Michael A. Scott

On linear independence of T-spline blending functions

This paper shows that, for any given T-spline, the linear independence of its blending functions can be determined by computing the nullity of the T-spline-to-NURBS transform matrix. The paper analyzes the class of T-splines for which no perpendicular T-node extensions intersect, and shows that the blending functions for any such T-spline are linearly independent.

Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis

Abstract In this paper hierarchical analysis-suitable T-splines (HASTS) are developed. The resulting spaces are a superset of both analysis-suitable T-splines and hierarchical B-splines. The additional flexibility provided by the hierarchy of T-spline spaces results in simple, highly localized refinement algorithms which can be utilized in a design or analysis context. A detailed theoretical formulation is presented. Bezier extraction is extended to HASTS simplifying the implementation of HASTS in existing finite element codes. The behavior of a simple HASTS refinement algorithm is compared to the local refinement algorithm for analysis-suitable T-splines demonstrating the superior efficiency and locality of the HASTS algorithm. Finally, HASTS are utilized as a basis for adaptive isogeometric analysis.

Analysis-suitable T-splines: Characterization, refineability, and approximation

We establish several fundamental properties of analysis-suitable T-splines which are important for design and analysis. First, we characterize T-spline spaces and prove that the space of smooth bicubic polynomials, defined over the extended T-mesh of an analysis-suitable T-spline, is contained in the corresponding analysis-suitable T-spline space. This is accomplished through the theory of perturbed analysis-suitable T-spline spaces and a simple topological dimension formula. Second, we establish the theory of analysis-suitable local refinement and describe the conditions under which two analysis-suitable T-spline spaces are nested. Last, we demonstrate that these results can be used to establish basic approximation results which are critical for analysis.

Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis

Abstract We introduce Bezier projection as an element-based local projection methodology for B-splines, NURBS, and T-splines. This new approach relies on the concept of Bezier extraction and an associated operation introduced here, spline reconstruction, enabling the use of Bezier projection in standard finite element codes. Bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global L 2 projection. Bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging, degree elevation and reduction, basis roughening and smoothing, and spline reparameterization. In fact, Bezier projection provides a quadrature-free approach to refinement and coarsening of splines. In this sense, Bezier projection provides the fundamental building block for h p k r -adaptivity in isogeometric analysis.

Volumetric T-spline construction using Boolean operations

In this paper, we present a novel algorithm for constructing a volumetric T-spline from B-reps inspired by constructive solid geometry Boolean operations. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into two groups represented, topologically, by either a cube or a torus. We perform two Boolean operations (union and difference) with the primitives and convert them into polycubes through parametric mapping. With these polycubes, octree subdivision is carried out to obtain a volumetric T-mesh, and sharp features detected from the input model are also preserved. An optimization is then performed to improve the quality of the volumetric T-spline. The obtained T-spline surface is C2 everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C0 to G1. Finally, we extract trivariate Bézier elements from the volumetric T-spline and use them directly in isogeometric analysis.

Adaptive sweeping techniques

This paper presents an adaptive approach to sweeping one-to-one and many-to-one geometry. The automatic decomposition of many-to-one geometry into one-to-one “blocks” and the selection of an appropriate node projection scheme are vital steps in the efficient generation of high-quality swept meshes. This paper identifies two node projection schemes which are used in tandem to robustly sweep each block of a one-to-one geometry. Methods are also presented for the characterization of one-to-one geometry and the automatic assignment of the most appropriate node projection scheme. These capabilities allow the sweeper to adapt to the requirements of the sweep block being processed. The identification of the two node projection schemes was made after an extensive analysis of existing schemes was completed. One of the node projection schemes implemented in this work, BoundaryError, was selected from traditional node placement algorithms. The second node projection scheme, SmartAffine, is an extension of simple affine transformations and is capable of efficiently sweeping geometry with source and/or target curvature while approximating the speed of a simple transform. These two schemes, when used in this adaptive setting, optimize mesh quality and the speed that swept meshes can be generated while minimizing required user interaction.

