Hierarchical B-splines Research Articles

Gradient-based optimization with B-splines on sparse grids for solving forward-dynamics simulations of three-dimensional, continuum-mechanical musculoskeletal system models.

Investigating the interplay between muscular activity and motion is the basis to improve our understanding of healthy or diseased musculoskeletal systems. To be able to analyze the musculoskeletal systems, computational models are used. Albeit some severe modeling assumptions, almost all existing musculoskeletal system simulations appeal to multibody simulation frameworks. Although continuum-mechanical musculoskeletal system models can compensate for some of these limitations, they are essentially not considered because of their computational complexity and cost. The proposed framework is the first activation-driven musculoskeletal system model, in which the exerted skeletal muscle forces are computed using 3-dimensional, continuum-mechanical skeletal muscle models and in which muscle activations are determined based on a constraint optimization problem. Numerical feasibility is achieved by computing sparse grid surrogates with hierarchical B-splines, and adaptive sparse grid refinement further reduces the computational effort. The choice of B-splines allows the use of all existing gradient-based optimization techniques without further numerical approximation. This paper demonstrates that the resulting surrogates have low relative errors (less than 0.76%) and can be used within forward simulations that are subject to constraint optimization. To demonstrate this, we set up several different test scenarios in which an upper limb model consisting of the elbow joint, the biceps and triceps brachii, and an external load is subjected to different optimization criteria. Even though this novel method has only been demonstrated for a 2-muscle system, it can easily be extended to musculoskeletal systems with 3 or more muscles.

Adaptive fitting with THB-splines: Error analysis and industrial applications

Automatic fitting techniques are required in many industrial applications, as for example, instrument calibration, data analysis, geometric modeling or reverse engineering. We present an error analysis of advanced techniques for scattered data approximation based on truncated hierarchical B-spline (THB-spline) constructions. The adaptive fitting framework is validated on data of industrial complexity related to geometric models of different aircraft engine parts. We demonstrate that the twofold adaptive nature of the method, which takes into account both the shape and the distribution of the data, is crucial to define an effective fully automatic scheme with suitable local refinement capabilities.

A non-rigid image registration method based on multi-level B-spline and L2-regularization

To solve the problem that the cost function of the classic free-form deformation (FFD) cannot simulate transformation field of images with large elastic deformation or local distortion in image registration better, and to increase the registration accuracy and robustness, a new non-rigid image registration method based on the classic hierarchical FFD is proposed. Since the smooth term has a significant influence on registration accuracy, and its coefficient is not easy to be controlled in the classic hierarchical B-spline based FFD, a L2-regularization term with faster and more stable optimization is introduced in the cost function of the proposed model. By coordinating the coefficients of this regularization term and the smooth term, this novel L2-regularized FFD model is able to solve the problem of low registration accuracy caused by strong smooth constraint while maintaining the images topologies. The introduced L2-regularization term can impose a spatial constraint on the control lattices transformation field, and the over-registration problem can be suppressed to a certain extent, so it can register the images with local large distortion. A series of registration experiments of natural images and medical images show that the new method has an obvious advantage over the classic model in registration accuracy measured by mean square error.

A stabilised immersed framework on hierarchical b-spline grids for fluid-flexible structure interaction with solid–solid contact

We present a robust and efficient stabilised immersed framework for fluid–structure interaction involving incompressible fluid flow and flexible structures undergoing large deformations and also involving solid–solid contact. The efficiency of the formulation stems from the use of second-order accurate sequential staggered solution scheme for resolving fluid–solid coupling. Mixed Galerkin formulation, along with SUPG/PSPG stabilisation, is employed to obtain the numerical solutions of the incompressible Navier–Stokes equations. The immersed formulation is based on hierarchical b-spline grids, with unsymmetric Nitsche method employed to impose boundary as well as interface conditions on the fluid domain, while ghost-penalty operators are applied to alleviate the numerical instabilities arising due to small cut cells. The solid is modelled using linear continuum elements with finite strain formulation to facilitate the modelling of large structural deformations, and the contact between solids is modelled using the normal frictionless node-to-segment contact elements with Lagrange multipliers. In order to deal with the issue of uncovering for cut-cell based numerical schemes, a simple mapping technique is also introduced. Spatial and temporal convergence studies of the proposed scheme are performed by studying a simple example of flow over a deformable beam in cross flow. The robustness and accuracy of the proposed scheme are demonstrated by studying the benchmark examples of an oscillating beam in two-dimensions and flutter of a flexible simplified bridge deck in three-dimensions. In order to demonstrate the applicability of the proposed framework to complex fluid–structure interaction problems, the proposed methodology is used to simulate the fluid–structure interaction of a check valve with flexible valve plate.

