Catmull-Clark Research Articles

Realistic aggregate based on rough textures with deep learning

Model construction is a critical element in simulating the mesoscale structure of concrete. This study introduces an advanced deep learning approach that excels in generating highly realistic aggregates with diverse and heterogeneous surface roughness. The methodology leverages a synergy between Gaussian Process Regression (GPR) and Convolutional Neural Networks (CNNs) to imbue textures with remarkable variations, all originating from a single source texture. Initially, aggregates are randomly generated through the cell fracture technique, and subsequently, the Catmull–Clark subdivision algorithm is applied to refine and smoothen their surfaces. The integration of displacement mapping, in conjunction with the generated texture, results in the creation of intricately textured rough surfaces. Significantly, this research presents an innovative technique for introducing a wide spectrum of surface roughness variations to aggregates, greatly enhancing our ability to construct highly accurate mesoscale models for concrete. This groundbreaking development not only enriches the domain of concrete modeling but also clears a path for elevated levels of precision and sophistication in the realm of concrete structure analysis.

Quadratic-attraction subdivision with contraction-ratio [formula omitted]

Classic generalized subdivision, such as Catmull–Clark subdivision, as well as recent subdivision algorithms for high-quality surfaces, rely on slower convergence towards extraordinary points for mesh nodes surrounded by n>4 quadrilaterals. Slow convergence corresponds to a contraction-ratio of λ>0.5. To improve shape, prevent parameterization discordant with surface growth, or to improve convergence in isogeometric analysis near extraordinary points, a number of algorithms explicitly adjust λ by altering refinement rules. However, such tuning of λ has so far led to poorer surface quality, visible as uneven distribution or oscillation of highlight lines. The recent Quadratic-Attraction Subdivision (QAS) generates high-quality, bounded curvature surfaces based on a careful choice of quadratic expansion at the central point and, just like Catmull–Clark subdivision, creates the control points of the next subdivision ring by matrix multiplication. But QAS shares the contraction-ratio λCC>1/2 of Catmull–Clark subdivision when n>4. For n=5,…,10, QAS+ improves the convergence to the uniform λ=12 of binary domain refinement and without sacrificing surface quality compared to QAS.

Uncertainty Quantification of Vibroacoustics with Deep Neural Networks and Catmull–Clark Subdivision Surfaces

This study proposes an uncertainty quantification method based on deep neural networks and Catmull–Clark subdivision surfaces for vibroacoustic problems. The deep neural networks are utilized as a surrogate model to efficiently generate samples for stochastic analysis. The training data are obtained from numerical simulation by coupling the isogeometric finite element method and the isogeometric boundary element method. In the simulation, the geometric models are constructed with Catmull–Clark subdivision surfaces, and meantime, the physical fields are discretized with the same spline functions as used in geometric modelling. Multiple deep neural networks are trained to predict the sound pressure response for various parameters with different numbers and dimensions in vibroacoustic problems. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

HSS-progressive interpolation for Loop and Catmull–Clark Subdivision Surfaces

Loop and Catmull–Clark subdivision schemes are the most popular and commonly employed approximation subdivision schemes. However, some features of the given mesh are lost in their refinement process. Thus, the subdivision surfaces of Loop and Catmull–Clark schemes does not interpolate the vertices of the given mesh and shrink in some cases. This paper proposed new progressive iterative approximation (PIA) formats based on Hermitian and skew-Hermitian splitting iteration technique (HSS). The proposed method named H-PIA and its weighted version called WH-PIA force the limit surface of Loop and Catmull–Clark subdivision schemes to interpolate the vertices of the given mesh. The approximate optimal weight of WH-PIA is given, and the convergence of H-PIA and WH-PIA are proved. Various test examples are provided to illustrate the efficiency and effectiveness of the proposed H-PIA and WH-PIA. Experimental results demonstrate that the rate of convergence of H-PIA and WH-PIA are faster than that of the PIA and weighted PIA (W-PIA), respectively.

