# Presentation: 301551

## Krylov Space Approximations for Materials Science Problems

**RESULTS:** Calculating the properties and behavior materials has a high computational demand, which currently makes analyzing larger systems prohibitively expensive. Often, increasing the number of elements in a computation can increase the computing time significantly. Our research deals with some specific components of the calculations: linear systems and least squares, eigenvalue problems, systems of ODEs, and approximations to functions of matrices. Our approach is to recycle information from earlier computations, to make subsequent computations substantially more efficient. Some simulations which previously required supercomputers now only require desktop machines.

**Advanced materials:** We have computed an optimal design for a material that becomes thicker when stretched in one direction. We investigate both the ideal shape of a single element of the material (inset) and the behavior of the whole material.

**Krylov Space Approximations ****for Materials Science Problems**

**WHY IT MATTERS:** Computer science plays a key role in modeling and creating materials that have unique properties and behavior. We designed, for example, a material that both widens and lengthens when stretched, and narrows when compressed — unlike almost all known materials. Potential applications include fasteners, such as ideal nails, and shock-absorbing material. We are also calculating the properties of important deep-earth materials, with unprecedented accuracy. At depths greater than 100 kilometers (60 miles), these materials can be difficult to obtain (some are found only in meteorites) and therefore cannot be evaluated with laboratory experiments. Computational mineral physics is part of a larger effort to predict Earth’s behavior and to understand how it became a habitable planet.

## NEXT STEPS: We plan to extend these models to the building blocks of a solid (at the scale of an individual atom and of electrons), and to perform even larger simulations. 

- For effective recycling, we consider the effects of small changes in the matrix: perturbations of solutions, invariant subspaces (or eigenvectors), and other relevant values, vectors, or spaces.
- Our primary objectives are:
- •  to determine when recycling is appropriate and efficient. For example, in simulating crack propagation, the linear solver is faster when low-frequency modes are removed from the problem. These modes do not change for a tiny propagation of the crack. Hence, they are appropriate for recycling and the next linear system can be solved faster.
- •  to create a solver that copes with continually changing systems. For example, after sufficient propagation of a crack, the low-frequency modes change drastically, but our solver+recycling method is able to cope with the rate of change.
- •  to take advantage of parallel computing.

**NEXT STEPS:**** **We plan to extend these models to the building blocks of a solid (at the scale of an individual atom and of electrons), and to perform even larger simulations.** **
**PUBLICATIONS ARISING FROM THIS RESEARCH:**
- •  Misha Kilmer and Eric de Sturler, “Recycling Subspace Information for Diffuse Optical Tomography”, *SIAM Journal on Scientific Computing* **27** (6), 2140-66, 27 pages (2006)
- •  Michael L. Parks, Eric de Sturler, Greg Mackey, Duane D. Johnson, and Spandan Maiti, “Recycling Krylov Subspaces for Sequences of Linear Systems”, *SIAM Journal on Scientific Computing*, 24 pages (accepted 2006)
- •  Shun Wang, Eric de Sturler, and Glaucio H. Paulino, “Large-Scale Topology Optimization using Preconditioned Krylov Subspace Methods with Recycling”, *International Journal for Numerical Methods in Engineering*, 28 pages (accepted 2006)
**PRINCIPAL INVESTIGATORS:**
- E. de Sturler (CS, Virginia Tech), G. Paulino (Civ. Eng, UIUC), D. Ceperley (Phys. and NCSA, UIUC), J. Kim (NCSA, UIUC); Students: Z. Cheng (CS, UIUC), G. Mackey (Sandia Nat’l Lab), M. Parks (Sandia Nat’l Lab), C. Siefert (CS, UIUC), S. Wang (CS, UIUC)

**Notes:**
Under primary objectives, the fast-coping solver can be described another way: The behavior of the material will change drastically, depending on the size of the crack.  The solver can handle when the problem as the length of the crack changes.

Update on student researchers:
Zhen Cheng graduated with a master’s degree from UIUC-CS, and is now at Morgan Stanley. She was directly supported by the MCC.
Greg Mackey graduated and is at Sandia National Lab.
Chris Siefert is now Sandia National Lab, and was supported by the geophysics grant under David Ceperley and Eric de Sturler.