# Subdivision for Modeling and Animation

... Mặc dù mặt cong phân mảnh đã trở nên phổ biến và cho phép biểu diễn bề mặt đa mức với hình dáng bất kỳ, nhƣng khó có thể tính toán chính xác vị trí của từng điểm trên bề mặt cũng nhƣ khó điều khiển hình dáng bề mặt một cách cục bộ. Với những đối tƣợng có hình dáng bất kỳ, chỉ đòi hỏi bề mặt mềm, mƣợt khi nhìn bằng mắt, không cần tƣơng tác hoặc xác định vị trí một cách chính xác lên bề mặt đối tƣợng thì mặt cong phân mảnh là một giải pháp hiệu quả [7]. Tuy nhiên, với những ứng dụng đòi hỏi tính toán chi tiết, chính xác và có thể truy cập đến vị trí bất kỳ, nhằm hỗ trợ cho khả năng tƣơng tác đến bề mặt của đối tƣợng thì mặt cong tham số là sự lựa chọn đúng đắn. ...

... Các phép biến đổi này thƣờng đƣợc mô tả bởi các mặt nạ hoặc các ma trận. Bắt đầu từ lƣới M 0 , bằng cách thực hiện liên tiếp các bƣớc phân mảnh thông qua các mặt nạ hay các ma trận biến đổi, các lƣới M 1 , M 2 , M 3 ,… lần lƣợt đƣợc sinh ra và dần hội tụ về một mặt cong trơn đƣợc gọi là mặt cong phân mảnh [7]. ...

... Các nghiên cứu về phân mảnh để làm mịn lƣới đa giác đã đƣợc công bố và phát triển, bao gồm các lƣợc đồ phân mảnh nhƣ [7]: Catmul-Clark, Doo-Sabin, Loop, Butterfly, Kobbelt,… Với những vùng lƣới đủ trơn mƣợt (kích thƣớc các mặt đều, đồng nhất, các góc giữa hai mặt liền kề đủ nhỏ), các lƣợc đồ phân mảnh khác nhau vẫn cho kết quả gần nhƣ khó phân biệt đƣợc. Phân mảnh Catmull-Clark, Doo-Sabin và Kobbelt thích hợp cho các lƣới tứ giác và không tốt đối với các lƣới đƣợc tứ giác hóa từ lƣới tam giác. ...

Mặt cong trên miền tham số tam giác kế thừa các ưu điểm của B-spline một biến và mặt cong tham số tứ giác, kết nối mềm dẻo và phù hợp cho mô phỏng các bề mặt có hình dạng phức tạp. Do đó, chúng đóng vai trò quan trọng và có triển vọng trong việc mô hình hóa hình học cũng như hỗ trợ thiết kế linh hoạt. Bài viết này đề xuất phương pháp tái tạo các mặt cong trên miền tham số tam giác có bậc thấp từ lưới tam giác mô phỏng bề mặt của đối tượng3D, cụ thể là các mặt cong Bézier tam giác, B-patch và B-spline tam giác. Hướng tiếp cận dựa trên tái hợp mảnh lưới để giảm bậc của mặt cong tham số tái tạo và sử dụng giải thuật xấp xỉ hình học nhằm tránh giải các hệ phương trình tuyến tính. Mô hình đề xuất gồm ba giai đoạn chính: tạo lưới điều khiển, dựng mặt cong và xấp xỉ hình học. Sự hội tụ của phương pháp đề xuất cũng được chứng minh về mặt toán học và thông qua các kết quả thực nghiệm. Hầu hết các mặt cong tham số được sử dụng trong thiết kế hình học có bậc thấp nên kết quả đạt được trong bài viết này có ý nghĩa thực tiễn.

... Mặc dù mặt cong phân mảnh đã trở nên phổ biến và cho phép biểu diễn bề mặt đa mức với hình dáng bất kỳ, nhƣng khó có thể tính toán chính xác vị trí của từng điểm trên bề mặt cũng nhƣ khó điều khiển hình dáng bề mặt một cách cục bộ. Với những đối tƣợng có hình dáng bất kỳ, chỉ đòi hỏi bề mặt mềm, mƣợt khi nhìn bằng mắt, không cần tƣơng tác hoặc xác định vị trí một cách chính xác lên bề mặt đối tƣợng thì mặt cong phân mảnh là một giải pháp hiệu quả [7]. Tuy nhiên, với những ứng dụng đòi hỏi tính toán chi tiết, chính xác và có thể truy cập đến vị trí bất kỳ, nhằm hỗ trợ cho khả năng tƣơng tác đến bề mặt của đối tƣợng thì mặt cong tham số là sự lựa chọn đúng đắn. ...

... Các phép biến đổi này thƣờng đƣợc mô tả bởi các mặt nạ hoặc các ma trận. Bắt đầu từ lƣới M 0 , bằng cách thực hiện liên tiếp các bƣớc phân mảnh thông qua các mặt nạ hay các ma trận biến đổi, các lƣới M 1 , M 2 , M 3 ,… lần lƣợt đƣợc sinh ra và dần hội tụ về một mặt cong trơn đƣợc gọi là mặt cong phân mảnh [7]. ...

... Các nghiên cứu về phân mảnh để làm mịn lƣới đa giác đã đƣợc công bố và phát triển, bao gồm các lƣợc đồ phân mảnh nhƣ [7]: Catmul-Clark, Doo-Sabin, Loop, Butterfly, Kobbelt,… Với những vùng lƣới đủ trơn mƣợt (kích thƣớc các mặt đều, đồng nhất, các góc giữa hai mặt liền kề đủ nhỏ), các lƣợc đồ phân mảnh khác nhau vẫn cho kết quả gần nhƣ khó phân biệt đƣợc. Phân mảnh Catmull-Clark, Doo-Sabin và Kobbelt thích hợp cho các lƣới tứ giác và không tốt đối với các lƣới đƣợc tứ giác hóa từ lƣới tam giác. ...

... Subdivision surface algorithms define a smooth surface as the limit of a sequence of successive refinements of a mesh or a control polygon of a parametric surface. Since the late 1970s, when the publication of the papers by Catmull and Clark [50] and Doo and Sabin [51] marked the beginning of subdivision for surface modeling, a number of subdivision schemes have been proposed and found applications in computer graphics and computer-assisted geometric design [52][53][54][55]. Moreover, these approaches are very easy to implement and very efficient [52]. ...

