Triangular Mesh Research Articles

A centroid-enriched strain-smoothed radial point interpolation method for nearly incompressible elastoplastic problems in solid mechanics

While edge-based (2D) and face-based (3D) smoothed radial point interpolation methods (ES-RPIM and FS-RPIM) have demonstrated excellent performance in solid mechanics, the volumetric locking problem limits their applications in nearly incompressible problems. This paper develops a centroid-enriched edge-based/face-based smoothed RPIM (CE-ES-RPIM and CE-FS-RPIM) to address the volumetric locking problem in nearly incompressible problems. The proposed method comprises the following elements: (1) a centroid-enriched scheme is creatively used and implemented into the traditional ES-RPIM and FS-RPIM framework, in which the centroids of triangular mesh are added as spatial discretization nodes; (2) the strain-smoothing technique is performed on triangle (2D) or tetrahedron (3D) mesh to maintain the linear exactness of the radial point interpolation method; and (3) the combination of the centroid-enriched scheme and strain smoothing technique makes the ratio of the number of displacement equations to the number of incompressible constraints reach the optimal value of 2, effectively eliminating volumetric locking. The accuracy, effectiveness and efficiency of the proposed method are validated on several 2D and 3D nearly incompressible benchmark examples. All results demonstrate that the proposed method can provide accurate numerical results under an incompressible limitation and provide better performance than conventional quadratic element methods.

Entropy Production Analysis in an Octagonal Cavity with an Inner Cold Cylinder: A Thermodynamic Aspect

Understanding fluid dynamics and heat transfer is crucial for designing and improving various engineering systems. This study examines the heat transfer characteristics of a buoyancy-driven natural convection flow that is laminar and incompressible. The investigation also considers entropy generation (Egen) within an octagonal cavity subject to a cold cylinder inside the cavity. The dimensionless version of the governing equations and their corresponding boundary conditions have been solved numerically using the finite element method, employing triangular mesh elements for discretization. The findings indicated that incorporating a cold cylinder inside the octagonal cavity resulted in a higher heat transfer (HT) rate than in the absence of a cold cylinder. Furthermore, using the heat flux condition led to a higher average Nusselt number (Nuavg) and a lower Bejan number (Be) than the isothermal boundary condition. The results also showed that HT and Egen were more significant in the Al2O3-H2O nanofluid than the basic fluids such as air and water, and HT increased as χ increased. The current research demonstrates that employing the heat flux condition and incorporating nanoparticles can enhance the rate of HT and Egen. Furthermore, the thermo-fluid system should be operated at low Ra to achieve greater HT effectiveness for nanofluid concerns.

Extended-resolution of a single-camera synthetic Schlieren method for measurement of free liquid surfaces

In a synthetic Schlieren method of measuring the height of a dynamic free liquid surface, the smallest wavelength of the measured surface topography is determined by the spatial resolution of the displacement field of markers. Currently, a displacement vector is obtained for each interrogation window, including a few markers, with a cross-correlation algorithm. In this study, the measurement resolution is extended by obtaining the displacement of individual markers. This simple and rational change, however, brings technical difficulties in numerically solving the governing equation over spatially randomly distributed markers. For this, the governing equation, which associates the surface height to be measured with the virtual displacements of the markers, is solved by the finite-volume scheme for the unstructured meshes, where a marker locates at the center of each triangular mesh face. The present method is examined with the synthetic generated data, in laboratory experiments of a transparent solid carved with a sinusoidal surface, and in experiments of water ripples. The measurement uncertainty is discussed.

Deep reinforcement learning spacecraft guidance with state uncertainty for autonomous shape reconstruction of uncooperative target

In recent years, space research has shifted heavily its focus towards enhanced autonomy on-board spacecrafts for on-orbit servicing activities (OOS). OOS and proximity operations include a variety of activities: the focal point of this work is the autonomous guidance of a chaser spacecraft for the shape reconstruction of an artificial uncooperative object. Adaptive guidance depends on the ability of the system to build a map of the uncertain environment, figuring out its location inside of it and accordingly determining the control law. Thus, autonomous navigation is framed as an active Simultaneous Localization and Mapping (SLAM) problem and modeled as a Partially Observable Markov Decision Process (POMDP). A state-of-the-art Deep Reinforcement Learning (DRL) method, Proximal Policy Optimization (PPO), is investigated to develop an agent capable of cleverly planning the shape reconstruction of the target. Starting from previous research on the topic, this work develops further proposing a continuous action space, such that the agent is no more forced to choose between a predefined set of possible discrete actions, fixed both in magnitude and direction. In this way any combination of the three-dimensional thrust vector components is available. The chaser spacecraft is a small satellite mounting an electric propulsion engine defining the action space range, in linearized eccentric relative motion with the selected uncooperative object. Through a rendered triangular mesh, the agent capabilities of geometry reconstruction and mapping are evaluated, considering the number of quality pictures made for each face. Extensive training tests are performed with random initial conditions to verify the generalizing capability of the DRL agent. The results are then validated in a comprehensive testing campaign, whose primary focus is the introduction of noisy measurements coming from navigation, affecting pose estimation. The sensitivity of the proposed method to this condition is analyzed and the efficiency of a retraining procedure is examined. The applicability of DRL methods and neural networks to support autonomous guidance in a close proximity scenario is corroborated and the technique employed is vastly tested and verified.

