.

SUNYSB-AMS-91-10

Sunčica Čanić (Suncica Canic) and Bradley J. Plohr

We present a new approach to characterizing admissible shock wave
solutions for systems of conservation laws. A shock wave is admissible
if it has a viscous profile, *i.e.*, its end states are joined by an orbit
for an associated dynamical system. The family of all such dynamical
systems is parameterized by the fundamental wave manifold *W*, which
provides a global setting for studying shock waves. The regions of
*W* comprising admissible shock waves are bounded by loci of
structurally unstable dynamical systems.

We give explicit formulae for many of the admissibility boundaries for
systems of two conservation laws with quadratic fluxes. These
boundaries include the loci associated with saddle-node, Hopf, and
certain heteroclinic bifurcations. Furthermore, we explore other
heteroclinic loci and the homoclinic locus using numerical methods.
One surprising observation is that the heteroclinic loci appear to form
ruled surfaces within *W*.

Anais do 18° Colóquio Brasileiro Matemática, ed. M. J. Pacífico, pp. 199-216, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, 1993.

SUNYSB-AMS-91-09

Kevin Zumbrun, Bradley J. Plohr, and Dan Marchesin

We study the stability and asymptotic behavior of transitional shock waves as solutions of a parabolic system of conservation laws. In contrast to classical shock waves, transitional shock waves are sensitive to the precise form of the parabolic term, not only in their internal structure but also in terms of the end states that they connect.

In our numerical investigation, these waves exhibit robust stability. Moreover, their response to perturbation differs from that of classical waves; in particular, the asymptotic state of a perturbed transitional wave depends on the location of the perturbation relative to the shock wave. We develop a linear scattering model that predicts behavior agreeing quantitatively with our numerical results.

Mat. Contemp., vol. 3, pp. 191-209, 1993.

SUNYSB-AMS-91-08

Eli Isaacson, Dan Marchesin, C. Frederico Palmeira, and Bradley J. Plohr

We introduce a unifying framework for treating all of the fundamental
waves occurring in general systems of *n* conservation laws.
Fundamental waves are represented as pairs of states satisfying the
Rankine-Hugoniot conditions; after trivial solutions have been
eliminated by means of a blow-up procedure, these pairs form an
(*n*+1)-dimensional manifold *W*, the fundamental wave manifold.
There is a distinguished *n*-dimensional submanifold of *W*
containing a single one-dimensional foliation that represents the rarefaction
curves for all families. Similarly, there is a foliation of *W*
itself that represents shock curves. We identify other *n*-dimensional
submanifolds of *W* that are naturally interpreted as boundaries of
regions of admissible shock waves. These submanifolds also have
one-dimensional foliations, which represent curves of composite waves.

This geometric framework promises to simplify greatly the study of the stability and bifurcation properties of global solutions of Riemann problems for mixed hyperbolic-elliptic systems. In particular, bifurcations of wave curves can be understood as resulting from loss of transversality between foliations and admissibility boundaries.

Commun. Math. Phys., vol. 146, pp. 505-552, 1992.

*last modified: March 14, 2008 by Bradley J. Plohr*
<plohr@*lanl.gov*>