... Since the late 1970s, when the publication of the papers by Catmull and Clark [50] and Doo and Sabin [51] marked the beginning of subdivision for surface modeling, a number of subdivision schemes have been proposed and found applications in computer graphics and computer-assisted geometric design [52][53][54][55]. Moreover, these approaches are very easy to implement and very efficient [52]. ...

... Over the improvement of fatigue life, the proposed approach brings several advantages in geometric modeling and data exchange for additive manufacturing. Besides being easy to implement and efficient [52], subdivision algorithms together with slicing tools are suitable for parallel computation, and some implementations at graphics processing unit (GPU) are available [72,73]. Moreover GPU developers had implemented subdivision algorisms in their hardware (e.g. ...

According to recent studies, a new paradigm in the geometric modeling of lattice structures based on subdivision surfaces for additive manufacturing overcomes the critical issues on CAD modeling highlighted in the literature, such as scalability, robustness, and automation. In this work, the mechanical behavior of the subdivided lattice structures was investigated and compared with the standard lattices. Five types of cellular structures based on cubic cell were modeled: struts based on squared or circular section, with or without fillets and cell based on the subdivision approach. Sixty-five specimens were manufactured by selective laser sintering technology in polyamide 12 and tensile and fatigue tests were performed. Furthermore, numerical analyses were carried out in order to establish the stress concentration factors. Results show that subdivided lattice structures, at the same resistant area, improve stiffness and fatigue life and reduce stress concentration while opening new perspectives in the development of lattice structures for additive manufacturing technologies and applications.

... The most popular representation of them is both the subdivision surface and the parametric surface [1]. Although the subdivision surfaces have become very popular and allow the representation of multiresolution surfaces having an arbitrary topology, they are difficult to evaluate accurately and control locally [2]. The parametric surface is not only for representing the smooth surfaces with high continuity, stability, flexibility, local modification properties but also for providing more effective and accurate differential operator evaluations. ...

... The Loop subdivision is an approximating face-split subdivision scheme for triangular mesh based on the triangular splines, which produces C 2 -continuous quartic triangular B-spline surfaces over triangular meshes [2]. In each step of this subdivision, each triangular face of a coarse mesh is split into four smaller triangular faces. ...

... For the boundary vertices, by applying the inverse masks for cubic B-spline, the inverse formula is [2]: ...

Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.

... In the late seventies, the publication of the papers by Catmull and Clark (1978) and Doo and Sabin (1978) marked the beginning of subdivision for surface modeling. Since then a number of subdivision surfaces schemes were proposed and found their way into wide applications in computer graphics and computer assisted geometric design (Zorin and Schröder, 2000, Peters and Reif, 1997, Dyn et al., 1990, Zorin et al., 1996, Loop, 1987, Vlachos et al., 2001, Kobbelt, 2000. A subdivision scheme defines a smooth surface as the limit of a sequence of successive refinements of a mesh. ...

... A subdivision scheme defines a smooth surface as the limit of a sequence of successive refinements of a mesh. This approach is very easy to implement and very efficient (Zorin and Schröder, 2000). In this section the potential of different subdivision schemes in cellular structure geometric modeling is proposed. ...

... These schemes include the Mid-Edge, an original approach proposed by the authors, the Doo-Sabin, the Catmull-Clark and the Bi-Quartic subdivision schemes. These schemes progressively increase the continuity degree of the subdivided mesh together with the computational cost (Zorin and Schröder, 2000). ...

Purpose
This paper aims to propose a consistent approach to geometric modeling of optimized lattice structures for additive manufacturing technologies.
Design/methodology/approach
The proposed method applies subdivision surfaces schemes to an automatically defined initial mesh model of an arbitrarily complex lattice structure. The approach has been developed for cubic cells. Considering different aspects, five subdivision schemes have been studied: Mid-Edge, an original scheme proposed by the authors, Doo–Sabin, Catmull–Clark and Bi-Quartic. A generalization to other types of cell has also been proposed.
Findings
The proposed approach allows to obtain consistent and smooth geometric models of optimized lattice structures, overcoming critical issues on complex models highlighted in literature, such as scalability, robustness and automation. Moreover, no sharp edge is obtained, and consequently, stress concentration is reduced, improving static and fatigue resistance of the whole structure.
Originality/value
An original and robust method for modeling optimized lattice structures was proposed, allowing to obtain mesh models suitable for additive manufacturing technologies. The method opens new perspectives in the development of specific computer-aided design tools for additive manufacturing, based on mesh modeling and surface subdivision. These approaches and slicing tools are suitable for parallel computation, therefore allowing the implementation of algorithms dedicated to graphics cards.

... Subdivision surfaces are now widely deployed in many computer graphics and geometric modeling tasks (for an overview see [34]). They are desirable for many modeling, animation, and simulation applications, in part because they efficiently and robustly generate smooth surfaces from an arbitrary topology control mesh. ...

... • i = 0 We assume that x 0 = [1, 1 . . . 1] T , as this is required for the subdivision surface to be affine invariant [34]. Then by the partition of unity property of the box spline basis, Bx 0 is unity and Bux 0 is zero, and from Equation 2 we conclude that φ 0 u (u, v) is zero everywhere and Φ 0 u = [0, 0]. ...

Subdivision surfaces are an attractive representation when modeling arbitrary-topology free-form surfaces and show great promise for applications in engineering design and computer animation. Interference detection is a critical tool in many of these applications. In this paper, we derive normal bounds for subdivision surfaces and use these to develop an efficient algorithm for (self-) interference detection.

... Subdivision 206 surfaces are mathematical instruments for repeated and converging implementations of rules for building 207 smooth surfaces (Fig. 7). This method not only overcomes the limitations of NURBS by defining smooth 208 and controllable surfaces that need no trimming for arbitrary topologies but is also computationally 209 efficient and suitable for complex geometry (Zorin et al. 2000). To explain subdivision surfaces, first, basic 210 concepts such as topology, mesh data (e.g., the positions of vertices) and shape should be clarified. ...