NURBS-based surface generation from 3D images: spectral construction and data-driven model selection

Abstract In this paper, we present a set of improved algorithms for recovering computer aided design (CAD-type) surface models from three-dimensional (3D) images. The goal of the proposed framework is to generate B-spline or non-uniform rational B-spline (NURBS) surfaces, which are standard mathematical representations of solid objects in digital engineering. To create a NURBS surface, we first compute a control network (a quadrilateral mesh) from a triangular mesh using the Marching Cubes algorithm and Discrete Morse theory. To create a NURBS surface, we first compute a triangular mesh using the Marching Cubes algorithm, then the control network (a quadrilateral mesh) is determined from the triangular mesh by using Discrete Morse theory. Discrete Morse theory uses the critical points of a specific scalar field defined over the triangulation to generate a quad mesh. Such a scalar field is obtained by solving a graph Laplacian eigenproblem over the triangulation. However, the resulting surface is not optimal. We therefore introduce an optimization algorithm to better approximate the geometry of the object. In addition, we propose a statistical method for selecting the most appropriate eigenfunction of the graph Laplacian to generate a control network that is neither too coarse nor too fine, given the precision of the 3D image. To do this, we set up a regression model and use an information criterion to choose the best surface. Finally, we extend our approach by taking into account both model and data uncertainty using probabilistic regression and sampling the posterior distribution with Hamiltonian Markov Chain Monte Carlo.

SpecTrHuMS: Spectral transformer for human mesh sequence learning

We present SpecTrHuMS, a Spectral Transformer for 3D triangular Human Mesh Sequence learning which combines known deep learning models with spectral mesh processing to capture characteristics of 3D shapes as well as temporal dependencies between the frames. Unlike previous works in this field, our approach is able to work directly with a compressed representation of the geometry, the spectral coefficients, rather than relying solely on skeleton joints that does not contain surface information. The vertices of each mesh of a sequence are first projected on the eigenvectors of the Graph Laplacian computed from the common triangulation. A convolutional encoder then encodes each frame into lower dimensional latent variables that preserve as much as possible the spectral information. These latent variables are next passed through a transformer architecture so that the model understands the context of the sequence and learns temporal dependencies between the frames. Each frame of the transformer’s output is then decoded by a convolutional decoder which aims to reconstruct the input spectral coefficients. Finally, all frames are transformed back into the spatial domain, resulting in a general process able to treat 4D surfaces with a constant connectivity. Our method is evaluated on a prediction task on AMASS, a dataset of human surface sequences, showing the ability of our model to produce realistic movements while preserving the identity of a subject, and showing that this work is a significant step towards efficient and high-quality representation of triangular mesh sequences with constant connectivity. Additional experiments show that our model can be easily extended to other tasks such as long term prediction, completion and that it is generalizable to other datasets with constant connectivity. This work opens up new possibilities for applications in the fields of animation, virtual reality, and computer graphics. Pretrained models, the code to train them and the code to create datasets will be made publicly available.

Modelling contractile ring formation and division to daughter cells for simulating proliferative multicellular dynamics.

Cell proliferation is a fundamental process underlying embryogenesis, homeostasis, wound healing, and cancer. The process involves multiple events during each cell cycle, such as cell growth, contractile ring formation, and division to daughter cells, which affect the surrounding cell population geometrically and mechanically. However, existing methods do not comprehensively describe the dynamics of multicellular structures involving cell proliferation at a subcellular resolution. In this study, we present a novel model for proliferative multicellular dynamics at the subcellular level by building upon the nonconservative fluid membrane (NCF) model that we developed in earlier research. The NCF model utilizes a dynamically-rearranging closed triangular mesh to depict the shape of each cell, enabling us to analyze cell dynamics over extended periods beyond each cell cycle, during which cell surface components undergo dynamic turnover. The proposed model represents the process of cell proliferation by incorporating cell volume growth and contractile ring formation through an energy function and topologically dividing each cell at the cleavage furrow formed by the ring. Numerical simulations demonstrated that the model recapitulated the process of cell proliferation at subcellular resolution, including cell volume growth, cleavage furrow formation, and division to daughter cells. Further analyses suggested that the orientation of actomyosin stress in the contractile ring plays a crucial role in the cleavage furrow formation, i.e., circumferential orientation can form a cleavage furrow but isotropic orientation cannot. Furthermore, the model replicated tissue-scale multicellular dynamics, where the successive proliferation of adhesive cells led to the formation of a cell sheet and stratification on the substrate. Overall, the proposed model provides a basis for analyzing proliferative multicellular dynamics at subcellular resolution.

