Mesh-based Methods Research Articles

A subdivision-stabilized B-spline mixed material point method

Subjected to external loadings, polymeric materials, e.g., biological tissues, hydrogels, and elastomers, may undergo extreme, nearly incompressible, (self-)contact deformations. For numerical modeling employing mesh-based techniques such as the finite element method (FEM), these deformations pose significant challenges due to large distortions in the deformed geometry, accuracy issues stemming from volumetric locking effects, and increased computational cost from complex contact searches. As an alternative to mesh-based methods, the material point method (MPM), a continuum-based particle technique, is gaining attention for its ability to handle extreme distortions and capture no-slip contact without added cost. For nearly incompressible material behaviors, while mixed formulations can address locking effects by treating displacements and pressure as independent fields, they can suffer from numerical instabilities close to the incompressibility limit due to the violation of the inf-sup condition, leading to inaccurate nodal pressure solutions. Here we propose an efficient and stable mixed B-spline material point method with highest achievable regularity for quasi-compressible polymeric materials. Using the two-scale relation of B-splines, we introduce a subdivision-stabilization for the two-field mixed MPM and obtain numerically stable, oscillation-free nodal solutions with equal-order interpolations with optimal regularity. Building on the Eulerian-Lagrangian nature of MPM, a previously-converged solution framework is adopted to mitigate issues related to cell-crossing and numerical fracture artifact present in standard MPM. We assess the stability and accuracy of the developed mixed MPM at large deformations for soft materials through the benchmark Cook’s membrane problem. Additionally, we test the robustness of the proposed MPM by modeling several examples, including the compression and indentation of a circular block into a quasi-compressible substrate and the twisting deformation of a rectangular block. The findings demonstrate the MPM’s capabilities for modeling practical soft material applications.

Prediction modeling of coastal sediment transport using accelerated smooth particle hydrodynamics approach

A GPU-accelerated 3D smooth particle hydrodynamics (SPH) scheme is developed and applied to a coastal multi-phase liquid-sediment interaction and sediment transport. The SPH scheme's meshless design and the sediment's particle structure enable the modeling of the waves' interactions with the sediment particles beyond the limitation of the mesh-based methods. A Newtonian constitutive model is used to model the liquid phase, and the sediment transport is formulated based on the Herschel-Bulkley-Papanastasiou (HBP) model. The yield characteristics of the sediment phase are estimated using the Drucker-Prager yield criterion. Due to the parallelization of the solution on graphics processing units, the 3D SPH scheme's performance, which uses millions of particles, is improved. Good correlations were observed in the SPH predictions and experimental measurements, with a maximum difference of 4.85 %. The validated scheme is applied to formulate forecasting models for the coastline sediment transport. It is found that erosion and scouring are expected at the coastline region inclined to the direction of the sea waves, with a predicted mass erosion of about 60e3 kg in four years. The wave's velocity is also established to be directly proportional to the sediment transport. The proposed multi-phase SPH methodology is proven effective for sediment transport prediction.

Cardiac Electrophysiology Meshfree Modeling through the Mixed Collocation Method

We present the meshfree mixed collocation method (MCM) for cardiac electrophysiology simulation. Capitalizing on the meshfree property of MCM, we introduce an immersed grid approach for automated generation of meshfree node grids from medical image data. This approach allows us to avoid the time-consuming mesh generation and processing that mesh-based methods like the finite element method (FEM) require. We employ the MCM to solve the cardiac monodomain model considering electrical propagation in 2D tissue sheets, 3D tissue slabs, and a realistic biventricular anatomy. We demonstrate that the solutions obtained by the MCM are in good agreement with the FEM, particularly when immersed grid is used. These findings confirm the suitability of the MCM for cardiac electrophysiology simulation and make the MCM a promising alternative to the FEM for cardiac electrical investigations.

Application of Cross model for granular flow and impact analysis using three-dimensional B-spline material point method

Granular flow and impact, such as landslides and debris flows, are typical problems involving large deformations of non-Newtonian free surfaces, which are of great interest in geological engineering and also pose challenges for traditional mesh-based numerical methods. As an enhancement of the traditional Material Point Method (MPM), the B-Spline Material Point Method (BSMPM) can not only effectively reduce mesh-crossing and quadrature errors, but also facilitate the tracking and capturing of free surfaces and contact interfaces. In this paper, the non-Newtonian general Cross model, the rheological parameters of which are determined by the Bingham model and the Mohr-Coulomb yield criterion, is introduced into the three-dimensional BSMPM (3D BSMPM) for simulating granular flow and impact problems. Interparticle effects of granular materials are represented by artificial damping forces. Dynamic simulations are carried out on the collapse of sand column and glass beads, the flow of bentonite suspensions, and the impact between granular mass and elastic obstacles. The overall evolution of granular flow and impact behavior are presented and compared with existing experimental and/or original MPM results. It is found that the proposed method well reproduces the overall process of granular flow and impact behavior, including its various stages of formation, acceleration, deceleration, and deposition, as well as the impact pressure. The results show that the granular flow and impact behavior can be well represented by the proposed 3D BSMPM with the Cross rheological model, demonstrating the effectiveness of the proposed method in simulating complicated environmental fluid dynamics.

JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities

We present a scalable strategy for development of mesh-free hybrid neuro-symbolic partial differential equation solvers based on existing mesh-based numerical discretization methods. Particularly, this strategy can be used to efficiently train neural network surrogate models of partial differential equations by (i) leveraging the accuracy and convergence properties of advanced numerical methods, solvers, and preconditioners, as well as (ii) better scalability to higher order PDEs by strictly limiting optimization to first order automatic differentiation. The presented neural bootstrapping method (hereby dubbed NBM) is based on evaluation of the finite discretization residuals of the PDE system obtained on implicit Cartesian cells centered on a set of random collocation points with respect to trainable parameters of the neural network. Importantly, the conservation laws and symmetries present in the bootstrapped finite discretization equations inform the neural network about solution regularities within local neighborhoods of training points. We apply NBM to the important class of elliptic problems with jump conditions across irregular interfaces in three spatial dimensions. We show the method is convergent such that model accuracy improves by increasing number of collocation points in the domain and preconditioning the residuals. We show NBM is competitive in terms of memory and training speed with other PINN-type frameworks. The algorithms presented here are implemented using JAX in a software package named JAX-DIPS (https://github.com/JAX-DIPS/JAX-DIPS), standing for differentiable interfacial PDE solver. We open sourced JAX-DIPS to facilitate research into use of differentiable algorithms for developing hybrid PDE solvers.

A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche’s method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.

Numerical study of wave-ice floe interactions and overwash by a meshfree particle method

The interaction between regular waves and ice floes under certain wave conditions involves overwash which is characterized by waves running up over the surface of an ice floe with large deformed free surfaces. An in-house moving particle semi-implicit (IMPS) method is employed to simulate the wave-ice floe interaction, especially for the specific behavior namely overwash due to a small freeboard of sea ice and a large body motion. In this work, a series of regular waves are generated using the Lagrangian particle method based on the Stokes wave theory. Models of the floating sea ice are simplified as circular disks without and with an edge barrier and a circular disk with a central hole, which can be used to study the influences of overwash. Hydrodynamic responses of the ice floe in regular wave conditions with different wave lengths and a constant wave height are analyzed. Compared with the mesh-based methods, the Lagrangian particle-based methods have advantages in treating highly deformed free surfaces during wave overwash processes. A piston-type wave maker and a sponge layer are applied for generating waves and avoiding the wave reflection from wall boundaries, respectively. Validations are presented including the circular ice floe with a diameter of 400 mm and a thickness of 15 mm interacting with regular waves with the wave length λ/D from 1.5 to 3.5. RAOs of the original circular disk are compared with those of the circular disks with an edge barrier and with a central hole. Overwash on the ice floe can be prevented by using an edge barrier. A new type of overwash which waves flow over the upper surface and then into the hole was proposed and simulated. The present results of numerical predictions are in good agreement with the experimental data and the published numerical results using the CFD code Flow-3D.

Enhanced Floating Isogeometric Analysis

The numerical simulation of additive manufacturing techniques promises the acceleration of costly experimental procedures to identify suitable process parameters. We recently proposed Floating Isogeometric Analysis (FLIGA), a new computational solid mechanics approach, which is mesh distortion-free in one characteristic spatial direction. FLIGA emanates from Isogeometric Analysis and its key novel aspect is the concept of deformation-dependent “floating” of individual B-spline basis functions along one parametric axis of the mesh. Our previous work showed that FLIGA not only overcomes the problem of mesh distortion associated to this direction, but is also ideally compatible with material point integration and enjoys a stability similar to that of conventional Lagrangian mesh-based methods. These features make the method applicable to the simulation of large deformation problems with history-dependent constitutive behavior, such as additive manufacturing based on polymer extrusion. In this work, we enhance the first version of FLIGA by (i) a novel quadrature scheme which further improves the robustness against mesh distortion, (ii) a procedure to automatically regulate floating of the basis functions (as opposed to the manual procedure of the first version), and (iii) an adaptive refinement strategy. We demonstrate the performance of enhanced FLIGA on relevant numerical examples including a selection of viscoelastic extrusion problems.