... 231Schröder andZorin (2000) noted that during the subdivision surfaces procedure, either faces can be split 232 into subfaces (primal), or vertices can be divided into multiple vertices (dual). In primal schemes, new 233 vertices are created based on either interpolation or approximation of the original vertices, which divides 234 this approach into two relevant categories: interpolation and approximation schemes.235 ...

Methods from the field of Computer Graphics are the foundation for the representation of geological structures in the form of geological models. However, as many of these methods have been developed for other types of applications, some of the requirements for the representation of geological features may not be considered and the capacities and limitations of different algorithms are not always evident. In this work, we, therefore, review surface-based geological modelling methods from both a geological and computer graphics perspective. Specifically, we investigate the use of NURBS (Non-Uniform Rational B-Splines) and subdivision surfaces, as two main parametric surface-based modelling methods, and compare the strengths and weaknesses of both approaches. Although NURBS surfaces have been used in geological modelling, subdivision surfaces as a standard method in the animation and gaming industries have so far received little attention – even if subdivision surfaces support arbitrary topologies and watertight modelling, two aspects that make them an appealing choice for complex geological modelling. Watertight modelling is a type of modelling in which the surfaces of the model have sealed interactions with all surrounding surfaces, resulting in the generation of closed volumes. Watertight models are, therefore, an important basis for subsequent process simulations based on these models.Many complex geological structures require a combination of smooth and sharp edges. Investigating subdivision schemes with semi-sharp creases is therefore an important part of this paper, as semi-sharp creases characterize the resistance of a mesh structure to the subdivision procedure. Moreover, non-manifold topologies, as a challenging concept in complex geological and reservoir modelling, are explored, and the subdivision surface method, which is compatible with non-manifold topology is described. Finally, solving inverse problems by fitting the smooth surfaces to complex geological structures is investigated with a case study. The fitted surfaces are watertight, controllable with control points, and topologically similar to the main geological structure. Also, the fitted model can reduce the cost of modelling and simulation by using a reduced number of vertices in comparison to the complex geological structure.

... The proposed approach allows enhancing automation and shape complexity of CAD tools compared to other mesh and NURBS approaches for modeling shell-like periodic surfaces, obtaining consistent polygon meshes. Indeed, the proposed approach and subdivision schemes are efficient, easy to implement [47], and suitable for parallel computation, as some implementations in graphics processing unit (GPU) show [48,49], also developed by the hardware producer itself (e.g. NVIDIA). ...

... Fig. 7 Example of the proposed approach integrated with topology optimization: a boundary conditions and density map by topology optimization, b iso-density surface adopted in the current standard modeling process (density 0.28), c mesh model obtained by the proposed approach in the whole design space and d limited to density higher than 0.05 The proposed approach makes it possible to easily model TPMSs, defining point by point the desired thickness or relative density, capacity that is currently unavailable in commercial software. The method can be easily implemented [47] in standard CAD tools automating the geometric modeling procedures for polygonal mesh, avoiding other complex mathematical definitions such as function-based (implicit functions) or volumetric modeling, which, however, must be converted in mesh for visualization, data exchange, and slicing operations. This results in the chance to fully exploit the modeling capability of CAD tools, allowing precise modeling of boundary needed in assembly management and in hybrid manufacturing. ...

Minimal surfaces are receiving a renewed interest in biomedical and industrial fields, due to the capabilities of additive manufacturing technologies which allow very complex shapes. In this paper, an approach for geometric modeling of variable thickness triply periodic minimal surfaces in a CAD environment is proposed. The approach consists of three main steps: the definition of an initial mesh, the adoption of a subdivision scheme and the assignment of a variable thickness by a differential offset. Moreover, the relationship between relative density and mesh thickness was established for two types of minimal surfaces: Schoen’s gyroid, Schwarz’ Primitive. The proposed method improves the main issues highlighted in literature in the modeling of cellular materials and allows to easily obtain a consistent polygonal mesh model satisfying functional requirements. Two test cases were presented: the first shows a gradient thickness gyroid; in the second the relative density obtained by topology optimization was adopted in our modeling approach using a Schwarz’ Primitive. In both cases, guidelines for selecting the geometric modeling parameters taking into account the specific additive manufacturing process constraints were discussed. The proposed method opens new perspectives in the development of effective CAD tools for additive manufacturing, improving the shape complexity and data exchange capacity in cellular solid modeling.

... Current focus remains on developing the interpolation subdivision techniques for predicting the points in the data set [10], [44], [74], [35], [34], [23], [16]. Since, a better tolerance range can be achieved with this approach; the development of these methods has been given a priority. ...

... A general classification for the different types of subdivision methods involved are given in the table 2.1 below. A complete reference to these method types can be found in [74]. In general it has been found that the approximation subdivision methods produce smoother surfaces than the interpolation methods. ...

Undergraduate thesis report titled 'Surface Reconstruction from Low-Quality Data' supervised by Prof. Puneet Tandon at IIITDM Jabalpur in 2015.

... to implement. The popularity of subdivision schemes is due to their applications in several different research areas like computer aided geometric design [1], geometric modeling [2,3], modeling and animation [4,5] and multiresolution analysis [6,7]. Depending on the refinement rules, they can be classified into 2 types: stationary subdivision schemes (i.e. ...

Trigonometric box splines can be considered as the multivariate generalizations of univariate B-splines. This means multivariate trigonometric box splines can be constructed from univariate trigonometric B-splines by suitable adaptive methods. Mainly, this paper is concerned with the construction of a new class of bivariate trigonometric box spline functions with the help of trigonometric B-splines through directional convolution method. These are refinable functions and some of their properties are also investigated. Secondly, a new effective non-stationary subdivision scheme is derived from the two-scale relationship of a particular trigonometric box spline. The non-stationary subdivision scheme is capable of producing such trigonometric box spline surfaces in regular regions. Some important properties along with the interactive modeling capability of this subdivision scheme are described in detail.

... Whereas extrinsic coarsening is essential for, e.g., adaptive visualization [Hoppe 1996], intrinsic coarsening is well-suited to multiresolution solvers such as geometric multigrid ]. Here, coarse-to-fine schemes, e.g. based on subdivision surfaces [Zorin et al. 2000], yield regular connectivity and principled prolongation operators based on subdivision basis functions Shoham et al. 2019]. However, without careful preprocessing [Eck et al. 1995;Hu et al. 2022] subdivision behaves poorly on coarse, low-quality meshes encountered in the wild [Zhou and Jacobson 2016]. ...