Prediction of the 3D shape of the L1 vertebral body from adjacent vertebrae.

The aim of treatments of vertebral fractures is the anatomical reduction to restore the physiological biomechanics of the spine and the stabilization of the fracture to allow bone healing. However, the three-dimensional shape of the fractured vertebral body before the fracture is unknown in the clinical setting. Information about the pre-fracture vertebral body shape could help surgeons to select the optimal treatment. The goal of this study was to develop and validate a method based on Singular Value Decomposition (SVD) to predict the shape of the vertebral body of L1 from the shapes of T12 and L2. The geometry of the vertebral bodies of T12, L1 and L2 vertebrae of 40 patients were extracted from CT scans available from the VerSe2020 open-access dataset. Surface triangular meshes of each vertebra were morphed onto a template mesh. The set of vectors with the node coordinates of the morphed T12, L1 and L2 were compressed with SVD and used to build a system of linear equations. This system was used to solve a minimization problem and to reconstruct the shape of L1. A leave-one-out cross-validation was performed. Moreover, the approach was tested against an independent dataset with large osteophytes. The results of the study show a good prediction of the shape of the vertebral body of L1 from the shapes of the two adjacent vertebrae (mean error equal to 0.51±0.11mm on average, Hausdorff distance equal to 2.11±0.56mm on average), compared to current CT resolution typically used in the operating room. The error was slightly higher for patients presenting large osteophytes or severe bone degeneration (mean error equal to 0.65±0.10mm, Hausdorff distance equal to 3.54±1.03mm). The accuracy of the prediction was significantly better than approximating the shape of the vertebral body of L1 by the shape of T12 or L2. This approach could be used in the future to improve the pre-planning of spine surgeries to treat vertebral fractures.

On the Approximation of the Bay of Bengal Domain to be Compatible for the Implementation of Finite Element Method

In this study, the Bay of Bengal domain has been approximated using triangular mesh so that the finite element method (FEM) can be employed on it. The area between 15º N and 23º N Latitudes and 85º E and 95º E Longitudes is considered as the physical domain. A MATLAB routine and the cubic spline interpolation have been used to extract the coordinates of the points on the boundary of the whole domain and the points on the boundary of the islands from a colour image of the domain. A C++ routine is used to generate an edp file for triangular mesh using the extracted coordinates. Then FreeFem++ is used to create triangular mesh for the whole domain from the generated edp file. All the major islands are also incorporated in the final mesh. The obtained triangular mesh can be used to develop a storm surge prediction model for the Bay of Bengal region implementing FEM.

Unsupervised Contrastive Representation Learning for 3D Mesh Segmentation (Student Abstract)

3D deep learning is a growing field of interest due to the vast amount of information stored in 3D formats. Triangular meshes are an efficient representation for irregular, non-uniform 3D objects. However, meshes are often challenging to annotate due to their high computational complexity. Therefore, it is desirable to train segmentation networks with limited-labeled data. Self-supervised learning (SSL), a form of unsupervised representation learning, is a growing alternative to fully-supervised learning which can decrease the burden of supervision for training. Specifically, contrastive learning (CL), a form of SSL, has recently been explored to solve limited-labeled data tasks. We propose SSL-MeshCNN, a CL method for pre-training CNNs for mesh segmentation. We take inspiration from prior CL frameworks to design a novel CL algorithm specialized for meshes. Our preliminary experiments show promising results in reducing the heavy labeled data requirement needed for mesh segmentation by at least 33%.

