Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes

In the present work we introduce a complete set of algorithms to efficiently perform adaptive refinement and coarsening by exploiting truncated hierarchical B-splines (THB-splines) defined on suitably graded isogeometric meshes, that are called admissible mesh configurations. We apply the proposed algorithms to two-dimensional linear heat transfer problems with localized moving heat source, as simplified models for additive manufacturing applications. We first verify the accuracy of the admissible adaptive scheme with respect to an overkilled solution, for then comparing our results with similar schemes which consider different refinement and coarsening algorithms, with or without taking into account grading parameters. This study shows that the THB-spline admissible solution delivers an optimal discretization for what concerns not only the accuracy of the approximation, but also the (reduced) number of degrees of freedom per time step. In the last example we investigate the capability of the algorithms to approximate the thermal history of the problem for a more complicated source path. The comparison with uniform and non-admissible hierarchical meshes demonstrates that also in this case our adaptive scheme returns the desired accuracy while strongly improving the computational efficiency.Comment: 20 pages, 12 figure

Discontinuity Detection by Null Rules for Adaptive Surface Reconstruction

We present a discontinuity detection method based on the so-called null rules, computed as a vector in the null space of certain collocation matrices. These rules are used as weights in a linear combination of function evaluations to indicate the local behavior of the function itself. By analyzing the asymptotic properties of the rules, we introduce two indicators (one for discontinuities of the function and one for discontinuities of its gradient) by locally computing just one rule. This leads to an efficient and reliable scheme, which allows us to effectively detect and classify points close to discontinuities. We then show how this information can be suitably combined with adaptive approximation methods based on hierarchical spline spaces in the reconstruction process of surfaces with discontinuities. The considered adaptive methods exploit the ability of the hierarchical spaces to be locally refined, and fault detection is a natural way to guide the refinement with low computational cost. A selection of test cases is presented to show the effectiveness of our approach

Kirchhoff-Love shell representation and analysis using triangle configuration B-splines

This paper presents the application of triangle configuration B-splines (TCB-splines) for representing and analyzing the Kirchhoff-Love shell in the context of isogeometric analysis (IGA). The Kirchhoff-Love shell formulation requires global $C^1$-continuous basis functions. The nonuniform rational B-spline (NURBS)-based IGA has been extensively used for developing Kirchhoff-Love shell elements. However, shells with complex geometries inevitably need multiple patches and trimming techniques, where stitching patches with high continuity is a challenge. On the other hand, due to their unstructured nature, TCB-splines can accommodate general polygonal domains, have local refinement, and are flexible to model complex geometries with $C^1$ continuity, which naturally fit into the Kirchhoff-Love shell formulation with complex geometries. Therefore, we propose to use TCB-splines as basis functions for geometric representation and solution approximation. We apply our method to both linear and nonlinear benchmark shell problems, where the accuracy and robustness are validated. The applicability of the proposed approach to shell analysis is further exemplified by performing geometrically nonlinear Kirchhoff-Love shell simulations of a pipe junction and a front bumper represented by a single patch of TCB-splines

Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties. We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class. The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces

Local (T)HB-spline projectors via restricted hierarchical spline fitting

International audienceThis paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O(p d) floating point operations and O(1) evaluations of the given function per degree of freedom, while providing essentially the same accuracy as global approximation. Our spline projectors are based on a unifying framework for quasi-interpolation in hierarchical spline spaces. We present a detailed comparison with the scheme of Speleers and Manni (2016)

THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis

International audienceLocal refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A selection of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework