Boundary Representation Research Articles

Evaluation of Topological Consistency in CityGML

Boundary representation models are data models that represent the topology of a building or city model. This leads to an issue in combination with geometry, as the geometric model necessarily has an underlying topology. In order to allow topological queries to rely on the incidence graph only, a new notion of topological consistency is introduced that captures possible topological differences between the incidence graph and the topology coming from geometry. Intersection matrices then describe possible types of topological consistency and inconsistency. As an application, it is examined which matrices can occur as intersection matrices, and how matrices from topologically consistent data look. The analysis of CityGML data sets stored in a spatial database system then shows that many real-world data sets contain many topologically inconsistent pairs of polygons. It was observed that even if data satisfy the val3dity test, they can still be topologically inconsistent. On the other hand, it is shown that the ISO 19107 standard is equivalent to our notion of topological consistency. In the case when the intersection is a point, topological inconsistency occurs because a vertex lies on a line segment. However, the most frequent topological inconsistencies seem to arise when the intersection of two polygons is a line segment. Consequently, topological queries in present CityGML data cannot rely on the incidence graph only, but must always make costly geometric computations if correct results are to be expected.

Structural shape optimization using Bézier triangles and a CAD-compatible boundary representation

A method for shape optimization using Bezier triangles is introduced. The proposed procedure takes as input a CAD-compatible boundary representation of the domain and outputs an optimal design while maintaining an exact geometry representation at each iteration. The use of a triangular discretization allows the modeling of complex geometric domains, including voids, using a single patch. Some topology changes, such as those resulting from merging boundaries, can also be easily considered. An automatic mesh generator based on a quadtree construction is used to create the mesh. A gradient-based optimization algorithm (the method of moving asymptotes) is employed together with a sensitivity propagation procedure. We apply the method to some standard benchmark problems commonly considered in the literature and show that the proposed method converges to an optimal shape in only a few iterations.

Slicing and support structure generation for 3D printing directly on B-rep models

Traditional 3D printing is based on stereolithography or standard tessellation language models, which contain many redundant data and have low precision. This paper proposes a slicing and support structure generation algorithm for 3D printing directly on boundary representation (B-rep) models. First, surface slicing is performed by efficiently computing the intersection curves between the faces of the B-rep models and each slicing plane. Then, the normals of the B-rep models are used to detect where the support structures should be located and the support structures are generated. Experimental results show the efficiency and stability of our algorithm.

HDG-NEFEM for two dimensional linear elasticity

This paper proposes a new methodology for the solution of two dimensional linear elastic problems in domains with curved boundaries. The proposed method exploits the advantages of the hybridisable discontinuous Galerkin method to obtain an accurate approximation of both the displacement and the stress fields by solving a global problem that only involves the displacement field on the element edges as unknown. In addition, the methodology incorporates the exact boundary representation of the domain by means of the so-called NURBS-enhanced finite element method. Numerical examples are used to illustrate the three main advantages of the proposed method, namely the reproducibility of polynomials in domains with curved boundaries, the super-convergence of the solution even for linear approximation and the effectiveness and reliability of degree adaptive processes driven by displacement or stresses.

Boundary correspondence of planar domains for isogeometric analysis based on optimal mass transport

Domain parameterization is the process of setting up a map from a parametric domain to a computational domain. It is a key step in isogeometric analysis and has regained much attention in recent years. A prerequisite for domain parameterization is that a correspondence between the boundary of the parametric domain and that of the computational domain should be established. The boundary correspondence has a significant influence on the subsequent parameterization and numerical simulation. Currently, such correspondence is generally provided manually by users, which is very cumbersome and subjects to trial and error. In this paper, we propose an automatic approach to compute a correspondence between the boundaries of a unit square and a planar domain based on the theory of optimal mass transport (OMT). Given the boundary representation of a planar domain, the problem becomes to select four corner points on the boundary such that the difference between the curvature measure of the boundary of the planar domain and that of the unit square is minimized. We formulate the problem into an optimization problem, the objective function of which includes the transport cost between two curvature measures and the length differences of the opposite edges of the planar domain. Minimizing the objective function is equivalent to maximizing the similarity between the unit square and the computational domain. We develop an efficient algorithm to solve the optimization problem by combining Sinkhorn’s algorithm with the L-BFGS method. Numerous examples show that our approach can produce satisfactory boundary correspondence results which are comparable to manually selected ones.

