Irregular Meshes Research Articles

On the numerical solution to space fractional differential equations using meshless finite differences

We derive a discretization of the Caputo and Riemann–Liouville spatial derivatives by means of the meshless Generalized Finite Difference Method, which is based on moving least squares. The conditional convergence of the method is proved and several examples over one dimensional irregular meshes are given.

Cross-gradient joint inversion and clustering of ERT and SRT data on structured meshes incorporating topography

Summary The combination of electrical resistivity and seismic refraction tomography is a common practice for the characterization of subsurface features. Presently, the cross-gradient inversion scheme stands out as one of the most robust joint approaches, and some authors modified it to manage complex topographies on unstructured meshes even if at the expense of introducing additional parameters in the inversion process. We propose in this work a cross-gradient algorithm for jointly inverting electrical and seismic tomographic data on structured meshes in cases with non-flat topography. The proposed approach preserves the benefit of the classical cross-gradient approach without the need to impose physical length scales, as for irregular meshes. The quality of the results is evaluated in comparison with independent inversion through a new standardized cross-gradient index and a fuzzy c-means analysis that provides an assessment of the reconstruction accuracy through the membership function. The proposed method was applied to both synthetic models and field-scale examples located in Central Italy, where an accurate geophysical reconstruction is needed for the rehabilitation of existing dams. For all cases, joint inversion yielded superior results compared to independent inversion, demonstrating better agreement with available borehole data. The effectiveness of the joint approach was also demonstrated by the post-inversion tools, where the new cross-gradient index highlighted changes in structural similarity whilst fuzzy c-means clustering allowed for a quantitative reconstruction (position and shape) of the main units at the sites, facilitating the detection of site layering modifications.

1D CNNs and face-based random walks: A powerful combination to enhance mesh understanding and 3D semantic segmentation

In this paper, we present a novel face-based random walk method aimed at addressing the 3D semantic segmentation issue. Our method utilizes a one-dimensional convolutional neural network for detailed feature extraction from sequences of triangular faces and employs a stacked gated recurrent unit to gather information along the sequence during training. This approach allows us to effectively handle irregular meshes and utilize the inherent feature extraction potential present in mesh geometry. Our study's results show that the proposed method achieves competitive results compared to the state-of-the-art methods in mesh segmentation. Importantly, it requires fewer training iterations and demonstrates versatility by applying to a wide range of objects without the need for the mesh to adhere to manifold or watertight topology requirements.

New pollen taxon Syncolpraedapollis angolensis nov. gen. sp. nov.: A noteworthy discovery reported in the preliminary investigation of the latest Eocene-latest Oligocene deposits in the Kwanza Basin, Angola

A palynostratigraphic study of the upper Cunga and lower Quifangondo deposits in the Cabo de São Brás section, Kwanza Basin, Angola, has revealed a new pollen named Syncolpraedapollis angolensis nov. gen. sp. nov. This finding was part of a wider survey in the upper Cunga and lower Quifangondo deposits of the Cabo de São Brás section, Kwanza Basin, Angola, covering the latest Eocene to the latest Oligocene.The novel pollen features unique characteristics, notably a 3-syncolporate structure with well-defined and distinctive pores. The pollen grain is adorned with a free but irregular reticulum with an irregular polygonal mesh. Syncolpraedapollis angolensis nov. gen. sp. nov. is sporadically but consistently observed within the latest Eocene-latest Oligocene interval, conspicuously absent in the underlying Eocene sediments (lower Cunga Formation) and occurring infrequently in the latest Oligocene sediments (lower Quifangondo Formation). Thus, it is plausible to infer a limited age range, likely restricted to the latest Eocene-latest Oligocene, as previous studies in the Kwanza Basin have not documented this pollen.

Laplacian2Mesh: Laplacian-Based Mesh Understanding.