Injective hierarchical free-form deformations using THB-splines

The free-form deformation (FFD) method deforms geometry in n-dimensional space by employing an n-variate function to deform (parts of) the ambient space. The original method pioneered by Sederberg and Parry in 1986 uses trivariate tensor-product Bernstein polynomials in R3 and is controlled as a Bézier volume. We propose an extension based on truncated hierarchical B-splines (THB-splines). This offers hierarchical and local refinability, an efficient implementation due to reduced supports of THB-splines, and intuitive control point hiding during FFD interaction. Additionally, we address the issue of fold-overs by efficiently checking the injectivity of the hierarchical deformation in real-time.

Improved conditioning of isogeometric analysis matrices for trimmed geometries

A stable basis for isogeometric analysis of trimmed models is obtained by combining extended B-splines with truncated hierarchical B-splines. While extended B-splines guarantee that the condition number of system matrices is independent of the location of a trimming curve, local refinement is used to improve the robustness of the procedure and the accuracy of the numerical results. The present extended B-spline construction works in the context of Galerkin and collocation methods. The paper focuses on the latter and introduces a new collocation scheme for truncated hierarchical B-splines. A proper transition between refinement levels is assured by a novel balancing algorithm that employs a simple criterion. The enhanced performance of the locally refined stabilization is verified by scalar Laplace and linear elasticity problems analyzed by a collocation based isogeometric boundary element method. The proposed approach yields excellent results and requires few refinement levels to improve the stabilization procedure and accuracy along trimming curves.

Projection and transfer operators in adaptive isogeometric analysis with hierarchical B-splines

We present projection methods and transfer operations required for adaptive mesh refinement/coarsening in problems with internal variables. We extend the results of Hennig et al. 2016 on Bézier extraction of truncated hierarchical B-splines and its application to adaptive isogeometric analysis. It is shown that isogeometric analysis improves the performance of transfer operations as already the coarsest mesh represents the exact geometry and the hierarchical structure allows for quadrature free projection methods. We propose two different local least squares projection methods for field variables and compare them to existing global and semi-local versions. We discuss the application of two different transfer operators for internal variables. An alternative new operator inspired by superconvergent patch recovery is also proposed. The presented projection methods and transfer operations are tested in benchmark problems and applied to phase-field modelling of spinodal decomposition and brittle and ductile fracture.

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.

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension [Formula: see text]. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

Coupling adaptively refined multi-patch spline discretizations via boundary compatibility

The present paper studies adaptive refinement on multi-patch domains in isogeometric analysis. In particular, we investigate the gluing construction for adaptively refined spline spaces to obtain discretizations that are C0 smooth across interfaces. We will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction.Given a spline basis (or, more generally, a generating system if linear independence is not guaranteed) on a d-dimensional box domain, there are two possibilities for constructing the spline basis on the domain boundary. Firstly, one can simply restrict the basis functions to the boundary. Secondly, one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh. The two constructions do not necessarily produce the same set of functions. If they do, then the spline bases are said to be compatible.We study this property for hierarchical (HB-) and truncated hierarchical B-splines (THB-splines) and identify sufficient conditions. These conditions are weaker for THB- than for HB-splines. Finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis, in particular when considering multi-patch domains.

Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines

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.

Multi-level Bézier extraction for hierarchical local refinement of Isogeometric Analysis

One of the main topics of research on Isogeometric Analysis is local refinement. Among the various techniques currently studied and developed, one of the most appealing, referred to as hierarchical B-Splines, consists of defining a suitable set of basis functions on different hierarchical levels. This strategy can also be improved, for example to recover partition of unity, resorting to a truncation operation, giving rise to the so-called truncated hierarchical B-Splines. Despite its conceptual simplicity, implementing the hierarchical definition of shape functions into an existing code can be rather involved. In this work we present a simple way to bring the hierarchical isogeometric concept closer to a standard finite element formulation. Practically speaking, the hierarchy of functions and knot spans is flattened into a sequence of elements being equipped with a standard single-level basis. In fact, the proposed multi-level extraction is a generalization of the classical Bézier extraction and analogously offers a standard element structure to the hierarchical overlay of functions. Moreover, this approach is suitable for an extension to non-linear problems and for a parallel implementation. The multi-level extraction is presented as a general concept that can be applied to different kinds of refinements and basis functions. Finally, few basic algorithms to compute the local multi-level extraction operator for knot insertion on spline spaces are outlined and compared, and some numerical examples are presented.

Patchwork B-spline refinement

Hierarchical splines allow to use representations with varying level of detail in different parts of a geometric model. However, the progression from coarse to fine scale is based on a single sequence of nested spline spaces. More precisely, each space defining a representation of some level must simultaneously be a subspace of all the higher level spaces and contain all the lower level ones. This requirement imposes severe restrictions on the available refinement strategies. We introduce the new framework of Patchwork B-splines (PB-splines), which alleviates these constraints and therefore increases the flexibility of the representations that are available in different parts of a geometric model. We derive the mathematical foundations of multivariate PB-splines, in particular focusing on the construction of a basis that forms a convex partition of unity. This generalizes the concept of truncated hierarchical (TH) B-splines to the novel framework. Moreover, we discuss the application of PB-splines to surface reconstruction with adaptive refinement. It is observed that the increased flexibility of the local representations provides significant advantages.

Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth

Moving interface problems are ubiquitous in science and engineering. To develop an accurate and efficient methodology for this class of problems, we present algorithms for local h-adaptivity of hierarchical B-splines to be utilized in isogeometric analysis. We extend Bézier projection, an efficient quadrature-free local projection technique, to the hierarchical setting. In this case, extraction operators may not be invertible. To address this issue we develop a multi-level reconstruction operator which maintains the locality properties of the projection. We also introduce a balance parameter to control the overlap of hierarchical functions leading to improved numerical conditioning. We apply our algorithms to the simulation of localized prostate cancer growth. We model this disease using the phase-field method and a set of diffusion–reaction equations to account for the dynamics of nutrients and a key biomarker termed Prostate Specific Antigen. Our results include examples on simple 2D and 3D domains and a more compelling tissue-scale, patient-specific simulation, which is run over a prostate anatomy extracted from medical images. Our methods for local h-adaptivity efficiently capture the evolving interface between the tumor and the neighboring healthy tissue with remarkable accuracy in all cases.

Phase-field guided surface reconstruction based on implicit hierarchical B-splines

Constructing smooth surface representations from point clouds is a fundamental problem in geometric modeling and computer graphics, and a wealthy of literature has focused on this problem. Among the many approaches, implicit surface reconstruction has been a central topic in the past two decades due to its ability to represent objects with complicated geometry and topology. Recently, the problem of reducing the storage requirement for implicit representations has attracted much attention. In this paper, we propose a phase-field guided implicit surface reconstruction method to tackle this problem. The implicit function of our method behaves like the phase-field of a binary system, in which it takes distinct values (i.e., −1 and 1) in each of the phases with a smooth transition between them. Given an unorganized point cloud, we present a method to construct a phase-filed function represented by a hierarchical B-spline whose zero level set approximates the point cloud as much as possible. Unlike previous approaches, our mathematical model avoids the use of the normal information of the point cloud. Furthermore, as demonstrated by experimental results, our method can achieve very compact representation since we mainly need to save the coefficients of the hierarchical B-spline function within a narrow band near the point cloud. The ability of our method to produce reconstruction results with high quality is also validated by experiments.

Adaptive scattered data fitting by extension of local approximations to hierarchical splines

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations.

Estimating non-simplified vine copulas using penalized splines

Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first non-parametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback-Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.

A stabilised immersed boundary method on hierarchical b-spline grids for fluid–rigid body interaction with solid–solid contact

An accurate, efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly-supported rigid bodies and incompressible fluid flow with topology changes and solid–solid contact. The solution of the incompressible Navier–Stokes equations is approximated by employing a stabilised formulation on Cartesian grids discretised with hierarchical b-splines. The solid is modelled as a rigid body and represented by linear segments along its boundary. Kinematic conditions along the fluid–rigid body interface are enforced weakly using Nitsche’s method, while ghost penalty operators are employed to avoid excessive ill-conditioning of the system matrix arising from small cut cells. A staggered scheme is used for resolving the coupled fluid–rigid body interaction. The contact between moving or moving and fixed solid bodies is modelled with Lagrange multipliers. The excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems. The examples cover the galloping phenomena, particulate flow, hydraulic check valves and a model turbine.

Multilevel Editing of B-Spline Curves with Robust Orientation of Details

We facilitate the editing of hierarchical B-spline curves at multiple resolutions by expressing a displacement function at each level in rotation minimizing frames (RMFs) on the curve at the next lower level. When the curve is edited at a particular level, RMFs at all the higher levels are updated, and the control points of the displacement function at each higher level are obtained from these new RMFs. This transfers details created at the current level to higher levels. Our method presents a hundred-fold faster way to reflect editing results compared to the traditional approach using Frenet frames. We demonstrate the effectiveness of our technique by showing several examples of editing curves with fine details.

A stabilised immersed boundary method on hierarchical b-spline grids

In this work, an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b-spline grid and employs the non-symmetric and penalty-free version of Nitsche’s method to enforce the boundary conditions. The strategy allows for h- and p-refinement and employs a so-called ghost penalty term to stabilise the cut cells. An effective procedure based on hierarchical subdivision and sub-cell merging, which avoids excessive numbers of quadrature points, is used for the integration of the cut cells. A basic Laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty-free Nitsche method as well as their impact on accuracy. The methodology is also applied to the incompressible Navier–Stokes equations, where the SUPG/PSPG stabilisation is employed. Simulations of the lid-driven cavity flow and the flow around a cylinder at low Reynolds number show the good performance of the methodology. Excessive ill-conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field.