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation - a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.

... Later on, subdivision was studied in its own right [5]. However, only recently have subdivision surfaces received broad attention and development in the computer graphics community (see [46] and the references therein). In particular, a broad theoretical understanding of the analytic properties of subdivision surfaces was missing for a long time. ...

Multi media data types such as digital sound, images, and video are now ubiquitous in all areas of computing and daily life. This wide impact was made possible by a number of factors. A key factor in the wide use of a given data type is the ease and economy of acquiring it. Using a rough time line one can observe that this was true for sound in the 70s, images in the 80s, and finally video in the 90s, roughly following the development of computing hardware with its ever increasing cpu and memory resources (Figure 1). Another key factor in the wide use of a given data type is the existence of efficient algorithms for creation, storage, transmission, editing and other manipulations of the data. The mathematical foundation for these algorithms has for a very long time rested on sampling and associated Fourier techniques. Even more recent developments, such as the use of wavelets for image and video compression still rest upon the foundation laid by Fourier analysis. As such, these methods now codified as “Digital Signal Processing” (DSP) have been extraordinarily successful impacting areas ranging from cheap consumer devices such as cell phones and MP3 players to high end scientific computing applications solving some of today’s most demanding PDEs, for example.

... Moreover, by changing the position of the control points, the generated mesh changes smoothly (Fig. 4d). The subdivision surfaces scheme not only overcomes the limitations of NURBS by defining smooth and controllable surfaces that need no trimming for arbitrary topologies, but is also computationally efficient and suitable for complex geometry (Zorin 2000). ...

Methods from the field of computer graphics are the foundation for the representation of geological structures in the form of geological models. However, as many of these methods have been developed for other types of applications, some of the requirements for the representation of geological features may not be considered, and the capacities and limitations of different algorithms are not always evident. In this work, we therefore review surface-based geological modelling methods from both a geological and computer graphics perspective. Specifically, we investigate the use of NURBS (non-uniform rational B-splines) and subdivision surfaces, as two main parametric surface-based modelling methods, and compare the strengths and weaknesses of the two approaches. Although NURBS surfaces have been used in geological modelling, subdivision surfaces as a standard method in the animation and gaming industries have so far received little attention—even if subdivision surfaces support arbitrary topologies and watertight boundary representation, two aspects that make them an appealing choice for complex geological modelling. It is worth mentioning that watertight models are an important basis for subsequent process simulations. Many complex geological structures require a combination of smooth and sharp edges. Investigating subdivision schemes with semi-sharp creases is therefore an important part of this paper, as semi-sharp creases characterise the resistance of a mesh structure to the subdivision procedure. Moreover, non-manifold topologies, as a challenging concept in complex geological and reservoir modelling, are explored, and the subdivision surface method, which is compatible with non-manifold topology, is described. Finally, solving inverse problems by fitting the smooth surfaces to complex geological structures is investigated with a case study. The fitted surfaces are watertight, controllable with control points, and topologically similar to the main geological structure. Also, the fitted model can reduce the cost of modelling and simulation by using a reduced number of vertices in comparison with the complex geological structure.
Graphical Abstract

... Each side of the square serves as an edge and all corners are connected to the central vertex. The refinement is done with loop subdivision [46] by splitting each triangular face of the mesh into four smaller triangles by connecting the midpoints of the edges. Figure 5 shows meshes at different levels of resolution. ...

Face recognition is a widely accepted biometric identifier, as the face contains a lot of information about the identity of a person. The goal of this study is to match the 3D face of an individual to a set of demographic properties (sex, age, BMI, and genomic background) that are extracted from unidentified genetic material. We introduce a triplet loss metric learner that compresses facial shape into a lower dimensional embedding while preserving information about the property of interest. The metric learner is trained for multiple facial segments to allow a global-to-local part-based analysis of the face. To learn directly from 3D mesh data, spiral convolutions are used along with a novel mesh-sampling scheme, which retains uniformly sampled points at different resolutions. The capacity of the model for establishing identity from facial shape against a list of probe demographics is evaluated by enrolling the embeddings for all properties into a support vector machine classifier or regressor and then combining them using a naive Bayes score fuser. Results obtained by a 10-fold cross-validation for biometric verification and identification show that part-based learning significantly improves the systems performance for both encoding with our geometric metric learner or with principal component analysis.

... Due to the intensive computations required by most existing subdivision algorithms for their evaluation, subdivision surfaces were not generally used in real time applications. Subdivision engines implementation uses a recursive process that inserts new vertices into the mesh, refines existing point positions, and updates the connectivity [6]. Four different approaches are suggested in the literature to evaluate and render subdivision surfaces, namely: (i) recursive evaluation [5] [4] [3], (ii) direct evaluation [7] [8], (iii) reduction to the regular setting [9] [10] and (iv) pre-tabulated basis function composition [1]. ...

... The basic idea of subdivision is to "define a smooth curve or surface as the limit of sequence of successive refinements" [Zorin et al. 2000]. This broad definition admits a wide variety or "zoo" of different subdivision schemes that would be outside the scope of this paper to cover thoroughly. ...

This paper introduces Neural Subdivision, a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.

... In addition, the proposed technique is tested under subdivision attacks, especially for two typical subdivision schemes, with one iteration: the simple midpoint scheme and the Loop scheme [26]. As depicted in Table XIII, the obtained results in terms of correlation and MRMS are encouraging. ...

Nowadays, three-dimensional meshes have been extensively used in several applications such as, industrial, medical, computer-aided design (CAD) and entertainment due to the processing capability improvement of computers and the development of the network infrastructure. Unfortunately, like digital images and videos, 3-D meshes can be easily modified, duplicated and redistributed by unauthorized users. Digital watermarking came up while trying to solve this problem. In this paper, we propose a blind robust watermarking scheme for three-dimensional semiregular meshes for Copyright protection. The watermark is embedded by modifying the norm of the wavelet coefficient vectors associated with the lowest resolution level using the edge normal norms as synchronizing primitives. The experimental results show that in comparison with alternative 3-D mesh watermarking approaches, the proposed method can resist to a wide range of common attacks, such as similarity transformations including translation, rotation, uniform scaling and their combination, noise addition, Laplacian smoothing, quantization, while preserving high imperceptibility.

