Fast and robust solvers for local/global domain parameterizations within G+Smo
Date:
Slides, Abstract, Photo 1, Photo 2, Conference Link
The first step in the isogeometric analysis (IGA) pipeline consists of constructing a high-quality, analysis-suitable domain parameterization from the boundary representation of a CAD model. This step is crucial for the efficiency and accuracy of all downstream tasks, e.g., analysis and optimization. While algebraic parameterization methods such as the discrete Coons method and the spring patch method are simple and efficient, they often result in invalid parameterizations, particularly for complex geometries. To overcome this limitation, more robust and sophisticated approaches have been proposed, which require solving nonlinear systems of equations or nonlinear optimization problems. Fast solvers for these problems therefore play a crucial role in the robust and efficient construction of analysis-suitable domain parameterizations. The Geometry + Simulation Modules (G+Smo) library [1] is a specialized, open-source C++ library that provides mathematical tools and operations for both geometric design and isogeometric simulation. In this talk, we present several recently developed fast and robust solvers for domain parameterization that have been integrated into the G+Smo library.
In the first part of the talk, we present solvers for generating parameterizations based on the elliptic grid generation paradigm. The underlying nonlinear systems of equations are solved by a standard Newton method or by our recently developed preconditioned Anderson acceleration method with dynamic preconditioning strategies [2], which reduces the frequent and costly computation of full Jacobian updates. In the second part of the talk, we showcase two recently developed parameterization methods based on nonlinear optimization. These methods are backed by fast and robust solvers such as LBFGS++, HLBFGS, and IPOPT. The optimization-based parameterization methods are also used to locally improve domain parameterizations produced by smoothing methods such as the D-Patch or Almost-C1 methods.
[1] Jüttler, B., Langer, U., Mantzaflaris, A., Moore, S. E., & Zulehner, W. (2014). Geometry + simulation modules: Implementing isogeometric analysis. PAMM, 14(1), 961–962.
[2] Ji, Y., Chen, K., Möller, M., & Vuik, C. (2023). On an improved PDE-based elliptic parameterization method for isogeometric analysis using preconditioned Anderson acceleration. Under review.
Based on joint work with Hugo M. Verhelst and Matthias Möller.
Keywords: Isogeometric Analysis, Domain Parameterization, G+Smo, Preconditioned Anderson Acceleration, Nonlinear Optimization