hyperbolic heat conduction and thermomechanical response

Hyperbolic thermal transport in acomposite microstructure: Thisanimation shows wave-like thermaltransport in a composite materialsystem; the circular inclusions havelower conductivity and highercapacitance than the matrix. Sharp,wave-like response is observed atearly times, but reflections anddiffusive effects lead to nearlyparabolic response at later times.Height and color indicate temperature.

Animations corresponding to thisresearch are available athttp://www.cpsd.uiuc.edu.

Hyperbolic Heat Conduction and ThermomechanicalResponseFaculty: Robert B. Haber (Mechanical Science & Engineering) and Duane Johnson (Materials Science& Engineering)Students: Scott Miller (Mechanical Science & Engineering) and Brent Kraczek (Materials Science &Engineering)

ObjectiveThe parabolic Fourier heat equation, although useful in manysituations, implies infinite propagation speed and is ineffective atthe very small length and time scales associated with nanoscalesystems. We seek an effective numerical implementation ofhyperbolic thermal and thermomechanical models to avoid theseproblems.

ApproachWe use a spacetime discontinuous Galerkin method for systems ofconservation laws to implement the hyperbolic Maxwell-Cattaneo-Vernotte thermal model and a coupled three-field modelfor elastodynamics. An adaptive spacetime solution procedureresolves multiscale response. Recent progress includes treatmentof bi-material interfaces for modeling composite microstructures.

ImpactThis high-resolution model can be applied to pulsed lasers incorneal surgery, nanotechnology (e.g., CPU overheating, phase-change data storage, micromachining of thin films with pulsedlasers), and thermomechanical dynamic fracture. A similar methodcan model the dynamics of phase transitions (generalized Cahn-Hilliard equation applied to shape memory alloys), biotransport aswell as chemisorption and hydrogen storage.

Center for Process Simulation and Design (UIUC), http://www.cpsd.uiuc.edu/.Principal investigators: Robert B. Haber, Duane D. Johnson, Jonathan A. Dantzig DMR-0121695. Copyright 2005-2007.

Top: a solidification front (green)approaching two cells. Bottom:example of the capture (small cell tothe right) or pushing (large cell to theleft) of cells, depending on the localinterface morphology and the cell sizeand properties.

Interaction of Biological Cells with Solidification FrontsFaculty: Jon Dantzig (Mechanical Sciences And Engineering)Students: A. Chang

ResearchThe goal of this project is to develop computational models forcryopreservation - the freezing of biological cells. The ice crystalsmay reject or engulf the cells, and the local composition profiledetermines whether the cells survive. We use an adaptive level setprocedure to track the moving interface and determine theinteraction of the cells with it. The image at right shows anexample of the capture or pushing of cells, depending on the localinterface morphology and the cell size and properties.

Broader ImpactsThis work represents an opportunity to use engineering principlesand techniques to examine biological problems. One studentreceived his PhD and is now a postdoctoral researcher at HarvardMedical School. The work has been published in two journalarticles.

Publications arising from this research

A. Chang, J. A. Dantzig, B. T. Darr, and A. Hubel. Modelingthe interaction of biological cells with a solidifyiing interface,J. Comp. Phys. 2007. In press.A. Chang, J. A. Dantzig, B. T. Darr, and A. Hubel. Cellpartitioning during the directional solidification of trehalosesolutions, Cryobiology. 2007. In press.


First transient study of fractureprocess zone size shows that zone sizeapproaches zero as crack-tip velocitynears the material’s Rayleigh wavespeed.

Spacetime finite element simulation ofdynamic fracture for mixed-modeshock loading. Adaptive spacetimemeshing tracks solution-dependentcrack path. Faster tensile pressurewave causes initial straight-linegrowth; slower shear waves causechange in direction. Quasi-singularvelocity spike appears as crackaccelerates. Color indicates strainenergy density; height shows velocitymagnitude.