... Modeling surfaces of general topology is quite difficult with patches. Subdivision surfaces are a generalization of splines that are popular for modeling surfaces of arbitrary topology (Zorin and Schröder, 2000). ...

... For a fast and accurate computation, the hemisphere must meet the following conditions: (1) little difference among all of the areas of the spherical polygons to obtain a more accurate calculation, and (2) the hemisphere must be convenient to find if a spherical polygon is obscured by the canopy facets to achieve a fast calculation. Since mesh subdivision (Schröder et al., 2000;Zorin, 1999) could create a hierarchical regular mesh, we adopted it to construct the hemisphere. In our mesh, the polygons are triangles. ...

Studying the detailed organization of plant canopy structure allows a better understanding of functional processes in functional-structural plant models (FSPM). Canopy gap fraction (CGF) is an important indicator describing the canopy structure and affects the way plants capture light to perform photosynthesis, especially the interception of diffuse light. Though efforts have been made to improve the accuracy and efficiency of CGF measurement and estimation, current technologies are usually position limited. Thus, this work developed new virtual methods for computing CGF and diffuse light interception in the 3D space of plant canopies. Five hierarchical hemispheres, containing 15, 40, 360, 1,440, and 5,760 triangle patches, respectively, were constructed by applying Sqrt-3 and Butterfly subdivision schemes on an original icosahedron. Compared with traditional hemisphere division strategies using solid angles or crossed arcs of latitude and longitude, the proposed hierarchical hemispheres provide more resolution choices for different accuracy demands. Most of the patches on the hemispheres are regular triangles with similar sizes, which improves the CGF and diffuse light calculation accuracy using the Turtle model. Acceleration mechanism was built when calculating the detection of plant facets in the canopy using the tree relationship between adjacent hemispheres. Two geometric models and six maize canopy geometric models with cultivar and density differences were constructed using measured data to validate the approach. The maximum CGF error was 1.39% for the two geometric models. The average error of the four derived CGFs from hemisphere photographs was 4.23%. The diffuse light distribution correlation coefficients R2 were 0.96, 0.98, and 0.90 for three different canopy densities. Our algorithm provides multi-scale hemisphere choices for CGF and diffuse light interception simulation. The study also paves the way for further investigation into plant canopy structure analysis and simulation of light dynamics in plant canopies.

... In the coarsest finite element mesh the nodes are placed on the gyroid surface given by (19). Finer meshes are obtained by successive refinement of the mesh using the Loop subdivision scheme (Zorin and Schröder, 2000). That is, the same basis functions φ i (x) are used for generating the geometry and the FE discretisation, which is also referred to as isogeometric FE analysis ( Hughes et al., 2005). ...

The finite element method (FEM) is one of the great triumphs of modern day applied mathematics, numerical analysis and algorithm development. Engineering and the sciences benefit from the ability to simulate complex systems with FEM. At the same time the ability to obtain data by measurements from these complex systems, often through sensor networks, poses the question of how one systematically incorporates data into the FEM, consistently updating the finite element solution in the face of mathematical model misspecification with physical reality. This paper presents a statistical construction of FEM which goes beyond forward uncertainty propagation or solving inverse problems, and for the first time provides the means for the coherent synthesis of data and FEM.

... Due to the large body of known sudivision schemes for simplicial or polytopal meshes of arbitrary manifold domains [85], any (nested or non-nested) hierarchy of meshes for which subdivision schemes exist could be used for our construction. However, regular (Cartesian) grids are undeniably simpler due to their tensor product nature, and often more amenable to efficient implementation. ...

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator , a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are -orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations.

... Our present high-order boundary reconstruction approach relies on implicitly defined smooth geometries. Most industrial CAD geometries consist however of trimmed NURBS (Non-Uniform Rational B-Spline) patches or other emerging geometry representations, like subdivision surfaces [59,60]. In addition, industrial geometries have usually many sharp features in form of corners and edges and contain many small geometry features, including chamfers, fillets and holes, that cannot be realistically resolved with a discretisation mesh. ...

We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non-boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are advanced in time with an explicit time marching scheme. The discretisation meshes may contain simplicial (triangular or tetrahedral) elements of different sizes and need not be structured. On the discretisation mesh the fluid domain boundary is represented with an implicit signed distance function. The cut-elements partially covered by the solid domain are integrated after tessellation with the marching triangle or tetrahedra algorithms. Two alternative techniques are introduced to overcome the excessive stable time step restrictions imposed by cut-elements. In the first approach the cut-basis functions are replaced with the extrapolated basis functions from the nearest largest element. In the second approach the cut-basis functions are simply scaled proportionally to the fraction of the cut-element covered by the solid. To achieve high-order accuracy additional nodes are introduced on the element faces abutting the solid boundary. Subsequently, the faces are curved by projecting the introduced nodes to the boundary. The proposed approach is verified and validated with several two- and three-dimensional subsonic and hypersonic low Reynolds number flow applications, including the flow over a cylinder, a space capsule and an aerospace vehicle.

... It is based on a remarkable property of uniform B-splines: their natural refinement by subdivision. For a univariate B-spline basis of degree p, the subdivision property leads to the following two-scale relation [78]: ...

This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche's formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin's strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.

... Modeling surfaces of general topology is quite difficult with patches. Subdivision surfaces are a generalization of splines that are popular for modeling surfaces of arbitrary topology (Zorin and Schröder, 2000). A subdivision surface is defined by a polygonal mesh and a refinement scheme. ...

This tutorial describes the geometry and algorithms for generating line drawings from 3D models, focusing on occluding contours. The geometry of occluding contours on meshes and on smooth surfaces is described in detail, together with algorithms for extracting contours, computing their visibility, and creating stylized renderings and animations. Exact methods and hardware-accelerated fast methods are both described, and the trade-offs between different methods are discussed. The tutorial brings together and organizes material that, at present, is scattered throughout the literature. It also includes some novel explanations, and implementation tips. A thorough survey of the field of non-photorealistic 3D rendering is also included, covering other kinds of line drawings and artistic shading.

