Engineering design and optimization is an iterative process, one in which preliminary concepts are continuously refined to develop the best final product per specifications. High performance computing (HPC) has become a key tool for simulating design options and product performance and more deeply understanding the underlying physics involved.

Simulating contact mechanics—how rigid objects touch, press, and push against each other and any resulting deformation—is an important piece of the engineering puzzle both in industry and for carrying out the Laboratory’s programmatic missions. Direct linear solvers are known for producing fast, accurate results for small, well-defined systems, but they are limited in their scalability, robustness, and accuracy when applied to design systems with complex contact nonlinearities on large or heavily refined meshes.

As part of a three-year Laboratory Directed Research and Development (LDRD) project, researchers in Lawrence Livermore’s Center for Applied Scientific Computing (CASC) have developed scalable numerical algorithms using algebraic multigrid (AMG) techniques to address the limitations in contact mechanics analyses and engineering design. Cosmin Petra, the project’s principal investigator says, “We have showed that scalable linear solvers using algebraic multigrid techniques can be implemented within scalable outer loop Newton continuation methods to solve realistically large contact problems.”

Libraries Support Scalability

In contact analyses, the underlying mathematical problems are systems of partial differential equations (PDEs) that are nonsmooth (change abruptly rather than gradually), highly nonlinear (behave unpredictably), and poorly conditioned (susceptible to errors). To overcome these challenges, the multidisciplinary project team—a collaboration of experts in Livermore’s Computing and Engineering directorates—focused on a technical approach to improve both algorithmic and computational scalability, focusing on how contact mechanics problems are set up and how to solve them more efficiency using many parallel processors. The work leveraged the Laboratory’s leading-edge HPC systems and numerical libraries, specifically the algebraic multigrid HYPRE solver; the MFEM software library, which handles finite element discretization of PDEs; and TRIBOL, a specialized library for generating scalable contact constraint equations.

These unique software capabilities underpin the team’s new homotopy Newton continuation algorithm, which solves for contact nonlinearities by executing a series of steps that continually refine the solution, starting from a more generalized, easier problem and iterating toward the more complex target problem of interest. The algorithm breaks down problems into two levels of iteration: outer loop nonlinear calculations that effectively manage, or “smooth,” the algorithm’s overall progress, and inner loop calculations that solve the linear equations involved with each step. To execute these functions reliably and efficiently, the algorithm uses trust regions to limit the size of each computational step and incorporates AMG filtering to validate whether a solution is good enough to advance the problem to the next stage. In addition, the algorithm is given leeway along the solution path to explore solutions that may not be ideal but that are still sufficiently plausible, significantly speeding time to solution.

Advances Enhance Mission

The team tested their new methodology on a suite of contact mechanics benchmarking and design optimization problems that varied in scale and complexity, involving bodies of different shapes and sizes, multiple contact surfaces, and displacement and rotational movements. The results showed strong robustness, efficiency, and scalability. Says Petra, “The homotopy algorithm and filtering AMG method reached unprecedented computational scales—up to hundreds of millions of unknown variables and thousands of computational cores on Livermore Computing’s Dane HPC cluster,” says Petra. “We set new standards in the community with these very large simulations.”

Computer codes containing both the homotopy outer loop and filtering AMG inner loop techniques have been integrated into the MFEM library, and researchers are working on transitioning the scalable iterative solvers into programmatic codes, such as Diablo. Says Petra, “The research will substantially increase the scalability of both current and next generation design and contact analysis codes and will enhance the Laboratory’s stockpile stewardship and national security activities.”

—Caryn Meissner