Patrick M. Knupp Distinguished Member Technical Staff Optimization and Uncertainty Estimation Dept.

Mathematical Information and Computational Sciences (MICS)

Title: R-Adaptive Mesh Quality Improvement Using the Target-Matrix Paradigm

Patrick M. Knupp, PI (Sandia National Laboratories)
Lori Freitag Diachin (Lawrence Livermore National Laboratories)
Todd Munson (Argonne National Laboratories)

Executive Summary:

Mesh optimization has long been used to to improve partial differential equation (PDE) solution efficiency and accuracy by adjusting mesh element size, shape, and orientation. There has been siginificant research on the development of a priori methods that optimize the mesh according to some predetermined criteria which may or may not be the best choice to minimize solution error. More recent investigations have begun to directly improve the mesh according to solution features or error indicators. Mesh optimization is difficult to do well and thus has unrealized potential. The authors believe that their established and widely recognized team is well-positioned to make significant progress towards realizing that potential. Their mathematical approach to mesh optimization has yielded signifcant research results and has identified many promising avenues for further research. Because their current (and future) MICS research is focused on the link between geometric mesh properties and physical solution characteristics, they expect to effect a paradigm shift in the way mesh optimization is done. To complete the paradigm shift, the authors will concentrate on r-adaptivity in which mesh vertices move according to solution characteristics or error indicators. By focusing their recent research results on the problem of r-adaptivity they will improve simulation accuracy and efficiency and thus impact a wide variety of applications of importance to the Office of Science and to DOE in general.

The bulk of our previous work in mesh improvement, particularly in the area of vertex movement, focused on improving geometric properties of the mesh and has resulted in
1. the initial development of a comprehensive mathematical theory of matrix-based mesh quality metrics,
2. a unifying paradigm that directly poses the mesh quality improvement problem as an optimization problem; and
3. a thorough investigation of several numerical optimization techniques for certain quality metrics and improvement goals.

This work has been widely recognized by the research community and positions this well-established team to make significant progress toward realizing the full potential of mesh optimization methods.

The research in this proposal will extend our earlier work by focusing on the link between geometric mesh properties and physical solution characteristics. We will concentrate on r-adaptivity techniques in which mesh vertices move according to solution features or error indicators to improve simulation accuracy and efficiency. In so doing, we expect to effect a paradigm shift in the way mesh optimization is done. In particular, we will analyze and extend the authors' target-matrix paradigm to r-adaptive quality metrics; we will develop robust and effective numerical optimization solvers tuned for different application classes; we will identify and compare different high-level strategies for solving the mesh optimization problems; and we will investigate the impact of r-adaptive mesh improvement on simulation accuracy and efficiency and compare this technique with other methods, such as h-refinement, to assess when and how to most effectively use each. New hybrid, adaptive algorithms, such as combined hr-adaptive methods, will also be developed that utilize the full power of the matrix-based r-adaptive formulation. These methods will improve the overall mesh quality while preserving desirable mesh characteristics, such as element size or alignment, obtained via the other techniques.

By addressing these topics systematically, we can realize the full potential of r-adaptive methods and impact many aspects of the mesh improvement problem. At the end of the three year time period we anticipate the following will have been accomplished:

• We will have developed an advanced mathematical theory for mesh quality improvement methods, particularly in the area of matrix-based quality metrics, and completed a systematic and comprehensive study of this topic.
• We will have applied this theory toward realization of the full potential of mesh optimization methods including: non-adaptive mesh smoothing (mesh untangling, smoothing, shape, size, and orientation control), adaptive mesh smoothing for deforming mesh applications, ALE rezoning, anisotropic mesh smoothing, and quality improvement for transitions between hybrid structured-unstructured meshes.
• We will have automated certain mesh-related algorithms such as ALE mesh rezoning (hydro- and MHD simulations at LANL, LLNL, and SNL) and deforming meshes (SLAC accelerator design effort).
• We will have developed prototype code in Mesquite to verify the theory and to perform preliminary tests that indicate effectiveness and robustness on problems of interest to the applications and research community with fast, effective solution algorithms for the mesh quality improvement problem using state-of-the-art optimization solvers.
• We will have impacted mesh generation codes such as Cubit, Overture, and NWGrid as well as improved the accuracy and efficiency for application simulations such as those found in accelerator design, fusion, climate, and biology.
• We will have augmented the suite of effective adaptivity tools through a thorough understanding of the role of r-adaptivity in the application solution process and through the development of hybrid hr-adaptive methods.
  

Year 1 Activities:

1. Implement the software in Mesquite needed to support research into the target-matrix paradigm, r-adaptivity, and other aspects of this reseearch,
2. Verify and explore the space of optimization capabilities within the target-matrix paradigm,
3. Investigate target-matrix construction algorithms for deforming domains, ALE and other moving mesh applications, and r-adaptivity,
4. Develop inexact Newton method with a trust region, block coordinate descent method, and limited-memory quasi-Newton algorithm, and efficient preconditioners for unconstrained mesh optimization problems. Adapt and extend current Mesquite solvers to work efficiently with the new target-based mesh quality metrics,
5. Described and analyze the high-level solution strategies suggested in the proposal,
6. Lay the groundwork and create initial test suite for studies on impact of mesh of solution accuracy and efficiency.

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