... Generating curves and surfaces using subdivision schemes is a popular approach in graphical modeling, animation and CAD/CAM [13,19,22,26] because of their stability in numerical computation and simplicity in coding. ...

In this paper we first derive a recursive relation of the generating functions of a family of dual 2n-point subdivision schemes. Based on the recursive relation we design repeated local operations for implementing the 2n-point subdivision schemes. Associated interpolation properties of the limit curve sequence of the dual 2n-point subdivision schemes when n tends to infinity are then investigated. Based on the repeated local operations, we further prove that the limit curves of the family of the dual 2n-point subdivision scheme sequence approach a circle that interpolates all initial control points as n approaches infinity, provided that the initial control points form a regular control polygon. Other interpolation properties show that the limit curve interpolates all closed initial control points with odd points or with even points but satisfying an extra condition, and interpolates all newly inserted vertices of an original closed polygon, when n approaches infinity. Some numerical examples are provided to illustrate the validity of our theoretic analyses.

... The cross product t 1 × t 2 is then calculated for vertex normal. (For more in details please refer to [16]) With this we can solve for an analytic gradient of the energy with a symbolic differentiation package such as sympy [1]. ...

We propose a technique to reconstruct a general 3D object using surface reflectance information from multiple viewpoints. Our core optimization framework uses multi-view normal integration, which can recovers water-tight surface of the object iteratively in a coarse to fine manner. The integration requires normal vector field from multiple viewpoints, which we can derive from surface reflectance. We then handle the topological changes if self-intersection occurs from the optimization. We also employ the idea of multi-resolution and weighted data heuristic which helps dealing with noisy data and improves both accuracy and optimization time. Our experiment shows that the framework is able to robustly recover 3D surface well with both synthetic and real data.

... More interestingly, the fine and coarse basis functions are further related via a refinement equation, also called the two-scale relation for B-splines. 47,64 For any polynomial degree p, this relation is given by ...

... However, the subdivision surface is considering as a bridge between a control mesh of parametric surface and a smooth limit surface through the repeated process of a fixed set of subdivision rules on a control mesh. As is well known, the subdivision is a process to create a finer mesh from an arbitrary coarse mesh by adding new vertices and new faces into [5], whereas the inverse subdivision aims at constructing a coarse mesh from a given dense one. In other words, the subdivision increases the resolution of an object while the inverse subdivision reduces the resolution of that object. ...

B-patch is the main block for creating the multivariate B-spline surfaces over triangular parametric domains. It has many interesting properties in the smooth surface construction with arbitrary topology. This paper proposes a new approach for reconstructing B-patch surfaces from triangular mesh based on a local geometric approximation, along with inverse subdivision scheme. The result B-patches with the low degree cross through most of the data points of the original meshes after some steps of the local geometric approximation. The accuracy of result surfaces can be carried out by changing the position of control points and adjusting knotclouds in each of the iterations. Some concrete experimental examples are alsoprovided to demonstrate the effectiveness of the proposed method. Because most of the low degree parametric curves and surfaces are often employed in CAGD, this result has practical significance, especially for mesh compression, inverse engineering, and virtual reality.

... In addition, the proposed technique is tested under subdivision attacks, especially for two typical subdivision schemes, with one iteration: the simple midpoint scheme and the Loop scheme [26]. As depicted in Table XIII, the obtained results in terms of correlation and MRMS are encouraging. ...

Nowadays, three-dimensional meshes have been extensively used in several applications such as, industrial, medical, computer-aided design (CAD) and entertainment due to the processing capability improvement of computers and the development of the network infrastructure. Unfortunately, like digital images and videos, 3-D meshes can be easily modified, duplicated and redistributed by unauthorized users. Digital watermarking came up while trying to solve this problem. In this paper, we propose a blind robust watermarking scheme for three-dimensional semiregular meshes for Copyright protection. The watermark is embedded by modifying the norm of the wavelet coefficient vectors associated with the lowest resolution level using the edge normal norms as synchronizing primitives. The experimental results show that in comparison with alternative 3-D mesh watermarking approaches, the proposed method can resist to a wide range of common attacks, such as similarity transformations including translation, rotation, uniform scaling and their combination, noise addition, Laplacian smoothing, quantization, while preserving high imperceptibility.

The analysis of electromagnetic scattering in the isogeometric analysis (IGA) framework based on Loop subdivision has long been restricted to simply connected geometries. The inability to analyze multiply connected objects is a glaring omission. In this paper, we address this challenge. IGA provides seamless integration between the geometry and analysis by using the same basis set to represent both. In particular, IGA methods using subdivision basis sets exploit the fact that the basis functions used for surface description are smooth (with continuous second derivatives) almost everywhere. On simply connected surfaces, this permits the definition of basis sets that are divergence-free and curl-free. What is missing from this suite is a basis set that is both divergence-free
and
curl-free, a necessary ingredient for a complete Helmholtz decomposition of currents on multiply connected structures. In this paper, we effect this missing ingredient numerically using random polynomial vector fields. We show that this basis set is analytically divergence-free and curl-free. Furthermore, we show that these basis recovers curl-free, divergence-free, and curl-free and divergence-free fields. Finally, we use this basis set to discretize a well-conditioned integral equation for analyzing perfectly conducting objects and demonstrate excellent agreement with other methods.

There is an urgent unmet need to develop a fully-automated image-based left ventricle mitral valve analysis tool to support surgical decision making for ischemic mitral regurgitation patients. This requires an automated tool for segmentation and modeling of the left ventricle and mitral valve from immediate pre-operative 3D transesophageal echocardiography. Previous works have presented methods for semi-automatically segmenting and modeling the mitral valve, but do not include the left ventricle and do not avoid self-intersection of the mitral valve leaflets during shape modeling. In this study, we develop and validate a fully automated algorithm for segmentation and shape modeling of the left ventricular mitral valve complex from pre-operative 3D transesophageal echocardiography. We performed a 3-fold nested cross validation study on two datasets from separate institutions to evaluate automated segmentations generated by nnU-net with the expert manual segmentation which yielded average overall Dice scores of 0.82±0.03 (set A), 0.87±0.08 (set B) respectively. A deformable medial template was subsequently fitted to the segmentation to generate shape models. Comparison of shape models to the manual and automatically generated segmentations resulted in an average Dice score of 0.93-0.94 and 0.75-0.81 for the left ventricle and mitral valve, respectively. This is a substantial step towards automatically analyzing the left ventricle mitral valve complex in the operating room.