Automatic generation of puzzle tile maps for spatial-temporal data visualization

Tile maps are a visualization tool to display geographic data without the accurate representation of geographic boundaries. Each region in a tile map is represented as a tile of identical shape and size. The tiles are fit in a regular grid at positions that approximate their geographic positions such that large regions do not dominate the map visualization, and information in small regions can be enhanced. In this study, the automatic generation of a tile map composed of puzzle tiles is proposed for spatial-temporal data visualization. A puzzle tile is an extension of a standard square tile. A sequence of connected and directional pieces in a puzzle tile is used to represent time-varying quantities in a geographic region. To generate a puzzle tile map, the proposed method includes algorithms for optimizing district-to-tile mapping according to not only geographic positions but also region orientations and for placing puzzle pieces in a tile. The proposed puzzle tile map can serve as a choropleth map in which the ordered pieces in a tile are shaded in proportion to the measurements of a statistical time variable, such as a time sequence of fertility rates, air pollution (PM2.5), or transfer of residential property, being displayed on a 2D map. Experimental demonstrations of various cases show that the proposed methods for district-to-tile mapping optimization and puzzle generation are feasible for automatic puzzle tile map generation. User studies show the capabilities of the puzzle tile map in terms of usability, readability, and comparability of spatial-temporal data visualization.

Isogeometric Segmentation via Midpoint Subdivision Suitable Solids

The framework of Isogeometric Analysis (IgA) makes frequent use of trivariate NURBS parameterizations (representing topological cuboids) of the computational domain. Several recent publications [1–6] describe methodologies that decompose a given three-dimensional solid in boundary representation into a collection of topological cuboids, or generate trivariate NURBS parameterizations for each of them in a subsequent step. The decomposition can be derived via a segmentation into sufficiently simple “base solids”, for which cuboidal multi-patch representations are readily available. Based on midpoint subdivision, we propose a new class of base solids. In addition, we establish the pre-processing step of face pre-segmentation, which simplifies the splitting operations and improves the shape of the resulting topological cuboids. Finally, we show how to realize the midpoint subdivision by a template mapping approach, which simultaneously generates parameterizations of the base solids as trivariate multi-patch NURBS volumes.

Generative Design Conversion to Editable and Watertight Boundary Representation

We present the first, to our knowledge, fully automatic method for the conversion of a general generative design to a watertight B-Rep composed of the exact residual geometry of the input solids and editable T-NURCC surfaces. The design can be synthesized by any combination of both additive and subtractive in-synthesis processes, and constrained by any number of keep-in and -out solids. Our method requires only the input generative setup solids, rudimentary data from the solve, and a boundary mesh of the optimized design.Leveraging the generative solve data, we augment the boundary mesh with “incidence” attributes linking it to the input solids, and partition it into organic and incident regions. The organic regions are parameterized and approximated with T-NURCCs. Then, their boundaries are “pulled” into the adjacent input solids to construct clean intersection curves. Finally, the organic surfaces are combined with the input solids to compose a watertight solid B-Rep of the generative design.Without such an automatic conversion, users seeking to exploit generative designs in downstream workflows (e.g. setting up assembly constraints, simulation analyses, performing parametric edits) have to manually convert the design to a B-Rep, slowing down the path to manufacturing and increasing the product time to market. As a result of our work, engineers can now quickly obtain useful CAD models of any number of generative designs computed by a variety of algorithms, settings and iterations.

Immersogeometric analysis of compressible flows with application to aerodynamic simulation of rotorcraft

We present an immersogeometric analysis (IMGA) approach for the simulation of compressible flows around complex geometries. In this method, compressible flow simulations are performed directly on various boundary representations (B-reps) of mechanical designs, circumventing the labor-intensive and time-consuming cleanup of complex geometric models. A new formulation for the weak imposition of essential boundary conditions in the context of non-body-fitted meshes is proposed. The formulation employs the non-symmetric Nitsche method, which yields good performance especially when the penalty parameters are difficult to estimate. We test the proposed immersogeometric formulation on benchmark problems for a wide range of Reynolds and Mach numbers, showing its robustness and accuracy. Finally, the methodology is applied to the simulation of a UH-60 Black Hawk helicopter in forward flight, illustrating the ability of the proposed approach to support the design of real-world engineering systems through high-fidelity aerodynamic analysis.