Computational instability analysis of inflated hyperelastic thin shells using subdivision surfaces

The inflation of hyperelastic thin shells is a highly nonlinear problem that arises in multiple important engineering applications. It is characterised by severe kinematic and constitutive nonlinearities and is subject to various forms of instabilities. To accurately simulate this challenging problem, we present an isogeometric approach to compute the inflation and associated large deformation of hyperelastic thin shells following the Kirchhoff–Love hypothesis. Both the geometry and the deformation field are discretized using Catmull–Clark subdivision bases which provide the required C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-continuous finite element approximation. To follow the complex nonlinear response exhibited by hyperelastic thin shells, inflation is simulated incrementally, and each incremental step is solved using the Newton–Raphson method enriched with arc-length control. An eigenvalue analysis of the linear system after each incremental step assesses the possibility of bifurcation to a lower energy mode upon loss of stability. The proposed method is first validated using benchmark problems and then applied to engineering applications, where the ability to simulate large deformation and associated complex instabilities is clearly demonstrated.

An extended tuned subdivision scheme with optimal convergence for isogeometric analysis

This paper presents an extended tuned subdivision scheme for quadrilateral meshes that recovers optimal convergence rates in isogeometric analysis while preserving high-quality limit surfaces. Convergence rates can be improved for subdivision surfaces through proper mask tuning with a lower subdominant eigenvalue λ. However, such a tuning might affect the quality of limit surfaces if the tuning parameters are restricted to one-ring refined control vertices only. In this work, we further relax Catmull–Clark subdivision rules for 2-ring refined vertices with additional degrees of freedom in tuning masks with limit surfaces towards curvature continuity at extraordinary positions. Curvatures are bounded near extraordinary positions of valences N≤7, and a relaxation approach is proposed to suppress the local curvature variation for N≥8. We compare the proposed tuned subdivision scheme with several state-of-the-art schemes in terms of both surface qualities and applications in isogeometric analysis. Laplace–Beltrami equations with prescribed test solutions are adopted for verification of analysis results, and the proposed scheme recovers optimal convergence rates in the L2-norm with considerably lower absolute solution error than other schemes. As to surface quality, the proposed scheme has significant improvements compared with other schemes with optimal convergence rates for isogeometric analysis. The proposed scheme produces tighter curvature bounds with comparable reflection lines to the prevalent Catmull–Clark subdivision.

Non-uniform subdivision schemes of ωB-spline curves and surfaces with variable parameters

In this paper, we propose a new non-uniform subdivision scheme that includes a tension parameters sequence. Each tension parameter of the sequence is assigned at each edge of the initial control polygon. The proposed scheme can produce limiting curves that are more consistent with the original data points and the control polygon. It has also the advantage of generating a wide variety of shapes for the limiting curves. Moreover, to produce smooth limit surfaces, the proposed scheme is firstly extended to tensor-product surfaces, then to meshes of arbitrary topology. The convergence and smoothness of the proposed scheme are proven. Numerical results that illustrate the advantages of the proposed non-uniform scheme are given. • This family of non-uniform subdivision schemes accompanied with algorithms for curves and surfaces can be used in many engineering applications. • It is able to generate a wide variety of shapes for the limiting curves and surfaces as it includes a tension parameters sequence. • It allows local control of the shape of the resulting limit curves and surfaces. • It is extended to meshes of arbitrary topology to generate complex surface subdivisions. • It turns out to be the uniform scheme, when the considered tension parameters in the sequence are all equals. • It turns out to be several well-known subdivision schemes, such as Doo–Sabin and Catmull–Clark subdivision, when all the tension parameters in the sequence are equals to 1.

Acoustic shape optimization based on isogeometric boundary element method with subdivision surfaces

This study proposes a new acoustic shape optimization approach by combining isogeometric subdivision surfaces with the boundary element method. The geometry of the structural boundary is constructed by the Catmull–Clark subdivision surfaces, which are able to provide multiresolution subdivision hierarchy models to meet different precision requirements. The acoustic propagation in the unbounded domain is simulated using the boundary element method. The control points in subdivision meshes are set as design variables, and the minimization of sound pressure at the observation points is selected as the design objective. The acoustic shape sensitivities are calculated by the sensitivity boundary integral equation based on the direct differentiation method. In each iteration the geometry is updated from a direct optimized shape, then a secondary processing is considered to obtain smooth surfaces. Compared with the conventional optimization methods, we eliminate jagged geometry by multiresolution approach without needing the cumbersome meshing procedure and volume parameterization. Several numerical experiments demonstrate the effectiveness and validity of the developed shape optimization approach.