Application extension of the meshless local Petrov-Galerkin method: Non-Newtonian fluid flow implementations

A computer code is developed to enhance the meshless local Petrov-Galerkin method capability to solve non-Newtonian fluid flow problems. In particular, the mesh-free method here, is to be empowered to simulate and solve the two-dimensional laminar incompressible power-law cases for the first time. The newly developed computer code expresses the appropriate governing equations in terms of the vorticity-stream function formulation and sustains a weighting function of unity. The local quadrature domain integrated by parts over the appropriate control volumes provides the local weak form of the considered governing equations. The extended scheme implements the moving least square interpolation technique to approximate the obtained field variables. The ability of the developed code to handle the Ostwald-de Waele (also known as the power-law) fluid flow occurrences were assessed through comparing the obtained results of the extended method to those of the conventional mesh-based methods for some benchmarking cases available in the paper. Based on this comparison, the extended code demonstrates a good capability to address the power-law fluid flow problems for different indices for one of the most popular non-Newtonian fluid flow models.

A Mesh-Free 3-D Deep Learning Electromagnetic Inversion Method Based on Point Clouds

Deep learning has been successfully used in the 2-D-mesh-based electromagnetic inverse scattering (EMIS) problems in the past. However, for 3-D EMIS problems, the massive amount of meshes generated by the mesh-based inversion methods makes deep learning implementation very challenging. Addressing this challenge will facilitate the further development of deep learning for 3-D EMIS problems. Inspired by the idea of point cloud, we propose a novel 3-D deep learning inversion methodology in this article. Different from previous work, the introduction of point clouds obviates the need to mesh the domain of interest (DOI) in this method, so we call it a mesh-free inversion method. The major advantage of this method compared with the mesh-based inversion methods is the low computational cost. In addition, the proposed method has other advantages. First, differently from prior deep learning inversion methods, the use of a reversible deep neural network (DNN) enables the proposed method to tackle both EMIS and electromagnetic forward scattering (EMFS) computations. Second, the inversion process becomes more controllable in this method, i.e., we can reconstruct merely a portion of the scatterer by simply shrinking the DOI. Third, the number of points used for the inversion can also be set as desired, without depending on the computational resources. Numerical and measured experiments demonstrate the feasibility and efficiency of the proposed method. This work opens a new vista on the application of deep learning toward 3-D EMIS problems. The code will be available at https://github.com/PKU-EMSensingLab/A-mesh-free-inversion-method-based-on-point-clouds.

Dynamic analysis of cracked plates based on a first-order shear deformation theory formulation by using an extended meshfree method

An extended meshfree method is employed in this paper for investigating the dynamic behaviour of cracked plates based on the first-order shear deformation theory (FSDT). The FSDT is a straightforward formulation with the assumption of first-order shear deformation as its name implies, which is appropriate for relatively thick plates. In this study, the meshfree method is chosen as an alternative to the conventional mesh-based methods to model plate structures. Among various meshfree formulations, Moving Kriging (MK) is a method that satisfies the Kronecker delta property, allowing for the easy imposition of essential boundary conditions. An extended MK formulation is proposed in this paper to model cracked plates without explicitly pre-defining the crack in the geometry domain. In the extended concept, the extrinsic enriched functions are employed to model the discontinuity due to the crack. Particularly, the Heaviside step function is employed to describe the discontinuity of the displacement fields on two sides of the crack surface. And the asymptotic enriched functions are used for stress singularity around the crack tip. In the dynamic analysis of cracked plates, one of the important factors that must be evaluated is the dynamic stress resultant intensity factor (DSRIF). In this paper, the DSRIFs are shown through many numerical examples and compared with analytical solutions and other numerical methods, showing the accuracy and efficiency of the present extended MK approach.

An improved equivalent magnetic network model of IPM motor in electric vehicles for dynamic simulation of transient effects

The existing electric motor model for electric vehicles cannot address the simulation of transient conditions. Moreover, it is difficult to balance the computation time and simulation accuracy. To address these shortcomings, an improved internal permanent magnet motor model containing nonlinear magnetic field factors was proposed based on the equivalent magnetic circuit theory. The effects of the magnetic circuit saturation, cogging torque, and magnetic field harmonics on the electromagnetic torque could be reflected. Furthermore, the electromagnetic vibration and rotor eccentricity could be considered. To improve the computational efficiency of the equivalent magnetic network method, lumped parameter or mesh-based methods were used for flux paths with different complexities. Second, the single-layer mesh-based method with local variable permeability parameters improved the efficiency of the air-gap dynamic analysis and computational efficiency while simulating 2D flux paths. The radial and tangential electromagnetic forces were calculated using the Maxwell tensor method and the action on the rotor. The accuracy of the nonlinear magnetic field modeling was verified by finite element analysis and experimental. In addition, the motion modeling method and dynamic characteristics of the model were studied. A reference was provided for dynamic simulations of electric vehicles and other using permanent magnet motors.

