LSMLIB Applications

Dynamics of dislocations and grain boundary migration

Kevin T. Chu and David Srolovitz's research group, Mechanical & Aerospace Engineering, Princeton University

Understanding the evolution of dislocations and grain boundaries in materials continues to present an important and interesting challenge for material scientists due to their role in plastic deformations. We are actively studying both of these microstructural entities using mesoscopic computational models based on the level set method.

In our model, dislocations are represented by the intersection of the zero level sets of two level set functions. An important consequence of this implicit representation of the dislocation line is that topological changes that occur when dislocation lines interact with obstacles and with each other take place automatically without requiring complicated ``surgical procedures.''

The image on the left (courtesy of Zhaoxuan Wu) shows six interacting dislocation lines. The image on the right shows the near equilibrium structure of a pure twist grain boundary corresponding to a straight-line description that satisfies Frank's formulas.

Investigating pore scale configurations of two immiscible fluids

Masa Prodanovic and Steven L. Bryant, Institute for Computational Engineering and Sciences, University of Texas at Austin

The interface between two immiscible fluids at equilibrium (constant pressure and surface tension) in a porous medium structure can be modeled as a constant mean curvature surface. The irregularity of the microstructure, however, makes it difficult to evaluate the surface curvature and corresponding geometric configuration of the two fluids. The figure on the left shows the fluid-fluid interface (blue surface) at a critical curvature in a porous medium "throat" delineated by three spheres (gray surface). Nonwetting fluid advances into the throat toward the viewer as the capillary pressure  (pressure difference between the fluids) is gradually increased. The figure on the right shows the fluid-fluid interface (red surface) at a slightly higher curvature (or equivalently, a higher capillary pressure). Notice that the stable meniscus no longer exists. Instead, the fluid-fluid interface (red surface) splits into disjoint rings around sphere contacts. This pore scale event is known as Haines jump.

LSMLIB was used to develop a code based on the constant curvature LSM equation. Level set visualization (isosurfacing) was done with the help of GTS and the output was viewed in Geomview.

Shape optimization for multifunctional material optimization

Youngjean Jung, Kevin T. Chu and Salvatore Torquato Princeton University

The level set method is naturally suited to shape optimization problems where the objective function and constraints are representable (possibly implicitly) as a functionals of the level set function. In these situations, it is possible to use variational calculus to devise a constrained, steepest descent optimization procedure. In addition to yielding a computational procedure, the variational level set method approach also provides us with a means to theoretically characterize optimal surfaces.

We have used a variational, level set-based shape optimization approach to study the properties of triply periodic microstructures. Specifically, we have explored the structure of triply periodic surfaces that minimize the total surface area while satisfying a constraint on the volume fractions of the two phases it separates. One of the main conclusions of this work is that surfaces that optimize the total surface area under a volume fraction constraint are precisely those possessing a constant mean curvature. Recently, we have begun to explore the material properties of composite materials whose microstructures are defined by this class of surfaces. Preliminary results indicate that these microstructures correspond to optima with respect to multi-functional optimization of effective material properties.

The top figure shows the Schwarz P surface at a volume fraction of 0.5. The six panels below that show the Schwarz P-family of surfaces that minimize the total surface area when the volume fraction of phase 1 is set to 0.25, 0.35, 0.45, 0.55, 0.65, and 0.75, respectively.

Two-phase Stokes Flow

Xiaohai Wan, Biomathematics Graduate Program, North Carolina State University

Relaxation of a two-dimensional, starfish-shaped fluid droplet under the influence of surface tension. The animation on the left shows the time-evolution of the starfish-shaped droplet to a circle. The image on the right shows the discontinuity of the pressure profile across the interface between the two fluids. The two-phase Stoke's flow problem was solved using the immersed interface method; the level set function used to implicitly represent the interface was updated using LSMLIB.

Crowd Flows

Yi Li, Robotic Algorithms and Motion Planning (RAMP) Lab, School of Engineering Science, Simon Fraser University

Physically accurate crowd models are widely used in computer graphics. Currently, most crowd models are agent-based. Because agent-based models separate global path planning and local collision avoidance, it is difficult to consistently produce realistic motion. Crowd models based on continuum dynamics, however, produce smooth flows that exhibit many of the features observed in real crowds. Moreover, continuum models may be simulated in real-time using efficient algorithms based on the fast marching method.

In the continuum model, we assume that agents in a virtual environment are grouped into a few groups, where agents in each group have a common destination. During each time step, we first solve an Eikonal equation for each group to construct a potential field given the terrain, the location of obstacles and the position of other agents in the environment. The speed function in the Eikonal equation is set so that it is faster in open areas and slower in regions with obstacles, other agents, or rough terrain. Next, we compute the velocity of each agent -- each agent moves in the direction of the gradient of the potential field with a speed that is a function of the topographical speed (which depends on the terrain) and the flow speed (which depends on other agents nearby). Finally, we update the position of each agent. Notice that by using the continuum model, we have eliminated the need for explicit collision avoidance, which makes it possible to handle thousands of agents at several frames per second at coarse discretizations of the virtual environment.

This real-time crowd model was proposed in "Continuum Crowds" by A. Treuille, S. Cooper, and Z. Popovic at SIGGRAPH 2006. The paper and its accompanied video are located at

In the following five figures (ordered from top to bottom), there are two groups of agents. Each group has three members. The grid size is 21 by 21. The line behind each agent indicates path that the agent has travelled from its starting position. The agents exhibit smoother motion and made fewer sudden changes in direction compared to agent-based approaches. They form also lanes while approaching each other. This emergent phenomena has been observed in real crowds.

The fast marching method functionality provided by LSMLIB is used to solve the Eikonal equations that arise during each time step.