Convolution Surfaces Research Articles

Block‐structured mesh generation from implicit geometries for cardiovascular applications

AbstractGeneration of suitable meshes for patient‐specific vascular domains remains a major obstacle in the context of numerical simulations relying on the finite element method, as well as many similar methods. Herein, we discuss a skeleton‐based structured meshing approach, revolving around block‐structures. We rely on an implicitly represented surface, either via a signed distance function, or a convolution surface. The mesh generation using the aforementioned concepts is performed through the proper utilisation of transfinite maps. Application‐wise, we consider healthy blood vessels, and extensions required to tackle aortas afflicted by aortic dissection. The high quality of the resulting meshes is affirmed through common mesh quality metrics. Our approach features large flexibility in terms of spatial resolution, as well as mesh order.

Higher-order block-structured hex meshing of tubular structures

Numerical simulations of the cardiovascular system are growing in popularity due to the increasing availability of computational power, and their proven contribution to the understanding of pathodynamics and validation of medical devices with in-silico trials as a potential future breakthrough. Such simulations are performed on volumetric meshes reconstructed from patient-specific imaging data. These meshes are most often unstructured, and result in a brutally large amount of elements, significantly increasing the computational complexity of the simulations, whilst potentially adversely affecting their accuracy. To reduce such complexity, we introduce a new approach for fully automatic generation of higher-order, structured hexahedral meshes of tubular structures, with a focus on healthy blood vessels. The structures are modeled as skeleton-based convolution surfaces. From the same skeleton, the topology is captured by a block-structure, and the geometry by a higher-order surface mesh. Grading may be induced to obtain tailored refinement, thus resolving, e.g., boundary layers. The volumetric meshing is then performed via transfinite mappings. The resulting meshes are of arbitrary order, their elements are of good quality, while the spatial resolution may be as coarse as needed, greatly reducing computing time. Their suitability for practical applications is showcased by a simulation of physiological blood flow modelled by a generalised Newtonian fluid in the human aorta.

A Modeling Method for Heterogeneous Lattice Structures based on Convolution Surface

Computer-Aided Design and Applications is an international journal on the applications of CAD and CAM. It publishes papers in the general domain of CAD plus in emerging fields like bio-CAD, nano-CAD, soft-CAD, garment-CAD, PLM, PDM, CAD data mining, CAD and the internet, CAD education, genetic algorithms and CAD engines. The journal is aimed at all developers and users of CAD technology to ptovide CAD solutions for various stages of design and manufacturing. The journal publishes all about Computer-Aided Design and Computer-Aided technologies.

Controllable blending of line and polygon skeleton-based convolution surfaces with finite support kernels

We present a novel approach to control the blending of line and polygon skeleton-based convolution surfaces using locally varying Ratio of Support radius and Thickness (RST). With our method, solutions for local convolution surface approximation with prescribed surface thickness and support radii can be derived analytically. In addition, iso-surface shrinkage can be avoided by offsetting the endpoints of line skeletons and the edges of polygon skeletons. Our RST-based blending for convolution surfaces is local and can generate desired blending effects while approximating shapes with a specified thickness. Moreover, our method is intuitive and users can control the blending by adjusting the skeletal radius or the support radius of the finite support kernel independently. As our blending utilizes convolution integration only without requiring any extra composition operators, it allows for successive convolution blending operations to create complex shapes.

Three-dimensional Motion Skeleton Reconstruction Algorithm for Gymnastic Dancing Movements

Aiming at the problem of inaccurate matching results in the traditional three-dimensional reconstruction algorithm of gymnastic skeleton, a three-dimensional motion skeleton reconstruction algorithm of gymnastic dance action is proposed. Taking the center of gravity of the human body as the origin, the position of other nodes in the camera coordinate system relative to the center point of the human skeleton model is calculated, and the human skeleton data collection is completed through action division and posture feature calculation. Polynomial density is introduced into the integration of convolution surface, and the human body model of convolution surface is established according to convolution surface. By using the method of binary parameter matching, the accuracy of the matching results is improved, and the three-dimensional skeleton of gymnastic dance movement is reconstructed. The experimental results show that the fitting degree between the proposed method and the actual reconstruction result is 99.8%, and the reconstruction result of this algorithm has high accuracy.

Memory-Efficient Modeling and Slicing of Large-Scale Adaptive Lattice Structures

Abstract Lattice structures have been widely used in various applications of additive manufacturing due to its superior physical properties. If modeled by triangular meshes, a lattice structure with huge number of struts would consume massive memory. This hinders the use of lattice structures in large-scale applications (e.g., to design the interior structure of a solid with spatially graded material properties). To solve this issue, we propose a memory-efficient method for the modeling and slicing of adaptive lattice structures. A lattice structure is represented by a weighted graph where the edge weights store the struts' radii. When slicing the structure, its solid model is locally evaluated through convolution surfaces and in a streaming manner. As such, only limited memory is needed to generate the toolpaths of fabrication. Also, the use of convolution surfaces leads to natural blending at intersections of struts, which can avoid the stress concentration at these regions. We also present a computational framework for optimizing supporting structures and adapting lattice structures with prescribed density distributions. The presented methods have been validated by a series of case studies with large number (up to 100M) of struts to demonstrate its applicability to large-scale lattice structures.

