Maximum-principle preserving space-time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions

Scan-based immersed isogeometric analysis

Scan-based simulations contain innate topologically complex three-dimensional geometries, represented by large data sets in formats which are not directly suitable for analysis. Consequently, performing high-fidelity scan-based simulations at practical computational costs is still very challenging. The main objective of this dissertation is to develop an efficient and robust scan-based simulation strategy by acquiring a profound understanding of three prominent challenges in scan-based IGA, viz.: i) balancing the accuracy and computational effort associated with numerical integration; ii) the preservation of topology in the spline-based segmentation procedure; and iii) the control of accuracy using error estimation and adaptivity techniques. In three-dimensional immersed isogeometric simulations, the computational effort associated with integration can be the critical component. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this dissertation we provide a thorough investigation of the accuracy and computational effort of the octree integration technique. We quantify the contribution of the integration error using the theoretical basis provided by Strang’s first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed IGA. To exploit the advantageous properties of IGA in a scan-based setting, it is important to extract a smooth geometry. This can be established by convoluting the voxel data using B-splines, but this can induce problematic topological changes when features with a size similar to that of the voxels are encountered. This dissertation presents a topology-preserving segmentation procedure using truncated hierarchical (TH)B-splines. A moving-window-based topological anomaly detection algorithm is proposed to identify regions in which (TH)B-spline refinements must be performed. The criterion to identify topological anomalies is based on the Euler characteristic, giving it the capability to distinguish between topological and shape changes. A Fourier analysis is presented to explain the effectiveness of the developed procedure. An additional computational challenge in the context of immersed IGA is the construction of optimal approximations using locally refined splines. For scan-based volumetric domains, hierarchical splines are particularly suitable, as they optimally leverage the advantages offered by the availability of a geometrically simple background mesh. Although truncated hierarchical B-splines have been successfully applied in the context of IGA, their application in the immersed setting is largely unexplored. In this dissertation we propose a computational strategy for the application of error estimation-based mesh adaptivity for stabilized immersed IGA. The conducted analyses and developed computational techniques for scan-based immersed IGA are interrelated, and together constitute a significant improvement in the efficiency and robustness of the analysis paradigm. In combination with other state-of-the-art developments regarding immersed FEM/IGA (\emph{e.g.}, iterative solution techniques, parallel computing), the research in this thesis opens the doors to scan-based simulations with more sophisticated physical behavior, geometries of increased complexity, and larger scan-data sizes.Scan-based simulations contain innate topologically complex three-dimensional geometries, represented by large data sets in formats which are not directly suitable for analysis. Consequently, performing high-fidelity scan-based simulations at practical computational costs is still very challenging. The main objective of this dissertation is to develop an efficient and robust scan-based simulation strategy by acquiring a profound understanding of three prominent challenges in scan-based IGA, viz.: i) balancing the accuracy and computational effort associated with numerical integration; ii) the preservation of topology in the spline-based segmentation procedure; and iii) the control of accuracy using error estimation and adaptivity techniques. In three-dimensional immersed isogeometric simulations, the computational effort associated with integration can be the critical component. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this dissertation we provide a thorough investigation of the accuracy and computational effort of the octree integration technique. We quantify the contribution of the integration error using the theoretical basis provided by Strang’s first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed IGA. To exploit the advantageous properties of IGA in a scan-based setting, it is important to extract a smooth geometry. This can be established by convoluting the voxel data using B-splines, but this can induce problematic topological changes when features with a size similar to that of the voxels are encountered. This dissertation presents a topology-preserving segmentation procedure using truncated hierarchical (TH)B-splines. A moving-window-based topological anomaly detection algorithm is proposed to identify regions in which (TH)B-spline refinements must be performed. The criterion to identify topological anomalies is based on the Euler characteristic, giving it the capability to distinguish between topological and shape changes. A Fourier analysis is presented to explain the effectiveness of the developed procedure. An additional computational challenge in the context of immersed IGA is the construction of optimal approximations using locally refined splines. For scan-based volumetric domains, hierarchical splines are particularly suitable, as they optimally leverage the advantages offered by the availability of a geometrically simple background mesh. Although truncated hierarchical B-splines have been successfully applied in the context of IGA, their application in the immersed setting is largely unexplored. In this dissertation we propose a computational strategy for the application of error estimation-based mesh adaptivity for stabilized immersed IGA. The conducted analyses and developed computational techniques for scan-based immersed IGA are interrelated, and together constitute a significant improvement in the efficiency and robustness of the analysis paradigm. In combination with other state-of-the-art developments regarding immersed FEM/IGA (\emph{e.g.}, iterative solution techniques, parallel computing), the research in this thesis opens the doors to scan-based simulations with more sophisticated physical behavior, geometries of increased complexity, and larger scan-data sizes

C1-continuous space-time discretization based on Hamilton's law of varying action

We develop a class of C1-continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1-continuity for the displacement field and C0-continuity for the velocity field. From the discrete virtual action we finally construct a class of one-step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one-dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate -- denoted as p2-scheme -- converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long-term energy behavior.Comment: slightly condensed the manuscript, added references, numerical results unchange

Isogeometric Analysis on V-reps: first results

Inspired by the introduction of Volumetric Modeling via volumetric representations (V-reps) by Massarwi and Elber in 2016, in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps. We develop tools for approximation and local re-parametrization of trimmed elements for three dimensional problems, and we provide a theoretical framework that fully justify our algorithmic choices. We validate our approach both on two and three dimensional problems, for diffusion and linear elasticity.Comment: 36 pages, 44 figures. Reviewed versio

Isogeometric Analysis in advection-diffusion problems: tension splines approximation

We present a novel approach, within the new paradigm of isogeometric analysis introduced by Hughes et al., to deal with advection dominated advection-diffusion problems. The key ingredient is the use of Galerkin approximating spaces of functions with high smoothness, as in IgA based on classical B-splines, but particularly well suited to describe sharp layers involving very strong gradients