G1 – Smooth biquintic approximation of Catmull-Clark subdivision surfaces

In this paper a construction of a globally G1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier patches, which are biquintic G1 around extraordinary vertices and bicubic elsewhere; the smoothness around regular vertices neighboring EV with even valence will be C1, while it results to be C2 in the other occurrences of regular vertices. Each Bézier point is computed using a locally defined mask around the neighboring mesh vertices. To define G1 conditions, we assign quadratic gluing data around extraordinary vertices that depend solely on their valence and we use degree five patches to satisfy these G1 constraints. We explore the space of possible solutions, considering several projections on the solution space leading to different explicit formulas for the masks. Certain control points are computed by means of degree elevation of the C0 scheme of Loop and Schaefer (2008), while for others, explicit masks are deduced by providing closed-form solutions of the G1 conditions, expressed in terms of the masks. We come up with four different schemes and conduct curvature analysis on an extensive benchmark in order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically. We demonstrate numerically that the resulting surfaces converge quadratically to the CC limit when the mesh is subdivided.

Structure simplification of planar quadrilateral meshes

In this paper, we present a structure simplification framework for planar all-quad meshes with open boundaries. Our simplification framework can handle quad meshes with complex structures (e.g., quad meshes obtained via Catmull–Clark subdivision of the triangle meshes) to produce simpler meshes while preserving the boundary features. To achieve that, we introduce a set of separatrix-based semi-global operations and combine them with existing local operations to develop a new simplification framework. Additionally, we organize and order the individual simplification operations into groups and employ ranking strategies for each group to sort these operations to produce quad meshes with better quality and simpler structure. We provide a comprehensive evaluation of our framework using different input parameters on a number of representative planar quad meshes with various boundary configurations. To demonstrate the advantages of our method, we compare it with a few existing frameworks. Our comparison shows that our simplification framework usually produces simpler structure with faster computation than the state-of-the-art methods.

CAD-Integrated Form-Finding of Structural Membranes Using Extended Catmull–Clark Subdivision Surfaces

Extended Catmull–Clark subdivision surfaces are implemented within a CAD-integrated analysis workflow for form-finding by using the updated reference strategy. In this context, form-finding determines the shape of a structure that is in equilibrium for a predefined stress state. Following the principles of isogeometric analysis, the geometry for the form-finding is described by the control mesh of the subdivision surface. Using the subdivision scheme, the coarse control mesh is recursively refined to a smooth limit surface. The shape functions are obtained from the refinement process by applying algorithmic differentiation to the subdivision algorithm. This approach also supports sharp features like creases and corners. In this way, the modeling functionalities of common CAD environments are covered which allows seamless integration of the analysis into the CAD system. A core-congruential element formulation enables the efficient combination of mechanical properties with different geometric discretizations. This allows an existing framework to be extended easily with subdivision surfaces. Within this work, form-finding is applied to prestressed cable and membrane structures. Weak boundary conditions allow the modeling of oriented edge supports. The CAD-integrated implementation within the framework of a visual programming language is outlined. Using selected numerical examples, the method is demonstrated and verified. • Form-finding of tensile structures based on extended Catmull–Clark surfaces. • FEM-based form-finding using the updated reference strategy. • Modular implementation within an interactive CAD-integrated IGA environment.

Iterative Interpolation based 3D Shape Generation Using General Catmull-Clark Subdivision Surfaces

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.

Fundamental functions for local interpolation of quadrilateral meshes with extraordinary vertices

The aim of this work is to present a general and simple strategy for the construction of compactly supported fundamental spline (piecewise-polynomial) functions for local interpolation, that are defined over quadrangulations of the real plane with extraordinary vertices. The proposed strategy — which extends the univariate framework introduced in Antonelli et al. (Adv Comput Math 40:945–976, 2014) and Beccari et al. (J Comput Appl Math 240:5–19, 2013)—consists in considering a suitable combination of bivariate polynomial interpolants with blending functions that are the natural generalization of odd-degree tensor-product B-splines. These blending functions are constructed as basic limit functions of the bivariate, primal subdivision schemes developed simultaneously in Stam (Comput Aided Geom Des 18:397–427, 2001) and Zorin et al. (Comput Aided Geom Des 18:483–502, 2001). As an application example of our constructive strategy we present the compactly supported C^2 fundamental functions for local interpolation that arise by considering as blending functions the basic limit functions of the celebrated Catmull–Clark subdivision scheme proposed in Catmull and Clark (Comput Aided Des 46:103–124, 2016).