Topology design optimization of conductive thermal problems subject to design-dependent load using density gradients

In this article, a design sensitivity analysis method is developed for topology optimization of steady-state conductive thermal problems subject to design-dependent thermal loads using density gradients–based boundary detection. In a real physical heat conduction problem, a design-dependent thermal load through the convective boundary is an important factor; however, it is not easy to impose boundary conditions in heat conduction problem due to the difficulties of exact boundary representation. Here, the detection of convection boundaries is available by the density distinction between void and materials, which is similar to a gradient operator in image processing. Applying density gradients and Dirac delta function of densities, we impose a design-dependent load effect on thermal analysis and perform the design sensitivity analysis for topology optimization. At each optimization process, it is noted that the results of thermal analysis and the design sensitivities are influenced by the design-dependent thermal load. Through demonstrative numerical examples, the proposed approach using density gradients is proven to work yielding meaningful optimization results. The developed approach is applicable to the plant engineering and nuclear industries where thermo-mechanical coupling occurs inevitably by the thermal convection.

Valuing American-style options under the CEV model: an integral representation based method

This article derives a new integral representation of the early exercise boundary for valuing American-style options under the constant elasticity of variance (CEV) model. An important feature of this novel early exercise boundary characterization is that it does not involve the usual (time) recursive procedure that is commonly employed in the so-called integral representation approach well known in the literature. Our non-time recursive pricing method is shown to be analytically tractable under the local volatility CEV process and the numerical experiments demonstrate its robustness and accuracy.

A contact algorithm for voxel-based meshes using an implicit boundary representation

Numerical methods operating on structured grids have become popular since they offer good run time performance and are able to process directly voxel-based digital data from image recordings. Hence, the general framework of these fast solvers presupposes an unfitted boundary approximation avoiding complicated meshing of bodies. This allows an efficient handling with geometrical issues. Nevertheless, contacts between deformable solids are hard to deal with in the presence of this boundary representation. For this difficulty we suggest the usage of an implicit boundary representation combined with a modified saddle point formulation, resembling Nitsche’s approach. Both ideas give an elegant approach for discretizing the contact terms and enable a simple contact detection. Moreover, we suggest an intermediate surface as new reference contact area which fits well into our proposed method. In the end we present numerical results and analyze the accuracy and convergence rate. Furthermore, we demonstrate the current application range of our approach for problems with multiple contacts. In this paper we focus only on frictionless contact with small deformations.

Construction of Analysis-Suitable Vascular Models Using Axis-Aligned Polycubes.

Image-based modeling is an active and growing area of biomedical research that utilizes medical imaging to create patient-specific simulations of physiological function. Under this paradigm, anatomical structures are segmented from a volumetric image, creating a geometric model that serves as a computational domain for physics-based modeling. A common application is the segmentation of cardiovascular structures to numerically model blood flow or tissue mechanics. The segmentation of medical image data typically results in a discrete boundary representation (surface mesh) of the segmented structure. However, it is often desirable to have an analytic representation of the model, which facilitates systematic manipulation. For example, the model then becomes easier to union with a medical device, or the geometry can be virtually altered to test or optimize a surgery. Furthermore, to employ increasingly popular isogeometric analysis (IGA) methods, the parameterization must be analysis suitable. Converting a discrete surface model to an analysis-suitable model remains a challenge, especially for complex branched structures commonly encountered in cardiovascular modeling. To address this challenge, we present a framework to convert discrete surface models of vascular geometries derived from medical image data into analysis-suitable nonuniform rational B-splines (NURBS) representation. This is achieved by decomposing the vascular geometry into a polycube structure that can be used to form a globally valid parameterization. We provide several practical examples and demonstrate the accuracy of the methods by quantifying the fidelity of the parameterization with respect to the input geometry.

Fast update of sliced voxel workpiece models using partitioned swept volumes of three-axis linear tool paths

This paper presents a new and effective method to achieve fast machining simulation via the frame-sliced voxel representation (FSV-rep) workpiece modeling environment. The FSV-rep workpiece modeling scheme has been demonstrated as an efficient and accurate geometric modeling format for simulating general milling operations. The method presented in this paper specifically targets linear tool paths in three-axis milling. Based on the boundary representation of the linear tool swept volume and the voxel space for modeling the workpiece, it partitions the swept volume into a set of elemental regions, named as the swept volume regions (SVRs). With SVRs, the best performance of FSV-rep workpiece model update in three-axis milling is achieved. In the representative case studies, up to an order of magnitude faster computational performance has been observed compared to the generic approach of approximating the tool swept volume as a set of sampled tool instances. In addition, the quality of the simulated machined surface is much improved due to the use of non-approximated tool swept volumes. The efficient way to use SVRs in FSV-rep workpiece model update and the specific qualities of SVRs enabling the observed superior performance are also discussed in the paper.

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

3D objects, modeled using Computer Aided Geometric Design (CAGD) tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain.A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task Xu et al. (2017). In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process, in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed Bézier trivariates, at all its internal knots. Then, each trimmed Bézier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples of the algorithm on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.