DMIS: Dynamic Mesh-Based Importance Sampling for Training Physics-Informed Neural Networks

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are widely-used. However, traditional numerical algorithms are computationally expensive and challenging in handling multiphysics systems. Recently, using neural networks to solve PDEs has made significant progress, called physics-informed neural networks (PINNs). PINNs encode physical laws into neural networks and learn the continuous solutions of PDEs. For the training of PINNs, existing methods suffer from the problems of inefficiency and unstable convergence, since the PDE residuals require calculating automatic differentiation. In this paper, we propose Dynamic Mesh-based Importance Sampling (DMIS) to tackle these problems. DMIS is a novel sampling scheme based on importance sampling, which constructs a dynamic triangular mesh to estimate sample weights efficiently. DMIS has broad applicability and can be easily integrated into existing methods. The evaluation of DMIS on three widely-used benchmarks shows that DMIS improves the convergence speed and accuracy in the meantime. Especially in solving the highly nonlinear Schrödinger Equation, compared with state-of-the-art methods, DMIS shows up to 46% smaller root mean square error and five times faster convergence speed. Code is available at https://github.com/MatrixBrain/DMIS.

Linear interpolation of shape operators for umbilical points through local parametrization

Curvature analysis is frequently employed in shape interrogation. Umbilical points are of particular interest in curvature analysis because of their identical normal curvatures in all tangential directions. Locating the umbilical points is the basis of geometric analysis. On the one hand, as singularities, umbilical points severely hinder the analysis (e.g., in nets of curvature lines). On the other hand, they provide qualitative information about the intrinsic shape of a surface and are therefore desirable quantities in some applications. In this study, we develop a straightforward and effective method to detect generic umbilical points on triangular meshes. This method is applicable to any type of admissible parametrization. We propose two local parametrization schemes–orthogonal projection and conformal transformation–to be used with the proposed method. Furthermore, we systematically analyze our method and prove its convergence behavior. The algorithm used in our approach is flexible and straightforward to implement for triangular meshes of arbitrary topology.

An improved discontinuity sensor for high-order weighted essentially non-oscillatory scheme on triangular meshes

In this paper, a new type of hybrid weighted essentially non-oscillatory (WENO) schemes with third-order and fourth-order accuracy is developed to simulate compressible flow problems on unstructured meshes. First, an improved discontinuity sensor is designed on unstructured meshes, which can automatically and accurately identify the troubled cells without manually adjusting the parameters related to the problem. Second, a hybrid unequal-sized WENO (US-WENO) scheme with different accuracy is developed based on the new hybridization strategy and the existing US-WENO scheme to reduce the computational overhead caused by the computation of nonlinear weights and local characteristic decompositions in WENO procedures. Finally, some benchmark examples are employed to verify the performance of the proposed method in terms of numerical dissipation, shock capture ability, and computational efficiency. Numerical experiments show that the proposed hybrid US-WENO schemes can inherit the excellent characteristics of the original US-WENO schemes while improving computational efficiency.

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

This paper introduces a 3D equilibrium-based finite element formulation for reinforced concrete. The stress unknowns in the proposed formulation strongly satisfy the equilibrium equations throughout the entire volume, with the exception of the mesh interfaces where the rebars intersect. The concrete stress tensor is interpolated using tetrahedral piece-wise linear elements with statically admissible discontinuities, while the rebars are considered as 1D elements embedded into the concrete, intersecting the triangular mesh. Thus, the global equilibrium is ensured by writing for each face of the mesh the traction continuity equation, while including the rebar stress contribution for the intersected triangular faces. The elastic equilibrium formulation is then written into an equivalent optimization problem, extended to the elastoplastic case by simply adding semi-definite matrix constraints on the concrete stress tensor, corresponding to a Rankine or a truncated Mohr–Coulomb criterion. As for the rebars, they are considered to obey a 1D perfectly elastoplastic behavior. The present formulation is also developed for limit analysis, to directly obtain an estimate of the lower bound of the collapse load. The resulting semi-definite programming optimization problems are solved using an in–house interior point algorithm.

Analysis of dissipation operators that damp spurious modes while maintaining discrete approximate geostrophic equilibriums for the B-grid staggered scheme on triangular meshes

We consider the lowest order - so called B-grid - discretization on triangular cells which is common in the ocean or atmosphere simulation science, with pressure field at the vertices of the triangles and the velocity field at the cell centers. It is known that this discretization presents spurious modes due to a too large number of velocities unknowns and that these modes should be damped; since preservation of approximate geostrophic equilibrium (GE) may be of primary importance in the simulations, we propose and analyze damping operators that, when applied to the linearized constant coefficient configuration, exactly preserve discrete versions of the GEs. We also analyze their effects in more general configurations and their possible coupling with more standard damping operators. These strategies cover triangular variants of the Low Froude (LF) scheme [1], the Apparent Topography (AT) scheme [2,3], as well as a new modification of it. The LF scheme has no numerical diffusion terms in the pressure equation and damps the normal velocity jumps between adjacent cells in the momentum equation. In the AT scheme, numerical diffusion is added in the pressure equation in such a way that it does not impact the discrete GE. However, since the AT scheme alone cannot be proved to be damping through energy estimates and does not preserve the space which is orthogonal to the discrete GE, we also study what we call the Modified Apparent Topography (MAT) that satisfies these important properties. We also suggest semi-implicit time discretizations that preserve these properties. Extensions to the full non-linear equations are also provided. Numerical experiments illustrate the performance of the new schemes.