On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary.

Numerical treatment of singular charges is a grand challenge in solving the Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface PB models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this work, a novel regularization formulation is proposed to capture the singularity analytically, which is the first of its kind for diffuse interface PB models. The success lies in a dual decomposition - besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. To validate the proposed regularization, a Gaussian convolution surface (GCS) is also introduced, which efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.

Barycentric convolution surfaces based on general planar polygon skeletons

Using barycentric coordinates for thickness interpolation, we present a novel polygonal skeleton based convolution surface approximation method with varying radii. Given the prescribed radii of an arbitrary planar polygonal skeleton, we first employ a smooth interior mean value coordinate interpolation approach to calculate the thickness at each projection position in the polygonal plane. Then a local thickness approximation method based on finite-support kernels is introduced to create an implicit surface with smoothly varying thickness. In addition, a polygon offset with different distances by winding numbers is employed to create local approximation at polygon boundaries. Our experiments show that the proposed uniform smooth thickness interpolation and local convolution approximation method can not only avoid surface wrinkles but also reduce computation cost. Moreover, our approach is insensitive to exterior thickness interpolation. Therefore smooth barycentric coordinates within a polygon can all be easily integrated into our approach.

Anisotropic convolution surfaces

Convolution surfaces with 1D skeletons have been limited to close-to-circular normal sections. The new formalism presented here allows for ellipsoidal normal sections. Anisotropy is prescribed on G1 skeletal curves, chosen as circular splines, by a rotation angle and the three radii of an ellipsoid at each extremity. This lightweight model creates smooth shapes that previously required tweaking the skeleton or supplementing it with 2D pieces. The scale invariance of our formalism achieves excellent radii control and thus lends itself to approximate a variety of shapes. The construction of a scaffold is extended to skeletons with G1 branches. It projects onto the convolution surface as a quad mesh with skeleton bound edge-flow.

Dynamic real-time star map simulation based on convolution surface

Dynamic real-time star map simulation based on convolution surface

Erratum to ”On Convolution surfaces in Euclidean spaces”

In the present study we give some corrections for our paper which published in the first volume of this journal.

On convolution surfaces in Euclidean 3-space

In the present paper we study with the convolution surface $C=M\star N$ of a paraboloid $M\subset \mathbb{E}^{3}$ and a parametric surface $N\subset \mathbb{E}^{3}$. We take some spacial surfaces for $N$ such as, surface of revolution, Monge patch and ruled surface and calculate the Gaussian curvature of the convolution surface $C$. Further, we give necessary and sufficient conditions for a convolution surface $C$ to become flat.

Brush2Model: Convolution surface‐based brushes for 3D modelling in head‐mounted display‐based virtual environments

AbstractEasy and efficient 3D modelling in virtual environments is an important and unsolved topic. This paper proposes a new modelling approach to tackle this issue. It invents convolution surface‐based brushes to directly draw 3D models in head‐mounted display‐based virtual environments. In order to maximize the efficiency, flexibility, and capacity of our proposed modelling approach, we propose three different skeleton‐based convolution surfaces to tackle different modelling tasks: point skeleton‐based convolution surfaces for metaball shapes, line skeleton‐based convolution surfaces for cylindrical shapes, and polygon skeleton‐based convolution surfaces for planar surfaces. Their combination makes 3D modelling more flexible and powerful. The high efficiency is further raised by our developed closed‐form solutions for point skeletons, ends of line skeletons, and edges of polygonal skeletons. Different user‐friendly sweeping schemes are provided to facilitate intuitive inputs for various complex shape generation. Unlike Google's Tilt Brush, which is used to create disconnected sheet‐like surfaces only, our proposed convolution surface‐based brushes can produce smoothly blended manifold surfaces, and novice users can easily learn and use them to create various interesting 3D models efficiently.

Heterogeneous Composition Adaptation With Material Convolution Control Features

Controlling and accomplishing the desired functional material composition in a heterogeneous object (HO) is a close loop process and requires frequent remodeling-and-analysis. Thus, flexibility and capability to efficiently modify the existing CAD model of a heterogeneous object are essential aspects of heterogeneous object modeling. The current work unfolds such capabilities of the developed material convolution surface approach. The geometric and material control features associated with the approach demonstrate the potential to modify existing material-distributions to remodel complex material variations and assure rapid heterogeneous composition adaptations. Convolution material primitives (CMPs), material potential functions, and heterogeneous and grading enclosure are manipulated to achieve desired material compositions across the heterogeneous region. The manipulation process for each control feature has been established. A few examples of modeling and modifying complex material-distributions have been reported for the validation of work.

Heterogeneous object modeling with material convolution surfaces

The possibility to attain diverse applications from heterogeneous objects calls for a generic and systematic modeling approach for design, analysis and rapid manufacturing of heterogeneous objects. The available heterogeneous object modeling techniques model simple material-distributions only and just a few of them are capable of modeling heterogeneous objects with complex geometries. Even these approaches have also, at time, shown some glitches while modeling complex objects with compound and irregular material variations. This paper unfolds the development of a stand-alone convolution surface-based modeling approach to model complex heterogeneous objects with multi-functional heterogeneities, entailing stratified sub-analytic boundary-representation, convolution material primitives, membership functions and material-potential functions. One-dimensional (associative and non-associative) and compound two-and three-dimensional material-distribution schemas are formulated and outlined to model simple, compound and irregular material-distributions in simple/complex geometry objects. The paper also illustrates a few examples of modeling complex heterogeneous objects by implementing the approach using specialized languages and software tools.

High-quality tree structures modelling using local convolution surface approximation

In this paper, we propose a local convolution surface approximation approach for quickly modelling tree structures with pleasing visual effect. Using our proposed local convolution surface approximation, we present a tree modelling scheme to create the structure of a tree with a single high-quality quad-only mesh. Through combining the strengths of the convolution surfaces, subdivision surfaces and GPU, our tree modelling approach achieves high efficiency and good mesh quality. With our method, we first extract the line skeletons of given tree models by contracting the meshes with the Laplace operator. Then we approximate the original tree mesh with a convolution surface based on the extracted skeletons. Next, we tessellate the tree trunks represented by convolution surfaces into quad-only subdivision surfaces with good edge flow along the skeletal directions. We implement the most time-consuming subdivision and convolution approximation on the GPU with CUDA, and demonstrate applications of our proposed approach in branch editing and tree composition.

SCALe‐invariant Integral Surfaces

AbstractExtraction of skeletons from solid shapes has attracted quite a lot of attention, but less attention was paid so far to the reverse operation: generating smooth surfaces from skeletons and local radius information. Convolution surfaces, i.e. implicit surfaces generated by integrating a smoothing kernel along a skeleton, were developed to do so. However, they failed to reconstruct prescribed radii and were unable to model large shapes with fine details. This work introduces SCALe‐invariant Integral Surfaces (SCALIS), a new paradigm for implicit modelling from skeleton graphs. Similarly to convolution surfaces, our new surfaces still smoothly blend when field contributions from new skeleton parts are added. However, in contrast with convolution surfaces, blending properties are scale‐invariant. This brings three major benefits: the radius of the surface around a skeleton can be explicitly controlled, shapes generated in blending regions are self‐similar regardless of the scale of the model and thin shape components are not excessively smoothed out when blended into larger ones.

Efficient polygonization of tree trunks modeled by convolution surfaces

We present an efficient polygonization approach for tree trunks modeled by line skeleton-based convolution surfaces. A quad-dominated non-convex bounding polyhedron is firstly created along the skeleton, which is then tetrahedralized and subdivided into the pre-defined resolution. After that, the iso-surface within each tetrahedron is extracted using marching tetrahedra. Our algorithm can generate polygons with adaptive edge lengths according to the thickness of the trunk. In addition, we present an efficient CUDA-based parallel algorithm utilizing the high parallelism of the tetrahedron subdivision, the potential field calculation, and the iso-surface extraction.

Convolution surfaces based on polygonal curve skeletons

This paper reviews and generalizes convolution surfaces, a technique used in computer graphics to generate smooth 3D volumes around skeletons that are lower dimensional or simpler geometric models of the shape to be created. Convolution surfaces are defined as level sets of a function obtained by integrating a kernel function along this skeleton. To allow interactive modeling, the technique has relied on closed form formulas for integration obtained through symbolic computation software.This paper provides new qualitative results and generalizations on the topic when the skeleton is a polygonal line. It is also an opportunity for us to introduce the field of convolution surfaces to the symbolic computation community, hoping that researchers well versed into integration techniques can bring additional contributions to this appealing shape representation.

Analytical solutions for sketch-based convolution surface modeling on the GPU

Convolution surfaces are attractive for modeling objects of complex evolving topology. This paper presents some novel analytical convolution solutions for planar polygon skeletons with both finite-support and infinite-support kernel functions. We convert the double integral over a planar polygon into a simple integral along the contour of the polygon based on Green’s theorem, which reduces the computational cost and allows for efficient parallel computation on the GPU. For finite support kernel functions, a skeleton clipping algorithm is presented to compute the valid skeletons. The analytical solutions are integrated into a prototype modeling system on the GPU (Graphics Processing Unit). Our modeling system supports point, polyline and planar polygon skeletons. Complex objects with arbitrary genus can be modeled easily in an interactive way. Resulting convolution surfaces with high quality are rendered with interactive ray casting.