Penalty function-based volumetric parameterization method for isogeometric analysis

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In isogeometric analysis, constructing bijective and low-distortion parameterizations is a fundamental task. Compared with the planar problem, the volumetric case is more challenging in terms of both robustness and efficiency. In this paper, we present a robust and efficient volumetric parameterization method based on the idea of penalty functions and the Jacobian regularization technique. The proposed method does not require a bijective initialization and thus avoids an extra foldover elimination step. The main contributions of this work are threefold. First, a new objective function that characterizes the volume distortion is established using the divergence theorem. Second, we employ a novel penalty function for the Jacobian regularization, and derive the full analytical gradient of the objective function to enhance the numerical stability of gradient-based optimization. Third, we develop a reduced numerical integration strategy to accelerate the new algorithm. Several numerical examples demonstrate that our method significantly outperforms competing state-of-the-art approaches in terms of both robustness and efficiency.

Keywords: Isogeometric Analysis, Volumetric Parameterization, Penalty Function, Jacobian Regularization, Reduced Numerical Integration