Numerical study of the splashing wave induced by a seaplane using mesh-based and particle-based methods

In recent years, forest fires and maritime accidents have occurred frequently, which have had a bad impact on human production and life. Thus, the development of seaplanes is an increasingly urgent demand. It is important to study the taxiing process of seaplanes for the development of seaplanes, which is a strong nonlinear fluid-structure interaction problem. In this paper, the smoothed particle hydrodynamics (SPH) method based on the Lagrangian framework is utilized to simulate the taxiing process of seaplanes, and the SPH results are compared with those of the finite volume method (FVM) based on the Eulerian method. The results show that the SPH method can not only give the same accuracy as the FVM but also have a strong ability to capture the splashing waves in the taxiing process, which is quite meaningful for the subsequent study of the effect of a splash on other parts of the seaplane.

Deep learning-based semantic segmentation of urban-scale 3D meshes in remote sensing: A survey

Semantic segmentation in 3D meshes is the classification of its constituent element(s) into specific classes or categories. Using the powerful feature extraction abilities of deep neural networks (DNNs), significant results have been obtained in the semantic segmentation of various remotely sensed data formats. With the increased utilization of DNNs to segment remotely sensed data, there have been commensurate in-depth reviews and surveys summarizing the various learning-based techniques and methodologies that entail these methods. However, most of these surveys focused on methods that involve popular data formats like LiDAR point clouds, synthetic aperture radar (SAR) images, and hyperspectral images (HSI) while 3D meshes hardly received any attention. In this paper, to our best knowledge, we present the first comprehensive and contemporary survey of recent advances in utilizing deep learning techniques for the semantic segmentation of urban-scale 3D meshes. We first describe the different approaches employed by mesh-based learning methods to generalize and implement learning techniques on the mesh surface, and then describe how the element-wise classification tasks are achieved through these methods. We also provide an in-depth discussion and comparative analysis of the surveyed methods followed by a summary of the benchmark large-scale mesh datasets accompanied with the evaluation metrics for assessing the segmentation performance of the methods. Finally, we summarize some of the contemporary problems of the field and provide future research directions that may help researchers in the community.

Optimal Point-to-Point geodesic path generation on point clouds

Computing the shortest path between two points or a set of points on a point cloud representing a manifold is a crucial problem in Computational Science, particularly in Computer Graphics and CAD/CAM. Most methods calculate geodesic paths on a polygonal model of the manifold, while only a few compute them directly on the point cloud. When the point cloud has noise, constructing a triangular mesh is challenging and results in inaccurate geodesic path calculations. This work calculates the geodesic path by minimizing the discrete geodesic curvature of the connecting curve, using an iterative Newton minimization and a projection procedure based on directed point-projection and Gabriel neighborhoods. The proposed algorithm has quadratic convergence and produces accurate results in synthetic and free-form point clouds, outperforming triangle mesh-based methods in noisy point clouds. The paper concludes with a demonstration of geodesic path computation in diverse free-form point-clouds and a case study in template-based digital surface reconstruction. The source code of the proposed method can be found in https://github.com/agalex1974/LibGeodesicOPPGC.

FAmesh: Generating Frequency Adaptive Meshes from Single Images under 2D Hole Constraints

Reconstructing 3D models from a single image has numerous applications in fields such as VR/AR, medical imaging, and gaming. However, most mesh-based methods are limited by the use of 0-genus initial templates, which makes it difficult to reconstruct 3D meshes with complex topologies. Additionally, existing methods often prioritize reconstructing the overall shape and neglect to study local meshes with varying curvatures, resulting in a lack of correct and detailed local features in the generated meshes. This paper proposes a 3D reconstruction framework that transitions from global to local and incorporates MLP and GCN. The framework introduces a mesh pruning strategy under a 2D hole constraint to restore the correct mesh topology. Moreover, the framework fine-tunes local details by separately learning corresponding mapping functions on high-frequency and low-frequency local extended patches. The experiment with the proposed network on the ShapeNet dataset shows that the network has a CD value of 1.763 and an F-score of 85.40. The results from extensive experiments demonstrate that our proposed method outperforms existing methods in topology correction and local detail reconstruction.

