In this article we introduce all the ingredients to develop adaptive isogeometric methods based on hierarchical B-splines. In particular, we give precise definitions of local refinement and ...coarsening that, unlike previously existing methods, can be understood as the inverse of each other. We also define simple and intuitive data structures for the implementation of hierarchical B-splines, and algorithms for refinement and coarsening that take advantage of local information. We complete the paper with some simple numerical tests to show the performance of the data structures and algorithms, that have been implemented in the open-source Octave/Matlab code GeoPDEs.
We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving ...fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampère–Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical tests in three dimensions.
•Discretization of the double de Rham complex with tensor product spaces of splines.•Strongly preserved discrete charge conservation laws in both sequences.•Conserved discrete energy with non-standard discrete Hodge star operators.•Fast time stepping through efficient inversion of Kronecker product matrices.
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 effective 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.
•A complete set of algorithms to perform adaptive refinement and coarsening.•Truncated hierarchical B-splines (THB-splines) defined on suitably graded meshes.•We apply the proposed algorithms to linear heat transfer problems.•THB-spline admissible solution delivers an effective discretization.•Our algorithms strongly improve computational efficiency.
The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic ...partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff–Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.
•Development of an error-driven isogeometric framework on trimmed surfaces.•Adaptive strategy recovers optimal rates of convergence in all trimmed benchmarks.•Error estimation based on an easy-to-compute, local, auxiliary problem.•Adaptive Kirchhoff–Love shell analysis performed on the B-pillar of a car.
We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) ...hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection of numerical examples validates the theoretical results and the performance of the preconditioner.
Despite advances in diagnosis and new treatments such as targeted therapies, breast cancer (BC) is still the most prevalent tumor in women worldwide and the leading cause of death. The principal ...obstacle for successful BC treatment is the acquired or de novo resistance of the tumors to the systemic therapy (chemotherapy, endocrine, and targeted therapies) that patients receive. In the era of personalized treatment, several studies have focused on the search for biomarkers capable of predicting the response to this therapy; microRNAs (miRNAs) stand out among these markers due to their broad spectrum or potential clinical applications. miRNAs are conserved small non-coding RNAs that act as negative regulators of gene expression playing an important role in several cellular processes, such as cell proliferation, autophagy, genomic stability, and apoptosis. We reviewed recent data that describe the role of miRNAs as potential predictors of response to systemic treatments in BC. Furthermore, upon analyzing the collected published information, we noticed that the overexpression of miR-155, miR-222, miR-125b, and miR-21 predicts the resistance to the most common systemic treatments; nonetheless, the function of these particular miRNAs must be carefully studied and further analyses are still necessary to increase knowledge about their role and future potential clinical uses in BC.
Abstract
We propose and analyse optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in ...Morgenstern & Peterseim (2015, Analysis-suitable adaptive T-mesh refinement with linear complexity. Comput. Aided Geom. Design, 34, 50–66) we can guarantee that the obtained T-meshes have a multilevel structure and that the associated T-splines are analysis suitable, for which we can define a dual basis and a stable projector. Taking advantage of the multilevel structure we develop two Bramble-Pasciak-Xu (BPX) preconditioners: the first on the basis of local smoothing only for the functions affected by a newly added edge by bisection and the second smoothing for all the functions affected after adding all the edges of the same level. We prove that both methods have optimal complexity and present several numerical experiments to confirm our theoretical results and also to compare the practical performance of the proposed preconditioners.
Abstract
Trimming consists of cutting away parts of a geometric domain, without reconstructing a global parametrization (meshing). It is a widely used operation in computer-aided design, which ...generates meshes that are unfitted with the described physical object. This paper develops an adaptive mesh refinement strategy on trimmed geometries in the context of hierarchical B-spline-based isogeometric analysis. A residual a posteriori estimator of the energy norm of the numerical approximation error is derived, in the context of the Poisson equation. The estimator is proven to be reliable, independently of the number of hierarchical levels and of the way the trimmed boundaries cut the underlying mesh. Numerical experiments are performed to validate the presented theory, and to show that the estimator’s effectivity index is independent of the size of the active part of the trimmed mesh elements.
We present an adaptive scheme for isogeometric phase-field modeling, to perform suitably graded hierarchical refinement and coarsening on both single- and multi-patch geometries by considering ...truncated hierarchical spline constructions which ensure C1 continuity between patches. We apply the proposed algorithms to the Cahn–Hilliard equation, describing the time-evolving phase separation processes of immiscible fluids. We first verify the accuracy of the hierarchical spline scheme by comparing two classical indicators usually considered in phase-field modeling, for then demonstrating the effectiveness of the grading strategy in terms of accuracy per degree of freedom. A selection of numerical examples confirms the performance of the proposed scheme to simulate standard modes of phase separation using adaptive isogeometric analysis with smooth hierarchical spline constructions.
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