Geometric deep learning has sparked a rising interest in computer graphics to perform shape understanding tasks, such as shape classification and semantic segmentation. When the input is a polygonal surface, one has to suffer from the irregular mesh structure. Motivated by the geometric spectral theory, we introduce Laplacian2Mesh, a novel and flexible convolutional neural network (CNN) framework for coping with irregular triangle meshes (vertices may have any valence). By mapping the input mesh surface to the multi-dimensional Laplacian-Beltrami space, Laplacian2Mesh enables one to perform shape analysis tasks directly using the mature CNNs, without the need to deal with the irregular connectivity of the mesh structure. We further define a mesh pooling operation such that the receptive field of the network can be expanded while retaining the original vertex set as well as the connections between them. Besides, we introduce a channel-wise self-attention block to learn the individual importance of feature ingredients. Laplacian2Mesh not only decouples the geometry from the irregular connectivity of the mesh structure but also better captures the global features that are central to shape classification and segmentation. Extensive tests on various datasets demonstrate the effectiveness and efficiency of Laplacian2Mesh, particularly in terms of the capability of being vulnerable to noise to fulfill various learning tasks.

Forecasting water resources from satellite image time series using a graph-based learning strategy

Abstract. In the context of climate change, it is important to monitor the dynamics of the Earth’s surface in order to prevent extreme weather phenomena such as floods and droughts. To this end, global meteorological forecasting is constantly being improved, with a recent breakthrough in deep learning methods. In this paper, we propose to adapt a recent weather forecasting architecture, called GraphCast, to a water resources forecasting task using high-resolution satellite image time series (SITS). Based on an intermediate mesh, the data geometry used within the network is adapted to match high spatial resolution data acquired in two-dimensional space. In particular, we introduce a predefined irregular mesh based on a segmentation map to guide the network’s predictions and bring more detail to specific areas. We conduct experiments to forecast water resources index two months ahead on lakes and rivers in Italy and Spain. We demonstrate that our adaptation of GraphCast outperforms the existing frameworks designed for SITS analysis. It also showed stable results for the main hyperparameter, i.e., the number of superpixels. We conclude that adapting global meteorological forecasting methods to SITS settings can be beneficial for high spatial resolution predictions.

An unstructured body-of-revolution electromagnetic particle-in-cell algorithm with radial perfectly matched layers and dual polarizations

A novel electromagnetic particle-in-cell algorithm has been developed for fully kinetic plasma simulations on unstructured (irregular) meshes in complex body-of-revolution geometries. The algorithm, implemented in the BORPIC++ code, utilizes a set of field scalings and a coordinate mapping, reducing the Maxwell field problem in a cylindrical system to a Cartesian finite element Maxwell solver in the meridian plane. The latter obviates the cylindrical coordinate singularity in the symmetry axis. The choice of an unstructured finite element discretization enhances the geometrical flexibility of the BORPIC++ solver compared to the more traditional finite difference solvers. Symmetries in Maxwell's equations are explored to decompose the problem into two dual polarization states with isomorphic representations that enable code reuse. The particle-in-cell scatter and gather steps preserve charge-conservation at the discrete level. Our previous algorithm (BORPIC+) discretized the E and B field components of TEϕ and TMϕ polarizations on the finite element (primal) mesh [1,2]. Here, we employ a new field-update scheme. Using the same finite element (primal) mesh, this scheme advances two sets of field components independently: (1) E and B of TEϕ polarized fields, (Ez,Eρ,Bϕ) and (2) D and H of TMϕ polarized fields, (Dϕ,Hz,Hρ). Since these field updates are not explicitly coupled, the new field solver obviates the coordinate singularity, which otherwise arises at the cylindrical symmetric axis, ρ=0 when defining the discrete Hodge matrices (generalized finite element mass matrices). A cylindrical perfectly matched layer is implemented as a boundary condition in the radial direction to simulate open space problems, with periodic boundary conditions in the axial direction. We investigate effects of charged particles moving next to the cylindrical perfectly matched layer. We model azimuthal currents arising from rotational motion of charged rings, which produce TMϕ polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