A methodology for quadrilateral finite element mesh coarsening

High fidelity finite element modeling of continuum mechanics problems often requires using all quadrilateral or all hexahedral meshes. The efficiency of such models is often dependent upon the ability to adapt a mesh to the physics of the phenomena. Adapting a mesh requires the ability to both refine and/or coarsen the mesh. The algorithms available to refine and coarsen triangular and tetrahedral meshes are very robust and efficient. However, the ability to locally and conformally refine or coarsen all quadrilateral and all hexahedral meshes presents many difficulties. Some research has been done on localized conformal refinement of quadrilateral and hexahedral meshes. However, little work has been done on localized conformal coarsening of quadrilateral and hexahedral meshes. A general method which provides both localized conformal coarsening and refinement for quadrilateral meshes is presented in this paper. This method is based on restructuring the mesh with simplex manipulations to the dual of the mesh. In addition, this method appears to be extensible to hexahedral meshes in three dimensions.

Conformal Refinement and Coarsening of Unstructured Hexahedral Meshes

This paper describes recently developed procedures for local conformal refinement and coarsening of all-hexahedral unstructured meshes. Both refinement and coarsening procedures take advantage of properties found in the dual or “twist planes” of the mesh. A twist plane manifests itself as a conformal layer or sheet of hex elements within the global mesh. We suggest coarsening techniques that will identify and remove sheets to satisfy local mesh density criteria while not seriously degrading element quality after deletion. A two-dimensional local coarsening algorithm is introduced. We also explain local hexahedral refinement procedures that involve both the placement of new sheets, either between existing hex layers or within an individual layer. Hex elements earmarked for refinement may be defined to be as small as a single node or as large as a major group of existing elements. Combining both refinement and coarsening techniques allows for significant control over the density and quality of the resulting modified mesh.

Tissue-scale, personalized modeling and simulation of prostate cancer growth

Significance We perform a tissue-scale, personalized computer simulation of prostate cancer (PCa) growth in a patient, based on prostatic anatomy extracted from medical images. To do so, we propose a mathematical model for the growth of PCa. The model includes an equation for the reference biomarker of PCa: the prostate-specific antigen (PSA). Hence, we can link the results of our model to data that urologists can easily interpret. Our model reproduces features of prostatic tumor growth observed in experiments and clinical practice. It also captures a known shift in the growth pattern of PCa, from spheroidal to fingered geometry. Our results indicate that this shape instability is a tumor response to escape starvation, hypoxia, and, eventually, necrosis. Recently, mathematical modeling and simulation of diseases and their treatments have enabled the prediction of clinical outcomes and the design of optimal therapies on a personalized (i.e., patient-specific) basis. This new trend in medical research has been termed “predictive medicine.” Prostate cancer (PCa) is a major health problem and an ideal candidate to explore tissue-scale, personalized modeling of cancer growth for two main reasons: First, it is a small organ, and, second, tumor growth can be estimated by measuring serum prostate-specific antigen (PSA, a PCa biomarker in blood), which may enable in vivo validation. In this paper, we present a simple continuous model that reproduces the growth patterns of PCa. We use the phase-field method to account for the transformation of healthy cells to cancer cells and use diffusion−reaction equations to compute nutrient consumption and PSA production. To accurately and efficiently compute tumor growth, our simulations leverage isogeometric analysis (IGA). Our model is shown to reproduce a known shape instability from a spheroidal pattern to fingered growth. Results of our computations indicate that such shift is a tumor response to escape starvation, hypoxia, and, eventually, necrosis. Thus, branching enables the tumor to minimize the distance from inner cells to external nutrients, contributing to cancer survival and further development. We have also used our model to perform tissue-scale, personalized simulation of a PCa patient, based on prostatic anatomy extracted from computed tomography images. This simulation shows tumor progression similar to that seen in clinical practice.

Robust Poisson Surface Reconstruction

We propose a method to reconstruct surfaces from oriented point clouds with non-uniform sampling and noise by formulating the problem as a convex minimization that reconstructs the indicator function of the surface’s interior. Compared to previous models, our reconstruction is robust to noise and outliers because it substitutes the least-squares fidelity term by a robust Huber penalty; this allows to recover sharp corners and avoids the shrinking bias of least squares. We choose an implicit parametrization to reconstruct surfaces of unknown topology and close large gaps in the point cloud. For an efficient representation, we approximate the implicit function by a hierarchy of locally supported basis elements adapted to the geometry of the surface. Unlike ad-hoc bases over an octree, our hierarchical B-splines from isogeometric analysis locally adapt the mesh and degree of the splines during reconstruction. The hierarchical structure of the basis speeds-up the minimization and efficiently represents clustered data. We also advocate for convex optimization, instead isogeometric finite-element techniques, to efficiently solve the minimization and allow for non-differentiable functionals. Experiments show state-of-the-art performance within a more flexible framework.