A CNN-supported Lagrangian ISPH model for free surface flow

As a popular method for modeling violent free surface flow, the incompressible Smoothed Particle Hydrodynamics (ISPH) based on the Lagrangian formulation has attracted a great attention worldwide. The Lagrangian ISPH solves the unsteady Navier-Stokes and continuity equations using the projection method, in which the pressure is obtained by solving the pressure Poisson's equation (PPE) that is the most time-consuming part in the ISPH procedure. In this paper, the Convolutional Neural Network (CNN) is combined with ISPH and used to predict the fluid pressure instead of solving the PPE directly. Although limited attempts of using CNN for solving the PPE in Eulerian formulation (referred to as the Eulerian CNN framework) in mesh-based methods are found in the public domain, the present model is the first ISPH model supported by CNN in a Lagrangian formulation. The proposed model overcome several challenges associated with combining CNN with ISPH, including selecting the input parameters, formulating the objective functions, producing the training dataset and dealing with boundary conditions. Two classic free surface problems, i.e. the dam breaking and the wave propagation, are simulated to evaluate the performance of the present model. Quantitative assessments of the numerical error in terms of both the free surface profile and the pressure field are carried out. The assessments show that the new model does not only give results with satisfactory accuracy, but also requires much less computation time for estimating pressure if the number of particles is large, e.g., 100 thousands particles that is usually required in the practical ISPH simulation for free surface flow.

Accurate RBF-FD meshless numerical simulation of thermo-fluid problems for generic 3D geometries

In contrast to traditional mesh-based methods for the numerical solution of boundary value problems, e.g., Finite Element (FEM) and Finite Volume (FVM), in the recent period many meshfree approaches have been proposed in order to avoid those typical issues due to the mesh. For example, the quality of the mesh greatly affects the reliability of the final solution in the case of CFD problems and the human intervention of a professional is often still needed when dealing with complex-shaped domains. This in turn increases both cost and time required for the reliable simulation of problems of engineering relevance. Meshless methods, on the other side, usually rely on a simpler distribution of nodes and do not require the storage of connectivity information. Among others, one of the most promising meshless methods in terms of accuracy and flexibility is the one based on the Radial Basis Function – Finite Difference (RBF-FD) scheme. RBF-FD methods, however, are usually affected by severe ill conditioning issues when Neumann boundary conditions are employed. This fact is the main responsible for the appearance of large discretization errors near the boundary and for the lack of stability of traditional time integration schemes. In order to address this issue, some new algorithms for the robust treatment of boundary conditions have been developed and successfully employed to solve fluid flow problems with heat transfer. Furthermore, it is well acknowledged that the efficient resolution of boundary layers arising in this class of problems requires an adequate spatial discretization in the neighbourhood of the boundary, i.e., increased node/mesh density along the direction of large gradients only. This result is achieved by employing anisotropic node distributions, which is a novelty in the context of the RBF-FD method to the best of the authors’ knowledge. The method described above is successfully employed for the accurate solution of a representative 3D heat transfer problem with incompressible fluid flow.

BI-GreenNet: Learning Green’s Functions by Boundary Integral Network

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green’s function is difficult to compute. The troubles arise in the following threefold. Firstly, compared with the original PDE, the dimension of Green’s function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green’s function usually contains singularities which increase in the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we develop a new framework for computing Green’s function leveraging the fundamental solution, boundary integral method and neural networks with reasonably high accuracy in this paper. We focus on Green’s function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz equations in domains with interfaces. Extensive numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s function. In addition, Green’s function provides the operator from the source term and boundary condition to the PDE solution. We apply Green’s function to solve PDEs with different sources, and obtain reasonably high-precision solutions, which shows the good generalization ability of our method. However, the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs, such as variable coefficient equations, which will be investigated in our future work.

An explicit total Lagrangian Fragile Points Method for finite deformation of hyperelastic materials

This research explored a novel explicit total Lagrangian Fragile Points Method (FPM) for finite deformation of hyperelastic materials. In contrast to mesh-based methods, where mesh distortion may pose numerical challenges, meshless methods are more suitable for large deformation modelling since they use enriched shape functions for the approximation of displacements. However, this comes at the expense of extra computational overhead and higher-order quadrature is required to obtain accurate results. In this work, the novel meshless method FPM was used to derive an explicit total Lagrangian algorithm for finite deformation. FPM uses simple one-point integration for exact integration of the Galerkin weak form since it employs simple discontinuous polynomials as trial and test functions, leading to accurate results even with single-point quadrature. The proposed method was evaluated by comparing it with FEM in several case studies considering both the extension and compression of a hyperelastic material. It was demonstrated that FPM maintained good accuracy even for large deformations where FEM failed to converge.