A task-driven network for mesh classification and semantic part segmentation

Given the rapid advancements in geometric deep-learning techniques, there has been a dedicated effort to create mesh-based convolutional operators that act as a link between irregular mesh structures and widely adopted backbone networks. Despite the numerous advantages of Convolutional Neural Networks (CNNs) over Multi-Layer Perceptrons (MLPs), mesh-oriented CNNs often require intricate network architectures to tackle irregularities of a triangular mesh. These architectures not only demand that the mesh be manifold and watertight but also impose constraints on the abundance of training samples. In this paper, we note that for specific tasks such as mesh classification and semantic part segmentation, large-scale shape features play a pivotal role. This is in contrast to the realm of shape correspondence, where a comprehensive understanding of 3D shapes necessitates considering both local and global characteristics. Inspired by this key observation, we introduce a task-driven neural network architecture that seamlessly operates in an end-to-end fashion. Our method takes as input mesh vertices equipped with the heat kernel signature (HKS) and dihedral angles between adjacent faces. Notably, we replace the conventional convolutional module, commonly found in ResNet architectures, with MLPs and incorporate Layer Normalization (LN) to facilitate layer-wise normalization. Our approach, with a seemingly straightforward network architecture, demonstrates an accuracy advantage. It exhibits a marginal 0.1% improvement in the mesh classification task and a substantial 1.8% enhancement in the mesh part segmentation task compared to state-of-the-art methodologies. Moreover, as the number of training samples decreases to 1/50 or even 1/100, the accuracy advantage of our approach becomes more pronounced. In summary, our convolution-free network is tailored for specific tasks relying on large-scale shape features and excels in the situation with a limited number of training samples, setting itself apart from state-of-the-art methodologies.

Визначення об’єму насипів методом GNSS та лазерного сканування

In the modern period, the calculation of volumes of embankments plays an important role in a number of industries, starting from construction and ending with mining. This calls for the improvement of methods and technologies to ensure accurate and operational calculations of embankment volumes. One of the promising directions in this context is the use of Triangulated Irregular Networks (TIN) and Mesh models based on data obtained during geodetic measurements. Surveying is a large amount of accurate spatial measurement data that is necessary to create detailed geometric models of objects. The obtained geodetic data allow creating TIN and Mesh models that reflect the relief of the land surface with the necessary accuracy and detail. The use of these models to calculate the volumes of embankments becomes a key element in improving the processes of design, construction and management of natural resources. The purpose of this work is to compare the accuracy of the calculation of embankment volumes based on the data obtained by the hand-held laser scanner Stonex X120GO and the GNSS receiver Stonex S700A. Method. The method of comparing equal-precision measurements was used. Since there were no reference volumes of embankments at the site of the works, the difference between the volumes calculated on the basis of the data obtained with the hand-held laser scanner Stonex X120GO and the GNSS receiver Stonex S700A was used. Also, comparative and visual analysis was used to compare the difference between Mesh models obtained on the basis of different data sources. The results. The accuracy of determining the volumes of embankments obtained by a handheld laser scanner Stonex X120GO and a GNSS receiver Stonex S700A was studied. The average error of the differences between the determined volumes was 4.62%. Point clouds and Mesh models themselves were also analyzed (number of points, triangles, surface shape). The advantages and disadvantages of using a GNSS receiver and a hand-held laser scanner to collect data for calculating the volumes of material mounds are identified. Scientific novelty and practical significance. A technique for checking the accuracy of determining the volume of embankments without known reference values is proposed. The influence of the embankment area error on the accuracy of volume determination was studied. According to the results of the study, it is possible to assert the advantages in the detail of the received data and the speed of shooting when using a manual laser scanner, and about some difficulties associated with the desired presence of a powerful computer, a large array of data, an increase in the volume of camera work in comparison with the data obtained under the time of application of the GNSS receiver and the RTK method for determining the volumes of embankments

An Eight-Node Non-Conforming Generalized Partial Hybrid Element and Its Application in Stress Analysis of Repaired Composite Laminate Structures

To ensure the overall continuity of displacement and out-of-plane stress in composite laminate structures and to quantitatively analyze the mechanical properties of composite materials after damage or repair, a finite element solution method is applied based on the modified generalized H–R variational principle. This method utilizes an eight-node non-conforming generalized partial hybrid element (NCGPME8). The partial hybrid model established with this hybrid element can accurately satisfy the out-of-plane stress boundary conditions of the structure, ensuring the continuity of out-of-plane stress. Numerical examples are used to validate that this hybrid model can effectively compute thick and thin laminate structures with high accuracy and rapid convergence of out-of-plane stress. Finally, considering the insensitivity to irregular meshes and the accuracy in calculating in-plane stress, this method is propagated by element coefficient deduction or element material replacement, then employed to analyze the in-plane and out-of-plane stress distributions of laminates with damage from stepwise grinding perforations, and laminates repaired in a stepwise fashion. Stress and displacement at different locations on the laminates are compared and analyzed, leading to a quantitative assessment of the impact of damage and repair on the stress distribution of the laminates.

