Spline-based methods with adaptive refinement for problems of acoustics and fracture mechanics of thin plates

Abstract

Both the CAD software and FEM software have a significant impact on engineering nowadays. Even though both are powerful tools for design and analysis, the main drawback is that CAD geometries and Finite Element models do not entirely match, which results in the necessity to re-parameterize the geometry many times during the solution cycle in FEM. Isogeometric Analysis (IGA) was proposed to fulfill this gap and create the direct link between the CAD design and FEM analysis. The main idea of IGA is to substitute the shape functions used in FEM by the shape functions used in the CAD software. In particular, one of the main drawbacks of NURBS basis functions, and therefore of IGA, is the lack of local refinement, which makes them computationally highly expensive in applications that demands a non-uniform refinement of the geometry. Polynomial splines over Hierarchical T-meshes (PHT-splines) were introduced by Deng et al. as a type of spline that allows local refinement and adaptability by means of a polynomial basis capable of parameterizing the geometry. In this work, we demonstrate the application of PHT-splines for two type of problems: time-harmonic acoustic problems, modeled by the Helmholtz equation, and fracture mechanics of thin plate problems, modeled by the Kirchhoff-Love theory. Solutions of the Helmholtz equation have two features: global oscillations associated with the wave number and local gradients caused by geometrical irregularities. The results show that after a sufficient number of degrees of freedom is used to approximate global oscillations, adaptive refinement can capture local features of the solution. The residual-based and recovery-based error estimators are compared and the performance of $p$-refinement is investigated. Moreover, an eXtended Geometry Independent Field approximaTion (XGIFT) formulation based on Polynomials Splines Over Hierarchical T-meshes (PHT-splines) for modeling both static and dynamic fracture mechanic problems for plates described by the Kirchhoff-Love theory is presented. Adaptive refinement is employed using a recovery-based error estimator. Results show that adaptive refinement can capture local features of the solution around the crack tip, improving results in both static and dynamic examples. In both cases, the simulations are done in the context of recently introduced Geometry Independent Field approximaTion (GIFT), where PHT-splines are only used to approximate the solution, while the computational domain is parameterized with NURBS. This approach builds on the natural adaptation ability of PHT-splines and avoids the re-parameterization of the NURBS geometry during the solution refinement process.