Poly-Spline Finite-Element Method

We introduce an integrated meshing and finite-element method pipeline enabling solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which contains a small number of star-shaped polyhedra, and build a set of high-order bases on its elements, combining triquadratic B-splines, triquadratic hexahedra, and harmonic elements. We demonstrate that our approach converges cubically under refinement, while requiring around 50% of the degrees of freedom than a similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate our approach solving Poisson’s equation on a large collection of models, which are automatically processed by our algorithm, only requiring the user to provide boundary conditions on their surface.

Groups acting on rooted trees and their representations on the boundary

We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of locally 2-transitive actions. Assuming local 2-transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents.The method can be used to study Gelfand pairs and enables us to answer a question of Grigorchuk. To provide examples, we analyse in detail the local 2-transitivity of GGS-groups. Moreover, our results can be used to determine explicit formulae for zeta functions of induced representations defined by Klopsch and the author.

Variational B-rep model analysis for direct modeling using geometric perturbation

Abstract The very recent CAD paradigm of direct modeling gives rise to the need of processing 3D geometric constraint systems defined on boundary representation (B-rep) models. The major issue of processing such variational B-rep models (in the STEP format) is that free motions of a well-constrained model involve more than just rigid-body motions. The fundamental difficulty lies in having a systematic description of what pattern these free motions follow. This paper proposes a geometric perturbation method to study these free motions. This method is a generalization of the witness method, allowing it to directly deal with variational B-rep models represented by the STEP format. This generalization is essentially achieved by using a direct, geometric representation of the free motions, and then expressing the free motions in terms of composites of several basis motions. To demonstrate the effectiveness of the proposed method, a series of comparisons and case studies are presented. Highlights A new method to analyze geometric constraint systems for direct modeling. A generalization of the witness configuration method. A new method to characterize the constraint states of variational B-rep models.

Structural shape optimization with meshless method and swarm-intelligence based optimization

This paper presents a distinctive numerical approach for shape optimization by coupling meshless method with stochastic swarm intelligence based optimization technique for two dimensional linear elastic problems. Element free Galerkin method has been used here for structural analysis to circumvent frequently encountered issues with traditional grid-based technique like FEM in shape optimization such as heavy reliance on quality mesh for accurate solutions needing remeshing due to initial mesh distortion in case of large shape changes, discontinuous secondary field variables across element boundaries needing post-processing techniques and mesh optimization to minimize computational errors. Another distinguishing feature of present work is deployment of gradient-free particle swarm optimization technique for obtaining near optimal solution in shape optimization which eradicates computational efforts and errors associated with sensitivity computation. In this work, for design boundary representation Akima spline has been used due to its better stability against outlier points during shape optimization which generates natural looking boundaries. The performance of proposed technique is validated through numerical examples of shape optimization with behavior constraints on displacement and stress. To demonstrate effectiveness of present technique, results obtained through the proposed technique are compared with other techniques of past literature. To ensure acceptable levels of solution accuracy during shape optimization, h-refinement for initial problem geometry has been carried out as some shapes generated during the optimization process may have low field node density.

A quadtree-polygon-based scaled boundary finite element method for image-based mesoscale fracture modelling in concrete

A quadtree-polygon scaled boundary finite element-based approach for image-based modelling of concrete fracture at the mesoscale is developed. Digital images representing the two-phase mesostructure of concrete, which comprises of coarse aggregates and mortar are either generated using a take-and-place algorithm with a user-defined aggregate volume ratio or obtained from X-ray computed tomography as an input. The digital images are automatically discretised for analysis by applying a balanced quadtree decomposition in combination with a smoothing operation. The scaled boundary finite element method is applied to model the constituents in the concrete mesostructure. A quadtree formulation within the framework of the scaled boundary finite element method is advantageous in that the displacement compatibility between the cells are automatically preserved even in the presence of hanging nodes. Moreover, the geometric flexibility of the scaled boundary finite element method facilitates the use of arbitrary sided polygons, allowing better representation of the aggregate boundaries. The computational burden is significantly reduced as there are only finite number of cell types in a balanced quadtree mesh. The cells in the mesh are connected to each other using cohesive interface elements with appropriate softening laws to model the fracture of the mesostructure. Parametric studies are carried out on concrete specimens subjected to uniaxial tension to investigate the effects of various parameters e.g. aggregate size distribution, porosity and aggregate volume ratio on the fracture of concrete at the meso-scale. Mesoscale fracture of concrete specimens obtained from X-ray computed tomography scans are carried out to demonstrate its feasibility.