The state of the art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low-frequency breakdown, dense-discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in the development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are continuous, but their normals are discontinuous at the boundary between triangular tessellations of control nodes, or patches, with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design is second order differentiable almost everywhere on the surfaces. Using these descriptions for analysis opens the door to several possibilities, and is the area we explore in this article. Our focus is on loop subdivision-based isogeometric methods. In this article, our goals are twofold: 1) development of computational infrastructure for isogeometric analysis of electrically large simply connected objects, and 2) introduction of the notion of manifold harmonic transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented.

This document describes a contribution to the Insight Toolkit intended to support the process of subdivision of triangle mesh. Four approaches, linear, Loop, modified butterfly and sqrt{3} s q r t 3 subdivision schemes were introduced.

We first revisit the mathematical modeling of the flexoelectric effect in the context of continuum mechanics at infinitesimal deformations. We establish and clarify the relation between the different formulations, point out theoretical and numerical issues related to the resulting boundary value problems, and present the natural extension to finite deformations. We then present a simple B-spline based computational technique to numerically solve the associated boundary value problems, which can be extended to handle unfitted meshes, hence allowing for arbitrarily-shaped geometries. Several numerical examples illustrate the flexoelectric effect in simple benchmark setups, as well as in new flexoelectric devices and metamaterials engineered for sensing or actuation.

The triangle meshes are often used to approximate smooth or piecewise smooth surfaces. Improving the uniformity of the approximation error distribution is crucial for the mesh quality. In order to achieve such improvement it is often necessary to increase the mesh resolution (by introducing new mesh vertices and triangles) where needed depending on a local estimator for the error distribution. Usually such estimators work per triangle producing an estimation for the local mesh curvature or the local deviation from the target surface.
Many methods for subdivision of triangle meshes have been developed in the recent decades. The major flaw in most of them is the tendency to produce poor quality subdivided meshes when working on (starting from) irregular meshes. Even starting from a regular mesh which unevenly approximates the target surface, most methods tend to produce poor quality result (bad triangles aspect ratios and/or highly irregular mesh connectivity in some regions). Our research shows that the main problem with most of the methods is that they don't control the aspect ratios of the newly produced triangles and their close vicinity during the subdivision. Another flaw in most of the existing approaches is that the interpolant they construct doesn't behave well on highly irregular meshes (highly constrained and/or low quality) which often produces bumpy meshes as a result. Here we take advantage of the 2.5D type of the meshes that we process which allows us to simplify the interpolant and make it more stable in the irregular cases.
In this paper we present a smooth interpolating subdivision approach for refinement of irregular 2.5D triangle meshes. It combines an edge-splitting subdivision with a Hermite-based smooth interpolation of the triangle mesh. Working on 2.5D meshes (defined in 2D domains) makes the task simpler so that the only thing required from the interpolant is to calculate the elevations on the target interpolation surface for the new mesh vertices introduced by the subdivision. The interpolant provides the data for the estimation of local approximation error at each subdivision step used to decide whether the target surface is locally well approximated (which is the stop criteria for the subdivision process). At the same time, the method estimates how well the local mesh quality is being maintained after each subdivision step, analyses the possibilities to improve the triangles aspect ratios and applies them whenever possible which also speeds up the convergence of the overall process.
The described approach produces multiresolution triangle mesh which uniformly approximates the smooth interpolation surface while maintaining the mesh quality.

This paper introduces Neural Subdivision , a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.

Probabilistic distribution models like Gaussian mixtures have shown great potential for improving both the quality and speed of several geometric operators. This is largely due to their ability to model large fuzzy data using only a reduced set of atomic distributions, allowing for large compression rates at minimal information loss. We introduce a new surface model that utilizes these qualities of Gaussian mixtures for the definition and control of a parametric smooth surface. Our approach is based on an enriched mesh data structure, which describes the probability distribution of spatial surface locations around each vertex via a Gaussian covariance matrix. By incorporating this additional covariance information, we show how to define a smooth surface via a nonlinear probabilistic subdivision operator based on products of Gaussians, which is able to capture rich details at fixed control mesh resolution. This entails new applications in surface reconstruction, modeling, and geometric compression.

Nowadays, topology optimization and lattice structures are being re-discovered thanks to Additive Manufacturing technologies, that allow to easily produce parts with complex geometries.
The primary aim of this work is to provide an original contribution for geometric modeling of conformal lattice structures for both wireframe and mesh models, improving previously presented methods. The secondary aim is to compare the proposed approaches with commercial software solutions on a piston rod as a case study.
The central part of the rod undergoes size optimization of conformal lattice structure beams diameters using the proposed methods, and topology optimization using commercial software tool. The optimized lattice is modeled with a NURBS approach and with the novel mesh approach, while the topologically optimized part is manually remodeled to obtain a proper geometry. Results show that the lattice mesh modelling approach has the best performance, resulting in a lightweight structure with smooth surfaces and without sharp edges at nodes, enhancing mechanical properties and fatigue life.

In this work, we present a new method for controlled deformation and detail addition to 3d shapes represented as variable resolution meshes. The input data is a surface with arbitrary genus, represented by a polygonal mesh, and a set of parameters for edition control: positional information, the level of resolution, mesh features and direction of propagation of the deformation. An adaptive hierarchical mesh structure is constructed using an iterative feature-sensitive simplification method that concomitantly generates the parameterization of the mesh. The coarsest level of the representation defines the base domain which stores the original geometry via a local parameterization process. We apply local modifications to the base domain according to predefined functions; a noise function for details or any geometric deformation. In the sequel, the deformation of the base mesh is propagated to the original mesh. Our main contribution is a method that relies on the power of adaptive hierarchical structures to generate details with a greater degree of control by using a set of operators that explore the data structure properties as well as the information extracted and computed from the mesh.