Spatio-temporal variability of leaf macronutrients in a conilon coffee crop

Studies focusing the understanding of spatio-temporal variability of soil and plant attributes may contribute to the rational use of agricultural inputs, enabling economic and environmental profits. The objective of this work was to determine the spatial and temporal variability of the foliar macronutrients in a Coffea canephora (Conilon coffee) plantation, in two sampling periods (pre-harvest and fruit growth). The study was performed in a Conilon coffee plantation with double spacing of 3.0 x 2.0 x 1.0m (4.000 plat ha-1) under drip irrigation system, in the county of São Mateus, Espírito Santo - Brazil. An irregular mesh with approximately 1.37 ha with 100 points, at a minimum distance of 2 m with each other, was installed. On each sampling point foliar tissue samples were collected in two distinctive periods, during pre-harvest and fruit growth and the levels of foliar macronutrients were determined. Results were submitted to descriptive analysis and geostatistics. A moderate spatial dependence structure was observed and verified for foliar contents of nitrogen, phosphorus, potassium and calcium in both sampling periods.

An efficient shear and bending‐locking‐free quadrilateral plate element using a modified Hellinger‐Reissner functional and the Bergan free formulation

AbstractThis paper presents a general variational principle as theoretical support for creating a simple and efficient quadrilateral plate finite element called BKWA, having only a single displacement and two rotations at corner nodes according to the Reissner‐Mindlin plate theory. The functional is a modified Hellinger‐Reissner in terms of the kinematic variables and independent transverse shear strains. Hence, we consider quadratic rotations, as in DKQ, and linear independent shear strains with constant tangential components on sides, as in MITC4. The quadratic rotation contributions and the mixed shear strain components are wiped out by static condensation. To adequately satisfy the constant curvature patch tests, we also consider the orthogonality conditions between the linear and quadratic bending modes, as in the Bergan free formulation, to avoid ‘spurious bending energy’ for thick plates (also called bending locking). The paper contains comprehensive patch tests and convergence tests with regular and irregular meshes (displacements, internal forces, and strain energy) of circular and square plates. The element is compared with other existing elements (MITC4, DKMQ). It is concluded that BKWA is an efficient element regarding several criteria.

Energy-Dependent, Self-Adaptive Mesh h(p)-Refinement of a Constraint-Based Continuous Bubnov-Galerkin Isogeometric Analysis Spatial Discretization of the Multi-Group Neutron Diffusion Equation with Dual-Weighted Residual Error Measures

Energy-dependent self-adaptive mesh refinement algorithms are developed for a continuous Bubnov-Galerkin spatial discretization of the multi-group neutron diffusion equation using NURBS-based isogeometric analysis (IGA). The spatially self-adaptive algorithms employ both mesh (h) and polynomial degree (p) refinement. Constraint-based equations are established across irregular interfaces with hanging-nodes; they are based upon master-slave relationships and the conservative interpolation between surface meshes. A similar Galerkin projection is employed in the conservative interpolation between volume meshes to evaluate group-to-group source terms over energy-dependent meshes; and to evaluate interpolation-based error measures. Enforcing continuity over an irregular mesh does introduce discretization errors. However, local mesh refinement allows for a better allocation of computational resources; and thus, more accuracy per degree of freedom. Two a posteriori interpolation-based error measures are proposed. The first heuristically minimizes local contributions to the discretization error, which becomes competitive for global quantities of interest (QoIs). However, for localized QoIs, over energy-dependent meshes, certain multi-group components may become under-resolved. The second employs duality arguments to minimize important error contributions, which consistently and reliably reduces the error in the QoI.

TGN: A Temporal Graph Network for Physics Prediction

Long-term prediction of physical systems on irregular unstructured meshes is extremely challenging due to the spatial complexityof meshes and the dynamic changes over time; namely, spatial dependence and temporal dependence. Recently, graph-based next-step prediction models have achieved great success in the task of modeling complex high-dimensional physical systems. However, due to these models ignoring the temporal dependence, they inevitably suffer from the effects of error accumulation. To capture the spatial and temporal dependence simultaneously, we propose a temporal graph network (TGN) to predict the long-term dynamics of complex physical systems. Specifically, we introduce an Encode-Process-Decode architecture to capture spatial dependence and create low-dimensional vector representations of system states. Additionally, a temporal model is introduced to learn the dynamic changes in the low-dimensional vector representations to capture temporal dependence. Our model can capture spatiotemporal correlations within physical systems. On some complex long-term prediction tasks in fluid dynamics, such as airfoil flow and cylinder flow, the prediction error of our method is significantly lower than the competitive GNN baseline. We show accurate phase predictions even for very long prediction sequences.