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

The finite element error and adaptive analysis are implemented in finite element procedures to increase the reliability of numerical analyses. In this paper, the mesh-free error-recovery technique based on moving least squares (MLS) interpolation is applied to recover the errors in the stresses and displacements of incompressible elastic finite element solutions and errors are estimated in energy norms. The effects of element types (triangular and quadrilateral elements) and the formation of patches (mesh-free patch, mesh-dependent element-based patch, and mesh-dependent node-based patch) for error recovery in MLS and conventional least-square interpolation-error quantification are also assessed in this study. Numerical examples of incompressible elasticity, including a problem with singularity, are studied to display the effectiveness and applicability of the mesh-free MLS interpolation-error recovery technique. The mixed formulation (displacement and pressure) is adopted for a finite element analysis of the incompressible elastic problem. The rate of convergence, the effectivity of the error estimation, and modified meshes for desired accuracy are used to assess the effectiveness of the error estimators. The error-convergence rates are computed in the original FEM solution, in the post-processed solution using mesh-free MLS-based displacement, stress recovery, mesh-dependent patch-based least-square-based displacement, and stress recovery (ZZ) as (0.9777, 2.2501, 2.0012, 1.6710 and 1.5436), and (0.9736, 2.0869, 1.6931, 1.8806 and 1.4973), respectively, for four-node quadrilateral, and six-node triangular meshes. It is concluded that displacement-based recovery was more effective in the finite element incompressible elastic analysis than stress-based recovery using mesh-free and mesh-dependent patches.

$$hp$$-Optimal Interior Penalty Discontinuous Galerkin Methods for the Biharmonic Problem

We prove $$hp$$ -optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions, and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global $$H^2$$ piecewise polynomial approximants with $$hp$$ -optimal approximation properties over the given meshes. The $$hp$$ -optimality is also discussed for $$\mathcal C^0$$ -IPDG in two and three dimensions, and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that $$p$$ -suboptimality occurs in presence of singular essential boundary conditions.

3D Reconstruction of Celadon from a 2D Image: Application to Path Tracing and VR

We present a straightforward approach for reconstructing 3D celadon models from a single 2D image. The celadon is a historical example of the surface of revolution. Our approach uses a surface of revolution technique to generate the basic shape of the celadon and then applies texture mapping to create a realistic appearance. The process involves detecting the contour and corners of the celadon image, determining an axis of revolution, generating a profile curve, and finally constructing a 3D celadon model. Additionally, we create models as triangular meshes at multiple resolutions, employing a B-spline curve as the profile curve. It enhances the adaptability of the models for various purposes. We render various scenes using a path tracer to assess the suitability of the generated 3D celadon models and generate a VR celadon museum with the models. Overall, our approach offers a simple and efficient solution for reconstructing a 3D celadon model, generating VR content, and demonstrating extensive applicability across numerous disciplines.

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

Application of immersed boundary method to the simulation of three-dimensional flow in solid rocket motors

The immersed boundary method (IBM) is currently utilized in the simulation of two-dimensional axisymmetric flow in solid rocket motors. In this paper, the IBM is applied to three-dimensional flow fields, keeping the grain surface fixed. Based on the Cartesian grid, a three-dimensional Euler flow solver is developed using the finite difference method. All boundaries of the flow field are processed using the IBM, including the slip walls, mass flow inlet, pressure outlet, and rotational periodic boundary. Specific implementation of these boundary conditions and the mesh generation process are described. Using the ray-casting approach and the alternating digital tree data structure, an efficient method is proposed to determine the intersection relationship between a rectangular volume grid cell and a triangular surface mesh element. The numerical results of Taylor–Culick flow verify that the developed solver has more than one-order accuracy in space. To conduct the validation of the established method, three typical grains are selected for flow simulations, namely, the perforated cylindrical grain with burning on both the ends and the inner surface, the end-slotted end-burning grain, and the finocyl grain, respectively. The simulated results are compared with those of the zero-dimensional interior ballistics, the two-dimensional axisymmetric IBM, and the body-fitted grid method, verifying the fidelity of the developed three-dimensional flow solver.