Probabilistic distribution models like Gaussian mixtures have shown great potential for improving both the quality and speed of several geometric operators. This is largely due to their ability to model large fuzzy data using only a reduced set of atomic distributions, allowing for large compression rates at minimal information loss. We introduce a new surface model that utilizes these qualities of Gaussian mixtures for the definition and control of a parametric smooth surface. Our approach is based on an enriched mesh data structure, which describes the probability distribution of spatial surface locations around each vertex via a Gaussian covariance matrix. By incorporating this additional covariance information, we show how to define a smooth surface via a nonlinear probabilistic subdivision operator based on products of Gaussians, which is able to capture rich details at fixed control mesh resolution. This entails new applications in surface reconstruction, modeling, and geometric compression.

In the paper we consider a treatment of Bernoulli type shape optimization problems in three dimensions by the combination of the boundary element method and the hierarchical algorithm based on the subdivision surfaces. After proving the existence of the solution on the continuous level we discretize the free part of the surface by a hierarchy of control meshes allowing to separate the mesh necessary for the numerical analysis and the choice of design parameters. During the optimization procedure the mesh is updated starting from its coarse representation and refined by adding design variables on finer levels. This approach serves as a globalization strategy and prevents geometry oscillations without any need for remeshing. We present numerical experiments demonstrating the capabilities of the proposed algorithm.

Under a variety of external stimuli, hydrogels can undergo coupled solid deformation and fluid diffusion and exhibit large volume changes. The numerical analysis of this process can be complicated by numerical instabilities when using mixed formulations due to the violation of the inf-sup condition. In addition, the large deformations produce complex instability patterns causing singularities in the underlying set of equations. For these reasons, the experimentally observed complex patterns remain elusive and poorly understood. Furthermore, a stability criterion suitable to detect critical conditions and predict post-instability patterns is lacking for hydrogel simulations. Here we investigate the stability criterion for coupled problems with a saddle point nature and propose a generic framework to study diffusion-driven swelling-induced instabilities of hydrogels. Adopting a numerically stable subdivision-based mixed isogeometric analysis, we show that the proposed framework for stability analysis accurately captures instability points during the transient swelling of hydrogels. The influence of geometrical and material parameters on the critical conditions are also presented in stability diagrams for two useful problems involving the buckling of hydrogel rods and the wrinkling on the surface of hydrogel bilayers. The results show that the short-time response of hydrogels immersed in water are highly unstable. We believe that this generic scheme provides a theoretical and computational foundation to study the morphogenesis in nature, and it also paves the way to create functional materials and design novel hydrogel devices through stability diagrams.

Cloth simulations, widely used in computer animation and apparel design, can be computationally expensive for real‐time applications. Some parallelization techniques have been proposed for visual simulation of cloth using CPU or GPU clusters and often rely on parallelization using spatial domain decomposition techniques that have a large communication overhead. In this paper, we propose a novel time‐domain parallelization technique that makes use of the two‐level mesh representation to resolve the time‐dependency issue and develop a practical algorithm to smooth the state transition from the corresponding coarse to fine meshes. A load estimation and a load balancing technique used in online partitioning are also proposed to maximize the performance acceleration. Our method achieves a nearly linear performance scaling on manycore clusters and outperforms spatial‐domain parallelization on a diverse set of benchmarks.

Cutting-Edge Techniques to Better Analyze and Predict Complex Physical Phenomena Geometric Modeling and Mesh Generation from Scanned Images shows how to integrate image processing, geometric modeling, and mesh generation with the finite element method (FEM) to solve problems in computational biology, medicine, materials science, and engineering. Based on the author's recent research and course at Carnegie Mellon University, the text explains the fundamentals of medical imaging, image processing, computational geometry, mesh generation, visualization, and finite element analysis. It also explores novel and advanced applications in computational biology, medicine, materials science, and other engineering areas. One of the first to cover this emerging interdisciplinary field, the book addresses biomedical/material imaging, image processing, geometric modeling and visualization, FEM, and biomedical and engineering applications. It introduces image-mesh-simulation pipelines, reviews numerical methods used in various modules of the pipelines, and discusses several scanning techniques, including ones to probe polycrystalline materials. The book next presents the fundamentals of geometric modeling and computer graphics, geometric objects and transformations, and curves and surfaces as well as two isocontouring methods: marching cubes and dual contouring. It then describes various triangular/tetrahedral and quadrilateral/hexahedral mesh generation techniques. The book also discusses volumetric T-spline modeling for isogeometric analysis (IGA) and introduces some new developments of FEM in recent years with applications.

B-patch surface is the main block to construct the triangular B-spline surfaces and has many interesting properties of the surfaces over a triangular parametric domain. This paper proposes a new method for reconstructing the low degree B-patch surfaces using inverse Loop subdivision scheme, along with geometric approximation algorithm. The obtained surfaces are the low degree B-patches over the triangular domain and almost cross through the data points of the original triangular meshes after several steps of the geometric approximating. Comparing with techniques use the original mesh as the surface control polyhedron, our method reconstructed B-patches with the degree reduces to 2i times after i steps of the inverse. The accuracy of the result B-patches can be improved by adjusting the location of control points and knotvectors in each step of iterations. Some experimental results demonstrate the efficacy of the proposed approach. Because most the low degree parametric surfaces are often employed in CAGD, mesh compression, inverse engineering and virtual reality, this result has practical significance.

As a representative method of model surface, triangular mesh, which commonly has the stereo lithography (STL) format, has recently been widely applied in the CAD/CAE/CAM field, due to its superior robustness and high efficiency in tool-path generation. 3D product model optimization is of great importance for improvement of function, reduction of production material, and improvement of the company's competition. Based on the circumstance of 3D modeling, simulation and optimization technologies, developing a more sustainable product and process become possible. However, there are many situations of the morphing, which are hard to be uniformed. Thus, developing a commonly used data-driven morphing method is difficult. In this paper, morphing situations are categorized into two classes, the algebraic morphing and the free-form morphing. Algebraic morphing patterns are developed, which can be adopted independently or combined together to complete complicated morphing operations. In the free-form morphing, control points are obtained by data mining, and then mesh subdivision is applied to refine surfaces smoothly. The proposed morphing method is applied to a truss core panel and a human head model, clarifying the robust function and high efficiency of the method proposed in this study to deal with complex 3D product model sustainable optimization.

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