Full-wave Image Reconstruction in Transcranial Photoacoustic Computed Tomography using a Finite Element Method.

Transcranial photoacoustic computed tomography presents challenges in human brain imaging due to skull-induced acoustic aberration. Existing full-wave image reconstruction methods rely on a unified elastic wave equation for skull shear and longitudinal wave propagation, therefore demanding substantial computational resources. We propose an efficient discrete imaging model based on finite element discretization. The elastic wave equation for solids is solely applied to the hard-tissue skull region, while the soft-tissue or coupling-medium region that dominates the simulation domain is modeled with the simpler acoustic wave equation for liquids. The solid-liquid interfaces are explicitly modeled with elastic-acoustic coupling. Furthermore, finite element discretization allows coarser, irregular meshes to conform to object geometry. These factors significantly reduce the linear system size by 20 times to facilitate accurate whole-brain simulations with improved speed. We derive a matched forward-adjoint operator pair based on the model to enable integration with various optimization algorithms. We validate the reconstruction framework through numerical simulations and phantom experiments.

Multi-material structural discrete variable topology optimization with minimum length scale control under mass constraint

Topology optimization of multi-material structure has a larger design space compared with single material optimization, and it also requires efficient material selection methods to provide references for designers. The multi-material additive manufacturing ensures the manufacturability of topology design of multi-material structures, but also puts forward new manufacturability requirements, such as the minimum size of multi-material and the connection interface. This paper focuses on the development of a topology optimization algorithm for multi-material structures under the total mass constraint based on discrete variables. Recursive multiphase materials interpolation (RMMI) scheme is utilized to establish the relationship between discrete design variables and element elastic modulus and density, and sequential approximate integer programming (SAIP) and canonical relaxation algorithms are constructed to update the discrete design variables. It is shown that this method can obtain multi-material topology design with clear interface display, and has the ability of free material coexistence and degradation, which indicates that the discrete variables have more advantages than the continuous variables in the RMMI scheme. Based on this, the material selection under mass constraints is discussed in depth, and it is pointed out that the traditional determination of material coexistence or degradation in design based on specific stiffness has limitations. Thus, the present work expands the necessary conditions of material coexistence. Then, a minimum size control method of multi-material design based on post-processing is proposed to solve the minimum size geometry constraint problem by SAIP. Numerical examples show that this method can not only precisely control the minimum size of each material phase (including void phase), but also resolve the problem of interface and boundary diffusion of the multi-material phase, and significantly improve the manufacturability of multi-material design. Finally, it is applied to optimization problems with complex design domain and irregular meshes which are closer to engineering design.

Domain independent post-processing with graph U-nets: applications to electrical impedance tomographic imaging⋆ ⋆SH and BH were supported by the National Institute Of Biomedical Imaging And Bioengineering of the National Institutes of Health under Award Number R21EB028064. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. AH

Objective. To extend the highly successful U-Net Convolutional Neural Network architecture, which is limited to rectangular pixel/voxel domains, to a graph-based equivalent that works flexibly on irregular meshes; and demonstrate the effectiveness on electrical impedance tomography (EIT). Approach. By interpreting the irregular mesh as a graph, we develop a graph U-Net with new cluster pooling and unpooling layers that mimic the classic neighborhood based max-pooling important for imaging applications. Main results. The proposed graph U-Net is shown to be flexible and effective for improving early iterate total variation (TV) reconstructions from EIT measurements, using as little as the first iteration. The performance is evaluated for simulated data, and on experimental data from three measurement devices with different measurement geometries and instrumentations. We successfully show that such networks can be trained with a simple two-dimensional simulated training set, and generalize to very different domains, including measurements from a three-dimensional device and subsequent 3D reconstructions. Significance. As many inverse problems are solved on irregular (e.g. finite element) meshes, the proposed graph U-Net and pooling layers provide the added flexibility to process directly on the computational mesh. Post-processing an early iterate reconstruction greatly reduces the computational cost which can become prohibitive in higher dimensions with dense meshes. As the graph structure is independent of ‘dimension’, the flexibility to extend networks trained on 2D domains to 3D domains offers a possibility to further reduce computational cost in training.