Improved non-uniform subdivision scheme with modified Eigen-polyhedron

In this study, a systematic refinement method was developed for non-uniform Catmull-Clark subdivision surfaces to improve the quality of the surface at extraordinary points (EPs). The developed method modifies the eigenpolyhedron by designing the angles between two adjacent edges that contain an EP. Refinement rules are then formulated with the help of the modified eigenpolyhedron. Numerical experiments show that the method significantly improves the performance of the subdivision surface for non-uniform parameterization.

Vibration analysis of piezoelectric Kirchhoff–Love shells based on Catmull–Clark subdivision surfaces

AbstractAn isogeometric Galerkin approach for analysing the free vibrations of piezoelectric shells is presented. The shell kinematics is specialized to infinitesimal deformations and follow the Kirchhoff–Love hypothesis. Both the geometry and physical fields are discretized using Catmull–Clark subdivision bases. This provides the required ‐continuous discretization for the Kirchhoff–Love theory. The crystalline structure of piezoelectric materials is described using an anisotropic constitutive relation. Hamilton's variational principle is applied to the dynamic analysis to derive the weak form of the governing equations. The coupled eigenvalue problem is formulated by considering the problem of harmonic vibration in the absence of external load. The formulation for the purely elastic case is verified using a spherical thin shell benchmark. Thereafter, the piezoelectric shell formulation is verified using a one dimensional piezoelectric beam. The piezoelectric effect and vibration modes of a transverse isotropic curved plate are analyzed and evaluated for the Scordelis–Lo roof problem. Finally, the eigenvalue analysis of a CAD model of a piezoelectric speaker shell structure showcases the ability of the proposed method to handle complex geometries.

Noise Pollution Reduction through a Novel Optimization Procedure in Passive Control Methods

This paper proposes a novel optimization framework in passive control techniques to reduce noise pollution. The geometries of the structures are represented by Catmull-Clark subdivision surfaces, which are able to build gap-free Computer-Aided Design models and meanwhile tackle the extraordinary points that are commonly encountered in geometric modelling. The acoustic fields are simulated using the isogeometric boundary element method, and a density-based topology optimization is conducted to optimize distribution of sound-absorbing materials adhered to structural surfaces. The approach enables one to perform acoustic optimization from Computer-Aided Design models directly without needing meshing and volume parameterization, thereby avoiding the geometric errors and time-consuming preprocessing steps in conventional simulation and optimization methods. The effectiveness of the present method is demonstrated by three dimensional numerical examples.

CONSTRUCTION OF CATMULL-CLARK SUBDIVISION SCHEME

Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

CONSTRUCTION OF CATMULL-CLARK SUBDIVISION SCHEME

Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

Bi-material topology optimization for fully coupled structural-acoustic systems with isogeometric FEM–BEM

This paper presents a novel method for topology optimization of vibrating structures interacted with acoustic wave for the purpose of minimizing radiated sound power level. We consider exterior acoustic fields and bi-material shell models without damping in this work. Within the isogeometric analysis framework, we employ Catmull–Clark subdivision surfaces to construct geometries and discretize physical fields. The isogeometric finite element method with Kirchhoff–Love shell elements is coupled with the isogeometric boundary element method in acoustics. The topology optimization is performed through density-based approaches, in which the sensitivities are evaluated with adjoint variable methods. Numerical experiments demonstrate the validity and effectiveness of the algorithm for topology optimization of structural-acoustic interaction systems.

Analyzing and Improving the Parameterization Quality of Catmull-Clark Solids for Isogeometric Analysis.

In the field of physically based simulation, high quality of the simulation model is crucial for the correctness of the simulation results and the performance of the simulation algorithm. When working with spline or subdivision models in the context of isogeometric analysis, the quality of the parameterization has to be considered in addition to the geometric quality of the control mesh. Following Cohen et al.'s concept of model quality in addition to mesh quality, we present a parameterization quality metric tailored for Catmull-Clark (CC) solids. It measures the quality of the limit volume based on a quality measure for conformal mappings, revealing local distortions and singularities. We present topological operations that resolve these singularities by splitting certain types of boundary cells that typically occur in interactively designed CC-solid models. The improved models provide higher parameterization quality that positively affects the simulation results without additional computational costs for the solver.