1. Priorities and Strategies Meeting, 02/07/05 1 Computational Science in LACSI  PDE discretization on unstructured meshes Current focus: polyhedral mimetic finite difference methods for the Laplacian operator in parabolic (diffusion) equations  Customer: Shavano Project (appropriate for many others) PI: Prof. Yuri Kuznetsov, UH Dept of Mathematics A successful collaboration with X-3 (Scott Runnels) and T-7 (Misha Shashkov, Konstantin Lipnikov)  Tools for code-based sensitivity, V&V Computing derivatives with automatic differentiation (AD) Evolution of “adifor” tool: AD for F90 with Truchas code and Ubiksolve linear solver library as initial target PI: Mike Fagan, Rice Dept of Computational and Applied Mathematics

2. Priorities and Strategies Meeting, 02/07/05 2 If G and D are transpose to each other then will be symmetric and positive definite. (A key goal) Discrete Diffusion Continuous Diffusion Steady-State Diffusion

3. Priorities and Strategies Meeting, 02/07/05 3 More detail: The above formula demonstrates that the inner products using the divergence and gradient operators are equal, which is the definition of adjointess. First a way to discretize the divergence is invented (i.e., we invent D ). Second, that D is inserted into the above equation, along with other approximations for the integrals, to derive a G. Theoretical Motivation

4. Priorities and Strategies Meeting, 02/07/05 4 Handling General Polyhedrons For general polyhedra, discretizing the volume integral presents difficulties. This was the major focus of the UH effort: How can the mimetic approach be extended to this case? Handling Polyhedra

5. Priorities and Strategies Meeting, 02/07/05 5 Break the polyhedron into tets. Establish new constraints to ensure 2 nd -order spatial convergence. This new idea is what is now being tested by T-7/X-3. Handling Polyhedra

6. Priorities and Strategies Meeting, 02/07/05 6 Polygonal Mesh AMR Mesh Results by Konstantin Lipnikov Order 2 for  Order 1.5 for flux Same Spatial Convergence: 2D Results: Tensor Diffusion

7. Priorities and Strategies Meeting, 02/07/05 7 Status Capabilities have been preserved for very general meshes. Accurate (2 nd order) Easy-to-solve matrix (SPD) Details  Developed in Shavano architecture  > 90 pages of documentation  All goals met: Parallel, transient, 3D, 2D Cartesian and r-z.  Strong SQA  Technical success and programmatically relevant No restrictions: Any polygon/polyhedron Any connectivity Source: Scott Runnels, X-3

8. Priorities and Strategies Meeting, 02/07/05 8 Cooperation and Information Flow Academic Investigation New, Risky Liaison Co-Development Academic Testing Application Requirements Feasible IdeasTechnology Guidance Keys to success  X-3 leadership who sets clear requirements and supports interaction (Burton)  An X-3 person dedicated to ensuring success (Runnels)  X-3 (Shavano) team buy-in and expertise (Kenamond, Gianakon, Berry)  CCS & T-7 support and integrated technology (Morel, Berndt)  A T-7 technical expert who contributes to the X-3 program (Lipnikov)  A T-7 person who guides and collaborates with UH (Shashkov)  An effective and responsive academician (Kuznetsov) UHT-7X-3 CCS LAMG linear solver Tight Collaboration with LANL

9. Priorities and Strategies Meeting, 02/07/05 9 Conserving and robust method for enforcing bound preservation More general grids and non- planar 3-D cell faces Cells of mixed materials B A Future Directions/Needs

10. Priorities and Strategies Meeting, 02/07/05 10 MFD Efforts: Success and Impact  Success metrics Evolution of algorithm accuracy/robustness  More efficient as well? Staff development  Students -> postdoc -> new LANL staff  Education/training of existing LANL staff  Impact Now: Implementation in Shavano Project software – available in FY05 releases Future: Truchas code, AMR mesh codes

11. Priorities and Strategies Meeting, 02/07/05 11 AD work at Rice  PI: Mike Fagan  Usefulness of AD? Code sensitivities: relation of output to input  Narrows focus to relevant models/algorithms Optimization: use derivatives for searching  Interface with SNL’s Dakota tool? Nonlinear methods: approximate the Jacobian? Verification: facilitating the use of MMS  Key question: Can an AD tool be used on the large ASC codes? If not, can it be applied on key kernels?  ASC is entering a phase where the codes are more mature and stable Therefore ready for this tool

12. Priorities and Strategies Meeting, 02/07/05 12 ASC Computational Science Needs  PDE discretization methods For unstructured and Eulerian/AMR meshes  V&V tools/methodologies; UQ, sensitivities  Methods for nonlinear multi-physics time integration  Linear and nonlinear solvers  Interface kinematics and dynamics Motion and physics around interfaces bounding N materials  Mesh management (for ALE, setup) Generation, motion, smoothing, remap/rezone  Methods for computational mechanics Material damage/failure on Eulerian meshes  Methods for turbulence/mix @ interfaces  Homogenization techniques for mixed materials  Transport methods: quicker, more efficient/accurate  Modeling unresolved scales

13. Priorities and Strategies Meeting, 02/07/05 13 LACSI FY06 Computational Science Efforts  Advanced Polyhedral Discretization Methods for Diffusion-Type Problems in Strongly Heterogeneous Media PI: Yuri Kuznetsov (UH) Continuation of FY05 work: presence of material discontinuities, monotonicity constraints, homogenization, performance  Application of AD tools PI: Mike Fagan (Rice) Targets: Telluride, …

14. Priorities and Strategies Meeting, 02/07/05 14 Computational Science Considerations  Metrics Staff recruitment, sabbatical opportunities, capability search/identification  Evaluation criteria Matching with requirements, approach for collaboration  Match with thrust areas (still TBD) on Weapons Science (WS) Foundation, e.g.: PDE discretization Interfacial physics models and methods Transport models and methods Multi-physics coupling V&V methodologies  Added value Have an Adv. Apps Project customer Close partnering with CS community (performance, SQE)  Find out about planned WS proposals from X/T/CCS