Identification of vortex in unstructured mesh with graph neural networks

Deep learning has been employed to identify flow characteristics from Computational Fluid Dynamics (CFD) databases to assist the researcher to better understand the flow field, to optimize the geometry design and to select the correct CFD configuration for corresponding flow characteristics. Convolutional Neural Network (CNN) is one of the most popular algorithms used to extract and identify flow features. However its use, without any additional flow field interpolation, is limited to the simple domain geometry and regular meshes which limits its application to real industrial cases where complex geometry and irregular meshes are usually used. Aiming at the aforementioned problems, we present a Graph Neural Network (GNN) based model with U-Net architecture to identify the vortex in CFD results on unstructured meshes. The graph generation and graph hierarchy construction using algebraic multigrid method from CFD meshes are introduced. A vortex auto-labeling method is proposed to label vortex regions in 2D CFD meshes. We precise our approach by firstly optimizing the input set on CNNs, then benchmarking current GNN kernels against CNN model and evaluating the performances of GNN kernels in terms of classification accuracy, training efficiency and identified vortex morphology. Finally, we demonstrate the adaptability of our approach to unstructured meshes and generality to unseen cases with different turbulence models at different Reynolds numbers.

Design of microstructure and performance testing of high-pressure ZnO grain boundary layer

Abstract This paper firstly constructs the microstructure of the ZnO grain boundary layer of a high voltage system fits: microstructure characteristics of the ZnO varistor and conductive mechanism of the ZnO varistor. The ZnO structure can be seen as an irregular three-dimensional mesh composed of ZnO grain-high resistance grain boundary layer-ZnO grain. Then the electric field distribution of the ZnO grain boundary layer is calculated by Poisson’s equation, and the relationship curve of the space charge distribution inside the ZnO grain boundary with time is further simulated. Finally, the ZnO grain boundary performance is tested based on the bipolar carrier transport model with a double Schottky barrier, and the results show that the negative charge density at the grain boundary interface gradually decreases from 5×1025m−3 to 4.2452×1025m−3 and the initial charge value in the grain boundary region is 0m−3, and the negative charge at the grain boundary interface gradually decreases, and the negative charge in the grain boundary region gradually increases. In this study, the internal space charge distribution, Schottky barrier height and grain boundary leakage current are systematically investigated, which provides important theoretical support for the in-depth research on the conductivity mechanism, aging mechanism and performance improvement related to ZnO varistors.

Isogeometric Convolution Hierarchical Deep-learning Neural Network: Isogeometric analysis with versatile adaptivity

We are witnessing a rapid transition from Software 1.0 to 2.0. Software 1.0 focuses on manually designed algorithms, while Software 2.0 leverages data and machine learning algorithms (or artificial intelligence) for optimized, fast, and accurate solutions. For the past few years, we have been developing Convolution Hierarchical Deep-learning Neural Network Artificial Intelligence (C-HiDeNN-AI), which enables the realization of Engineering Software 2.0 by opening the next-generation neural network-based computational tools that can simultaneously train data and solve mechanistic equations. This paper focuses on solving partial differential equations with C-HiDeNN. Still, the same neural network can be used for training and calibration with experimental data, which will be discussed in a separate paper. This paper presents a computational framework combining the C-HiDeNN theory with isogeometric analysis (IGA), called Convolution IGA (C-IGA). C-IGA has five key features that advance IGA: (1) arbitrarily high-order smoothness and convergence rates without increasing degrees of freedom; (2) a Kronecker delta property that enables direct imposition of Dirichlet boundary conditions; (3) automatic and flexible global/local mesh-adaptivity with built-in length scale control and adjustable radial basis functions; (4) ability to handle irregular meshes and triangular/tetrahedral elements; and (5) GPU implementation that speeds up the program as fast as finite element method (FEM). Mathematically, we prove that both IGA and C-IGA mappings are equivalent, and by taking a special design and modified anchors as nodes, C-IGA degenerates to IGA. We demonstrate the accuracy, convergence rates, mesh-adaptivity, and performance of C-IGA with several 1D, 2D, and 3D numerical examples. The future applications of C-IGA from topology optimization to product manufacturing with multi-GPU programming are discussed.