Isogeometric Methods for Numerical Simulation

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method.

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Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

Falini

D’Inverno

Sampoli

et al. 2023

Mathematics

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The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which can be computed as the solution of an elliptic problem, equipped with suitable Dirichlet boundary conditions. In recent years, Physics-Informed Neural Networks (PINNs) have been proved to be a powerful tool to compute the solution of Partial Differential Equations (PDEs) replacing standard numerical models, based on Finite Element Methods and Finite Differences, with deep neural networks; PINNs can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In this work, we exploit the PINN model in order to solve the PDE associated to the differential problem of the parameterization on both convex and non-convex planar domains, for which the describing PDE is known. The final continuous model is then provided by applying a Hermite type quasi-interpolation operator, which can guarantee the desired smoothness of the sought parameterization. Finally, some numerical examples are presented, which show that the PINNs-based approach is robust. Indeed, the produced mapping does not exhibit folding or self-intersection at the interior of the domain and, also, for highly non convex shapes, despite few faulty points near the boundaries, has better shape-measures, e.g., lower values of the Winslow functional.

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Isogeometric Methods for Numerical Simulation

Cited by 4 publications

References 58 publications

Automatic image-based analyses using a coupled quadtree-SBFEM/SCM approach

Automatic image-based analyses using a coupled quadtree-SBFEM/SCM approach

IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

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