Multi-scale Spacetime Simulation of Dynamic FractureFaculty: Robert Haber (Mechanical Science & Engineering)Students: Reza Abedi, Morgan Hawker (Mechanical Sciencse & Engineering)

ObjectiveDynamic fracture governs important modes of material failure aswell as the mechanics of earthquakes at much larger length scales.We develop new numerical methods to simulate dynamic fracturewith unprecedented levels of detail. We discovered unexpectedevidence of singular velocity response at cohesive crack tips; weseek deeper understanding of this phenomenon.

ApproachWe embed a cohesive failure model within an elastodynamicspacetime finite element simulation. Adaptive analysis techniquesguarantee very high-resolution solutions that can track arbitrarycrack paths. Post-simulation visualizations reveal quasi-singular(non-singular core) velocity response when the fracture processzone is sufficiently small; i.e., in high-strength materials and inweaker materials with faster crack velocities.

ImpactDiscovery of quasi-singular velocity response led to a rethinkingof fracture kinetics. Unprecedented resolution supports studies ofmicrocracking and other multi-scale phenomena for betterprediction of dynamic fractures in engineered materials and alongfault lines in earthquakes.


The images show the intersection offour crystals growing from the melt(top), along with part of the adaptedgrid (bottom).

Renormalization group methods for multiscale materialspattern formationFaculty: Nigel Goldenfeld (Physics), Jon Dantzig (MechSE)Students and Post docs: B. Athreya, P. Chan, Z. Huang

ResearchThe goal of this project is to develop multiscale methods forsimulating the development of materials microstructure. Ourapproach is based upon a continuum representation of atomicdensity, obeying diffusive dynamics. During the course of thisproject we have developed analytical methods to describe thecoarse-grained dynamics of the atomic density, suitable forsolution on adaptive grids. This year, we have extended themethod from 2D to 3D. The image shows the intersection of fourcrystals growing from the melt, along with part of the adaptedgrid.

Broader ImpactsThis work was performed by an interdisciplinary team ofmechanical engineers and theoretical physicists. Three studentshave been associated with this project, including two who havegraduated and moved to industry. We have disseminated thefindings through several keynote talks at conferences, as well asnumerous publications.


Sharp-interface coupling: solidmodeled by atomistic region betweentwo continuum zones.

Example below: Snapshots of right-traveling wave in a 1-D systemcomprised of linear-spring atomisticmodel coupled to elastic continuum.Wave starts in continuum zone;dispersion develops correctly inatomistic zone, as expected.

Above: non-optimal coupling -– energyand momentum balance, but left-traveling reflection is evident.

Above: Optimal coupling -- a 3-atomtrace operator in SDG model largelysuppresses spurious reflection.


Spacetime, balance-law formulation of coupled atomisticand continuum dynamics for solidsFaculty: Duane Johnson (Materials Science & Engineering), Robert Haber (Mechanical Science &Engineering)Students: Brent Kraczek (Physics)

ObjectiveReliable atomistic-continuum coupling method for dynamics ofsolids. While other methods use kinematic coupling inoverlapping continuum and atomistic zones, we seek a sharp-interface model that enforces momentum and energy balance, aswell as kinematic compatibility.

IssuesContinuum and atomistic models use distinct mathematicaldescriptions (local vs. non-local, continuous vs. discrete) forenergy and momentum transfer in solids; we seek a couplingmethod that bridges these differences. Most coupling methodssuffer spurious wave reflections at the continuum-atomisticinterface (or damp them, removing energy from the system).

ApproachWe use Spacetime Discontinuous Galerkin formulation withbalance-law coupling as unifying mathematical framework. Ourmethod balances energy and momentum – both globally andlocally at coupling interface. We can use any standard linear ornon-linear potential to define the atomistic model. We optimize anew atomistic trace operator to suppress spurious reflections.

ImpactComputer simulations play central role in analysis, prediction anddesign of structural, energy and biomaterials. Our method willpermit long-duration and accurate simulations of coupled systemswhen implemented within an adaptive SDG software framework.

PublicationsA paper about this work was submitted to the Journal ofComputational Physics (2007).


hyperbolic heat conduction and thermomechanical response

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